Quantum computational method, control layout for a quantum computer, method of determining same, and apparatus for quantum computation

Abstract

A quantum computational method is provided. The quantum computational method includes providing a physical quantum system (100) comprising constituents (50). The quantum computational method includes performing an encoded quantum computation on the physical quantum system. Performing the encoded quantum computation includes preparing at least a portion of the physical quantum system in an initial quantum state. Performing the encoded quantum computation includes evolving at least a portion of the physical quantum system to a final quantum state. Performing the encoded quantum computation includes measuring at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits (10) into code qubits (20). The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem (50, 350) of the physical quantum system, wherein the subsystem comprises one or more constituents. During a first portion of the encoded quantum computation, the code qubits include a first code qubit duplication set (1350, 1450, 1550, 1651-1653, 1750) associated with a first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit. The quantum computational method further includes performing a first sequence of physical quantum operations on the physical quantum system during the first portion of the encoded quantum computation. The first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting at least on the first logical qubit.

Description

QUANTUM COMPUTATIONAL METHOD, CONTROL LAYOUT FOR A QUANTUM COMPUTER, METHOD OF DETERMINING SAME, AND APPARATUS FOR QUANTUM COMPUTATION FIELD

[0001] Embodiments described herein relate to methods and apparatuses for performing an encoded quantum computation. In an encoded quantum computation, the state of the quantum system corresponds to a quantum error-correction code that encodes logical qubits into code qubits. The code qubits are acted upon to perform a quantum computation in an encoded form. Embodiments described herein specifically relate to methods and apparatuses for performing an encoded quantum computation that is fault-tolerant. BACKGROUND

[0002] Quantum computing devices are computing devices which make use of quantum mechanical effects to solve computational problems. In a quantum computing device, or quantum computer, information is carried by quantum systems, such as e.g. quantum bits (“qubits”). This is in contrast to conventional computers, which operate with classical bits, i.e. 0 and 1. During a quantum computation, quantum bits can be processed by evolving the quantum system. For example, groups of qubits of the quantum system can be coupled to each other according to a specified interaction. By evolving the quantum system, the information carried by the quantum system can be processed in order to carry out a computation, i.e. in order to solve a computational problem. In many cases, a quantum computer can be assisted by a classical computer, i.e. a computer operating with classical bits. The classical computer can provide instructions to the quantum computer as to how the qubits in the system are to be processed by the quantum computer.

[0003] It is known that quantum computers can be vulnerable to noise. When a quantum system interacts in an uncontrolled manner with the environment, the information stored in the quantum system may be corrupted, or in the worst case destroyed. In addition, the quantum operations (e.g. unitary operations and / or measurements) carried out during a quantum computation can themselves be imperfect, and hence introduce errors into the quantum system, which can further corrupt the information contained in the quantum system.

[0004] To fight the detrimental effects of noise, quantum error-correction codes have been designed. Using a quantum error-correction code, a system of (logical) qubits can be encoded into a larger system of qubits which contains the quantum information in an encoded form, so that errors can be corrected. The information contained in the quantum system during a quantum computation can be protected by performing an encoded quantum computation. Further, methods have been designed to construct encoded quantum computations that are fault-tolerant, i.e. quantum computations that can be reliably performed even when the quantum operations performed during the course of the computation are imperfect.

[0005] Whereas the theory of quantum error-correction and fault tolerance provide a proof of principle that quantum computations of arbitrary length can be reliably performed, several challenges remain for a practically feasible implementation of fault-tolerant quantum computers. For example, in existing encoding schemes, the overhead in terms of the total number of qubits needed to construct a quantum error-correction code and / or a fault-tolerant circuit is very large, so that even quantum computations initially acting on a relatively small number of (logical) qubits will, when encoded in a fault-tolerant manner, need to operate on a very large number of qubits - in many cases beyond what is realistically achievable in the foreseeable future. Further, in many situations, the encoded quantum circuits are inherently sequential, i.e. have a large circuit depth, which is a further obstacle for an effective implementation thereof.

[0006] Therefore, there is a need for improved methods and devices for performing an encoded quantum computation. SUMMARY

[0007] According to an embodiment, a quantum computational method is provided. The quantum computational method includes providing a physical quantum system comprising constituents. The quantum computational method includes performing an encoded quantum computation on the physical quantum system. Performing the encoded quantum computation includes preparing at least a portion of the physical quantum system in an initial quantum state. Performing the encoded quantum computation includes evolving at least a portion of the physical quantum system to a final quantum state. Performing the encoded quantum computation includes measuring at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state ofat least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits into code qubits. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem of the physical quantum system, wherein the subsystem comprises one or more constituents. During a first portion of the encoded quantum computation, the code qubits include a first code qubit duplication set associated with a first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit. The quantum computational method further includes performing a first sequence of physical quantum operations on the physical quantum system during the first portion of the encoded quantum computation. The first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation acting at least on the first logical qubit.

[0008] According to a further embodiment, a quantum computational method is provided. The quantum computational method includes providing a physical quantum system comprising constituents. The quantum computational method includes performing an encoded quantum computation on the physical quantum system. Performing the encoded quantum computation includes preparing at least a portion of the physical quantum system in an initial quantum state. Performing the encoded quantum computation includes evolving at least a portion of the physical quantum system to a final quantum state. Performing the encoded quantum computation includes measuring at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits into code qubits. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem of the physical quantum system, wherein thesubsystem comprises one or more constituents. The physical implementation has a dominant qubit error, wherein at least one qubit error different from the dominant qubit error is suppressed as compared to the dominant qubit error or is correctable by performing one or more error- correcting operations on the subsystem. The dominant qubit error is correctable by the parity code in one or more subsystems of the physical quantum system. The quantum computational method further includes performing a first sequence of physical quantum operations on the physical quantum system during a first portion of the encoded quantum computation. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation acting on one or more logical qubits. The first sequence of physical operations is a fault-tolerant implementation of the first logical quantum operation.

[0009] According to a further embodiment, a method of determining a control layout for a quantum computer is provided. The method includes determining a layout of a parity code that encodes logical qubits into code qubits. The layout is determined based on a specification of a logical quantum computation acting on the logical qubits. The logical quantum computation includes a first logical quantum operation acting at least on a first logical qubit. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. The code qubits include a first code qubit duplication set associated with the first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit.

[0010] According to a further embodiment, a control layout for a quantum computer is provided. The control layout includes a layout of an encoding of logical qubits into code qubits corresponding to a parity code. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. The code qubits include a first code qubit duplication set associated with a first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit.

[0011] According to a further embodiment, a data carrier or data carrier signal carrying information representing a control layout as described herein is provided.

[0012] According to a further embodiment, an apparatus for quantum computation is provided. The apparatus includes a physical quantum system comprising constituents. The apparatus includes a quantum processing system for evolving at least some of the constituents. The apparatus includes a measurement system for measuring one or more of the constituents. The apparatus includes a classical computing system connected to the quantum processing system and to the measurement system. The classical computing system is configured to instruct at least one of the quantum processing system and the measurement system to perform an encoded quantum computation on the physical quantum system. The encoded quantum computation includes preparing, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system in an initial quantum state. The encoded quantum computation includes evolving, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system to a final quantum state. The encoded quantum computation includes measuring, using the measurement system, at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits into code qubits. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem of the physical quantum system, wherein the subsystem comprises one or more constituents. During a first portion of the encoded quantum computation, the code qubits include a first code qubit duplication set associated with a first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit. Evolving at least a portion of the physical quantum system to a final quantum state includes performing a first sequence of physical quantum operations on the physical quantum system during the first portion of the encoded quantum computation. The first sequence of physical quantum operations acts at least on the subsystems of the physical quantum systemcorresponding to the code qubits in the first code qubit duplication set. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation acting at least on the first logical qubit.

[0013] According to a further embodiment, an apparatus for quantum computation is provided. The apparatus includes a physical quantum system comprising constituents. The apparatus includes a quantum processing system for evolving at least some of the constituents. The apparatus includes a measurement system for measuring one or more of the constituents. The apparatus includes a classical computing system connected to the quantum processing system and to the measurement system. The classical computing system is configured to instruct at least one of the quantum processing system and the measurement system to perform an encoded quantum computation on the physical quantum system. The encoded quantum computation includes preparing, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system in an initial quantum state. The encoded quantum computation includes evolving, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system to a final quantum state. The encoded quantum computation includes measuring, using the measurement system, at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits into code qubits. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem of the physical quantum system, wherein the subsystem comprises one or more constituents. The physical implementation has a dominant qubit error. At least one qubit error different from the dominant qubit error is suppressed as compared to the dominant qubit error or is correctable by performing one or more error-correcting operations on the subsystem. The dominant qubit error is correctable by the parity code in one or more subsystems of the physical quantum system. Evolving at least a portion of the physical quantum system to a final quantum state includes performing a first sequence of physical quantum operations on the physical quantum system during a first portion of the encoded quantum computation. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation acting on one or more logical qubits. The first sequence ofphysical quantum operations provides a fault-tolerant implementation of the first logical quantum operation.

[0014] Embodiments are also directed to methods for operating the systems described herein, and to the use of the systems to perform the methods according to the embodiments described herein.

[0015] Further advantages, features, aspects and details that can be combined with embodiments described herein are evident from the dependent claims, the description and the drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] A full and enabling disclosure to one of ordinary skill in the art is set forth more particularly in the remainder of the specification including reference to the accompanying drawings wherein: FIG.1 illustrates an encoding of logical qubits into code qubits according to a quantum error-correction code; FIG.2 illustrate that each code qubit may be physically implemented in a constituent of a physical quantum system; FIGs.3-4 illustrate that each code qubit may be physically implemented in a subsystem of a physical quantum system, wherein the subsystem includes several constituents; FIG.5 shows a logical quantum circuit acting on logical qubits; FIG.6 shows an encoded quantum circuit acting on code qubits; FIG.7 illustrate an encoding of logical qubits into code qubits using a parity code, where the code qubits are physically implemented in subsystems of the physical quantum system having a dominant qubit error that is correctable by the parity code; FIG.8 illustrates a parity code that encoded logical qubits into code qubits;FIGs.9-10 illustrate the notion of fault-tolerant error propagation of a quantum circuit; Fig.11 shows an encoded quantum circuit providing a fault-tolerant implementation of a Z gate; Fig.12 shows an encoded quantum circuit providing a fault-tolerant implementation of an X gate; Fig.13 shows an encoded quantum circuit providing a fault-tolerant implementation of a CNOT gate; Fig.14 shows an encoded quantum circuit providing a fault-tolerant implementation of a two-qubit ^ / 2 rotation; Fig.15 shows an encoded quantum circuit providing a fault-tolerant implementation of an S gate; Fig.16 shows an encoded quantum circuit providing a fault-tolerant implementation of a controlled-Z gate; Fig.17 shows an encoded quantum circuit providing a fault-tolerant implementation of a Hadamard-transformed T gate; Fig.18 shows an encoded quantum circuit providing a fault-tolerant implementation of a Hadamard gate; and Fig.19 shows an apparatus for quantum computation. DETAILED DESCRIPTION

[0017] Reference will now be made in detail to the various exemplary embodiments, one or more examples of which are illustrated in each figure. Each example is provided by way of explanation and is not meant as a limitation. For example, features illustrated or described as part of one embodiment can be used on or in conjunction with other embodiments to yield yet further embodiments. It is intended that the present disclosure includes such modifications and variations.

[0018] Within the description of the drawings, the same reference numbers refer to the same or similar components. Generally, only the differences with respect to the individual embodiments are described. The structures shown in the drawings are not necessarily depicted true to scale, and may contain details drawn in an exaggerated way to allow for a better understanding of the embodiments.

[0019] A physical quantum system as described herein is a physical system exhibiting quantum effects. That means, the physical quantum system is a real-world object. The physical quantum system includes constituents. The constituents are physical quantum entities themselves, and can be regarded as smaller quantum systems that jointly form the physical quantum system. Embodiments described here are not limited to any particular type of constituents. For example, the constituents may be atoms, ions, superconducting qubits (e.g. transmons), photons, electrons, electromagnetic or mechanical resonators (realizing an oscillator), and the like.

[0020] The physical quantum system can be in different quantum states, such as an initial quantum state (in which the physical quantum may be prepared at the beginning of a quantum computation) and a final quantum state (in which the physical quantum system may end up due to the quantum computation). The physical quantum system can be evolved from the initial quantum state to the final quantum state, e.g. by performing sequences of unitary operators and / or measurements. Such an evolution is a real-world process, and particularly a controlled technical process (quantum computation) which brings the physical quantum system from the initial quantum state to an a priori unknown final quantum state that contains information about the solution to a computational problem. This information can be revealed by measuring the physical quantum system or a part thereof, i.e., one or more of its constituents. The act of measuring is also a physical / technical process. Measurements allow to obtain a read-out of the physical quantum system. A read-out of a physical quantum system is a set of one or more measurement values obtained by measurements of constituents of the physical quantum system, involving physical interactions with the constituents.

[0021] The physical quantum system may include K constituents, wherein K may be at least 100, at least 1.000 or at least 10.000. K may be from 100 to 10.000, or from 100 to 100.000, but K may be larger than 100.000. It shall be understood that the quantum systems shown in the figures and described in examples may be much smaller for illustrative and explanatory purposes, but shall not be understood to provide any limitation.

[0022] In any realistic system, at least a small amount of noise is always present. Accordingly, quantum states cannot be realized with 100% accuracy. Likewise, operations performed on a physical quantum system, such as unitary operators and measurements, are always subject to at least some noise, and are not realized with 100% accuracy. It shall be understood that the quantum states and operations described herein encompass states and operations that are subject to small amounts of noise.

[0023] A qubit can be understood as a basic unit of quantum information. A qubit corresponds to a two-level quantum system. A qubit is mathematically represented by a two-dimensional vector space (Hilbert space). A (pure) quantum state of a qubit can be described by a two-dimensional vector |^^^ = a|0^ + b|1^. Therein, |0> and |1> are quantum basis states forminga quantum basis of the qubit, and a and b are complex numbers. The quantum states |0> and |1> may be called computational basis states, and the corresponding quantum basis may be called the computational basis.

[0024] A qubit can be physically implemented (also called “physically realized”) in a corresponding constituent of the physical quantum system. Alternatively, a qubit can be physically implemented in a group of several constituents. In other words, a qubit can be physically implemented in a subsystem of the physical quantum system, wherein the subsystem may include one or more constituents (this is detailed further below with respect to Figs.2-4). Physically implementing the qubit in the subsystem can include physically engineering the subsystem in a manner such that the subsystem behaves as the qubit. The subsystem may be engineered such that the quantum state of the subsystem is described by a 2-dimensional vector|^^^ = a|0^ + b|1^ of the qubit in question. Particularly, a first quantum state of the subsystemmay be identified with the basis state |0> and a second quantum state of the subsystem may be identified with the basis state |1>. A qubit can be physically implemented in a subsystem that is by its nature two-dimensional, or in a subsystem that is d-dimensional with d > 2 by occupying only two quantum levels (e.g. energy levels), which are decoupled from the remaining quantum levels and the environment, allowing coherent dynamics of the qubit at least during the time needed to perform the quantum computation. Two of the quantum levels may be identified with a quantum basis (computational basis) of the qubit. The quantum levels, e.g. energy levels, may be electronic, fine structure or hyperfine structure energy levels in atoms or ions. The quantum levels may be superpositions of energy levels of a quantum harmonic oscillator or a Kerr-nonlinear oscillator, which may for instance be realized in resonators insuperconducting circuits. The foregoing are merely examples of physical implementations, and it shall be understood that other examples can be given.

[0025] Further, operations on the qubit, such as unitary operations or measurements, can be physically realized by performing corresponding physical quantum operations (e.g. laser pulses, microwaves, application of voltages, and the like) to the subsystem in which the qubit is physically implemented.

[0026] A qubit can be physically implemented in a subsystem in various ways. According to the present disclosure, a particular type of physical implementation is provided, namely a physical implementation having a dominant qubit error. A physical implementation of a qubit is said to have a dominant qubit error if at least one qubit error different from the dominant qubit error is suppressed as compared to the dominant qubit error or is correctable by performing one or more error-correcting operations on the subsystem. A physical implementation having a dominant qubit error may be described to have a “noise bias”.

[0027] For example, the physical implementation may be engineered such that qubit errors of a first type, such as Z errors (phase flip errors), are suppressed (e.g. exponentially suppressed by increasing a control parameter of the physical implementation) as compared to qubit errors of a second type, such as X errors (bit flip errors). Therein, X stands for the σxPauli operator and Z stands for the σzPauli operator. The physical implementation in question may offer a passive protection against the first type of qubit errors. Due to the physical properties of the subsystem in question, the probability that an error of the first type occurs may be less likely (e.g. exponentially less likely) than the probability of an error of the second type occurring, so that there might be effectively no need to actively correct errors of the first type. Accordingly, the second type of qubit error may be considered the dominant qubit error in this physical implementation.

[0028] In another example, the physical implementation can be engineered such that the qubit errors of the first type, while potentially not being suppressed, can at least be corrected by actively performing quantum-error correction on the subsystem. Such a situation may, e.g., occur when the qubit is physically implemented in a group of several constituents according to a quantum error-correction code that allows correction of errors of the first type. For example, a qubit may be physically implemented in at least three constituents according to a (Hadamard-transformed) repetition code, which allows correcting Z errors by performing stabilizer measurements (syndrome measurements) and error-correcting unitary operations.

[0029] Embodiments described herein involve encoded quantum computations, wherein a logical quantum computation acting on logical qubits is encoded into an encoded quantum computation acting on code qubits.

[0030] A logical quantum computation acting on a set of logical qubits is considered. The logical quantum computation may include a sequence of logical quantum operations acting on the logical qubits. The sequence of logical quantum operations may include quantum gates, i.e. unitary operators, wherein each quantum gate acts on one or more logical qubits. The sequence of logical quantum operations may include measurements, e.g. performed in between some of the quantum gates, wherein each measurement acts on one or more logical qubits. The sequence of logical quantum operations evolves the set of logical qubits from an initial quantum state to a final quantum state of the logical qubits. The logical quantum computation may include performing at least one measurement of at least one logical qubit to provide a read-out of the logical quantum computation.

[0031] In any realistic situation, a quantum system is subject to noise, which can corrupt or even destroy the information carried by the quantum system, and hence annihilate any computational benefits that might be provided by solving a computational problem using a quantum computer rather than a classical computer. To counteract the detrimental effects of noise, a quantum error-correction code may be used. Instead of operating on the logical qubits directly, said logical qubits are encoded using the quantum error-correcting code. The set of logical qubits, forming a first quantum system, is mapped to, or encoded in, a second quantum system comprising a second set of qubits, called code qubits herein. The second quantum system is typically larger than the first quantum system, so that the second quantum system introduces a redundancy in the information carried by the logical qubits, thus allowing a correction of errors.

[0032] Fig.1 shows a set of logical qubits 10 that is encoded into a set of code qubits 20 using a quantum error-correction code. The logical qubits 10 are indicated by dashed circles. The code qubits 20 are indicated by solid circles.

[0033] When using a quantum error-correction code to encode logical qubits into code qubits, the code qubits may be physically realized, whereas the logical qubits are normally not. In other words, in such a case the logical qubits may be regarded as abstract entities that are used to understand the meaning of the quantum computation that one intends to carry out, yet the logical qubits themselves might not be implemented in a physical quantum system. The code qubits, on the other hand, may be physically implemented in respective subsystems of the physical quantum system. In particular, the code qubits may be physically implemented using a physical implementation having a dominant qubit error as described herein.

[0034] Fig.2 shows a physical quantum system 100 including constituents 50. In the example illustrated in Fig. 2, each code qubit 20 is physically implemented in a corresponding constituent 50. In other words, each code qubit 20 is physically implemented in a corresponding subsystem of the physical quantum system 100, wherein the subsystem consists of a single constituent 50.

[0035] Alternatively, each code qubit 20 may be physically implemented in a subsystem comprising more than one constituent 50, as illustrated in Figs.3-4. As illustrated in Fig.3, the constituents 50 may be grouped into subsystems 350, indicated by the dashed squares, of the physical quantum system 100. Each subsystem 350 may include two or more constituents 50. In the example shown in Fig.3, the subsystems 350 each include four constituents 50 in total, yet the disclosure is not limited thereto. A subsystem 350 may include fewer or more constituents 50. As shown in Fig. 3, the subsystems 350 may be mutually disjoint. Each constituent 50 of the physical quantum system 100 may be part of at most one subsystem 350. As illustrated in Fig.4, each code qubit 20 may be physically implemented in a corresponding subsystem 350 of the physical quantum system 100, wherein the subsystem 350 includes several constituents 50 (four constituents in the present example). For the sake of clarity, the constituents 50 are not drawn in Fig.4.

[0036] The physical implementation used for implementing the code qubits 20 into the respective subsystems may be a physical implementation having a dominant qubit error, as described herein.

[0037] A logical quantum computation acting on the logical qubits may be mapped, via the quantum error correcting code, to an encoded quantum computation acting on the code qubits. Particularly, each logical quantum operation of the logical quantum computation acting on oneor more logical qubits may be mapped to another quantum operation, or to a sequence of quantum operations, acting on one or more code qubits. The quantum operation or sequence of quantum operations acting on the respective code qubit(s) provide an encoded realization of the logical quantum operation. The (sequence of) quantum operations acting on the one or more code qubits may be physically implemented as a (sequence of) physical quantum operation(s) acting on the respective one or more subsystems that provide the physical implementation of the one or more code qubits.

[0038] Fig. 5 shows an example of a logical quantum computation 500. The logical quantum computation 500 includes a sequence of logical quantum operations, e.g. unitary operations (gates) and / or measurements, acting on the logical qubits 10. In the exemplary computation shown in Fig.5, the logical quantum computation 500 includes logical quantum operations 501, 502, 503, 504 and 505. The logical quantum operation 505 acts on a single logical qubit, the logical quantum operations 501, 503 and 504 each act jointly on a respective group of two logical qubits, and the logical quantum operation 502 acts jointly on three logical qubits. In in the illustrative example shown in Fig.5, the number of logical operations in the logical quantum computation 500, the ordering of said logical operations, and the groups of logical qubits on which they act are purely exemplary and the disclosure shall not be limited thereto. The logical quantum computation 500 may be an arbitrary quantum computation.

[0039] Fig. 6 shows an encoded quantum computation 600 that corresponds to the logical quantum computation 500 by applying a quantum error-correction code, particularly a parity code as described herein. The encoded quantum computation 600 consists of a sequence of quantum operations acting on the code qubits 20. Said quantum operations, which are indicated by the rectangular boxes in Fig. 6, may include e.g. unitary operations and / or measurements. Each quantum operation of the encoded quantum computation 600 may act on one or more code qubits 20. Each logical quantum operation of the logical quantum computation 500 in Fig. 5 may be mapped, using the quantum error-correction code, to a corresponding quantum operation, or a corresponding sequence of quantum operations, of the encoded quantum computation 600 in Fig. 6. Specifically, the logical quantum operations 501 through 505 are mapped to blocks 601 through 605, respectively. Each of said blocks includes a quantum operation, or a sequence of quantum operations, of the encoded quantum computation 600. For example, logical quantum operation 501 is mapped, by the quantum error correction code, to block 601 consisting of a sequence of two quantum operations, while logical quantum operation505 is mapped to block 605 consisting of a single quantum operation. Each block shown in Fig. 6 constitutes an encoded realization (also called encoded version) of the corresponding logical quantum operation. That is to say, the quantum error correction code defines a mapping from the logical qubits 10 to the code qubits 20. Said mapping from the logical qubits to the code qubits induces a mapping from any logical quantum operation acting on a given set of one or more logical qubits 10 to a corresponding encoded quantum operation acting on a set of one or more code qubits 20. Each block shown in Fig. 6, when viewed as a whole, realizes such an encoded quantum operation. Conversely, the action of each block on the code qubits corresponds to the action of the respective logical quantum operation on the logical qubits by inverting the mapping defined by the quantum error correction code.

[0040] In the illustrative example shown in Fig. 6, the number of quantum operations in the encoded quantum computation 600, the ordering of said quantum operations, the groups of code qubits on which they act, as well as the grouping of quantum operations into respective blocks are purely exemplary and the disclosure shall not be limited thereto.

[0041] Embodiments described herein involve a particular type of quantum error-correction code, namely a parity code. A parity code encodes a set of logical qubits into a set of code qubits. The code qubits include parity qubits. Each parity qubit is associated with a respective subset of logical qubits. The parity qubit encodes the parity of the subset in question. The parity codes form a family of codes, depending (inter alia) on how the logical qubits are grouped into subsets and correspondingly mapped to associated parity qubits.

[0042] A parity code is an instance of a stabilizer code. A stabilizer code, and hence also a parity code, has stabilizer operators acting on the code qubits. When a quantum state of the logical qubits is encoded into an encoded quantum state of the code qubits using the parity code, the resulting encoded quantum state is an eigenvector with eigenvalue 1 of the stabilizer operators of the parity code. The stabilizer operators can be understood as constraints that are fulfilled by the encoded quantum state.

[0043] A parity code, in view of the fact that the information contained in the logical qubits is encoded into parity qubits, allows correcting certain errors acting on the code qubits. In an example, the parity code may allow correcting X errors (bit flip errors) acting on the code qubits. For example, at least X errors on individual code qubits may be correctable. The errors can be corrected, for example, by measuring one or more stabilizer operators of the parity code;such measurements may be called syndrome measurements. Based on the measurement outcome(s) thereof (called the syndrome(s)), it can be determined whether an error has occurred. By performing a suitable operation on the affected code qubit(s), the error can be corrected.

[0044] The code qubits (e.g. the code qubits 20 shown in Fig.6) are physically implemented in respective subsystems of the physical quantum system. As described above, a particular physical implementation is provided, namely an implementation that has a dominant qubit error. According to embodiments described herein, the dominant qubit error associated with the selected physical implementation is an error that is correctable by the parity code according to which the logical qubits are encoded into the code qubits. For example, it may be the case that the dominant qubit error is the X error, and that the X error is correctable by the parity code under consideration.

[0045] Accordingly, embodiments described herein provide for an encoding scheme that allows to correct arbitrary errors. Due to the particular physical implementation that is selected, errors other than the dominant qubit error are either suppressed (passive protection) or correctable (active protection). Further, the dominant qubit error itself is correctable by the parity code. This situation is further illustrated in Fig. 7. The logical qubits 10 are encoded into the code qubits 20 using a parity code, indicated at 710. The parity code allows correcting a first type of errors, for example X errors. Further, the code qubits 20 are implemented in subsystems 350 of the physical quantum system 100 (where each subsystem 350 is a single constituent or includes multiple constituents) using a physical implementation indicated at 720, wherein the physical implementation may have, as a dominant qubit error, the error type that is correctable by the parity code (being X errors in the present example). Qubits errors other than the dominant qubit error are suppressed or correctable by the physical implementation. Accordingly, arbitrary qubit errors may be corrected.

[0046] The type of error that can be corrected by the parity code can be adjusted by performing an individual basis change (local unitary rotation) of the parity qubits (and more generally of the code qubits, which may include qubits other than parity qubits). In other words, depending on the chosen quantum basis, a parity code can be configured to correct a different type of qubit error. In one example, X errors can be corrected by the parity code if the parities are defined with respect to the eigenbasis of the Z operator, i.e. the basis {|0>, |1>}. In another example, abasis change from the basis {|0>, |1>} to the basis {|+>, |->} can be performed, where |+^ =(|0^ + |1^) / √2 and |−^ = (|0^ − |1^) / √2; such a basis change can be realized by applying aHadamard operation to each parity qubit (or more generally to each code qubit). The resulting parity code, i.e. in the new quantum basis, may allowing for the correction of Z errors (phase flip errors). In such a case, a physical implementation of the code qubits can be chosen where the dominant qubit error is a Z error. Accordingly, the dominant qubit error of the physical implementation is correctable by the parity code in question.

[0047] More generally, a quantum basis of the code qubits may be considered (e.g. the basis {|0>, |1>}, the basis {|+>, |->}, or any other basis) such that the parities associated with the subsets of logical qubits are defined with respect to said quantum basis. The quantum basis can be an eigenbasis of a first qubit operator (e.g. the operator Z, the operator X, or another operator). Further, the dominant qubit error associated with the physical implementation of the code qubits may be represented by a second qubit operator. That the dominant qubit error is correctable by the parity code is mathematically reflected in the property that the commutator of the first qubit operator and the second qubit operator is different from zero. In some cases, the first qubit operator may anti-commute with the second qubit operator. For example, the first qubit operator may be Z and the second operator may be X (or vice versa), where it is known that the commutator of X and Z is different from zero, in particular X and Z are anticommuting operators.

[0048] Embodiments described herein thus provide a method for performing an encoded quantum computation in a manner that allows correcting arbitrary types of errors on the code qubits (e.g. both bit flips, represented by X, and phase flips, represented by Z, can be correctable) by combining a parity code with a physical implementation having a dominant qubit error, where the dominant qubit error is correctable by the parity code. Accordingly, the detrimental effects of noise can be counteracted.

[0049] Further, in light of the fact that the parity code is used for encoding the logical qubits into the code qubits (and not some other quantum error-correction code), further advantages are provided.

[0050] When using a parity code, the resulting encoded quantum computation may involve short-range quantum operations only, i.e. quantum operations (e.g. unitary operators) that only act on groups of constituents that are spaced apart from each other by a small distance (e.g.nearest neighbor or next-nearest neighbor interactions in a two-dimensional arrangement of the constituents). Likewise, the stabilizer operators of a parity code are short-range operators, so that quantum error correction is facilitated. The fact that operations are short-range provides a significant benefit for a realistic implementation of the encoded quantum computations.

[0051] Further, a parity code allows a high degree of parallelizability of the encoded quantum computation. When using a parity code, logical quantum operations acting on the logical qubits are mapped to encoded versions thereof acting on the code qubits. As illustrated above, an encoded realization of a logical quantum operation may include a sequence of several quantum operations acting on the code qubits. Yet, the parity codes offer the benefit that the resulting sequence, for several instances of the logical quantum operation that is to be encoded, has a low circuit depth, and is thus highly parallelizable. Particularly, the encoded versions of even complex logical quantum gates, such as controlled-phase gates, can be parallelized to a constant circuit depth.

[0052] Further, in a parity code, less code qubits are needed as a function of the code distance, as compared to other error-correction codes.

[0053] A further aspect of the present application is that the encoding schemes described herein allow performing fault-tolerant quantum computation. Specifically, a fault-tolerant encoding of a universal set of quantum gates is described herein. This allows performing any quantum computation, e.g. arbitrary sequences of quantum gates realizing an arbitrary unitary operator, in a fault-tolerant manner. A fault-tolerant encoding of a quantum gate can be understood as an encoded quantum circuit realizing the quantum gate (where the encoding involves the parity code combined with the physical implementation having the dominant qubit error, as described herein) that is designed in a particular manner which is resilient to errors, even in a setting where the quantum operations themselves – namely, the unitary operations and measurements that form the encoded quantum circuit - are imperfect, i.e. are subject to possible failure. Having imperfect quantum operations reflects the real-life circumstances under which a quantum computer will have to reliably operate. As described herein, the inventors have devised encoded quantum circuits that have a particular circuit structure which prevents errors (which may either occur due to e.g. decoherence or which may be introduced by the quantum operations performed during the course of the encoded quantum circuit) from propagating uncontrollably to a plurality of qubits. By keeping such a potential cascading of errors to a minimum, a correctionof the errors (e.g. using syndrome measurements, as described herein) is feasible throughout the entire quantum computation, so that fault-tolerant quantum circuits are obtained. As mentioned above, the present disclosure provides a fault-tolerant encoding of a universal set of quantum gates, and hence the possibility to perform an arbitrary quantum computation in a fault-tolerant manner.

[0054] It is nontrivial to determine a fault-tolerant encoding of a logical quantum operation, for several reasons. First, as mentioned above, the parity codes form an (infinitely large) family of quantum error-correction codes. Given a specific logical quantum operation, it needs to be determined which particular parity code is suitable for encoding the logical quantum operation in a fault-tolerant manner, e.g. which particular groupings of logical qubits are to be selected for defining the associated parity qubits, and which code qubits, if any, are determined to be data qubits (the definition of data qubits is provided below). Second, it shall be determined, given a particular parity code, how a logical quantum operation acting on the logical qubits can be mapped to an encoded quantum circuit, which will typically include a sequence of multiple quantum operations acting on the code qubits, in a manner such that the encoded quantum circuit is fault-tolerant. Finding such a suitable sequence of quantum operations acting on the code qubits is a difficult problem, since again there are a priori many possibilities to be considered, i.e. there are many potential candidate sequences. In this respect, it is noted that a particular parity code that is used to encode a given logical quantum operation in a fault-tolerant manner may not be suitable for encoding another logical quantum operation fault-tolerantly. In other words, different logical quantum operations may require different parity codes for achieving a fault-tolerant realization of the logical operation in question. In connection thereto, the parity code may be modified along the course of the quantum computation (“re-coding”), as described herein. A third difficultly is that the parity code is used in conjunction with a physical qubit implementation having a dominant qubit error. The parity code is “responsible” for correcting the dominant qubit error. Yet, by applying quantum operations to the code qubits, a basis change may occur, and hence the initial dominant qubit error may be transformed into another dominant qubit error, which may no longer be correctable by the parity code. It is therefore beneficial to operate on the code qubits only with quantum operations that can be realized by physical operations that preserve the dominant qubit error (i.e. by “bias-preserving” operations, also called “dominant-error-conserving” operations). This further complicates thetask of finding a fault-tolerant encoding, since the available set of quantum gates that can be used is thereby restricted to the set of bias-preserving operations.

[0055] The inventors have overcome all of the above difficulties and have determined a fault- tolerant realization of a universal set of quantum gates. That is to say, for each logical quantum gate in a universal gate set, a corresponding parity code and an associated encoded, fault- tolerant quantum circuit acting on the code qubits has been determined.

[0056] The inventors have found that a particular structure of the parity codes is beneficial to provide fault-tolerant implementations of logical quantum operations. Specifically, a parity code may be considered where the code qubits include a code qubit duplication set. The latter is a set of multiple code qubits (at least three code qubits) that can be regarded as copies, or duplications, of one another. Each code qubit in a code qubit duplication set contains the same information, so that a code qubit duplication set can be considered as a repetition error- correction code which is combined with the parity code. For example, a code qubit duplication set can consist of a plurality of parity qubits each having the same label “ij” for a fixed “i” and “j” (so that these parity qubits all represent the parity of a same subset {i, j} of logical qubits), or a plurality of data qubits each having the same label “i” (so that these data qubits all correspond to a same logical qubit “i”). The presence of a code qubit duplication set allows to perform the quantum operations of the encoded quantum circuit in a distributed manner that prevent potential errors from propagating to a large number of qubits. The inventors have found that this ability to restrict error propagation is particularly useful to construct fault-tolerant encoded quantum circuits based on the parity codes in question. Parity codes

[0057] A parity code is a quantum error-correction code that maps logical qubits to code qubits. The code qubits include parity qubits. Each parity qubit represents the parity of an associated subset of logical qubits.

[0058] More specifically, let |0> and |1> be quantum basis states. The quantum basis in question (computational basis) is the eigenbasis of the Z operator. A subset of logical qubits S = {l1, l2, ..., lr} is mapped, via the parity code, to a corresponding parity qubit p (being one of the codequbits). If |s1^l1 , … , |sr^lr denote the quantum basis states of the respective logical qubits, withs1, ..., sr^ {0,1}, then the corresponding quantum basis state of the parity qubit p is|s1 + ... + sr>p, where the sum s1 + ... + sr is computed modulo 2. In other words, when the parity of the set of values {s1, ..., sr} is even or odd, the corresponding basis state of the associated parity qubit is |0> or |1>, respectively.

[0059] With the above-described mapping, a Z operator acting on the parity qubit p corresponds to a product Zl1... Zlr, where Zliis a Z operator acting on the i-th logical qubit li.

[0060] A plurality of K subsets of the logical qubits can be considered. Some of the subsets may share common elements and / or some of the subsets may be disjoint from each other. Each subset may include one, two, three, four or more logical qubits, yet the disclosure is not limited thereto, and the number of elements in a subset may be arbitrarily large. Each such subset may be mapped to a corresponding parity qubit in the manner described above. Accordingly, a plurality of K parity qubits, each representing the parity of an associated subset of logical qubits, may be provided.

[0061] The code qubits may further include one or more data qubits. A data qubit is associated with a corresponding logical qubit. The data qubit may represent a quantum state of said logical qubit. For example, a logical qubit li may be mapped to a data qubit di. If|si^lidenotes the quantum basis state (computational basis state) of the logical qubit li, with si^ {0,1}, then the corresponding quantum basis state of the data qubit di is|si^di. In other words, the state of the logical qubit is copied into the data qubit.

[0062] Fig.8 shows an example of an arrangement of code qubits of a parity code. The parity code in question maps logical qubits l1, l2, ..., lnto code qubits. In the present example n = 6, yet the value of n may in general be arbitrary. For convenience, in Fig.8 the logical qubits are represented by dashed circles labeled as 0, 1, 2, 3, 4 and 5, respectively. The code qubits, which include parity qubits 802 and data qubits 804, are represented as solid circles. Each parity qubit is labelled by an index pair “ij”. The parity qubit “ij” is associated with the subset {i, j} consisting of two logical qubits “i” and “j” and represents the parity thereof, according to the mapping described above. For example, the parity qubit “01” represents the parity of the set of logical qubits {0, 1}. Further, the data qubits are represented by solid circles having a single label “0”, “1”, “2”, “3”, “4” or “5” therein. Each data qubit “i” corresponds to the logical qubit(dashed circle) with the same label “i”, by copying the basis state of the logical qubit as described above. For example, the data qubit “0” corresponds to the logical qubit 0.

[0063] A parity code is an instance of a stabilizer code. A parity code has a set of stabilizer operators. A stabilizer operator is an operator, in particular a tensor product of Pauli operators, possibly supplemented with an overall phase factor, acting on a subset of code qubits. The stabilizer operators of a parity code may mutually commute. Each stabilizer operator of the parity code may have the property that any quantum state |ψ〉codeof the code qubits which is obtained by encoding a quantum state of the logical qubits by means of the parity code (i.e. each quantum state |ψ〉codethat is in the code space of the parity code) is an eigenvector with eigenvalue 1 of all stabilizer operators of the parity code. That is to say, M|ψ〉code= |ψ〉codefor every stabilizer operator M of the parity code.

[0064] According to embodiments, a parity code has a generating set of stabilizer operators that are short-range operators. Particularly, it may be the case that each stabilizer operator from this generating set only acts within a group of code qubits that are spaced apart from each other by a distance of at most d, where d is a constant independent of the total number of code qubits, or at least d is much smaller than the total number of code qubits. For example, the code qubits may be arranged according to a mesh having a plurality of cells. The mesh may be a two- dimensional lattice, and at least some of the cells may correspond to plaquettes of the lattice. The parity code may have a generating set of stabilizer operators, wherein each stabilizer operator in the generating set acts within a group of at most four code qubits located on the vertices of a cell (e.g. a plaquette) of the mesh.

[0065] For example, in Fig. 8, the code qubits, i.e. the parity qubits and the data qubits, are arranged according to a two-dimensional lattice. Each code qubit is located at a vertex (or node) of the lattice. The parity code corresponding to the configuration shown in Fig. 8 has a generating set of stabilizer operators that each act according to a plaquette of the lattice, i.e. either a 4-body operator (such as the 4-body plaquette formed by the four code qubits “12” – “02” – “03” – “13”, or a 3-body operator (such as the 3-body plaquette formed by three code qubits “3” – “34” – “4”).

[0066] The stabilizer operators of a parity code may be used to determine whether an error has occurred on one or more code qubits during an (encoded) quantum computation, namely by measuring one or more stabilizer operators. Specifically, by measuring a stabilizer operator M(syndrome measurement), it can be verified whether the condition M|ψ〉code = |ψ〉code issatisfied. If not, the state |ψ〉codelies outside of the code space of the parity code, which implies that an error has occurred. Further, by measuring multiple stabilizer operators, it can be determined on which code qubit(s) the error has occurred. Further, it suffices to measure a generating set of stabilizer operators. Thus, if a generating set of short-range stabilizer operators is available, as described above, the syndrome measurements can be performed by measuring short-range operators only.

[0067] A stabilizer operator of a parity code can be measured by providing an ancillary qubit. The ancillary qubit may be coupled, e.g. by one or more CNOT operations, to at least some of the code qubits on which the stabilizer operator acts. Subsequently, the ancilla qubit may be measured, thereby effectively realizing a measurement of the stabilizer operator. For example, where the code qubits are arranged according to a two-dimensional lattice (as e.g. shown in Fig. 8), an ancillary qubit may be arranged inside each plaquette of the lattice, as indicated by the dots in between the solid circles in Fig.8. A measurement of a stabilizer operator corresponding to a plaquette may be realized by coupling the ancillary qubit inside the plaquette in question to all code qubits located on the nodes of the plaquette by a CNOT operation, followed by a measurement of the ancillary qubit. The measurement result indicates if the constraint defined by the stabilizer operator is still fulfilled or if an error (namely a bit-flip error) has occurred.

[0068] Given a quantum state|ψ〉logicalof the logical qubits, this quantum state is mapped to (i.e. encoded in) a quantum state |ψ〉codeof the code qubits corresponding to the parity code. This kind of encoding can be achieved in several ways. Particularly, for stabilizer codes, of which the parity codes are particular examples, encoding schemes are known. For example, a sequence of unitary operators can be applied to map|ψ〉logicalinto|ψ〉code, thus providing a unitary encoding scheme. Alternatively, the encoding can be measurement-based, wherein the encoding can be achieved by performing measurements, namely measurements of a generating set of stabilizer operators.

[0069] In the example shown in Fig.8, a (unitary) encoding scheme can be provided as follows. Assume, for the sake of concreteness, that|ψ〉logicalis a computational basis state |x1 ... xn > with x1, ..., xn ^ {0,1}, where n is the total number of logical qubits (in Fig.8, we have n = 6).

[0070] First, the data qubits (bottom row of qubits in Fig.8) are initialized in the quantum state |x1... xn > i.e. the same bit configuration as |ψ〉logical. All parity qubits are initialized in the state |0>.

[0071] Second, for each parity qubit labelled “ij” in the second row of qubits (i.e. the row of parity qubits directly above the row of data qubits), a CNOT operation is applied between said parity qubit ij and the data qubit i, and another CNOT operation is applied between said parity qubit ij and the data qubit j. For example, a CNOT operation is applied between the parity qubit “12” (target qubit) and the data qubit “1” (control qubit), and also between the parity qubit “12” (target qubit) and the data qubit “2” (control qubit).

[0072] Third, for each parity qubit labelled “ij” in the third row of qubits, a CNOT operation is applied between said parity qubit “ij” and a parity qubit in the second row having an index of the form “ik”, and another CNOT operation is applied between said parity qubit “ij” and a parity qubit in the second row having an index of the form “kj”, for some k. For example, a CNOT operation is applied between the parity qubit “13” and the parity qubit “12”, and between the parity qubit “13” and the parity qubit “23”.

[0073] Fourth, for each parity qubit labelled “ij” in the fourth row of qubits, a CNOT operation is applied between said parity qubit “ij” and a parity qubit in the third row having an index of the form “ik”, another CNOT operation is applied between said parity qubit “ij” and a parity qubit in the third row having an index of the form “lj”, and another CNOT operation is applied between said parity qubit “ij” and a parity qubit in the second row having an index of the form “kl”, for some k and some l. For example, a CNOT operation is applied between the parity qubit “03” and the parity qubit “02”, between the parity qubit “03” and the parity qubit “13”, and between the parity qubit “03” and the parity qubit “12”.

[0074] Continuing in the manner described in the previous paragraph for each subsequent (i.e., fifth or sixth) row of parity qubits in Fig.8 results in an encoding of the logical qubits into the parity qubits according to the parity code.

[0075] By reversing the above sequence of operations, decoding can be performed.

[0076] Instead of at least some of the CNOT operations between a first and a second code qubit, two subsequent CNOT operations may be applied between the first code qubit and an ancillaryqubit and between the ancillary qubit and the second code qubit. Ancillary qubits are shown as dots in Fig.8. The ancillary qubit may be the ancillary qubit at the center of a plaquette (e.g., dot within the dashed triangle or square in Fig.8).

[0077] It shall be understood that the example provided in Fig.8 is one possible example of a parity code, and the disclosure shall not be limited thereto. For example, the subsets of logical qubits that are mapped to parity qubits can include more than two qubits, and can in fact include an arbitrary number of qubits. Further, the cardinalities of the subsets of logical qubits to which the respective parity qubits are associated can vary. That is to say, a first parity qubit can be associated with a subset of k1 logical qubits, a second parity qubit can be associated with a subset of k2logical qubits (with k1different from k2), and so on. Further, the total number of data qubits can be different from the total number of logical qubits. For example, it may be the case that not each logical qubit has a corresponding data qubit and / or that, for a given logical qubit, there are several data qubits associated with said logical qubit. Further, the locations of the data qubits may be different from what is shown in Fig. 8. Further, the geometrical arrangement of the code qubits can be different from the layout shown in Fig. 8. For example, the code qubits can be arranged according to a different two-dimensional layout, or even a three-dimensional layout.

[0078] In the above discussion, a parity code is described with respect to the computational basis {|0>, |1>}. That is to say, the quantum state |s1 + ... + sr>p of a parity qubit p, which represents the parity of a subset of logical qubits, is a computational basis state. Such a parity code will be referred to herein as a Z-type parity code. In a Z-type parity code, the stabilizer operators are products of Z operators. Further, the parity code is in this case configured for correcting bit flip errors, i.e. X errors. The disclosure is not limited to Z-type parity codes. By performing a change of basis (local unitary rotation) of one or more code qubits, a different parity code is obtained, which can be capable of correcting a different type of errors. For example, a Hadamard operation H can be applied to each code qubit, resulting in another parity code, called herein X-type parity code. The stabilizer operators of an X-type parity code are products of X operators (since the Hadamard operator maps Z to X and vice versa). Further, an X-type parity code is capable of correcting Z errors on the code qubits. It shall be understood that the Hadamard operation is only an example, and that arbitrary basis changes can be considered.Bosonic constituents

[0079] At least some constituents of the physical quantum system may be bosonic constituents. A bosonic constituent includes a plurality of quantum levels of a quantum oscillator. The quantum oscillator may be a quantum harmonic oscillator or a quantum anharmonic oscillator.

[0080] Let |n^Fdenote the Fock states of the quantum oscillator, where n ranges over all nonnegative integers, i.e. n = 0, 1, 2, ... A coherent state |α^c, where α is a complex coefficient, may be defined by the formulaThe coherent state |α^cis an eigenstate of the (bosonic) annihilation operator â with eigenvalueα, in other words â|α^c = α|α^c. Coherent states and their properties are studied, for example,in the field of quantum optics. In many respects, a coherent state|α^^^resembles a macroscopic (classical) field with complex amplitude ^^ when the average boson number |^^|2is substantially larger than one, for instance larger than three, larger than five, or larger than ten.

[0081] With respect to the complex coefficient α, the associated “cat states”|Cα+^and |Cα−^ are given byThe following superpositions of the above cat states, namely(exp2)), (|C+α^ − |C−α^) / √2 = |−α^c + ^^(exp(−2|α|2)) approximately correspond to the coherent states|α^cand|−α^c, respectively, when α is large in absolute value.

[0082] A qubit (e.g. any code qubit or any ancillary qubit as described herein) may be physically implemented in a bosonic constituent by suitably identifying two quantum states ofthe bosonic constituent that can serve as a quantum basis of a qubit. The two quantum states of the bosonic constituent can be orthogonal states or quasi-orthogonal states. This may be achieved in several possible ways.

[0083] In a first example (“Example C1”), a quantum basis (computational basis) {|0>, |1>} of the qubit may be defined as (−2|α|2)) (Equation 1)(exp(−2|α|2)). (Equation 2) In a second example (“Example C2”), the quantum basis {|0>, |1>} may be defined as |0^ = |Cα+^ (Equation 3) |1^ = |Cα−^. (Equation 4) In both examples C1 and C2, the quantum states |0> and |1> are orthogonal to each other at least approximately (i.e. the inner product is zero or at least very small) and can hence be used to define a qubit, which is thereby physically realized in the bosonic constituent in question.

[0084] In light of the above, a qubit (particularly, any code qubit or ancillary qubit) can be physically implemented as a superposition of quantum basis states of a bosonic constituent, wherein the quantum basis states are coherent states or superpositions thereof, at least approximatively. For example, in both Examples C1 and C2 the quantum basis states are superpositions of coherent states (since each cat state is a superposition of coherent states). Further, in Example C1 each basis state is approximately equal to an individual coherent state. In Examples C1 and C2, each quantum basis state is a cat state (Example C2) or a superposition of cat states (Example C1).

[0085] The disclosure shall not be limited to the above examples. For example, quantum states similar to the states|Cα+^and |Cα−^ can be provided by taking linear combination of more than two, e.g. four, coherent states, particularly with mutually different phases of their respective complex coefficient. The resulting quantum states are also called cat states. A computational basis can be defined based thereon, similar to the definitions provided above in Examples C1 and C2. Additionally or alternatively, squeezed coherent states may be used instead of the coherent states defined above.

[0086] A qubit that is physically implemented as a superposition of quantum basis states of a bosonic constituent, wherein the quantum basis states are coherent states or superpositions thereof, is referred to as a cat qubit implementation, or cat qubit for short. More specifically, the quantum basis states may be cat states or superpositions thereof, as shown above in Examples C1 and C2. According to embodiments, each code qubit of at least a subset of the code qubits may be physically implemented in a bosonic constituent as a cat qubit.

[0087] For technically realizing the physical implementation of a qubit in a bosonic constituent as a cat qubit, several possible hardware systems may be used. Generally, cat qubit implementations can be realized in circuit quantum electrodynamics (cQED) systems (superconducting qubit systems), which involve photonic resonators or nano-mechanical phononic resonators, both of which are typically operated in the microwave / gigahertz regime. The resonators are typically coupled to superconducting circuits for providing control of the qubits.

[0088] For example, some hardware systems include a resonator with at least one bosonic resonator mode, wherein the resonator is an electromagnetic resonator (also called photonic resonator). An electromagnetic resonator may include a (coplanar) waveguide resonator. The electromagnetic resonator may be coupled to a superconducting circuit for control (e.g. stabilization of the cat states and performing quantum operations). The waveguide resonator might include a sputtered niobium film on a silicon substrate.

[0089] Further possible hardware systems include mechanical resonators. A mechanical resonator (also called acoustic or phononic resonator) may include a crystalline piezoelectric material. The mechanical resonator may be coupled to a superconducting circuit via the piezoelectric effect for control. Specifically, the acoustic resonator may be formed as a one- dimensional phononic-crystal-defect resonator (PCDRs) made of a crystalline piezoelectric material (such as lithium niobate).

[0090] In the hardware systems described above, a cat qubit implementation may be realized by a driven-dissipative method using a harmonic or anharmonic oscillator. Specifically, a two- photon loss channel and a two-photon drive, or a two-phonon loss channel and a two-phonon drive (depending on whether photons or phonons are used as constituents) can be engineered. A phononic or photonic resonator storing the qubit in one of its modes may be strongly coupled to a (photonic) buffer resonator with high decay rate. The decay of phonons / photons from thestorage resonator via the buffer resonator may be engineered to happen mostly in pairs. A two- phonon / photon drive to the storage resonator may be realized by driving the buffer resonator. The buffer resonator may include a non-linear superconducting element, especially an Asymmetrically Threaded SQUID (standing for “Superconducting Quantum Interference Device”).

[0091] In a further example, a cat qubit implementation can be provided using a Kerr method. A non-linear Kerr oscillator, which is an anharmonic oscillator, with a two-photon drive is engineered. The non-linear Kerr oscillator may store the qubit. For example, for photonic modes, this can be realized by a superconducting non-linear resonator placed in a three- dimensional microwave cavity. Phonon implementations can likewise be provided.

[0092] It shall be understood that the foregoing are merely examples of possible hardware systems for realizing cat qubits. The disclosure shall not be limited thereto, and further examples of hardware systems can be provided. Physical implementations with dominant qubit error

[0093] As described herein, a qubit (specifically a code qubit or ancillary qubit) is physically implemented in a subsystem of the physical quantum system according to a physical implementation having a dominant qubit error. The subsystem may consist of a single constituent or may include several constituents. Different types of physical implementations with dominant qubit error may be considered, such as (a) error-suppressing physical implementations, which provide an “automatic”, i.e. passive, suppression of one or more qubit errors due to the particular physical properties of the subsystem, or (b) physical implementations that are configured to allow active error correction of at least one qubit error different from the dominant qubit error. Examples of physical implementations having a dominant qubit error are described in the following.

[0094] In a first example, a qubit may be physically implemented in a bosonic constituent as described herein. In this case, the subsystem in which the qubit is implemented may consist ofa single bosonic constituent (corresponding to the situation depicted in Fig.2). Specifically, the qubit may be physically implemented as a superposition of quantum basis states of the bosonic constituent, wherein each quantum basis state may be a coherent state or a superposition of coherent states, particularly a cat qubit implementation as described herein.

[0095] Cat qubit implementations are instances of physical implementations having a dominant qubit error by way of error-suppression, i.e. category (a) described above. That is to say, in a cat qubit, certain qubit errors are automatically suppressed, i.e. less likely to occur, as compared to the dominant qubit error, due to the specific design of the system, without a need for active error correction. Especially, increasing a control parameter of the cat qubit implementation may linearly increase the dominant error rate but exponentially suppress a certain error. The control parameter may be an amplitude of the drive of the buffer resonator as described above.

[0096] In a bosonic constituent, a main source of error may be a single-photon loss or single- phonon loss, depending on which type of constituent is used. Such loss is mathematically described by the action of the annihilation operator â. In Example C1 described above, where the computational basis states are approximately the coherent states|α^cand|−α^c, the dominant qubit error is represented by the phase flip operator Z, since a photon / phonon loss induces a phase flip onto the coherent state |−α^c. In this example, the bit flip noise (represented by the operator X) is exponentially suppressed, and an active error correction of bit flip errors is hence substantially unnecessary. In Example C2, where the computational basis states are the cat states|Cα+^and |Cα−^, the dominant qubit error is represented by the bit flip operator X, since a photon / phonon loss induces a change from even to odd and vice versa. In example C2, the phase flip noise (represented by the operator Z) is exponentially suppressed, and an active error correction of phase flip errors is hence substantially unnecessary.

[0097] In other words, depending on the specific physical implementation that is selected for implementing a qubit in a bosonic constituent, the dominant qubit errors may be X errors or Z errors.

[0098] In a second example of a qubit implementation with dominant qubit error, a qubit may be physically implemented in a superconducting two-level constituent, especially a flux constituent. This kind of implementation is another instance of a physical implementations having a dominant qubit error by way of error-suppression, i.e. category (a) described above.Further, in this example, a single constituent may be used to implement the qubit therein (i.e. is again an example of the situation depicted in Fig.2).

[0099] For example, a flux constituent may include computational basis states which correspond to distinct circulating-current orientations, i.e., clockwise or anticlockwise circulating currents, or superpositions thereof. Superconducting flux constituents may have a dominant error. When the computational basis states correspond to distinct circulating-current orientations, the dominant error may be the phase-flip error, while bit-flip errors are suppressed. When the computational basis states correspond to equally weighted superpositions of the distinct circulating-current orientation states, the dominant error may be the bit-flip error, while phase-flip errors are suppressed.

[0100] In a third example of a qubit implementation having a dominant qubit error, a qubit may be physically implemented in a subsystem that includes several constituents (corresponding to the discussion relating to Figs.3-4). This is an example of an implementation where at least one qubit error other than the dominant qubit error can be corrected via active error correction on the subsystem, i.e. category (b) as described above. The idea is to encode the qubit into the subsystem by means of an error-correction code, e.g. a repetition code (or any other error correction code), that allows (actively) correcting one type of qubit errors (e.g. X errors or Z errors, as the case may be). The error type that cannot be corrected by the error-correction code is then the dominant qubit error of the physical implementation in question.

[0101] For example, the subsystem may consist of three constituents, where each of the three constituents is configured for implementing a single qubit therein. The physical nature of these constituents is not particularly critical, in other words they may be any type of constituent that allows realizing a qubit (atoms, ions, photon, phonons, and the like). Each of the three constituents allows defining a respective qubit computational basis {|0>, |1>}, so that three computational bases are provided in total. A qubit q, e.g. any code qubit or ancillary qubit as described herein, may then be physically implemented in the subsystem consisting of the threeconstituents by defining the computational basis {|0^^^ , |1^^^} of the qubit q within the three-constituent subsystem as follows:|1^^^ = |1^|1^|1^where, in each of the above expressions, the three computational basis states on the right are the basis states of the three constituents as considered above. In other words, this is an example where the qubit q is physically implemented in a subsystem of three constituents by means of a three-qubit repetition code. Such a code allows correcting bit flip errors (X errors) by active error-correction, which may include performing one or more syndrome measurements (stabilizer measurements) on the subsystem and, based on the outcomes of the syndrome measurements, performing error-correcting operations on the subsystem in order to remove the error. Since bit flip errors can thereby be corrected, phase flip errors (Z errors) are in this example the dominant qubit errors, since these errors cannot be corrected using the above repetition code. It is noted that, in this example, the repetition code is a second error-correction code that is used in conjunction with the parity code (code concatenation). The function of the repetition code is, for example, to allow correction of X errors, and the role of the parity code is to allow correction of the Z errors (or vice versa, depending on the chosen local basis), since the Z errors are the dominant qubit error, which cannot be corrected (in this example) by the repetition code.

[0102] The above example of a repetition code is merely one possible example. Other repetition codes can be provided, e.g. by considering more than three constituents, or by performing a local basis change in each constituent. A change of basis may affect the type of errors that can be corrected by the repetition code. E.g., by performing a Hadamard rotation to each constituent, the resulting repetition code may be configured to correct Z errors, so that X errors are in this case the dominant qubit errors. Further, it is not necessary to restrict to repetition codes, and other codes (e.g. surface codes, in particular asymmetric surface codes) may likewise be used. Bias-preserving quantum operations

[0103] The code qubits are physically implemented in subsystems of the physical quantum system according to a physical implementation having a dominant qubit error. The code qubits are acted upon by an encoded quantum circuit to provide an encoded version of one or more logical quantum operations acting on the logical qubits. The quantum operations (e.g. unitary operators and measurements) that form the encoded quantum circuit are in turn physically realized as physical operations acting on the physical quantum system, specifically acting on the subsystems in which the code qubits are physically realized. It is beneficial that the physicalquantum operations acting on the physical quantum system are bias-preserving quantum operations with respect to the dominant qubit error of the physical implementation at hand.

[0104] A bias-preserving quantum operation (synonymously called “dominant-error- conserving operation” herein) can be understood as an operation that does not transform the dominant qubit error into an error of a different type. The term “bias-preserving” stems from the property that the dominant qubit error, i.e. the “noise bias”, is preserved by the physical quantum operations in question. For example, if the physical implementation under consideration has the X operator (bit flip error) as the dominant qubit error, a bias-preserving physical quantum operation maps the operator X to a transformed error operator Xtransf, where the transformed error operator Xtransfis not, and does not contain, a Z error. A special case of a bias-preserving physical quantum operation arises when the physical quantum operation commutes with X, in which case Xtransf is simply equal to X. If a physical quantum operation is not bias-preserving, the dominant qubit error may be transformed by the physical quantum operation into a qubit error having a different error type than the dominant qubit error (e.g. X may be transformed into Z), and said transformed qubit error might no longer be correctable by the parity code. In other words, the quantum system would, after application of the physical quantum operation, have a dominant qubit error against which the parity code cannot protect. Conversely, bias-preserving physical quantum operations prevent such a situation from occurring, and ensure that the dominant qubit error is preserved after each physical operation acting on the physical quantum system, so that this error is correctable by the parity code throughout the computation.

[0105] Mathematically, a bias-preserving physical operation can be an operation that maps the dominant qubit error into an error that commutes with the dominant qubit error. For example, this condition is fulfilled if the physical quantum operation itself commutes with the dominant qubit error. Accordingly, any physical quantum operation that commutes with the dominant qubit error is bias-preserving. Yet there also exist bias-preserving physical operations that do not commute with the dominant qubit error. A physical operation U acting on a single codequbit may transform the dominant qubit error X to Xtransf= UXU+. The physical operation Umay be bias-preserving if UXU+commutes with X (which is trivially fulfilled when U commutes with X). The discussion can be straightforwardly extended to operations U acting jointly on multiple code qubits. In such a case, U may be bias-preserving if, for any code qubitsk and j, Xtransf,k= UXk^^+commutes with the dominant qubit error Xj(where Xkand Xjcorrespond to an X error occurring on qubit k and j, respectively).

[0106] A bias-preserving physical quantum operation may be physically realized such that the dominant qubit error is preserved at almost all times during the evolution of the physical quantum operation. For physical implementations of subsystems with error suppression, error suppression may be effective at almost all times during the application of the physical quantum operation to ensure dominant-qubit-error conservation. For physical implementations of subsystems with active error correction, the code distance may stay the same, or at least not decrease below a minimum value, during the application of the bias-preserving physical quantum operation to ensure dominant-qubit-error conservation.

[0107] For several physical implementations, it can be shown that one or more operations (unitary operations and / or measurements) acting on the code qubits can be realized in a bias- preserving way, i.e. using bias-preserving physical operations only. For example, in the case of a physical implementation of qubits into bosonic constituents with dominant qubit error X as described herein (such as a cat qubit implementation), it is known (see e.g. GUILLAUD et al. “Repetition Cat Qubits for Fault-Tolerant Quantum Computation”, Phys. Rev. X 9, 041053 (2019)) that each of the following operations (O1)-(O6) can be physically realized by bias- preserving physical operations acting on the bosonic constituents: (O1) preparation of a qubit in any one of the states |0>, |1>, |+> or |->, i.e. the eigenstates of Z and X, respectively; (O2) Z measurements and X measurements; (O3) the Z gate; (O4) the X gate; (O5) the CNOT gate; and (O6) the two-qubit X rotation gate with arbitrary angle.

[0108] When a qubit is measured according to a Z measurement, the qubit is projected onto either the quantum state |0> (corresponding to the measurement outcome “0”) or the quantum state |1> (corresponding to the measurement outcome “1”). When a qubit is measured accordingto an X measurement, the qubit is projected onto either the quantum state |+> (corresponding to the measurement outcome “0”) or the quantum state |-> (corresponding to the measurement outcome “1”).

[0109] Specifically, in a driven-dissipative cat qubit implementation, state preparations as per (O1) can be performed in a bias-preserving manner by ramping up the amplitude of the drive of the buffer resonator from a value close to zero to a final value, and thereby transforming a vacuum state of a mode of the storage resonator to the even cat state |0>. The state |1> may be obtained from the |0> state by applying a bias-preserving X gate (as detailed below), and the |+> or |-> state may be obtained from the |0> state by applying an X measurement (as detailed below) followed by a Z gate (as detailed below) which is applied when the measurement outcome is 1 or 0, respectively. Further, Z measurements (see (O2)) can be performed in a bias- preserving way by dispersively coupling the mode of the storage resonator (or an intermediate read-out mode of a read-out resonator to which the quantum state was transferred) to a two- level constituent (e.g., a transmon), and measuring the two-level constituent. Further, X measurements (see (O2)) can be performed in a bias-preserving manner by transferring the quantum state from the mode of the storage resonator to a mode of the buffer resonator, followed by a homodyne measurement on the buffer mode. A Z operation, or Z gate (see (O3)), can be realized in a bias-preserving manner by tuning the phase of the drive on the buffer resonator from 0 to ^ in an adiabatic manner (since the phase of the coherent state is fixed by the phase of the driving field). An X operation, or X gate (see (O4)), may be implemented in a bias-preserving way by adding an additional weak resonant (i.e. single-photon / phonon) pump for a fixed time. Further, a CNOT gate (see (O5)) between a first and a second qubit can be implemented in a bias-preserving way by adding a further dissipation channel, where dissipation of a mode of the storage resonator of a first qubit is engineered to be conditioned on the state of the mode of the storage resonator of a second qubit, e.g., by coupling both storage resonators to a same buffer resonator. A two qubit X rotation by an arbitrary angle ^^ may be written as ^^^^^^^^^^^^^^ / 2(see (O6)) and can be implemented by applying a weak beam-splitter Hamiltonian in the presence of the two-photon / phonon driven dissipation.

[0110] In light of the above, for any arbitrary sequence of operations taken from the above list (O1)-(O6), a bias-preserving physical implementation (with respect to the dominant qubit error X) acting on the bosonic constituents is available. Specifically, in the examples of fault-tolerant gate implementations described below (see Examples G1-G8 and M1-M2), each encodedquantum circuit is built from operations (O1)-(O6). Hence, each of the encoded quantum circuits in question can be realized in a bias-preserving manner when the code qubits (and any ancillary qubits) are physically implemented into bosonic constituents.

[0111] When the physical implementation involves a (bit-flip correcting) repetition code on a subsystem of at least three constituents as described above, phase-flips are the dominant errors and hence the Hadamard-rotated versions of the operations (O1)-(O6) need to be implementable (hence, instead of the ^^^^^^^^^^^^^^ / 2gate, the ^^^^^^^^^^^^^^ / 2gate needs to be implemented). State preparations as per (O1) can be performed in a bias-preserving manner by preparing all constituents of the subsystem in the state |0>, which leads to the qubit state |0^^^of the subsystem. The qubit state|1^^^of the subsystem can be prepared by additionally applying an X operation on each constituent. The qubit states |+^^^and |−^^^of the subsystem can be prepared by additionally applying CNOT gates between each constituent of the subsystem and an ancillary constituent in the |+> state, performing an X measurement on the ancillary constituent, and applying a Z operation when the outcome is 1 or 0, respectively, to one of the constituents of the subsystem (wherein the procedure may be repeated for guaranteeing fault- tolerance on the level of the physical implementation). A bias-preserving X or Z measurement (see (O2)) can be performed by measuring all constituents in the X or Z basis, respectively, and calculating the parity or majority vote, respectively, of all measurement outcomes. In said parity, the outcomes, which can be 0 or 1, are mapped to a single bit obtained from the sum of all measurement outcomes modulo two. In said majority vote, the outcomes, which can each be 0 or 1, are mapped to a single bit having the value of the majority of measurement outcomes. A bias-preserving Z gate (see (O3)) can be performed by applying a physical Z gate to one of the constituents of the subsystem. A bias-preserving X gate (see (O4)) may be performed by applying a physical X gate to each of the constituents of the subsystem. A CNOT gate (see (O5)) can be performed by applying physical CNOT gates between each of the constituents of two different subsystems in a transversal manner. A two-qubit Z rotation (Hadamard-rotated version of (O6)) can be performed by applying a physical two-qubit Z rotation to one constituent of a first subsystems and one constituent of a second subsystem. Fault-tolerant implementation of universal set of quantum gates

[0112] A quantum computation acting on the logical qubits (as e.g. depicted in Fig. 5) may include a sequence of quantum operations (e.g. unitary operators and / or measurements) that areapplied one after the other to the logical qubits. A quantum operation acting on one or more logical qubits is referred to herein as a logical quantum operation.

[0113] As described herein, the logical qubits are encoded in the code qubits using a parity code. Correspondingly, each logical quantum operation is mapped, via the parity code, to an encoded quantum circuit providing an encoded realization of the logical quantum operation, said encoded quantum circuit acting on the code qubits. The encoded quantum circuit can be (and often is) a sequence of multiple quantum operations acting on the code qubits (cf. the blocks 601-605 in Fig. 6), or can in some cases consist of a single quantum operation. The encoded quantum circuit, as a whole, provides an encoded realization of the logical quantum operation in question.

[0114] In Examples G1-G8 and M1-M2 described below, a number of logical quantum operations are considered. In each example, it is demonstrated how a particular logical quantum operation is mapped to an encoded quantum circuit. In each example, a suitable parity code is described that may be used for providing the encoding of the logical quantum operation in question. Further, it is described how the parity code in question can be obtained, i.e., which additional quantum operations (called re-coding operations, or code deformation operations), if needed, may be performed to modify an initial parity code in order to encode the logical qubits according to the parity code in question. For the sake of concreteness, but without limitation, the examples described below are based on the parity code shown in Fig.8. Depending on the logical quantum operation under consideration, this particular parity code is used as such, or is further re-coded, as the case may be.

[0115] In Examples G1-G8 and M1-M2, the qubits (code qubits as well as any ancillary qubits) are physically implemented using a physical implementation having a dominant qubit error. For the sake of concreteness, but without limitation, a physical implementation of the qubits in bosonic constituents having the X operator (bit flip error) as dominant qubit error is considered (such as a cat qubit implementation). Further, all parity codes in the examples below are Z-type parity codes. It is recalled that such parity codes are capable of correcting X errors, i.e. the dominant qubit error of the physical implementation in question, so that error-correction of arbitrary errors is possible in each example. As described herein, this particular choice of the physical implementation and the parity code is exemplary and does not entail a limitation, since a local basis change of the code qubits allows to change the type of error that can be correctedby the parity code. By performing a basis change, the parity code can also be combined with a physical implementation that has, for example, Z errors as the dominant qubit errors, or any other errors.

[0116] In each of the examples below, the encoded quantum circuit provides a fault-tolerant implementation of the logical quantum operation in question. The notion of fault-tolerance is described in the following.

[0117] The notion of a fault-tolerant quantum circuit refers to a realistic situation wherein errors on the qubits may occur at any time during the encoded quantum circuit (e.g. due to an unwanted interaction of the quantum system with the environment) and where, moreover, every quantum operation that is performed throughout an encoded quantum circuit can itself be faulty, i.e. can introduce further errors into the computation. Quantum operations that may be faulty include state preparation operations, unitary operations that are part of the encoded quantum circuit, measurements that are part of the encoded quantum circuit, and the like. The encoded quantum circuit is said to be fault-tolerant if, provided that the error rate (the probability that a quantum operation is faulty) is below a given threshold, errors can be corrected with high probability, in spite of the fact that the quantum operations can themselves be erroneous. In other words, a quantum circuit is fault-tolerant if it can be reliably performed even when its elementary components are subject to failure. If a fault-tolerant quantum circuit is an encoded realization of a particular logical quantum gate, the quantum circuit is said to be a fault-tolerant implementation of the logical quantum gate.

[0118] In Examples G1-G8 and M1-M2, fault-tolerance is achieved due to a combination of multiple factors. A detailed discussion of why the circuits in question are fault-tolerant is provided below for each particular example. Still, the following remarks identify common ingredients that are present in all examples.

[0119] In each of the examples below, the encoded quantum circuit includes only quantum operations (unitary operators and / or measurements) that can be realized by a bias-preserving physical quantum operations within the physical implementation under consideration i.e. the qubits being implemented in bosonic constituents. In particular, each encoded quantum circuit contains only quantum operations from the list (O1)-(O6) described above, which can be realized in a bias-preserving way when the qubits are physically implemented in bosonic constituents. Since all physical operations applied to the physical quantum system are bias-preserving, the dominant qubit error (being the X operator in the present examples) is conserved throughout the encoded quantum circuit. Since the Z error is suppressed by virtue of the chosen physical implementation, this means that, throughout the encoded quantum circuit, only the dominant qubit error can realistically occur, and hence this is the only error that needs to be corrected.

[0120] Since a parity code is designed to offer protection against X errors (and the same is true for a repetition code, which is used in some of the examples for encoding ancillary qubits), an X error occurring on a sufficiently small set of code qubits (the size of said set depends on the distance of the parity code) can be corrected with high success probability by performing quantum error-correction, e.g. by measuring one or more stabilizer operators of the parity code (syndrome measurements), possibly followed by performing one or more error-correction operations. It is beneficial to perform such quantum error-correction at regular intervals throughout the encoded quantum circuit, e.g. directly before and directly after each quantum operation of the encoded quantum circuit is performed.

[0121] Since errors occurring simultaneously on large sets of qubits are unlikely (the error probability decays exponentially as a function of the number of qubits), only errors occurring on small sets of qubits need to be correctable by the parity code. However, since an encoded quantum circuit may involve interactions between the qubits, the quantum operations applied during the encoded quantum circuit could in principle cause an uncontrolled spreading, i.e. propagation, of the error. Accordingly, an error that initially occurred on a small set of qubits (e.g. at the start of the encoded quantum circuit) can be converted, through an application of the subsequence quantum operations, into an error on a larger set of qubits, which might not be correctable by the parity code. To prevent such a cascading of errors, the encoded quantum circuits described in the examples below have a particular fault-tolerant structure, which ensures that the propagation of X errors (being the dominant qubit error under consideration) is restricted. Particularly, an X error occurring on any qubit at any time during an encoded quantum circuit can be propagated to at most a few additional qubits (or even at most a single additional qubit) by the quantum operations of the encoded quantum circuit. It is said that the encoded quantum circuit has a fault-tolerant error propagation of X errors. Notably, since an X error (or more generally the dominant qubit error) is the only error that can realistically occur, only the error propagation of X errors needs to be considered.

[0122] The notion of fault-tolerant error propagation is further illustrated in Figs.9-10. Fig.9 shows a quantum circuit acting on n qubits 901 through 906, where n = 5 in the illustrated example, yet it is considered that n may be a much larger number. A sequence of n-1 CNOT operations is applied, where each CNOT operation acts on the same qubit 901 (control qubit) and one respective additional qubit (target qubit). If an X error occurs on qubit 901 at the start of the quantum circuit, i.e. before the first CNOT gate is applied, this X error will spread to all other qubits 902-906 in the system, since the qubit 901 is coupled to each other qubit by a respective CNOT gate. Thus, at the end of the quantum circuit, i.e. after the last CNOT gate has been applied, the X error, which was initially a single-qubit error, has propagated to all remaining qubits 902-906, and is now an error acting on all n qubits simultaneously, which might not be correctable by a parity code. Accordingly, Fig. 9 is an example of a quantum circuit that does not have a fault-tolerant error propagation of the X error.

[0123] Fig.10 shows a quantum circuit acting on n qubits 1001 through 1010, where n = 10 in the illustrated example, yet it is again considered that n may a large number. It may be considered that qubits 1001 through 1005 form a first encoded block and qubits 1006 through 1010 form a second encoded block. For example, an encoded block may consist of data qubits that are all associated with a same logical qubit “i” (e.g. data qubits each labelled “3”), may consist of parity qubits that are all dependent on the same logical qubit “i” (e.g. parity qubits each of the form “3j” or “j3”, where j is arbitrary), or may consist of ancillary qubits. It may be the case that the two encoded blocks do not relate to a same logical qubit, in other words that the two encoded blocks are “logically independent” from each other.

[0124] A sequence of five CNOT gates (or more generally n / 2 CNOT gates) is applied between the first encoded block and the second encoded block in the manner shown, where each CNOT gate acts on a qubit “i” belonging to the first encoded block and qubit “i + 5” belonging to the second encoded block. For example, the first CNOT gate acts on qubits 1001 and 1006, the second CNOT gate acts on qubits 1002 and 1007, and so on. The present quantum circuit is an example of a quantum circuit having a fault-tolerant propagation of the X error. If an X error occurs, for example, on qubit 1001 directly before the first CNOT gate is applied, this error can spread at most to qubit 1006, since qubit 1001 is only coupled to qubit 1006 (namely, by the first CNOT gate) and not to any other qubit, and moreover qubit 1006 itself is not coupled to any other qubits. Particularly, the remaining CNOT gates act neither on qubit 1001 nor on qubit 1006, and can hence do not cause a further propagation of the error to other qubits. Thus, at theend of the quantum circuit, an X error that occurred initially on qubit 1001 will have propagated to yield an error on at most on two qubits 1001 and 1006, i.e. only one additional qubit is affected. Particularly, the initial X error is propagated to one qubit error in the first encoded block and one qubit error in the second encoded block. The same reasoning applies to any single-qubit X error occurring on any qubit at any time during the quantum circuit: such an error can propagate to at most one additional qubit, namely the qubit that is coupled to the qubit in question by the respective CNOT gate. Particularly, an X error occurring on any single qubit can be propagated to at most one qubit error in each encoded block.

[0125] Due to the design of the circuit, the fact that an X error can propagate to at most one additional qubit is independent of the total number n of qubits on which the quantum circuit acts. That is to say, if the number n is increased, the conclusion remains the same, namely that an X error is converted into at most an error on two qubits. A circuit having a structure as shown in Fig. 10 is referred to herein as a transversal quantum circuit, or a circuit providing a transversal application of CNOT gates. The notion of transversal quantum circuits is of course not limited to CNOT gates, and can involve any other quantum gate.

[0126] The majority of examples G1-G8 and M1-M2 involve parity codes having a particular structure that allows to achieve fault-tolerance of the respective encoded quantum circuit, and in particular to achieve a fault-tolerant propagation of the dominant qubit error. In the parity codes in question, a redundancy is introduced such that the information in the relevant code qubits (namely the code qubits associated with the logical qubits on which the logical gate acts) is copied, i.e. duplicated, into a plurality of code qubits. For example, if a logical gate acts on logical qubits “i” and “j”, then the parity code used for realizing an encoded version of the logical gate may have a plurality of parity qubits each representing the parity of the set {i, j}. Each of these parity qubits contains the same information, and it is this duplication of information that allows constructing an encoded quantum circuit that has a fault-tolerant propagation of the dominant qubit error. A set of code qubits that are duplications of a same code qubit is referred to herein as a code qubit duplication set. As described above, a code qubit duplication set may include several duplicated parity qubits, but may in other examples likewise contain a plurality of duplicated data qubits, which all correspond to a same logical qubit. For the sake of simplicity, the code qubit duplication sets included in the examples below include three qubits. Yet the disclosure is not limited thereto, and a code qubit duplication set can include five, ten, twenty or even more qubits. Nevertheless, a code qubit duplication setconsisting of a total of three code qubits already has the desired effect of creating a redundancy that allows the encoded quantum circuit to have a fault-tolerant propagation of the dominant qubit error (being the X error in the present case).

[0127] A further aspect regards the effect of the dominant qubit error on any measurements included in the encoded quantum circuits. When a qubit is measured according to a Z measurement, the qubit is projected onto either the quantum state |0> (corresponding to the measurement outcome “0”) or the quantum state |1> (corresponding to the measurement outcome is “1”). If an X error (dominant qubit error) occurs directly before the measurement, the quantum states |0> and |1> are interchanged by the X operator, so that the outcome of the Z measurement is erroneous (e.g. outcome “0” is obtained where the outcome should have been “1”). Accordingly, it is possible that the outcomes of Z measurements are erroneous. Accordingly, such errors need to be corrected. In the examples below, this is achieved by performing Z measurements in parallel on multiple qubits that contain the same information (namely code qubits in a code qubit duplication set, or ancillary qubits that are similarly duplicated), and to remove any erroneous measurement outcomes by performing classical error- correction on the measurement outcomes, particularly in the form of majority voting. Thus, also here, the duplication of information achieved by the code qubit duplication sets allows correcting errors, and thus achieve fault-tolerance.

[0128] In contrast, it is noted that X measurements commute with X errors, so that such measurements are not affected these errors. Further, errors that could in principle affect the outcome of X measurements, namely Z errors, are suppressed due to the chosen physical implementation. Hence, there is no need to consider the case of erroneous X measurements since any errors that could disturb X measurements are suppressed.

[0129] Examples G1-G8 include a set of logical gates (where “gate” is in this context used synonymously with “unitary operator”) that forms a universal gate set for quantum computation. This means that an arbitrary quantum computation (arbitrary unitary operator) can be obtained by performing quantum circuits of gates taken from the universal gate set. For example, the CNOT gate, the TH gate and the Hadamard gate (Examples G3, G7 and G8, respectively) form a universal gate set. Thus, a fault-tolerant implementation of a universal gate set is provided. Since each logical gate in a universal gate set is implemented in a fault-tolerantmanner, any arbitrary quantum computation can be performed fault-tolerantly. Embodiments described herein thus allow for universal fault-tolerant quantum computation. Examples G1-G8 and M1-M2

[0130] In the following, examples of encoded, fault-tolerant implementations Gencodedof several logical quantum operations Glogical are provided.

[0131] For the sake of concreteness, the examples described below are based on the parity code shown in Fig.8. Depending on the logical quantum operation under consideration, this particular parity code is used as such, or is further re-coded, as the case may be.

[0132] It is noted that the dots shown in Fig. 8 in between the code qubits may represent additional qubits. These additional qubits together with the code qubits form a two-dimensional lattice, where a qubit is disposed on each lattice site. The additional qubits on the lattice sites represented by the dots may initially be in a default initial state, e.g. the state |0>. If a re-coding of the parity code is performed, one or more of these qubits may be coupled into the parity code, so that they become additional parity qubits or data qubits in a re-coded version of the parity code. Alternatively, one or more of the qubits in question may serve as ancillary qubits, which might not be part of the parity code but which are useful to realize certain quantum gates fault- tolerantly, e.g. by means of a gate teleportation protocol.

[0133] It is further considered, but not shown in the figures, that error-correction may be performed at any time during the encoded quantum circuit, e.g. directly before and after each quantum operation of the encoded quantum circuit, to correct X errors. This regards error- correction with respect to the parity code, and also with respect to the repetition code according to which the ancillary qubits, if present, are encoded. The additional qubits shown in Fig.8 may serve as measurement qubits for error correction.

[0134] It is further repeated that each of the examples below involves only quantum operations that can be realized in a bias-preserving way within the physical implementation under consideration, namely an implementation of the qubits in bosonic constituents, particularly a cat qubit implementation. Specifically, only CNOT gates, X and Z gates and X and Z measurements are used in the encoded quantum circuits, constituting the list of quantum operations (O1)-(O6) described above. Further, any magic states that are used as inputs to theencoded quantum circuits can be prepared in a bias-preserving manner as well, as will be described in more detail below.

[0135] In Example G1, the logical quantum gate Glogicalis the gate Z, i.e. the phase flip operator. The gate Z acts on a single logical qubit according to the action |0> → |0> and |1> → -|1>. The gate Z acts, say, on the i-th logical qubit. For realizing an encoded version of the logical quantum gate Z, a parity code is used that includes a data qubit corresponding to the i-th logical qubit. Recoding may not be needed. The encoded quantum circuit Gencoded corresponding to the Z gate is also a Z gate, namely the Z gate acting on the data qubit in question. This is a direct consequence of the definition of a data qubit, namely a data qubit “copies” the basis state of the associated logical qubit. Hence, a logical Z operator acting on the logical qubit corresponds to a Z operator acting on the data qubit. Since Gencodedthus acts on a single code qubit, Gencodedcannot cause a propagation of a single-qubit error to multiple qubits. Accordingly, this implementation has a fault-tolerant error propagation of X errors (and of any qubit errors). Particularly, the implementation in question is fault-tolerant.

[0136] Fig. 11 provides an illustration of Example G1. As shown in part a) of Fig. 11, the logical quantum gate Glogical is the gate Z acting on the logical qubit labelled “2” (dashed circle), where the logical qubit “2” is merely chosen for the sake of concreteness. As shown in parts b) and c) of Fig.11, in the system of code qubits (solid circles), the encoded circuit Gencodedis also a Z gate, now acting on the corresponding data qubit, i.e. the date qubit labelled “2”.

[0137] In Example G2, the logical quantum gate Glogical is the gate X, i.e. the bit flip operator. The gate X acts on a single logical qubit according to the action |0> → |1> and |1> → |0>. The gate X acts, say, on the i-th logical qubit. For realizing an encoded version of the logical quantum gate X, a parity code is used that includes a data qubit corresponding to the i-th logical qubit. Recoding may not be needed. The encoded quantum circuit Gencoded is a (tensor) product of several operators X acting on a subset of the code qubits. Specifically, an X operator is applied to (1) the data qubit corresponding to the i-th logical qubit, and (2) each parity qubit where the subset of logical qubits associated with the parity qubit contains the i-th logical qubit. Since Gencoded consists of single-qubit operators only, an X error occurring on any single qubit cannot spread to any other qubits. Thus, this quantum circuit has a fault-tolerant error propagation of X errors (and any other errors). Particularly, the implementation in question is fault-tolerant.

[0138] Fig. 12 provides an illustration of Example G2. As shown in part a) of Fig. 12, the logical quantum gate Glogical is the gate X acting on the logical qubit labelled “2”. As shown in parts b) and c) of Fig.12, the encoded circuit Gencodedthat acts on the code qubits is a product of X operators, where an X operator acts on (1) the data qubit labelled “2” (being the data qubit that corresponds to the logical qubit on which Glogical acts), and (2) all parity qubits “ij” where one of the indices i and j is equal to “2”, namely the parity qubits “02”, “12”, “23”, “24” and “25”. The logical quantum gate Glogicalcorresponds to a logical bit-flip, i.e., a change of the bit value of the computational basis states, |0>|0>. Hence, all data qubits representing the state of logical qubit “2” (those with index “2”) shall change their bit value as well, i.e., an X operator is applied to each data qubit in Gencoded. In Fig. 12, there is one data qubit with index “2”, consequently an X operator is applied to said data qubit. Further, a change of a bit value of a logical qubit with index “2” flips any parity value which depends on the bit value. Thus, an X operator is applied to each parity qubit comprising the index “2” in Gencoded.

[0139] In Example G3, the logical quantum gate Glogicalis the CNOT (controlled-not) gate. The CNOT gate acts on two logical qubits according to the action |x, y> → |x, x+y>, where x, y ^ {0, 1} and the sum x + y is computed modulo 2. In the definition of the CNOT gate, the first qubit (whose state is indicated by x) is called the control qubit, and the second qubit (whose state is indicated by y) is the target qubit. The CNOT gate has, say, the i-th logical qubit and the j-th logical qubit as control qubit and target qubit, respectively.

[0140] Fig. 13 further illustrates a fault-tolerant implementation of the logical gate Glogical = CNOT. As shown in part a) of Fig.13, in the present example, the CNOT gate 1310 acts on the logical qubit “3” (being the control qubit) and the logical qubit “2” (being the target qubit). These particular logical qubits are of course merely considered for the sake of illustration and the disclosure is not limited thereto.

[0141] For realizing an encoded version of the CNOT gate, a recoding of the parity code may be performed. The recoding may include removing some of the parity qubits from the parity code and adding data qubits to the parity code, as illustrated in part b) of Fig.11. Specifically, the parity qubits contained in the dashed rectangle (i.e. the parity qubits with labels “03”, “04”, “05”, “13”, “14”, “15”, “23”, “24” and “25”) may be removed from the parity code. Removing a qubit from the parity code can be understood in the sense that the qubit is disentangled from other code qubits (yet the qubit as such may continue to be present in the system). For example,the quantum state of the qubit in question may be set to a default state, such as |0>. Removing a qubit from the parity code can be performed, for example, by performing one or more CNOT operations (or another unitary operation) to disentangle the qubit from the remaining code qubits, or by measuring the qubit in question and applying correction operations depending on the measurement outcome to code qubits on the parity code, or a combination thereof.

[0142] By removing the code qubits in the dashed rectangle, two sets of code qubits remain in the parity code, namely the code qubits inside the two dashed triangles. The left triangle consists of all parity qubits and data qubits that include a label “0”, “1” or “2” while not containing the label “3”, “4” or “5”. Conversely, the right triangle consists of all parity qubits and data qubits that include a label “3”, “4” or “5” while not containing the label “0”, “1” or “2”. Notably, the dashed rectangle, i.e. the code qubits that are removed from the parity code, are those parity qubits “ij” where one of the labels “i” and “j” is in the set {“0”, “1”, “2”} and the other one of the labels “i” and “j” is in the set {“3”, “4”, “5”}. The removal of the code qubits in the dashed rectangle amounts to splitting the parity code into two groups, namely a first group (left triangle) associated to the target qubit “2” of the CNOT gate as well as all labels smaller than “2” (in this example: “0” and “1”) and a second group (right triangle) associated to the control qubit “3” as well as all labels larger than “2” (in this example: “4” and “5”).

[0143] The two triangles may be moved closer to each other. For example, the right-hand triangle may be moved one lattice spacing to the left (or vice versa, the left-hand triangle may be moved to the right). This may be achieved by suitably swapping the qubits (e.g. by performing SWAP gates, or by physically interchanging the qubits). As a result, the two triangles are adjacent to each other, i.e. on neighboring lattice sites, as shown in the bottom portion of part b) of Fig.11. Particularly, the data qubits labeled “2” and “3” (corresponding to the target qubit and the control qubit, respectively, of the CNOT gate under consideration) are adjacent to each other.

[0144] Further, additional data qubits corresponding to the control qubit “3” may be added to the parity code, forming a code qubit duplication set 1350, as shown in part b) of Fig. 13. A respective additional data qubit labelled “3” may be provided adjacent to each parity qubit containing the label “2” (i.e. the label of the target qubit of the CNOT gate). In the present example, this means that two additional data qubits with label “3” are added, namely a data qubit adjacent to parity qubit “02” and another data qubit adjacent to parity qubit “12”. Togetherwith the data qubit “3” that was already present initially (in the bottom row of the parity code, adjacent to data qubit “2”), this means that in the present example a total of thee data qubits labeled “3” are provided in the code qubit duplication set 1350, where each data qubit is adjacent to a corresponding parity qubit or data qubit containing the label “2”. As shown, the data qubits “3” in the code qubit duplication set 1350 may be arranged on a line. The code qubit duplication set 1350 can be viewed a 3-qubit repetition code forming part of the parity code.

[0145] Adding a data qubit to the parity code can be understood in the sense that a qubit that was initially separated (disentangled) from the parity code is coupled into the parity code, by performing an interaction between the qubit in question and one or more code qubits that are already part of the parity code, so as to transform said qubit into a data qubit with the desired properties - in the present example, a data qubit with label “3” corresponding to the logical qubit “3”. The addition of a data qubit with label “3” to the parity code can achieved, for example, by (1) initializing the data qubit in an initial state, such as |0>, and (2) performing a CNOT operation acting on said data qubit (target qubit) and the data qubit labeled “3” (control qubit) on the bottom row of the parity code. Alternatively, the data qubit in question can be added by (1) initializing said data qubit in an initial state, such as |+>, (2) performing a first CNOT operation on said data qubit and an additional qubit (prepared in the |0> state; denoted by a black dot in Fig.13), and performing a second CNOT operation on the data qubit labeled “3” and the additional qubit, (3) performing a Z measurement on the additional qubit, respectively, and (4) applying an X operation to said data qubit when the measurement outcome of the Z measurement is 1 (i.e., the measured state is |1>).

[0146] In the above discussion, some steps can be omitted. For example, the removal of the code qubits in the dashed rectangle may be convenient, as it allows reducing the number of data qubits that need to be added to the parity code, but may not be necessary, or at least a fewer number of code qubits may be removed. Further, moving the two remaining triangles closer together helps to reduce the distance over which the qubits need to interact with each other, but is also an optional step.

[0147] Part c) of Fig.13 shows the encoded circuit Gencoded for the CNOT gate 1310. The circuit Gencoded includes a sequence of CNOT gates including a respective CNOT gate between each data qubit labeled “3” (i.e. each qubit in the code qubit duplication set 1350) and the data qubit or parity qubit adjacent to said data qubit “3”, wherein the latter data qubit or parity qubitincludes the label “2”. In each of these CNOT gates, the data qubit “3” is the control qubit. In the example shown in Fig.13, Gencoded includes: (1) a CNOT gate acting on the data qubit “2” and the adjacent data qubit “3”, (2) a CNOT gate acting on the parity qubit “12” and the adjacent data qubit “3”, and (3) a CNOT gate acting on the parity qubit “02” and the adjacent data qubit “3”. 1

[0148] The CNOT gate Glogical can be expressed in terms of Pauli matrices as 2(^^^^ + ^^^^)^^^^ +1 2(^^^^ − ^^^^)^^^^, where the index i denotes the control qubit and the index j denotes the target qubitand where ^^^^denotes the identity operation applied to qubit i, ^^^^denotes the Z operation on qubit i, and ^^^^denotes the X operation on qubit j. The CNOT gate amounts to applying an X operation to the target qubit conditioned on the state of the control qubit in the Z basis, whereinthe operators (^^^^ + ^^^^) and (^^^^ − ^^^^) are projectors onto the (logical) |0> and |1> state (state ofthe control qubit in the Z basis), respectively. According to Fig.12 and as described above, a logical X operation acting on logical qubit j can be physically realized by applying an X operation to each data qubit or parity qubit including the label j. In the example from Fig.13, the CNOT operations of Gencoded are thus applied to the data qubit “2” and the parity qubits “12” and “02” with its target end (“X end”). The logical state of the control qubit in the Z basis may be probed by probing any single data qubit with label i. To provide a fault-tolerant encoded CNOT gate (and not for logical reasons), the state of the control qubit in the Z basis is probed from a different single data qubit with label i for each individual CNOT operation, which has the same effect as probing only one data qubit “3” when there is no error. In the example from Fig.13, each of the CNOT operations is thus applied to a different data qubit “3” with its control end (“Z end”).

[0149] The encoded quantum circuit shown in part c) of Fig.13 is transversal (the code qubit duplication set 1350 forms a first encoded block and the three code qubits dependent on logical qubit “2” form a second encoded block), similar to Fig.10, and hence has a fault-tolerant error propagation of X errors, and in fact of any errors. The encoded quantum circuit provides a fault- tolerant implementation of the CNOT gate 1310.

[0150] In the example shown in Fig.13, the CNOT gate acts on two consecutive qubits “2” and “3”, so that the corresponding data qubits are neighboring data qubits in the system shown in part b) of Fig.13. The disclosure is not limited thereto, and an encoded version of a CNOT gate acting on an arbitrary pair of qubits “i” and “j” can be provided. Suppose, for example, that theCNOT gate acts on logical qubits “2” and 5”. Then, instead of the parity code of Fig. 8, an alternative parity code can be used where the data qubit “5” is disposed on the position of data qubit “3” in Fig.8 (and thus adjacent to data qubit “2”), and where likewise each parity qubit containing the label “5” is disposed on a line adjacent to the line “2” – “12” – “02”, thus replacing the qubits “23”, “13” and “03” in Fig. 8. In other words, a parity code can be constructed where the code qubits involving the label “5” are disposed suitably close to the code qubits involving the label “2”. Then the operations described with respect to Fig.13 can be carried out analogously, thus providing an encoded and fault-tolerant implementation of a CNOT gate acting on logical qubits “2” and “5”.

[0151] In Example G4, the logical quantum gate Glogical is the ^^^^π^^^^Z^^ / 4gate, also called “two-qubit ^ / 2 rotation”. With respect to the basis {|00^, |01^, |10^, |11^}, this gate has the matrixrepresentationThe two-qubit ^ / 2 rotation acts, say, on logical qubits i and j.

[0152] A fault-tolerant implementation of the two-qubit ^ / 2 rotation is further explained with respect to Fig.14. For the sake of concreteness, the example of a two-qubit ^ / 2 rotation 1410 acting on logical qubits “2” and “5” is considered, as shown in part a) of Fig.14.

[0153] For realizing an encoded version of the two-qubit ^ / 2 rotation, a re-coding of the parity code may be performed. The recoding may include adding (at least) two parity qubits with label “25” to the parity code, as shown in part b) of Fig.14. Accordingly, a code qubit duplication set 1450 is provided having at least three parity qubits, namely the added parity qubits “25” as well as the original parity qubit “25”. More generally, if a two-qubit ^ / 2 rotation acting on logical qubits i and j is considered, at least two parity qubits with label “ij” are added to the parity code to form a code qubit duplication set. The additional parity qubits labeled “ij” are defined in the same way as the parity qubit “ij” that is already present in the parity code. Namely, a parity qubit “ij” represents the parity of the set of logical qubits {i, j}, as described herein. Returning to Fig. 14, the code qubit duplication set 1450 can be spatially arranged according to a line of (at least) three qubits. Each additional parity qubit “25” can be prepared,for example, by providing the qubit in question in an initial state, e.g. |0>, and by applying a CNOT gate (or other unitary operation) between said qubit and the parity qubit “25” that was already present in the initial parity code (or a parity qubit “25” that was added in a previous step). Alternatively, a parity qubit “25” can be added by (1) initializing said parity qubit in an initial state, such as |+>, (2) performing a first CNOT operation on said parity qubit and an additional qubit (prepared in the |0> state; denoted by a black dot in Fig.14), and performing a second CNOT operation on the parity qubit “25” that was already present in the initial parity code (or a parity qubit “25” that was added in a previous step) and the additional qubit, (3) performing a Z measurement on the additional qubit, and (4) applying an X operation to said parity qubit when the measurement outcome of the Z measurement is 1 (i.e., the measured state is |1>).

[0154] Further, at least three ancillary qubits may be provided. These ancillary qubits are indicated with label “A” in part b) of Fig.14. For example, the ancillary qubits may form a line parallel and adjacent to the line formed by code qubit duplication set 1450. The ancillary qubits are initially decoupled from the parity code (and not part of the code qubits). The ancillary qubits are physically implemented according to the same physical implementation as the code qubits, i.e. a physical implementation having a dominant qubit error. This applies likewise to the ancillary qubits considered in the other examples described herein.

[0155] In the present example, the parity qubit “25” lies on the boundary of the arrangement of code qubits (i.e. on an edge of the triangle of code qubits shown in part b) of Fig.14), so that the additional parity qubits “25” and the ancillary qubits “A” can conveniently be disposed on outwardly extending lines as shown in part b) of Fig.14. It is noted that this situation does not involve a loss of generality since, for each arbitrary pair of qubits i and j on which the two-qubit ^ / 2 rotation might act, a suitably parity code can be provided where the parity qubit “ij” lies on the boundary of the arrangement of code qubits. A similar remark also applies to other examples described below.

[0156] In order to implement a fault-tolerant version of the two-qubit ^ / 2 rotation, a fault- tolerant gate teleportation protocol is performed. Gate teleportation may be used for performing a single-qubit unitary gate G (i.e. a gate G that commutes with Z) on a qubit without directly applying the gate G. Instead, a magic state G|+^ may be prepared via a state distillation scheme on a set of ancillary qubits, and a sequence of CNOT gates, measurements and a conditionalGXG†operation are performed. The fault-tolerant version uses redundant encoding of the parity qubits and the ancillary qubits in a repetition code (at least three of each two types of qubits) and transversal application of the CNOT gates.

[0157] A two-qubit ^ / 2 rotation acting on a pair of logical qubits u and v is mapped, by the parity code, to an S gate acting on the parity qubit “uv”. An S gate acts as |0> → |0> and |1> → i |1>, where i is in this case the imaginary unit. An S gate shall not be performed directly on a code qubit, since the S gate cannot be performed in a bias-preserving manner in the physical implementation under consideration, i.e. qubits that are physically implemented in bosonic constituents with dominant qubit error X. Accordingly, a gate teleportation protocol may be performed for indirectly realizing the S gate in a bias-preserving way. Particularly, the unitary operator G discussed above with respect to the gate teleportation protocol is in this case the S gate.

[0158] The three ancillary qubits labelled “A” are initially prepared in a magic state |m(S)^ =(|^^^ +The magic state |m(S)^ is a joint stateof the three ancillary qubits and represents a logical single-qubit magic state corresponding to a 3-qubit repetition code, wherein the repetition code has the states |000^ and |111^ as the logical |0> and |1> states, respectively. Due to the encoding according to a repetition code, the ancillary qubits are protected against X errors (i.e. such errors can be corrected, for example, by performing stabilizer measurements and subsequent error-correcting operations), so that the repetition code plays a similar role as the parity code. Further, since the ancillary qubits are also physically implemented according to a physical implementation that suppresses Z errors, the latter errors effectively do not occur.

[0159] A fault-tolerant preparation of the magic state |m(S)^ can be achieved in a bias- preserving way by an available fault-tolerant state preparation (state distillation) scheme. See e.g. the reference Webster et al., “Reducing the overhead for quantum computation when noise is biased”, Phys. Rev. A 92, 062309 (2015), which provides a fault-tolerant preparation of the magic state |m(S)^ using operations that are bias-preserving with respect to the Z qubit error operator, as opposed to the X operator in the present example. When performing the magic state preparation protocol in a rotated basis, i.e. by applying a qubit rotation that maps Z to X (i.e. a Hadamard operation H, since HZH = X) to each ancillary qubit, the rotated protocol is bias- preserving with respect to X, as desired. The resulting state prepared by the rotated statepreparation protocol is ^^|m(S)^ = ^^(|^^^ + i |^^^) / √2, where H represents a logicalHadamard operation (with respect to the repetition code), which maps |^^^ → (|^^^ + |^^^) / √2and |^^^ → (|^^^ − |^^^) / √2. The state ^^|m(S) ^ may be transformed to |m(S)^ = (|^^ ^ +i |^^ ^) / √2 by applying the (logical) Z operation, since ^^^^|m(S) ^ = (|^^^ + i |^^^) / √2 =|m(S)^, where the first equality sign is to be understood up to a global phase, and where Zrepresents a logical Z operator (again with respect to the repetition code), which maps |^^^ →|^^^ and |^^^ → −|^^^. The operator Z can be realized by applying a single-qubit Z operator toone of the ancillary qubits, since such a single-qubit Z operator maps |000^ → |000^ and|111^ → Since a Z operation is a single-qubit operation, this operation is triviallyfault-tolerant, in particular cannot cause propagation of errors.

[0160] After the ancillary qubits “A” are prepared (fault-tolerantly and in a bias-preserving way) in the magic state|m(S)^, a quantum circuit as shown in part c) of Fig.14 is performed. The circuit is described in the following.

[0161] First, (at least) three CNOT gates are applied, wherein each CNOT gate acts on a parity qubit “25” (target qubit) in the code qubit duplication set 1450 and a neighboring ancillary qubit “A” (control qubit). Then, each of the parity qubits “25” in the code qubit duplication set 1450 is measured in the Z basis (i.e. the basis {|0>, |1>}), yielding an outcome 0 or 1. This measurement is indicated as “Mz” in Fig.14. Then, a Y operator (which is, up to a global phase, a product of X and Z, as indicated in part c) of Fig.14), is conditionally applied to one of the ancillary qubits “A”. The Y operator plays the role of the operator GXG†described above, sincein the present case we have G = S and thus GXG†= SXS† = Y (up to a global phase). Theapplication of the Y gate is conditioned on the outcomes of the Z measurements performed in the previous step. Specifically, the Y gate is conditioned on a majority vote of the three measurement outcomes. In said majority vote, the three outcomes, which can each be 0 or 1, are mapped to a single bit as defined by the majority mapping, namely 000 → 0, 001 → 0, 010 → 0, 100 → 0, 011 → 1, 110 → 1, 101 → 1, 111 → 1. If the majority vote returns the bit 1, the Y gate is applied to the ancillary qubit, while the Y gate is not applied if the bit 0 is obtained from the majority vote. Then, an X measurement may be performed (measurement in the basis {|+>, |->}, where |+> may be assigned the measurement outcome “0” and |-> may be assigned the measurement outcome “1”) on the two other ancillary qubits labelled “A”. This measurement is indicated as “Mx” in Fig. 14. Conditioned on the measurement outcome ofthese two X measurements, a Z operator may be applied to the remaining ancillary qubit “A” (i.e. the lowermost “A” qubit in part c) of Fig.14, which is not subjected to an X measurement). Specifically, the Z operator is applied subject to the condition that the parity of the outcomes of the two X measurement outcomes is equal to 1. Therein, the parity of two outcomes a and b, where a, b ^ {0, 1}, is equal to 0 if a = b and is equal to 1 otherwise. Then, the ancillary qubit “A” to which the Z operator is applied may optionally be swapped with the adjacent parity qubit “25” (e.g. by applying a SWAP gate, or by physically interchanging the two qubits in question).

[0162] The quantum circuit shown in part c) of Fig.14 thus performs a gate teleportation of the S gate onto a parity qubit “25”, and hence realizes an encoded version of a two-qubit ^ / 2 rotation 1410 acting on logical qubits “2” and “5”.

[0163] The encoded circuit shown in Fig. 14 is fault-tolerant, for the following reasons. As described above, the preparation of the magic state|m(S)^can be performed fault-tolerantly with bias-preserving operations. Further, the sequence of CNOT gates is applied transversally (the ancillary qubits “A” form a first encoded block and the code qubit duplication set 1450 forms a second encoded block), and thus has a fault-tolerant error propagation of X errors. As regards the subsequent Z measurements, a Z measurement can in principle be corrupted by an X error, as described above. However, such errors are addressed in Fig.14 by performing three Z measurements in parallel on the qubits in the code qubit duplication set 1450, followed by majority voting of the measurement outcomes. The majority voting of the measurement outcomes ensures that a possibly erroneous outcome is corrected (with high probability). This means that the outcome of the majority voting will, with high probability, be equal to the measurement outcome of an error-free Z measurement, so that any X error affecting the Z measurement is accounted for. Accordingly, the XZ operation is applied based on the correct condition. The XZ operation itself is a single-qubit operator and hence cannot cause a propagation of errors. Further, the two X measurements are unaffected by X errors, since these operations commute. The subsequent Z operator is a single-qubit operator and hence does not propagate an error to other qubits. The SWAP operation may propagate an X error at most to the remaining parity qubit, i.e. at most to one additional qubit. Accordingly, the quantum circuit under consideration has a fault-tolerant propagation of X errors. A fault-tolerant implementation of the two-qubit ^ / 2 rotation 1410 is provided.

[0164] In Example G5, the logical quantum gate Glogical is the S gate. Fig. 15 shows a fault tolerant implementation of the S gate, where, for the sake of concreteness, the S gate 1510 acts on logical qubit “3”, as shown in part a) of Fig. 15. An S gate acting on logical qubit “3” corresponds, in the parity encoding, to an S gate acting on the corresponding data qubit “3”. Accordingly, a fault-tolerant implementation of the S gate acting on data qubit “3” may be provided similarly to the encoded version of the two-qubit ^ / 2 rotation described in Example G4, yet now the aim is to perform an S gate on the data qubit “3” instead of the parity qubit “25”. Accordingly, instead of adding at least two parity qubits “25”, now at least two data qubits “3” are added to the parity code to provide a code qubit duplication set 1550 consisting of at least three data qubits “3”, as shown in part b) of Fig.15. At least three ancillary qubits “A” are provided adjacent (especially adjacent along a diagonal direction of the 2D grid) to these data qubits, similar to the ancillary qubits “A” in Example G4. These ancillary qubits may be prepared (fault-tolerantly & in a bias-preserving way) in the magic state |m(S)^ as described above. Then, as shown in part c) of Fig. 15, the same quantum circuit as shown in part c) of Fig.14 may be applied to the ancillary qubits “A” and the data qubits “3” (instead of the parity qubits “25”), yielding a fault-tolerant implementation of the S gate 1510.

[0165] In Example G6, the logical quantum gate Glogical is the CZ (controlled-Z) gate. The CZ gate acts on two logical qubits according to the action |00> → |00>, |01>|01>, |10> → |10> and |11> - |11>. That is to say, CZ is a diagonal gate in the computational basis, wherein each basis state except |11> is mapped to itself, and the state |11> is mapped to -|11>. The CZ gate may act on the i-th and j-th logical qubit.

[0166] A fault-tolerant implementation of the CZ gate is further discussed with respect to Fig.16. For the sake of concreteness, a CZ gate 1610 is considered that acts on logical qubits “2” and “5”, as shown in part a) of Fig.16.

[0167] A re-coding of the parity code may be performed. At least two data qubits labelled “2” and at least two data qubits labelled “5” may be added to the parity code (using similar operations as those described above, e.g. using CNOT operations or measurements), so that at least three data qubits “2” forming a first code qubit duplication set 1651 and at least three data qubits “5” forming a second code qubit duplication set 1652 are provided in total, arranged according to two lines, as shown in part b) of Fig.16. Likewise, at least two parity qubits “25”may be added to the parity code, so that in total at least three parity qubits “25” forming a third code qubit duplication set 1653 are provided, also arranged according to a line.

[0168] Further, at least three ancillary qubits “A”, at least three ancillary qubits “B” and at least three ancillary qubits “C”, are provided on lines adjacent to the data qubits “2”, the data qubits “5” and the parity qubits “25”, respectively. Each of these three sets of ancillary qubits is prepared in an initial state, being a magic state|m(S)^as described herein.

[0169] A CZ gate acting on logical qubits “i” and “j” can be decomposed as a product of three unitary operators U, V and W, that is to say CZ = UVW (where the equality is up to a global phase factor), wherein U is the operator ZS (product of Z and S gate) acting on logical qubit “i”, V is a two-qubit ^ / 2 rotation acting on logical qubits “i” and “j” and W is the operator ZS acting on logical qubit “j”. In Examples G4 and G5, fault-tolerant implementations of the two- qubit ^ / 2 rotation and the S gate were provided. Since the operator V is a two-qubit ^ / 2 rotation, this hence provides a fault-tolerant implementation of V. Further, the operators U and W only differ from an S gate by a multiplication of a Z operator, which can easily be accommodated by adding a final single-qubit Z operation in the circuit for realizing the encoded version of the S gate, namely the circuit shown in part c) of Fig.15 in connection to Example G5. This yields a fault-tolerant implementation of the operators U and W. By concatenating the three implementations and U, V and W, a fault-tolerant implementation of CZ = UVW is provided.

[0170] Specifically, with respect to the present example, involving a CZ gate 1610 being applied to logical qubits “2” and “5”, the quantum circuits for realizing U, V and W are shown in part c) of Fig.16, namely: • A first circuit for realizing the operator U: the circuit acts on the data qubits “2” (first code qubit duplication set 1651) and the neighboring ancillary qubits “A” and corresponds to the gate teleportation circuit of Example G5 for fault-tolerant implementation of an S gate (cf. Fig.15), supplemented with a final Z operation on the lowermost data qubit “2”; • A second circuit for realizing the operator W: the circuit acts on the data qubits “5” (second code qubit duplication set 1652) and the neighboring ancillary qubits “B” and also corresponds to the gate teleportation circuit of Example G5 for fault-tolerantimplementation of an S gate, and is also supplemented with a final Z operation, now on the lowermost data qubit “5”; and • A third circuit for realizing the operator V: the circuit acts on the parity qubits “25” (third code qubit duplication set 1653) and the neighboring ancillary qubits “C” and corresponding to the gate teleportation circuit of Example G4 for fault-tolerant implementation of a two-qubit ^ / 2 rotation (cf. Fig.14).

[0171] In Example G7, the logical quantum gate Glogicalis the Hadamard-transformed T gate, referred to herein as TH. The T gate as such maps |0> → |0> and |1>The Hadamard- transformed T gate THis equal to HTH, where H is the Hadamard gate. The gate THis equal to ^^^^π^^ / 8up to a global phase.

[0172] A fault-tolerant implementation Gencoded of the Hadamard-transformed T gate is further discussed with respect to Fig.17. For the sake of concreteness, a Hadamard-transformed T gate 1710 is considered that acts on logical qubit “1” as shown in part a) of Fig.17.

[0173] For providing a fault-tolerant implementation of the Hadamard-transformed T gate, the parity code may be recoded. At least two additional data qubits “1” can be added to the parity code (using similar techniques as described above), so that there are at least three data qubits “1” in total, forming a code qubit duplication set 1750. Further, all parity qubits that contain the label “1” (i.e. all parity qubits of the form “1j” or “i1”) may be changed into other parity qubits, or into data qubits, that do not include the label “1”. This can be achieved, for example, by applying CNOT gates. In the example shown in part b) of Fig.17, this regards the code qubits in the dashed boxes, namely the data qubits “0” and “2” and the parity qubits “23”, “34” and “45”. None of these qubits include the label “1”. In comparison, in the initial parity code (as shown in Fig. 8), these qubits were “01”, “12”, “13”, “14” and “15”, respectively, each including the label “1”. In other words, the label “1” is effectively removed from all parity qubits, so that this label is only present in the at least three data qubits “1”. In this way, the logical qubit “1” is protected against errors (error correction may be done by virtue of the at least three data qubits, which effectively provide a repetition code) and at the same isolated from the other logical qubits (since the label “1” does not appear in any of the parity qubits).

[0174] In the present example, the qubits in the dashed boxes are exemplarily transformed into the data qubits “0” and “2” and the parity qubits “23”, “34” and “45”, yet the disclosure is notlimited thereto, and other encodings may be chosen of code qubits that do not include the label “1”. The encoding may be chosen such that “logical lines”, i.e., lines connecting code qubits with a common label, are continuous.

[0175] Further, at least three ancillary qubits “A” may be provided on a line adjacent to the code qubit duplication set 1750, as shown in part b) of Fig. 17. The ancillary qubits “A” areprepared in an initial state, which may be the magic state |m(HTH)^ = ^^(|^^^ + eiπ / 4|^^^) / √2.As before, the ancillary qubits are encoded using a repetition code. A fault-tolerant and bias- preserving preparation of the magic state |m(HTH)^ may be achieved by an available state preparation scheme, e.g. using the above-mentioned reference Webster et al., “Reducing the overhead for quantum computation when noise is biased”, Phys. Rev. A 92, 062309 (2015). Said reference describes a fault-tolerant state preparation scheme of a magic state correspondingto a T gate, i.e., the magic state |m(T)^ = (|^^^ + eiπ / 4|^^^) / √2. Said preparation scheme isbias-preserving with respect to the dominant error operator Z, as opposed to the dominant error X considered in the present example. When performing a unitary rotation that maps Z to X, i.e. a Hadamard operation H, on each ancillary qubit, the resulting rotated preparation scheme is bias-preserving with respect to X, as desired. The magic state that is prepared by this rotatedscheme is ^^|m(T)^ = ^^(|^^^ + eiπ / 4|^^^) / √2, which is equal to |m(HTH)^. Thus, a fault-tolerant and bias-preserving (with respect to X) preparation of the magic state |m(HTH)^ is provided.

[0176] After the ancillary qubits “A” have been prepared in the magic state |m(HTH)^, a quantum circuit for performing gate teleportation is applied as illustrated in part c) of Fig.17. At least three CNOT gates are applied, where each CNOT gate acts on a data qubit “1” (target qubit) in the code qubit duplication set 1750 and the neighboring ancillary qubit “A” (control qubit). Then, each of the ancillary qubits “A” may be measured in the X basis, indicated as “Mx” in part c) of Fig.17. The parity of the three measurement outcomes is determined (if the three measurement outcomes are denoted by a, b, c ^ {0,1}, their parity is a + b + c ^ {0,1}, where the sum is computed modulo 2). If the parity is 0, the quantum circuit ends. Conditioned on the parity being equal to 1, a subsequent sequence of operations as shown in the dashed boxis performed. Specifically, the three ancilla qubits are prepared in the magic state ^^|m(S)^ =^^(|^^^ + ^^|^^^) / √2. Again three CNOT gates are applied as before, i.e. each CNOT gate actson a data qubit “1” (target qubit) in the code qubit duplication set 1750 and the neighboringancillary qubit “A” (control qubit). Then, each of the ancillary qubits “A” may be measured in the X basis, indicated as “Mx”. The parity of the three measurement outcomes is determined (if the three measurement outcomes are denoted by a, b, c ^ {0,1}, their parity is a + b + c ^ {0,1}, where the sum is computed modulo 2). Conditioned on said parity being 1, an X operator is applied to each data qubit “1” (and if the parity is 0, no further operations are applied). The sequence of operations in the dashed box provides a fault-tolerant implementation of a Hadamard-transformed SH gate, equal to HSH, as a logical quantum gate.

[0177] The circuit shown in Fig. 17 c) is an alternative embodiment of a gate teleportation protocol for an operator F. As opposed to the embodiments described in Figs.14 c), 15 c) and 16 c), the gate teleportation protocol of Fig. 17 c) may be used for performing a single-qubit unitary gate which commutes with X (instead of Z). The magic state for this protocol corresponds to the state F|0> (instead of G|+>). X measurements (instead of Z measurements) are performed.

[0178] Further as opposed to the embodiments described in Figs. 14 c), 15 c) and 16 c), the gate teleportation protocol of Fig. 17 c) is an equivalent protocol wherein the gate is directly applied to the data qubits, rendering the final SWAP gate of the previous embodiments obsolete. The conditional operation applied depending on the parity of the measurement outcomes of the X measurements for the equivalent protocol is, for this type of protocol, ^^^^^^+^^.

[0179] In Fig.17 c), F = TH and hence the magic state is F|0>=|m(HTH)^ and the conditionaloperation is ^^^^^^+^^ = ^^^^^^ = ^^^^. The ^^^^ gate commutes with X, and hence a similar gateteleportation protocol as for the TH can be applied (dashed box), the only difference being the magic state in which the ancilla qubits are prepared and the conditional operation. In the dashed box of Fig. 17 c), F=^^^^and hence the magic state is ^^^^|0>= ^^|m(S)^, and the conditionaloperation is ^^^^^^+^^ = ^^ (up to a global phase).

[0180] The encoded circuit shown in Fig.17 is fault-tolerant for essentially the same reasons as described with respect to the examples above. In short, the preparation of the magic states is known to be fault-tolerant. Further, the first sequence of CNOT gates is applied transversally. Further, the three X measurements are unaffected by X errors, since these operations commute. The box of operations that is applied conditioned on the parity of the measurement outcomes of the three X measurements contains similar components, in particular fault-tolerant magic state preparation followed by a transversal sequence of CNOT gates and X measurements.Further, the X measurements are not affected by potential X errors. Further, the final X rotations are single-qubit operations and hence trivially transversal. A fault-tolerant implementation of the Hadamard-transformed T gate 1710 is provided.

[0181] In Example G8, the logical quantum gate Glogicalis the Hadamard gate H. The Hadamardgate maps |0^ → (|0^ + |1^) / √2 and |1^ → (|0^ − |1^) / √2. A fault-tolerant implementationGencoded of the Hadamard gate is further discussed with respect to Fig.18. For the sake of illlustration, a Hadamard gate 1810 is considered that acts on logical qubit “1” as shown in part a) of Fig.18.

[0182] For providing a fault-tolerant implementation of the Hadamard gate, the parity code may be re-coded in the same manner as for the TH gate, as shown in part b) of Fig.17.

[0183] The Hadamard gate can be decomposed as H = S+^ SHS^ S+, where S is the S gate andthe superscript “+” denotes the Hermitian conjugate. In part b) of Fig.18, a circuit is provided that is a sequence of three sub-circuits providing fault-tolerant implementations of S+, SHS and again S+, respectively. Each sub-circuit realizes a fault-tolerant (and bias-preserving) gate teleportation protocol similar to the protocols described above.

[0184] In the sub-circuit for realizing the S+gate, the three ancillary qubits labelled “A” areinitially prepared in a magic state |m(S+)^ = (|^^^ − i |^^^) / √2. This state can be prepared byfirst preparing the state |m(S)^ = (|^^^ + i |^^^) / √2 using a fault-tolerant and bias-preservingstate preparation protocol as described above, yielding ^^|m(S)^ = ^^(|^^^ + i |^^^) / √2, whichis equivalent to|m(S+)^up to a global phase. After |m(S+)^ has been prepared, three CNOT operations are applied (where in this case the data qubits “1” in the code qubit duplication set 1750 are the control qubits and the ancillary qubits “A” are the target qubits), followed by a Z measurement on each ancillary qubit “A”. Conditioned on the majority vote of the three measurement outcomes being equal to 1, a Z operation is applied to one of the data qubits.

[0185] The circuit described above is a fault-tolerant gate teleportation protocol in the form of the equivalent protocol in Fig.17 c) (a version without SWAP gate). As opposed to Fig.17 c), the protocol is designed for an operator G which commutes with Z (such as the operator in question S+). The ancilla qubits are prepared in a magic state G|+>, which corresponds to |m(S+)^ for G=S+. The operation conditioned on the majority vote of the measurement outcomes of the Z measurement is ^^^^^^+^^, which corresponds to a Z operation for G=S+.

[0186] In the sub-circuit for realizing the SHS gate, the three ancillary qubits labelled “A” areinitially prepared in the magic state |m(S)^ = (|^^^ + i |^^^) / √2. This state can be prepared bya fault-tolerant and bias-preserving state preparation protocol as described above. After |m(^^)^ has been prepared, three CNOT operations are applied (where in this case the data qubits “1” in the code qubit duplication set 1750 are the target qubits and the ancillary qubits “A” are the control qubits), followed by an X measurement on each ancillary qubit “A”. Conditioned on the parity of the three measurement outcomes being equal to 1, an X operation is applied to each of the data qubits.

[0187] The circuit described above is a fault-tolerant gate teleportation protocol in the form of the equivalent protocol in Fig. 17 c) (a version without SWAP gate). As in Fig. 17 c), the protocol is designed for an operator F which commutes with X (such as the operator in question SHS). The ancilla qubits are prepared in a magic state F|0>, which corresponds to |m(S)^ for F=SHS. The operation conditioned on the parity of the measurement outcomes of the X measurement is ^^^^^^+^^, corresponding to an X operation for F=SHS.

[0188] The encoded quantum circuit in Fig. 18 is fault-tolerant since each of the three sub- circuits is fault-tolerant (for essentially the same reasons as described with respect to the examples above, which will not be repeated here). Accordingly, a fault-tolerant implementation of the Hadamard gate 1810 is provided.

[0189] Examples G1 to G8 described a fault-tolerant implementation of a plurality of logical unitary gates Glogical acting on one or more logical qubits. In a similar fashion, a fault-tolerant implementation of logical measurements Mlogicalacting on one or more logical qubits can be provided. Two examples are provided in the following.

[0190] In Example M1, let Mlogical be a Z measurement acting on a logical qubit - say, logical qubit “i”. A recoding of the parity code, using similar techniques as described above, can be performed so that the resulting parity code includes a code qubit duplication set having at least three data qubits with label “i”. An encoded implementation Mencoded of the logical Z measurement may then be provided by performing a Z measurement on each such data qubit, followed by a majority voting of the three measurement outcomes. The outcome of the majority voting, which is either 0 or 1, constitutes the measurement outcome of the encoded measurement. Due to the majority voting, the procedure is fault-tolerant.

[0191] In Example M2, let Mlogical be an X measurement acting on a logical qubit - say, logical qubit “i”. The encoded implementation Mencoded may be provided (in analogy with Example G2) by performing an X measurement on (1) the data qubit corresponding to the i-th logical qubit, and (2) each parity qubit where the subset of logical qubits associated with the parity qubit contains the i-th logical qubit. The measurement outcome of Mencoded is determined from the parity of the individual outcomes of all these measured qubits. As described above, X measurements are not affected by X errors in the present physical implementation, so that the procedure is fault-tolerant. If the logical qubits other than logical qubit “i” should still be usable after the measurement, quantum operations conditioned on the individual measurement outcomes of Mencodedmay be applied to at least some of the code qubits representing logical qubits other than logical qubit “i”. If an individual measurement result of an X measurement of a code qubit is 0, no conditional operation is applied due to this individual measurement. If an individual measurement result of an X measurement of a code qubit is 1, a Z operation is applied to each code qubit which had a stabilizer operator of the parity code in common with said code qubit.

[0192] Examples G1-G8 and M1-M2 are possible examples of fault-tolerant implementations of the gates and measurements considered, yet the disclosure is not limited thereto. The examples in question can be modified in several different ways. For example, instead of starting from the parity code shown in Fig. 8, another parity code can be taken as a starting point. Generally, a certain degree of freedom for modification of the parity code is available, especially in relation to code qubits that do not involve the logical qubit(s) that is / are currently acted upon by the logical gate under consideration. Some of the parity qubits may be chosen differently (e.g. some parity qubits may represent subsets of more than two logical qubits, such as three, four or more logical qubits) or even omitted. Likewise, some of the data qubits may be omitted from the code. Further, the recoding of the parity code can be performed in a different manner than the examples provided herein, e.g. by using parity qubits that represent subsets of more than two logical qubits, by using different types of ancillas or a different number of them, and so on. Correspondingly, also the quantum circuits Gencoded (or the encoded measurements Mencoded) are not limited to the form shown in the examples herein. One the one hand, if a different parity code is used, this will map to a different form for the quantum circuits. Further, the circuits Gencodedshown in the examples can also as such be modified without affecting the logical gate that they implement. Particularly, in a quantum circuit, ultimately onlythe overall working of the entire sequence of gates / measurements in the circuit is of interest, not the form of the individual gates. This again provides freedom to modify the individual gates without affecting the fact that the circuit implements a particular logical gate. For example, it is straightforward to design alternative circuits, different from the circuits shown in the examples, that also realize a gate teleportation protocol. Embodiments

[0193] According to an embodiment, a quantum computational method is provided. The quantum computational method includes providing a physical quantum system comprising constituents. The quantum computational method includes performing an encoded quantum computation on the physical quantum system. Performing the encoded quantum computation includes preparing at least a portion of the physical quantum system in an initial quantum state. Performing the encoded quantum computation includes evolving at least a portion of the physical quantum system to a final quantum state. Performing the encoded quantum computation includes measuring at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits into code qubits. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem of the physical quantum system, wherein the subsystem comprises one or more constituents. During a first portion of the encoded quantum computation, the code qubits include a first code qubit duplication set associated with a first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit. The quantum computational method further includes performing a first sequence of physical quantum operations on the physical quantum system during the first portion of the encoded quantum computation. The first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set. The first sequence of physical quantumoperations is an encoded realization, via the parity code, of a first logical quantum operation acting at least on the first logical qubit.

[0194] The first code qubit duplication set may refer, for example, to any of the code qubit duplication sets 1350, 1450, 1550, 1651-1653, 1750 in Examples G3-G8 or the code qubit duplication set in Example M1. The first sequence of physical quantum operations may refer, for example, to a sequence of physical quantum operations, particularly a sequence of bias- preserving quantum operations, that physically realizes any of the encoded quantum circuits in Examples G3-G8 and M1. The first logical quantum operation can, for example, refer to any of the logical quantum operations described in Examples G3-G8 and M1.

[0195] The initial quantum state can be a default, non-encoded quantum state. Alternatively, the initial quantum state can already be an encoded quantum state corresponding to the parity code.

[0196] Evolving at least a portion of the physical quantum system to a final quantum state can include performing one or more unitary operations and / or performing one or more measurements. Evolving at least a portion of the physical quantum system to a final quantum state can, e.g. if the initial quantum state is not yet encoded, include performing physical quantum operations to prepare an encoded quantum state corresponding to the parity code. Evolving at least a portion of the physical quantum system to a final quantum state can include performing one or more sequences of physical quantum operations, wherein each sequence implements an encoded quantum circuit, e.g. an encoded quantum circuit corresponding to a respective logical quantum operation. Evolving at least a portion of the physical quantum system to a final quantum state can include performing one or more sequences of re-coding operations, wherein each sequence of re-coding operations changes one instance of the parity code into another instance of the parity code.

[0197] Measuring at least a portion of the physical quantum system to provide a read-out can be understood as a performing a measurement to obtain information regarding a solution of a computational problem that the quantum computation aims to solve. The read-out may contain information regarding a solution of the computational problem.

[0198] Aspects of the present disclosure involve a parity code that encodes logical qubits into code qubits.

[0199] The code qubits include parity qubits, such as three or more, ten or more, or one hundred or more parity qubits. Each parity qubit represents the parity of an associated subset of logical qubits. Said subset of logical qubits may include two or more, three or more or four or more logical qubits. For a plurality of parity qubits, the associated subset of logical qubits may include four or less, three or less, e.g. two logical qubits.

[0200] The parity qubits may include one or more, or a plurality, of two-element-parity qubits, wherein a two-element-parity qubit represents the parity of an associated subset consisting of two logical qubits. For example, the parity qubits in Examples G1-G8 and M1-M2 are two- element-parity qubits.

[0201] The parity qubits may include a first parity qubit, a second parity qubit and a third parity qubit forming a correlated set of parity qubits. The first parity qubit may be a two-element- parity qubit representing the parity of a first subset {i, j} consisting of a first logical qubit i and a second logical qubit j. The second parity qubit may be a two-element-parity qubit representing the parity of a second subset {j, k} consisting of the second logical qubit j and a third logical qubit k. The third parity qubit may be a two-element-parity qubit representing the parity of a third subset {i, k} consisting of the first logical qubit i and the third logical qubit k. The parity qubits may include a plurality of correlated sets of parity qubits that are defined analogously. For example, in Fig. 8, the parity qubits “12”, “23” and “13” form a correlated set of parity qubits.

[0202] The notion that a parity qubit represents the parity of an associated subset of logical qubits may include that a label of a quantum basis state of the parity qubit corresponds to (e.g. is equal to) the parity of the labels of quantum basis states of the logical qubits in the associated subset. The label of a quantum basis state may, for example, be “0” or “1”, or “1” and “-1”, and the like.

[0203] The code qubits may include one or more data qubits, such as such as three or more, ten or more, or one hundred or more data qubits. A data qubit represents a quantum state of an associated logical qubit. The notion that a data qubit represents the quantum state of an associated logical qubit may include that a label of a quantum basis state of the data qubit corresponds to (e.g. is equal to) a label of a quantum basis state of the associated logical qubit.

[0204] It may be the case that the logical qubits are not physically implemented in a subsystem of the physical quantum system.

[0205] The first logical quantum operation, and likewise any other logical quantum operation described herein, may be a unitary operation or a measurement. The first logical quantum operation may act on one or more logical qubits. It may be the case that, for each logical qubit that is acted upon by the first logical quantum operation, (i) the code qubits include a parity qubit representing the parity of a subset of logical qubits, wherein the subset includes the logical qubit, and / or (ii) the code qubits include a data qubit representing the quantum state of the logical qubit. In other words, for each logical qubit that is acted upon by the first logical quantum operation, it may be the case that the code qubits include a parity qubit dependent on the logical qubit and / or a data qubit corresponding to the logical qubit. The same property may apply analogously to any other logical quantum operation described herein.

[0206] A code qubit duplication set as described herein includes at least three code qubits. A code qubit duplication set may include at least 10, at least 50, at least 100, or even more code qubits. The larger the size of a code qubit duplication set, the more effective the code qubit duplication set can be used to correct errors and prevent error propagation.

[0207] The first code qubit duplication set, and likewise any other code qubit duplication set described herein, may include at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a two-element-parity qubit representing the parity of a same first subset consisting of two logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit.

[0208] Aspects of the present disclosure involve physical quantum operations.

[0209] A physical quantum operation as described herein may act on one or more constituents of the physical quantum system. A physical quantum operation may be a single-body operation acting on a single constituent or an entangling operation acting jointly on two or more constituents. A physical quantum operation may be a unitary operation, a measurement, a dissipative operation, an adiabatic variation of a parameter of an observable of the physical quantum system, and the like. For example, the first sequence of physical quantum operations(and likewise any other sequence of physical quantum operations described herein) may include one or more unitary operations and / or one or more measurements.

[0210] The first sequence of physical quantum operations, and likewise any other sequence of physical quantum operations described herein, may include L physical quantum operations, where L is at least two. A sequence of physical quantum operations can be understood as an ordered set of physical quantum operations O1, O2, ..., OL applied to the physical quantum system in this order. That the physical operations form a sequence does not necessarily imply that each operation Oi+1 in the sequence should be applied directly after its predecessor Oi. It is possible that one or more further physical quantum operations (which are not part of the sequence in question) are applied in between two subsequent physical quantum operations Oiand Oi+1.

[0211] The first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set. This can be understood in the sense that, for each of the subsystems in question, at least one physical quantum operation in the first sequence acts on (at least) said subsystem. In other words, for each code qubit in the first code qubit duplication set, there is at least one physical quantum operation in the first sequence that acts (at least) on the subsystem corresponding to said code qubit.

[0212] The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation. The parity code encodes the logical qubits into the code qubits and correspondingly maps the first logical quantum operation, which defines a quantum evolution of one or more logical qubits, to an encoded quantum evolution of one or more code qubits (e.g. an encoded quantum circuit acting on one or more code qubits). The first sequence of physical quantum operations acting on the physical quantum system provides a physical implementation of the encoded quantum evolution.

[0213] Each physical quantum operation in the first sequence of physical quantum operations may be a short-range physical quantum operation. Likewise, each physical quantum operation in the second sequence of physical quantum operations described below, or in any other sequence of physical quantum operations described herein, may be a short-range physical quantum operation.

[0214] The terminology of a short-range physical quantum operation (e.g. a unitary operation or measurement) refers to a quantum operation acting only on a subgroup of the constituents, wherein any two constituents within the subgroup are distanced from each other by a distance of at most an interaction cut-off distance of the physical quantum system. The short-range physical quantum operation does not act on any constituent outside of the subgroup of constituents. The subgroup of constituents may include one or more constituents.

[0215] An interaction cut-off distance may be a constant distance. The interaction cut-off distance may be much smaller than a maximal constituent distance between the constituents of the physical quantum system. For example, the interaction cut-off distance may be 30% or less of the maximal constituent distance, in particular 20% or less, more particularly 10% or less.

[0216] For a plurality of constituents arranged according to a lattice, a short-range physical quantum operation may be an r-range physical quantum operation. For example, r may be five or less, particularly four or less, e.g. r = ^2, 2, 3, 4 or 5. An r-range physical quantum operation may be a physical quantum operation acting only on a subgroup of constituents of the plurality of constituents, wherein any two constituents within the subgroup of constituents are distanced from each other by a distance of at most r times an elementary distance (lattice constant) of the lattice.

[0217] For example, in Fig. 8, assuming that each code qubit is physically implemented in a single constituent (as in Fig.2), and that the dots in the figure represent ancillary qubits that are likewise each implemented in a single constituent, a two-dimensional lattice of constituents is provided, wherein a constituent is disposed on each node (lattice site) of the lattice. The elementary distance or lattice constant of the two-dimensional lattice may be given by the distance between two nodes. Accordingly, the distance between, e.g., the constituents corresponding to qubits “13” and “24” is two times the elementary distance, and the distance between the constituents corresponding to qubits “13” and “14” is ^2 times the elementary distance. Accordingly, a physical quantum operation acting only on the constituents corresponding to qubits “13” and “24” is a 2-range physical quantum operation and a physical quantum operation acting only on the constituents corresponding to qubits “13” and “14” is a ^2-range physical quantum operation.

[0218] For example, in each of the Examples G1-G8, the encoded circuits consist entirely of (i) CNOT gates acting on qubits that are either nearest neighbors (r = 1), such as e.g. in Fig.13,or qubits distanced from each other by ^2 times the elementary distance (so that r = ^2), such as e.g. in Fig.15, and (ii) single-qubit operations, which are trivially short-range. Accordingly, the encoded quantum circuits consist of short-range operations only, and hence the corresponding physical quantum operations are all short-range physical quantum operations as well.

[0219] Aspects of the present disclosure involve a physical implementation having a dominant qubit error.

[0220] It may be the case that each code qubit of at least a subset of the code qubits is physically implemented in a corresponding subsystem by a physical implementation having a dominant qubit error. Particularly, it may be the case that each code qubit in the first code qubit duplication set is physically implemented in a corresponding subsystem by a physical implementation having a dominant qubit error.

[0221] At least one qubit error different from the dominant qubit error may be suppressed as compared to the dominant qubit error or may be correctable by performing one or more error- correcting operations on the subsystem. The dominant qubit error may be correctable by the parity code in one or more subsystems of the physical quantum system.

[0222] Suppression of a qubit error can be understood as a passive protection against the qubit error, wherein the physical implementation may be engineered such that the qubit error occurs at a decreased rate, without a need for error-correction. The alternative that a qubit error is correctable can be understood as an active protection against the qubit error, wherein the qubit error is allowed to occur but the physical implementation is engineered to encode information in a redundant manner so that error-correction (by e.g. syndrome measurements and subsequent error-correction operations) of the qubit error in question is possible.

[0223] The at least one qubit error different from the dominant qubit error may be a qubit error that does not commute, particularly anti-commutes, with the dominant qubit error. For example, the at least one qubit error may be a Z error and the dominant qubit error may be an X error, or vice versa.

[0224] The aspect that, in a given subsystem of the physical quantum system, at least one qubit error is suppressed as compared to the dominant qubit error may include that a probability p1 that the at least one qubit error occurs in the subsystem is smaller than a probability p2 that thedominant qubit error occurs in the subsystem. For example, the ratio p1 / p2 may be 10−3or less, particularly 10−5or less, more particularly 10−8or less. In some cases, the probability p1 may be approximately zero.

[0225] It may be the case that each code qubit of at least a subset of the code qubits is physically implemented in a corresponding subsystem that consists of a constituent. The code qubit may be encoded into the constituent according to a quantum error-suppression code. The quantum state of at least a portion of the quantum system may be an encoded quantum state corresponding to a concatenated quantum code being a concatenation of the parity code and the quantum error-suppression code. At least one qubit error different from the dominant qubit error may be suppressed by the quantum error-suppression code as compared to the dominant qubit error. For example, a code qubit may be implemented in a bosonic constituent according to a cat qubit implementation, which is an example of a quantum error-suppression code. Another example of an error-suppressing code regards a physical implementation of a qubit into a superconducting constituent, as described herein.

[0226] The term “quantum error-suppression code” can be used synonymously with “error- suppression physical implementation”. An encoded quantum state corresponding to a concatenated quantum code being a concatenation of the parity code and the quantum error- suppression code can be understood in the sense that the logical qubits are encoded into the code qubits according to the parity code (providing a “first level” of encoding), and each code qubits is in turn encoded, or physically implemented, in a subsystem of the physical quantum system according to the quantum error-suppression code (providing a “second level” of encoding, which is, hence, concatenated with the first level of the encoding).

[0227] The aspect that, in a given subsystem of the physical quantum system, at least one qubit error different from the dominant qubit error is correctable by performing one or more error- correcting operations on the subsystem may include that a code qubit is encoded into the subsystem according to a further quantum error-correction code (additional to the parity code that encodes logical qubits into code qubits). The subsystem may include several constituents. Accordingly, a code qubit may be encoded, using the further quantum error-correction code, into a subsystem including several constituents. The further quantum error-correction code may, for example, be a repetition code, a surface code, or any other quantum error correction code. The further quantum error-correction code may be configured to correct the at least onequbit error different from the dominant qubit error by performing one or more error-correction operations on the subsystem, e.g. syndrome measurements and unitary operations in response thereto.

[0228] The dominant qubit error may be correctable by the parity code in one or more subsystems of the physical quantum system. For example, the dominant qubit error may be correctable by the parity code in each subsystem that is acted upon by the first sequence of physical quantum operations.

[0229] The aspect that the dominant qubit error is correctable by the parity code may be mathematically characterized as follows. For each parity qubit of at least a subset of the parity qubits, the parity of the subset of logical qubits associated with the parity qubit may be defined with respect to a quantum basis of the parity qubit. The quantum basis may be an eigenbasis of a first qubit operator (e.g. the Z operator, in the case of Z-type parity codes). The dominant qubit error of the physical implementation may be represented by a second qubit operator (e.g. the X operator). The commutator of the first qubit operator and the second qubit operator may be different from zero. For example, the first qubit operator and the second qubit operator may anti-commute with each other.

[0230] Aspects of the present disclosure involve dominant-error-conserving quantum operations.

[0231] One or more, particularly all, physical quantum operations in the first sequence of physical quantum operations may be dominant-error-conserving quantum operations that conserve the dominant qubit error of the physical implementation. A dominant-error- conserving operation is synonymously called “bias-preserving” operation herein. The property that a dominant-error-conserving physical quantum operation conserves the dominant qubit error of the physical implementation can include that the physical quantum operation maps the dominant qubit error into an error that commutes with the dominant qubit error. It may be the case that a dominant-error-conserving physical quantum operation, when applied to the physical quantum system, leaves the error type of the dominant qubit error invariant. The dominant-error-conserving physical quantum operation may transform the dominant qubit error into a transformed error operator which may have a same error type as the dominant qubit error. For example, if the dominant qubit error is a bit flip error, represented by an X operator, the transformed error operator may be an X-type error operator, namely an operator which, whenexpanded in the Pauli basis, only contains X operators and (tensor) products thereof. Similar examples are provided by considering the Z operator, the Y operator or any other qubit operator as the dominant qubit error.

[0232] As described above, each of the encoded quantum circuits of Examples G1-G8 and M1- M2 can be physically realized using dominant-error-conserving physical quantum operations only.

[0233] Aspects of the present disclosure involve fault-tolerant quantum computation.

[0234] The first sequence of physical quantum operations may provide a fault-tolerant implementation of the first logical quantum operation. A sequence of physical quantum operations (such as the first sequence of physical quantum operations, the second sequence of physical quantum operations described below, or any other sequence of physical quantum operations described herein) that is fault-tolerant can be a sequence which is configured such that errors occurring during the sequence are correctable by performing quantum error- correction on the physical quantum system, provided that the error rate according to which any physical quantum operation in the sequence can be faulty lies below a threshold value. The errors can occur due to noise acting on the physical quantum system and / or due to one or more physical quantum operations in the sequence being faulty. The errors may be correctable by performing quantum error-correction based at least on the parity code. Quantum error- correction based on a further code may also be performed (e.g. the repetition code used to encode ancillary qubits, if the latter are present).

[0235] The first sequence of physical quantum operations may have a fault-tolerant error propagation of the dominant qubit error. A sequence of physical quantum operations (such as the first sequence of physical quantum operations, the second sequence of physical quantum operations described below, or any other sequence of physical quantum operations described herein) has a fault-tolerant error propagation of the dominant qubit error if, at any time during the sequence, the dominant qubit error acting on any single qubit is transformed, due to the action of the physical quantum operations in the sequence, into a transformed error, wherein the transformed error acts on each qubit of a set of one or more affected qubits, wherein the set of one or more affected qubits includes at most one qubit in each encoded block of a plurality of encoded blocks. It may be the case that the encoded blocks are disjoint sets of qubits. An encoded block can be a set consisting of qubits that are all dependent on a same logical qubitor on a same subset of logical qubits or can be a set consisting of ancillary qubits. It may be the case that different encoded blocks do not contain qubits that are dependent on a same logical

[0236] For example, with respect to the implementation of a CNOT gate, the encoded quantum circuit in part c) of Fig.13 acts on a first encoded block consisting of the three data qubits “3” and a second encoded block consisting of the code qubits “2”, “12” and “02”. Each qubit in the first encoded block is dependent on logical qubit “3”. Each qubit in the second encoded block is dependent on logical qubit “2” (but not on logical qubit “3”). The sequence of CNOT gates in part c) of Fig. 13 is transversally applied between the first encoded block and the second encoded block. Any single-qubit X error (being the dominant qubit error in Examples G1-G8 and M1-M2) can be propagated to at most a single error in the first encoded block and a single error in the second encoded block. Accordingly, the (sequence of physical quantum operations corresponding to) the encoded quantum circuit in part c) of Fig. 13 has a fault-tolerant error propagation of the dominant qubit error.

[0237] In another example, with respect to the implementation of a two-qubit ^ / 2 rotation, the encoded quantum circuit in part c) of Fig. 14 acts on a first encoded block consisting of the three ancillary qubits “A” and a second encoded block consisting of the three parity qubits “25”. The sequence of CNOT gates in part c) of Fig. 14 is again transversally applied between the first encoded block and the second encoded block. Further, error propagation of erroneous Z measurements is prevented due to the majority voting, as described above. Accordingly, any single-qubit X error can be propagated to at most a single error in the first encoded block and a single error in the second encoded block. Thus, the (sequence of physical quantum operations corresponding to) the encoded quantum circuit in part c) of Fig. 14 has a fault-tolerant error propagation of the dominant qubit error. For similar reasons, the encoded quantum circuits in Figs.15-18 all have a fault-tolerant propagation of the dominant qubit error.

[0238] Aspects of the present disclosure involve bosonic constituents.

[0239] It may be the case that at least some of the constituents are bosonic constituents. Each bosonic constituent may include a plurality of quantum levels of a quantum oscillator. It may be the case that each code qubit of at least a subset of the code qubits is physically implemented in a corresponding subsystem that consists of a bosonic constituent. It may be the case that each code qubit of at least a subset of the code qubits is physically implemented as a superposition of quantum basis states of a bosonic constituent, wherein the quantum basis states may becoherent states or superpositions thereof. The quantum basis states of the bosonic constituent may include a first quantum basis state and a second quantum basis state forming a computational basis of a qubit. For example, each of the quantum basis states may at least approximately be a coherent state, as e.g. described in Example C1. In another example, each of the quantum basis states may be a superposition of coherent states, as e.g. described in Example C2. More specifically, each of the quantum basis states may be a cat state (see e.g. Example C2) or a superposition of cat states (see e.g. Example C1). Each code qubit of at least a subset of the code qubits may be physically implemented in a corresponding subsystem that consists of a bosonic constituent according to a cat qubit implementation.

[0240] Aspects of the present disclosure involve a variable parity code.

[0241] The parity code may be a variable code that is changed during the encoded quantum computation. During the first portion of the encoded quantum computation, the quantum state of at least a portion of the physical quantum system may be an encoded quantum state corresponding to a first instance of the parity code. In an illustrative example (which is merely provided for assisting the reader to better understand the concepts involved, without implying any limitation), the first portion of the encoded quantum computation can refer to a portion wherein an encoded implementation of e.g. a CNOT gate is performed, which makes use of the parity code shown in part b) of Fig.13. The latter parity code can be understood as an example of a first instance of a parity code.

[0242] During a second portion of the encoded quantum computation, the quantum state of at least a portion of the physical quantum system may be an encoded quantum state corresponding to a second instance of the parity code. The second portion may be before or after the first portion. Continuing the above illustrative example, the second portion of the encoded quantum computation can refer to a portion wherein an encoded implementation of e.g. a two-qubit ^ / 2 rotation is performed, which makes use of the parity code shown in part b) of Fig.14. The latter parity code can be understood as an example of a second instance of the parity code.

[0243] The first instance of the parity code is different from the second instance of the parity code. That the two instances are different can be understood in the sense that the layout of code qubits (parity qubits and any data qubits) is different for the second instance of the parity code as compared to the first instance of the parity code. For example, the first instance of the parity code may have a first parity qubit that represents the parity of a first subset of logical qubits,wherein the parity of the first subset of logical qubits is not represented by any parity qubit of the second instance of the parity code. Additionally or alternatively, the second instance of the parity code may have a second parity qubit that represents the parity of a second subset of logical qubits, wherein the parity of the second subset of logical qubits is not represented by any parity qubit of the first instance of the parity code.

[0244] During the second portion of the encoded quantum computation, the code qubits of the second instance of the parity code may include a second code qubit duplication set associated with a second logical qubit. The second logical qubit may be different from the first logical qubit or equal to the first logical qubit. The second code qubit duplication set may include at least three code qubits, wherein either (a) each code qubit in the second code qubit duplication set is a parity qubit representing the parity of a same subset of logical qubits that includes the second logical qubit or (b) each code qubit in the second code qubit duplication set is a data qubit representing a quantum state of the second logical qubit. It may be the case that the second code qubit duplication set is not part of the first instance of the parity code and / or that the first code qubit duplication set is not part of the second instance of the parity code. That the second code qubit duplication set is not part of the first instance of the parity code can include that at least one code qubit in the second code qubit duplication set is not a code qubit in the first instance of the parity code. That the first code qubit duplication set is not part of the second instance of the parity code can include that at least one code qubit in the first code qubit duplication set is not a code qubit in the second instance of the parity code. Continuing the above illustrative example, the first logical qubit may refer e.g. to the logical qubit “3” in part a) of Fig.13. The first code qubit duplication set of the first instance of the parity code can refer e.g. to the code qubit duplication set 1350. The second logical qubit may refer e.g. to the logical qubit “2” in part a) of Fig.14. The second code qubit duplication set of the second instance of the parity code can refer e.g. to the code qubit duplication set 1450. As evident from the figures, the code qubit duplication set 1350 is not part of the second instance of the parity code and the code qubit duplication set 1450 is not part of the first instance of the parity code.

[0245] It may be the case that each code in the second code qubit duplication set is physically implemented in a corresponding subsystem of the physical quantum system by a physical implementation having a dominant qubit error.

[0246] A second sequence of physical quantum operations may be performed on the physical quantum system during the second portion of the encoded quantum computation. The second sequence of physical quantum operations may act at least on the subsystems of the physical quantum system corresponding to the code qubits in the second code qubit duplication set. The second sequence of physical quantum operations may be an encoded realization, via the second instance of the parity code, of a second logical quantum operation acting at least on the second logical qubit. Continuing the above illustrative example, the first sequence of physical quantum operations can refer e.g. to a sequence of physical quantum operations that realizes the encoded quantum circuit shown in part c) of Fig. 13, which provides an encoded implementation of a CNOT gate acting on logical qubits “2” and “3”. The second sequence of physical quantum operations can refer e.g. to a sequence of physical quantum operations that realizes the encoded quantum circuit in part c) of Fig.14, which provides an encoded implementation of a two-qubit ^ / 2 rotation acting on logical qubits “2” and “5”.

[0247] One or more, particularly all, physical quantum operations in the second sequence of physical quantum operations may be dominant-error-conserving quantum operations. The second sequence of physical quantum operations may have a fault-tolerant error propagation of the dominant qubit error. The second sequence of physical quantum operations may provide a fault-tolerant implementation of the second logical quantum operation.

[0248] One or more re-coding operations may be performed on the physical quantum system to evolve the physical quantum system from an encoded quantum state that corresponds to the first instance of the parity code to an encoded quantum state that corresponds to the second instance of the parity code. The one or more re-coding operations may be performed after the first sequence of physical quantum operations and before the second sequence of physical quantum operations. It may be the case that the one or more re-coding operations do not provide an encoded realization of a logical quantum operation acting on the logical qubits. At least some of the re-coding operations may be physical implementations of one or more controlled-NOT (CNOT) operations. Continuing the above illustrative example, the one or more re-coding operations may include physical operations that transform (a physical implementation of) the parity code shown in part b) of Fig.13 to (a physical implementation of) the parity code shown in part b) of Fig.14. The first instance of the parity code may be transformed directly into the second instance of the parity code, or the first instance may first be transformed into a standardparity code (as e.g. shown in Fig.8), which may subsequently be transformed into the second instance.

[0249] Aspects of the present disclosure involve a universal set of quantum gates.

[0250] Embodiments described herein provide a fault-tolerant implementation of a universal set of quantum gates. Accordingly, an arbitrary quantum computation can be realized in a fault- tolerant manner. A universal set of quantum gates can be understood as a set of unitary operations configured such that an arbitrary unitary quantum operation can be expressed, either exactly or approximately, as a product of unitary operations taken from the universal gate set. In this context, a unitary quantum operation is called unitary gate.

[0251] It may be the case that N sequences of physical quantum operations are performed on the physical quantum system during N respective portions of the encoded quantum computation. The number N may be two or larger, three or larger, or ten or larger. During the i-th portion of the encoded quantum computation for each i ranging from 1 to N, the code qubits may include an i-th code qubit duplication set associated with an i-th logical qubit. The i-th code qubit duplication set may include at least three code qubits, wherein either (a) each code qubit in the i-th code qubit duplication set is a parity qubit representing the parity of a same subset of logical qubits that includes the i-th logical qubit or (b) each code qubit in the i-th code qubit duplication set is a data qubit representing a quantum state of the i-th logical qubit. The N code qubit duplication sets can all be different from each other or some of the N code qubit duplication sets can be equal to each other. The i-th sequence of physical quantum operations of the N sequences of physical quantum operations may act at least on the subsystems of the physical quantum system corresponding to the code qubits in the i-th code qubit duplication set. The i-th sequence of physical quantum operations may be an encoded realization, via the parity code (more particularly via an i-th instance of the parity code), of an i-th logical quantum operation of N logical quantum operations. The i-th logical quantum operation may act at least on the i-th logical qubit. Particularly, the i-th sequence of physical quantum operations may provide a fault-tolerant implementation of the i-th logical quantum operation. Each quantum gate of a universal set of quantum gates may be at least one of the N logical quantum operations. In other words, the N logical quantum operations may include each quantum gate of the universal set of quantum gates at least once.

[0252] It may be the case that each code in the i-th code qubit duplication set, where i ranges from 1 to N, is physically implemented in a corresponding subsystem of the physical quantum system by a physical implementation having a dominant qubit error. One or more, particularly all, physical quantum operations in the i-th sequence of physical quantum operations may be dominant-error-conserving quantum operations. The i-th sequence of physical quantum operations may provide a fault-tolerant implementation of the i-th logical quantum operation.

[0253] Aspects of the present disclosure involve encoded implementations of single-qubit logical quantum operations.

[0254] The first logical quantum operation may act solely on the first logical qubit. Each code qubit in the first code qubit duplication set may be a data qubit representing a quantum state of the first logical qubit. The first logical quantum operation may be a unitary operation or a measurement. In an illustrative example (without implying any limitation), the first logical quantum operation may be an S gate as shown in Fig.15, an HTH gate as shown in Fig.17, or an H gate as shown in Fig.18. Correspondingly, the first code qubit duplication set may be the code qubit duplication set 1550 (S gate) or 1750 (HTH gate or H gate), each of which includes data qubits only. In a further example, the first logical quantum operation may be a Z measurement, as e.g. described in Example M1.

[0255] The first sequence of physical quantum operations may act at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set and on at least three ancillary subsystems of the physical quantum system. A respective ancillary qubit may be physically implemented in each of the at least three ancillary subsystems, particularly according to a physical implementation having a dominant qubit error. For example, in Fig.15 (relating to the S gate), Fig.17 (relating to the HTH gate) and Fig.18 (relating to the H gate), each ancillary qubit is denoted by “A”.

[0256] It may be the case that an ancillary qubit is not a code qubit. An ancillary qubit can be a qubit that does not contain information regarding the logical qubits. These properties apply to any ancillary qubit described herein.

[0257] The first logical quantum operation may be an S gate, a Hadamard-transformed S-gate, a T gate, a Hadamard-transformed T-gate or a Hadamard gate acting on the first logical qubit.

[0258] Performing the first sequence of physical quantum operations may include preparing the at least three ancillary subsystems in an initial quantum state. Particularly, the initial quantum state may correspond to a repetition quantum error-correction code of the ancillary qubits. The initial quantum state may be an entangled quantum state of the at least three ancillary subsystems.

[0259] Performing the first sequence of physical quantum operations may include performing at least three physical quantum operations. Each of the at least three physical quantum operations may act jointly on at least a respective first subsystem and a respective second subsystem of the physical quantum system. Each first subsystem may be one of the at least three ancillary subsystems and each second subsystem may be a subsystem corresponding to a code qubit of the first code qubit duplication set. The at least three physical quantum operations may realize a transversal quantum circuit acting on the code qubits of the first code qubit duplication set and on the ancillary qubits. In an illustrative example (but without implying any limitation), the at least three physical quantum operations can be understood as including a sequence of physical quantum operations that realizes the transversal application of the three CNOT gates in part c) of Fig.15 (S gate), or a sequence of physical quantum operations that realizes any of the transversally applied CNOT gates in part c) of Fig.17 (HTH gate) or part b) of Fig.18 (H gate).

[0260] Performing the first sequence of physical quantum operations may include measuring at least one subsystem of the at least three ancillary subsystems and / or at least one subsystem corresponding to a code qubit of the first code qubit duplication set. In an illustrative example (but without implying any limitation), the measurements in question can be understood as including any of the X measurements or Z measurements in Fig. 15 (S gate), Fig. 17 (HTH gate) or Fig.18 (H gate).

[0261] Aspects of the present disclosure involve encoded implementations of logical quantum operations acting on at least two logical qubits.

[0262] The first logical quantum operation may act at least on the first logical qubit and a second logical qubit. Each code qubit in the first code qubit duplication set may be a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit and the second logical qubit. Particularly, the first subset of logical qubits may consist of the first logical qubit and the second logical qubit. The first logical quantum operation may bea unitary operation, particularly an entangling unitary operation. In an illustrative example (without implying any limitation), the first logical quantum operation may be a two-qubit ^ / 2- rotation as shown in Fig.14 or a CZ gate as shown in Fig.16. Correspondingly, the first code qubit duplication set may be the code qubit duplication set 1450 (two-qubit ^ / 2-rotation) or the code qubit duplication set 1653 (CZ gate), each of which includes two-element-parity qubits only.

[0263] The first sequence of physical quantum operations may act at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set and on at least three ancillary subsystems of the physical quantum system. A respective ancillary qubit may be physically implemented in each of the at least three ancillary subsystems, particularly according to a physical implementation having a dominant qubit error. For example, in Fig.14 (relating to the two-qubit ^ / 2-rotation) each ancillary qubit is denoted by “A”. In Fig.16 (relating to the CZ gate), the ancillary qubits can include the ancillary qubits denoted by “C”.

[0264] The first logical quantum operation may be a two-qubit ^ / 2-rotation or a controlled-Z gate acting on the first logical qubit and the second logical qubit.

[0265] Performing the first sequence of physical quantum operations may include preparing the at least three ancillary subsystems in an initial quantum state. Particularly, the initial quantum state may correspond to a repetition quantum error-correction code of the ancillary qubits. The initial quantum state may be an entangled quantum state of the at least three ancillary subsystems.

[0266] Performing the first sequence of physical quantum operations may include performing at least three physical quantum operations. Each of the at least three physical quantum operations may act jointly on at least a respective first subsystem and a respective second subsystem of the physical quantum system. Each first subsystem may be one of the at least three ancillary subsystems and each second subsystem may be a subsystem corresponding to a code qubit of the first code qubit duplication set. The at least three physical quantum operations may realize a transversal quantum circuit acting on the code qubits of the first code qubit duplication set and on the ancillary qubits. In an illustrative example (but without implying any limitation), the at least three physical quantum operations can be understood as including a sequence ofphysical quantum operations that realizes the transversal application of the three CNOT gates in part c) of Fig.14 (two-qubit ^ / 2-rotation), or a sequence of physical quantum operations that realizes the transversally applied CNOT gates in the lowermost circuit in part c) of Fig.16 (CZ gate).

[0267] Performing the first sequence of physical quantum operations may include measuring at least one subsystem of the at least three ancillary subsystems and / or at least one subsystem corresponding to a code qubit of the first code qubit duplication set. In an illustrative example (but without implying any limitation), the measurements in question can be understood as including any of the X measurements or Z measurements in part c) of Fig. 14 or in the lowermost circuit in part c) of Fig.16 (CZ gate).

[0268] The first sequence of physical quantum operations may include a quantum gate teleportation protocol. For example, in any of Figs. 14-18, at least one quantum gate teleportation protocol is performed.

[0269] The first logical quantum operation may act at least on the first logical qubit and a second logical qubit. Each code qubit in the first code qubit duplication set may be a data qubit representing a quantum state of the first logical qubit. The code qubits may include at least three code qubits that are dependent on the second logical qubit. A code qubit is dependent on the second logical qubit if the code qubit is a data qubit representing a quantum state of the second logical qubit or if the code qubit is a parity qubit representing the parity of a subset of logical qubits that includes the second logical qubit. The first logical quantum operation may be a unitary operation, particularly an entangling unitary operation. It may be the case that the at least three code qubits that are dependent on the second logical qubit are not dependent on the first logical qubit. In an illustrative example (without implying any limitation), the first logical quantum operation may be a CNOT gate as shown in Fig.13. Correspondingly, the first code qubit duplication set may be the code qubit duplication set 1350, which consists of data qubits. Specifically, in the present example, the first logical qubit and the second logical qubit can be understood as the logical qubit “3” and the logical qubit “2”, respectively. The at least three code qubits that are dependent on the second logical qubit can, in this example, be understood as the code qubits “2”, “12” and “02” (see part c) of Fig.13), which are all dependent on logical qubit “2” since the label “2” occurs in each of these code qubits.

[0270] It may be the case that each code of the at least three code qubits that are dependent on the second logical qubit is physically implemented in a corresponding subsystem of the physical quantum system by a physical implementation having a dominant qubit error.

[0271] The first sequence of physical quantum operations may act at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set and on the subsystems of the physical quantum system corresponding to the at least three code qubits that are dependent on the second logical qubit. Particularly, the first logical quantum operation may be a CNOT gate acting on the first logical qubit and the second logical qubit.

[0272] Performing the first sequence of physical quantum operations may include performing at least three physical quantum operations. Each of the at least three physical quantum operations may act jointly on at least a respective first subsystem and a respective second subsystem of the physical quantum system. Each first subsystem may be a subsystem corresponding to a code qubit of the first code qubit duplication set and each second subsystem may be a subsystem corresponding to one of the at least three code qubits that are dependent on the second logical qubit. The at least three physical quantum operations may realize a transversal quantum circuit acting on the code qubits of the first code qubit duplication set and on the at least three code qubits that are dependent on the second logical qubit. In an illustrative example (but without implying any limitation), the at least three physical quantum operations can be understood as including a sequence of physical quantum operations that realizes the transversal application of the three CNOT gates in part c) of Fig.13.

[0273] According to a further embodiment, a quantum computational method is provided. The quantum computational method includes providing a physical quantum system comprising constituents. The quantum computational method includes performing an encoded quantum computation on the physical quantum system. Performing the encoded quantum computation includes preparing at least a portion of the physical quantum system in an initial quantum state. Performing the encoded quantum computation includes evolving at least a portion of the physical quantum system to a final quantum state. Performing the encoded quantum computation includes measuring at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to aquantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits into code qubits. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem of the physical quantum system, wherein the subsystem comprises one or more constituents. The physical implementation has a dominant qubit error, wherein at least one qubit error different from the dominant qubit error is suppressed as compared to the dominant qubit error or is correctable by performing one or more error- correcting operations on the subsystem. The dominant qubit error is correctable by the parity code in one or more subsystems of the physical quantum system. The quantum computational method further includes performing a first sequence of physical quantum operations on the physical quantum system during a first portion of the encoded quantum computation. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation acting on one or more logical qubits. The first sequence of physical operations is a fault-tolerant implementation of the first logical quantum operation. The quantum computational method can include any features, either alone or in combination, of a quantum computational method as described above.

[0274] The first logical quantum operation may act on at least two logical qubits and / or it may be the case that the first logical quantum operation is not a Pauli operator. A Pauli operator can be understood as an operator that is, up to a global phase, a tensor product of X, Y and Z operators (where Y denotes the σ^^Pauli matrix). The first logical quantum operation may be a unitary operator. If the first logical quantum operation acts on at least two logical qubits, the first logical quantum operation may be an entangling operation. The first logical quantum operation may be a CNOT gate, a two-qubit ^ / 2 rotation, an S gate, a CZ gate, an HTH gate, or an H gate. For example, the first logical quantum operation may be any of the logical gates described with respect to Figs.13-18.

[0275] It may be the case that N sequences of physical quantum operations are performed on the physical quantum system. The number N may be two or larger. For i from 1 to N, the i-th sequence of physical quantum operations of the N sequences may be an encoded realization, via the parity code, of an i-th logical quantum operation of N logical quantum operations acting on the logical qubits. The i-th sequence of physical quantum operations may provide a fault- tolerant implementation of the i-th logical quantum operation. Each quantum gate of a universal set of quantum gates may be at least one of the N logical quantum operations.

[0276] Aspects of the present disclosure involve a control layout for a quantum computer.

[0277] According to a further embodiment, a method of determining a control layout for a quantum computer is provided. The method includes determining a layout of a parity code that encodes logical qubits into code qubits. The layout is determined based on a specification of a logical quantum computation acting on the logical qubits. The logical quantum computation includes a first logical quantum operation acting at least on a first logical qubit. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. The code qubits include a first code qubit duplication set associated with the first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit. The method may be a computer-implemented method, which may, for example, be carried out by a classical computing system as described herein. The method may include any features, either alone or in combination, described in relation to the above quantum computational methods.

[0278] A control layout for a quantum computer (and likewise a control layout of an encoded quantum computation) can include information that is to be transmitted to a controller of a quantum computer. The control layout can include control instructions for the quantum computer or information that allows to determine control instructions therefrom. A control layout can be a control layout of an entire quantum computation from start (preparation of initial quantum state) to finish (measurement(s) to provide read-out), or can alternatively be a control layout for a portion of a quantum computation. A control layout can specify how the qubits shall be encoded during at least a portion of the quantum computation. Specifically, a control layout can specify a layout of a parity code according to which the qubits will be encoded during at least a portion of the quantum computation.

[0279] A specification of a logical quantum computation acting on the logical qubits can include a specification, or description, of one or more logical quantum operations (e.g. unitary operations and / or measurements) that make up the logical quantum computation or a portion thereof. The specification may contain information that allows an operator or system to determine which quantum operations are performed in the logical quantum computation (or aportion thereof), on which logical qubits one or more quantum operations act, in which order the quantum operations are performed, and the like.

[0280] A layout of a parity code can include a specification of the code qubits, e.g. one or more parity qubits and / or one or more data qubits, of at least a portion of the parity code. The layout may include information that allows an operator or system to determine which parity qubits are part of the parity code (or a portion thereof), to which subset of logical qubits each parity qubit is associated, which data qubits (if any) are part of the parity code (or a portion thereof), to which logical qubits each data qubit is associated, optionally how the code qubits shall be spatially arranged, and the like.

[0281] The method of determining a control layout for a quantum computer may include acquiring (e.g. receiving, reading, calculating, and the like) the specification of the logical quantum computation. The method may include determining the layout of the parity code in response to acquiring the specification of the logical quantum computation. The method may include transmitting the determined layout of the parity code, e.g. to an operator or system, such as a classical computing system.

[0282] A control layout of an encoded quantum computation may be determined. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system may be an encoded quantum state corresponding to the parity code. The encoded quantum computation may include a first sequence of quantum operations that acts at least on the code qubits in the first code qubit duplication set. The first sequence of quantum operations may be an encoded realization, via the parity code, of the first logical quantum operation. The control layout of the encoded quantum computation may include the determined layout of the parity code. The encoded quantum computation can include any features, either alone or in combination, of an encoded quantum computation as described herein.

[0283] According to a further embodiment, a control layout for a quantum computer is provided. The control layout includes a layout of an encoding of logical qubits into code qubits corresponding to a parity code. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. The code qubits include a first code qubit duplication set associated with a first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubitsthat includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit.

[0284] The control layout for the quantum computer may include a control layout of an encoded quantum computation. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system may be an encoded quantum state corresponding to the parity code. The encoded quantum computation may include a first sequence of quantum operations that acts at least on the code qubits in the first code qubit duplication set. The first sequence of quantum operations may be an encoded realization, via the parity code, of a first logical quantum operation acting on the first logical qubit.

[0285] According to a further embodiment, a data carrier or data carrier signal carrying information representing a control layout as described herein is provided.

[0286] Aspects of the present disclosure involve an apparatus for quantum computation.

[0287] Fig.19 shows an apparatus 1900 for quantum computation according to embodiments described herein. The apparatus includes a physical quantum system 100 comprising constituents. The physical quantum system 100 includes subsystems 350, which may each include one or more constituents. Code qubits 20 of a parity code may be physically implemented in the subsystems 350. The apparatus 1900 includes a quantum processing system 1910 for evolving at least some of the constituents. The apparatus 1900 includes a measurement system 1920 for measuring one or more of the constituents. The apparatus 1900 includes a classical computing system 1930 connected to the quantum processing system 1910 and to the measurement system 1920. The classical computing system 1930 is configured to instruct at least one of the quantum processing system 1910 and the measurement system 1920 to perform an encoded quantum computation on the physical quantum system as described herein.

[0288] Any physical quantum operation as described herein may be performed by the quantum processing system 1910 and the measurement system 1920.

[0289] The quantum processing system 1910 may be configured for performing a unitary evolution of the physical quantum system, particularly a unitary evolution according to a short- range unitary operator. The quantum processing system 1910 may be configured for performing any physical quantum operation described herein that is a unitary operator. Depending on the physical implementation at hand, the quantum processing system 1910 can include one or morelasers, (micro-)wave generators, magnetic field generators, electrical signal generators, and the like.

[0290] The measurement system 1920 may be configured for performing any physical quantum operation described herein that is a measurement. Depending on the physical implementation at hand, the measurement system 1920 can include one or more sensors, (microwave) frequency interrogators, magnetometers, electrometers, photodetectors, and the like.

[0291] A classical computing system, or classical computer, can be understood as a computing system that processes information using only classical information carriers, such as classical bits. The term “classical” can in this context be understood as “not quantum”. A classical computing system can include, for example, a personal computer or a network of personal computers. The classical computing system 1930 can be or be part of a controller for controlling the operation of the quantum processing system 1910 or the measurement system 1920, e.g. based on a control layout as described herein.

[0292] According to a further embodiment, an apparatus for quantum computation is provided. The apparatus includes a physical quantum system comprising constituents. The apparatus includes a quantum processing system for evolving at least some of the constituents. The apparatus includes a measurement system for measuring one or more of the constituents. The apparatus includes a classical computing system connected to the quantum processing system and to the measurement system. The classical computing system is configured to instruct at least one of the quantum processing system and the measurement system to perform an encoded quantum computation on the physical quantum system. The encoded quantum computation includes preparing, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system in an initial quantum state. The encoded quantum computation includes evolving, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system to a final quantum state. The encoded quantum computation includes measuring, using the measurement system, at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits into code qubits. The code qubits include parity qubits, wherein each parity qubit represents the parity of anassociated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem of the physical quantum system, wherein the subsystem comprises one or more constituents. During a first portion of the encoded quantum computation, the code qubits include a first code qubit duplication set associated with a first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit. Evolving at least a portion of the physical quantum system to a final quantum state includes performing a first sequence of physical quantum operations on the physical quantum system during the first portion of the encoded quantum computation. The first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation acting at least on the first logical qubit. The apparatus may be configured for performing any operations, either alone or in combination, that are part of the quantum computational methods described herein.

[0293] According to a further embodiment, an apparatus for quantum computation is provided. The apparatus includes a physical quantum system comprising constituents. The apparatus includes a quantum processing system for evolving at least some of the constituents. The apparatus includes a measurement system for measuring one or more of the constituents. The apparatus includes a classical computing system connected to the quantum processing system and to the measurement system. The classical computing system is configured to instruct at least one of the quantum processing system and the measurement system to perform an encoded quantum computation on the physical quantum system. The encoded quantum computation includes preparing, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system in an initial quantum state. The encoded quantum computation includes evolving, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system to a final quantum state. The encoded quantum computation includes measuring, using the measurement system, at least a portion of the physical quantum system to provide a read-out. During at least a portion of the encoded quantum computation, the quantum state of at least a portion of thequantum system is an encoded quantum state corresponding to a quantum error-correction code. The quantum error-correction code is a parity code that encodes logical qubits into code qubits. The code qubits include parity qubits, wherein each parity qubit represents the parity of an associated subset of logical qubits. Each code qubit is physically implemented in a corresponding subsystem of the physical quantum system, wherein the subsystem comprises one or more constituents. The physical implementation has a dominant qubit error. At least one qubit error different from the dominant qubit error is suppressed as compared to the dominant qubit error or is correctable by performing one or more error-correcting operations on the subsystem. The dominant qubit error is correctable by the parity code in one or more subsystems of the physical quantum system. Evolving at least a portion of the physical quantum system to a final quantum state includes performing a first sequence of physical quantum operations on the physical quantum system during a first portion of the encoded quantum computation. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation acting on one or more logical qubits. The first sequence of physical quantum operations provides a fault-tolerant implementation of the first logical quantum operation. The apparatus may be configured for performing any operations, either alone or in combination, that are part of the quantum computational methods described herein.

[0294] While the foregoing is directed to embodiments, other and further embodiments may be devised without departing from the scope determined by the claims.

Claims

CLAIMS 1. A quantum computational method, comprising: providing a physical quantum system (100) comprising constituents (50); performing an encoded quantum computation on the physical quantum system, comprising: preparing at least a portion of the physical quantum system in an initial quantum state; evolving at least a portion of the physical quantum system to a final quantum state; and measuring at least a portion of the physical quantum system to provide a read-out, wherein, during at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code, wherein the quantum error-correction code is a parity code that encodes logical qubits (10) into code qubits (20), wherein the code qubits include parity qubits (802), wherein each parity qubit represents the parity of an associated subset of logical qubits, wherein each code qubit is physically implemented in a corresponding subsystem (50, 350) of the physical quantum system, wherein the subsystem comprises one or more constituents, wherein, during a first portion of the encoded quantum computation, the code qubits include a first code qubit duplication set (1350, 1450, 1550, 1651-1653, 1750) associated with a first logical qubit, wherein the first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit, wherein the quantum computational method further comprises:performing a first sequence of physical quantum operations on the physical quantum system during the first portion of the encoded quantum computation, wherein the first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set, wherein the first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting at least on the first logical qubit.

2. The quantum computational method of claim 1, wherein each code qubit of at least a subset of the code qubits is physically implemented in the corresponding subsystem by a physical implementation having a dominant qubit error, wherein at least one qubit error different from the dominant qubit error is suppressed as compared to the dominant qubit error or is correctable by performing one or more error-correcting operations on the subsystem, wherein the dominant qubit error is correctable by the parity code in one or more subsystems of the physical quantum system.

3. The quantum computational method of claim 2, wherein one or more physical quantum operations in the first sequence of physical quantum operations are dominant-error-conserving quantum operations that conserve the dominant qubit error of the physical implementation.

4. The quantum computational method of any of the preceding claims, wherein the first sequence of physical quantum operations provides a fault-tolerant implementation of the first logical quantum operation.

5. The quantum computational method of any of claims 2 to 4, wherein each code qubit of at least a subset of the code qubits is physically implemented in a corresponding subsystem that consists of a constituent (50), wherein the code qubit is encoded into the constituent according to a quantum error-suppression code, wherein at least one qubit error different from the dominant qubit error is suppressed by the quantum error-suppression code as compared to the dominant qubit error.

6. The quantum computational method of any of the preceding claims, wherein at least some of the constituents are bosonic constituents, wherein each bosonic constituent includes a plurality of quantum levels of a quantum oscillator, wherein each code qubit of at least a subset of the code qubits is physically implemented in a corresponding subsystem that consists of a bosonic constituent, particularly wherein each code qubit of at least a subset of the code qubits is physically implemented as a superposition of quantum basis states of a bosonic constituent, wherein the quantum basis states are coherent states or superpositions thereof.

7. The quantum computational method of any of the preceding claims, wherein the parity code is a variable code that is changed during the encoded quantum computation, wherein, during the first portion of the encoded quantum computation, the quantum state of at least a portion of the physical quantum system is an encoded quantum state corresponding to a first instance of the parity code, wherein, during a second portion of the encoded quantum computation, the quantum state of at least a portion of the physical quantum system is an encoded quantum state corresponding to a second instance of the parity code, wherein the first instance of the parity code is different from the second instance of the parity code, particularly wherein the first instance of the parity code has a first parity qubit that represents the parity of a first subset of logical qubits, wherein the parity of the first subset of logical qubits is not represented by any parity qubit of the second instance of the parity code, and / or wherein the second instance of the parity code has a second parity qubit that represents the parity of a second subset of logical qubits, wherein the parity of the second subset of logical qubits is not represented by any parity qubit of the first instance of the parity code.

8. The quantum computational method of claim 7,wherein, during the second portion of the encoded quantum computation, the code qubits of the second instance of the parity code include a second code qubit duplication set (1350, 1450, 1550, 1651-1653, 1750) associated with a second logical qubit, wherein the second logical qubit may be different from the first logical qubit or equal to the first logical qubit, wherein the second code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the second code qubit duplication set is a parity qubit representing the parity of a same subset of logical qubits that includes the second logical qubit or (b) each code qubit in the second code qubit duplication set is a data qubit representing a quantum state of the second logical qubit, wherein the second code qubit duplication set is not part of the first instance of the parity code and / or wherein the first code qubit duplication set is not part of the second instance of the parity code.

9. The quantum computational method of claim 8, further comprising: performing a second sequence of physical quantum operations on the physical quantum system during the second portion of the encoded quantum computation, wherein the second sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the second code qubit duplication set, wherein the second sequence of physical quantum operations is an encoded realization, via the second instance of the parity code, of a second logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting at least on the second logical qubit.

10. The quantum computational method of any of claims 7 to 9, further comprising: performing one or more re-coding operations on the physical quantum system to evolve the physical quantum system from an encoded quantum state that corresponds to the first instance of the parity code to an encoded quantum state that corresponds to the second instance of the parity code.

11. The quantum computational method of any of the preceding claims, wherein the quantum computational method includes performing N sequences of physical quantumoperations on the physical quantum system during N respective portions of the encoded quantum computation, wherein N is two or larger, wherein, during the i-th portion of the encoded quantum computation for each i ranging from 1 to N, the code qubits include an i-th code qubit duplication set (1350, 1450, 1550, 1651- 1653, 1750) associated with an i-th logical qubit, wherein the i-th code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the i-th code qubit duplication set is a parity qubit representing the parity of a same subset of logical qubits that includes the i-th logical qubit or (b) each code qubit in the i-th code qubit duplication set is a data qubit representing a quantum state of the i-th logical qubit, wherein the N code qubit duplication sets can all be different from each other or some of the N code qubit duplication sets can be equal to each other, wherein the i-th sequence of physical quantum operations of the N sequences of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the i-th code qubit duplication set, wherein the i-th sequence of physical quantum operations is an encoded realization, via the parity code, of an i-th logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) of N logical quantum operations, wherein the i-th logical quantum operation acts at least on the i-th logical qubit, particularly wherein the i-th sequence of physical quantum operations provides a fault-tolerant implementation of the i-th logical quantum operation, wherein each quantum gate of a universal set of quantum gates is at least one of the N logical quantum operations.

12. The quantum computational method of any of the preceding claims, wherein the first logical quantum operation (1510, 1710, 1810) acts solely on the first logical qubit, and wherein each code qubit in the first code qubit duplication set (1550, 1750) is a data qubit representing a quantum state of the first logical qubit.

13. The quantum computational method of claim 12, wherein the first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set and on at least threeancillary subsystems of the physical quantum system, wherein a respective ancillary qubit is physically implemented in each of the at least three ancillary subsystems, particularly wherein the first logical quantum operation is an S gate (1510), a Hadamard- transformed S-gate, a T gate, a Hadamard-transformed T-gate (1710) or a Hadamard gate (1810) acting on the first logical qubit.

14. The quantum computational method of claim 13, wherein performing the first sequence of physical quantum operations includes: preparing the at least three ancillary subsystems in an initial quantum state, particularly wherein the initial quantum state corresponds to a repetition quantum error-correction code of the ancillary qubits; performing at least three physical quantum operations, wherein each of the at least three physical quantum operations acts jointly on at least a respective first subsystem and a respective second subsystem of the physical quantum system, wherein each first subsystem is one of the at least three ancillary subsystems and each second subsystem is a subsystem corresponding to a code qubit of the first code qubit duplication set; and measuring at least one subsystem of the at least three ancillary subsystems and / or at least one subsystem corresponding to a code qubit of the first code qubit duplication set.

15. The quantum computational method of claims 1 to 11, wherein the first logical quantum operation (1410, 1610) acts at least on the first logical qubit and a second logical qubit, wherein each code qubit in the first code qubit duplication set (1450, 1651-1653) is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit and the second logical qubit, particularly wherein the first subset of logical qubits consists of the first logical qubit and the second logical qubit.

16. The quantum computational method of claim 15, wherein the first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set and on at least threeancillary subsystems of the physical quantum system, wherein a respective ancillary qubit is physically implemented in each of the at least three ancillary subsystems, particularly wherein the first logical quantum operation is a two-qubit ^ / 2-rotation (1410) or a controlled-Z gate (1610) acting on the first logical qubit and the second logical qubit.

17. The quantum computational method of claim 16, wherein performing the first sequence of physical quantum operations includes: preparing the at least three ancillary subsystems in an initial quantum state, particularly wherein the initial quantum state corresponds to a repetition quantum error-correction code of the ancillary qubits; performing at least three physical quantum operations, wherein each of the at least three physical quantum operations acts jointly on at least a respective first subsystem and a respective second subsystem of the physical quantum system, wherein each first subsystem is one of the at least three ancillary subsystems and each second subsystem is a subsystem corresponding to a code qubit of the first code qubit duplication set; and measuring at least one subsystem of the at least three ancillary subsystems and / or at least one subsystem corresponding to a code qubit of the first code qubit duplication set.

18. The quantum computational method of any of claims 1 to 11, wherein the first logical quantum operation (1310) acts at least on the first logical qubit and a second logical qubit, wherein each code qubit in the first code qubit duplication set (1350) is a data qubit representing a quantum state of the first logical qubit, wherein the code qubits include at least three code qubits that are dependent on the second logical qubit, wherein a code qubit is dependent on the second logical qubit if the code qubit is a data qubit representing a quantum state of the second logical qubit or if the code qubit is a parity qubit representing the parity of a subset of logical qubits that includes the second logical qubit.

19. The quantum computational method of claim 18, wherein the first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set and on the subsystems of the physical quantum system corresponding to the at least three code qubits that are dependent on the second logical qubit, particularly wherein the first logical quantum operation is a CNOT gate (1310) acting on the first logical qubit and the second logical qubit.

20. The quantum computational method of claim 19, wherein performing the first sequence of physical quantum operations includes: performing at least three physical quantum operations, wherein each of the at least three physical quantum operations acts jointly on at least a respective first subsystem and a respective second subsystem of the physical quantum system, wherein each first subsystem is a subsystem corresponding to a code qubit of the first code qubit duplication set and each second subsystem is a subsystem corresponding to one of the at least three code qubits that are dependent on the second logical qubit.

21. The quantum computational method of any of the preceding claims, wherein the first sequence of physical quantum operations includes a quantum gate teleportation protocol.

22. A quantum computational method, comprising: providing a physical quantum system (100) comprising constituents (50); performing an encoded quantum computation on the physical quantum system, comprising: preparing at least a portion of the physical quantum system in an initial quantum state; evolving at least a portion of the physical quantum system to a final quantum state; andmeasuring at least a portion of the physical quantum system to provide a read-out, wherein, during at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code, wherein the quantum error-correction code is a parity code that encodes logical qubits (10) into code qubits (20), wherein the code qubits include parity qubits (802), wherein each parity qubit represents the parity of an associated subset of logical qubits, wherein each code qubit is physically implemented in a corresponding subsystem (50, 350) of the physical quantum system, wherein the subsystem comprises one or more constituents, wherein the physical implementation has a dominant qubit error, wherein at least one qubit error different from the dominant qubit error is suppressed as compared to the dominant qubit error or is correctable by performing one or more error-correcting operations on the subsystem, wherein the dominant qubit error is correctable by the parity code in one or more subsystems of the physical quantum system, wherein the quantum computational method further comprises: performing a first sequence of physical quantum operations on the physical quantum system during a first portion of the encoded quantum computation, wherein the first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting on one or more logical qubits, wherein the first sequence of physical operations is a fault-tolerant implementation of the first logical quantum operation.

23. The quantum computational method of claim 22, wherein the first logical quantum operation acts on at least two logical qubits and / or wherein the first logical quantum operation is not a Pauli operator.

24. The quantum computational method of claim 22 or 23, wherein the quantum computational method includes performing N sequences of physical quantum operations on the physical quantum system, wherein N is two or larger, wherein, for i from 1 to N, the i-th sequence of physical quantum operations of the N sequences is an encoded realization, via the parity code, of an i-th logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) of N logical quantum operations acting on the logical qubits, wherein the i-th sequence of physical quantum operations provides a fault- tolerant implementation of the i-th logical quantum operation, wherein each quantum gate of a universal set of quantum gates is at least one of the N logical quantum operations.

25. A method of determining a control layout for a quantum computer, comprising: determining a layout of a parity code that encodes logical qubits (10) into code qubits (20), wherein the layout is determined based on a specification of a logical quantum computation acting on the logical qubits, the logical quantum computation including a first logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting at least on a first logical qubit, wherein the code qubits include parity qubits (802), wherein each parity qubit represents the parity of an associated subset of logical qubits, wherein the code qubits include a first code qubit duplication set (1350, 1450, 1550, 1651-1653, 1750) associated with the first logical qubit, wherein the first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit.

26. The method of claim 25, further comprising: determining a control layout of an encoded quantum computation, wherein, during at least a portion of the encoded quantum computation, the quantum state of at least a portion ofthe quantum system is an encoded quantum state corresponding to the parity code, wherein the encoded quantum computation includes a first sequence of quantum operations that acts at least on the code qubits in the first code qubit duplication set, wherein the first sequence of quantum operations is an encoded realization, via the parity code, of the first logical quantum operation.

27. A control layout for a quantum computer, comprising: a layout of an encoding of logical qubits (10) into code qubits (20) corresponding to a parity code, wherein the code qubits include parity qubits (802), wherein each parity qubit represents the parity of an associated subset of logical qubits, wherein the code qubits include a first code qubit duplication set (1350, 1450, 1550, 1651-1653, 1750) associated with a first logical qubit, wherein the first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit.

28. The control layout of claim 27, further comprising: a control layout of an encoded quantum computation, wherein, during at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to the parity code, wherein the encoded quantum computation includes a first sequence of quantum operations that acts at least on the code qubits in the first code qubit duplication set, wherein the first sequence of quantum operations is an encoded realization, via the parity code, of a first logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting on the first logical qubit.

29. A data carrier or data carrier signal carrying information representing the control layout of claim 27 or 28.

30. An apparatus (1900) for quantum computation, comprising:a physical quantum system (100) comprising constituents (50); a quantum processing system (1910) for evolving at least some of the constituents: a measurement system (1920) for measuring one or more of the constituents; and a classical computing system (1930) connected to the quantum processing system and to the measurement system, wherein the classical computing system is configured to instruct at least one of the quantum processing system and the measurement system to perform an encoded quantum computation on the physical quantum system, wherein the encoded quantum computation comprises: preparing, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system in an initial quantum state; evolving, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system to a final quantum state; and measuring, using the measurement system, at least a portion of the physical quantum system to provide a read-out, wherein, during at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code, wherein the quantum error-correction code is a parity code that encodes logical qubits (10) into code qubits (20), wherein the code qubits include parity qubits (802), wherein each parity qubit represents the parity of an associated subset of logical qubits, wherein each code qubit is physically implemented in a corresponding subsystem (50, 350) of the physical quantum system, wherein the subsystem comprises one or more constituents, wherein, during a first portion of the encoded quantum computation, the code qubits include a first code qubit duplication set (1350, 1450, 1550, 1651-1653, 1750) associated with a first logical qubit, wherein the first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit,wherein evolving at least a portion of the physical quantum system to a final quantum state includes performing a first sequence of physical quantum operations on the physical quantum system during the first portion of the encoded quantum computation, wherein the first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set, wherein the first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting at least on the first logical qubit.

31. An apparatus (1900) for quantum computation, comprising: a physical quantum system (100) comprising constituents (50); a quantum processing system (1910) for evolving at least some of the constituents: a measurement system (1920) for measuring one or more of the constituents; and a classical computing system (1930) connected to the quantum processing system and to the measurement system, wherein the classical computing system is configured to instruct at least one of the quantum processing system and the measurement system to perform an encoded quantum computation on the physical quantum system, wherein the encoded quantum computation comprises: preparing, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system in an initial quantum state; evolving, using at least one of the quantum processing system and the measurement system, at least a portion of the physical quantum system to a final quantum state; and measuring, using the measurement system, at least a portion of the physical quantum system to provide a read-out, wherein, during at least a portion of the encoded quantum computation, the quantum state of at least a portion of the quantum system is an encoded quantum state corresponding to a quantum error-correction code, wherein the quantum error-correction code is a parity code that encodes logical qubits (10) into code qubits (20), wherein the code qubits include parityqubits (802), wherein each parity qubit represents the parity of an associated subset of logical qubits, wherein each code qubit is physically implemented in a corresponding subsystem (50, 350) of the physical quantum system, wherein the subsystem comprises one or more constituents, wherein the physical implementation has a dominant qubit error, wherein at least one qubit error different from the dominant qubit error is suppressed as compared to the dominant qubit error or is correctable by performing one or more error-correcting operations on the subsystem, wherein the dominant qubit error is correctable by the parity code in one or more subsystems of the physical quantum system, wherein evolving at least a portion of the physical quantum system to a final quantum state includes performing a first sequence of physical quantum operations on the physical quantum system during a first portion of the encoded quantum computation, wherein the first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting on one or more logical qubits, wherein the first sequence of physical quantum operations provides a fault-tolerant implementation of the first logical quantum operation.

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