Quantum information processing using asymmetric error channels

Abstract

Techniques for performing quantum information processing using asymmetric error channels are provided. According to some aspects, the quantum information processing includes a data qubit and an auxiliary qubit, the auxiliary qubit having an asymmetric error channel. The data qubit is coupled to the auxiliary qubit. The auxiliary qubit can be driven using a stable microwave field to create an asymmetric error channel.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the benefit under 35 U.S.C. §119(e) of U.S. Provisional Patent Application No. 62 / 692,243, filed on June 29, 2018, entitled “FAULT TOLERANTMEASUREMENTS AND GATES FOR QUANTUM INFORMATION PROCESSING,” which is hereby incorporated by reference in its entirety.

[0003] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT

[0004] This invention was made with government support under 1609326 from the National Science Foundation, FA9550-15-0029 from the U.S. Air Force Office of Scientific Research, N00014-16-2270 from the U.S. Naval Research Institute, and W911NF-14-1-0011 and W911NF-16-1-0349 from the U.S. Army Research Institute. The government has certain rights in this invention. Technical Field

[0005] The technology described herein generally relates to quantum information systems. Specifically, the present application is directed to systems and methods for performing quantum information processing (QIP) using at least one qubit with an asymmetric error channel. Background Art

[0006] QIP uses quantum mechanical phenomena such as energy quantization, superposition, and entanglement to encode and process information in ways that are not exploited by conventional information processing. For example, it is known that certain computational problems can be solved more efficiently using quantum computing (rather than conventional classical computing). However, in order to become a viable computing option, quantum computing requires the ability to precisely control a large number of quantum bits (called "qubits") and the interactions between these qubits. In particular, qubits should: have long coherence times, be able to be manipulated individually, be able to interact with one or more other qubits to implement multi-qubit gates, be able to be efficiently initialized and measured, and be scalable to a large number of qubits.

[0007] A qubit can be formed from any physical quantum mechanical system with at least two orthogonal states. The two states of a system used to encode information are called a "computational basis". For example, photon polarization, electron spin, and nuclear spin are two-level systems that can encode information and can therefore be used as qubits for QIP. Different physical implementations of qubits have different advantages and disadvantages. For example, while photon polarization benefits from long coherence times and simple single-qubit manipulation, it is limited by the inability to create simple multi-qubit gates.

[0008] The above examples of qubits are physical two-level systems. However, quantum information can also be stored in logical qubits, which are formed by multiple physical two-level systems or quantum systems with more than two states. For example, the states of quantum mechanical oscillators (in which there are infinite energy eigenstates) can also be used to form a computational basis for QIP. For example, coherent states of quantum mechanical oscillators that are fully displaced from each other in phase space are quasi-orthogonal states and can be used as a computational basis. In addition, states that are superpositions of coherent states called "cat states" can be completely orthogonal to each other and are used to form a computational basis.

[0009] Different types of superconducting qubits using Josephson junctions have been proposed, including: "phase qubits", in which the computational basis is the quantized energy state of Cooper pairs in the Josephson junction; "flux qubits", in which the computational basis is the direction of the flow of circulating current in a superconducting loop; and "charge qubits", in which the computational basis is the presence or absence of Cooper pairs on a superconducting island. Superconducting qubits are a favorable choice for qubits because the coupling between the two qubits is strong, making it relatively easy to implement a two-qubit gate, and superconducting qubits are scalable because they are mesoscopic components that can be formed using conventional electronic circuit technology. In addition, superconducting qubits exhibit strong nonlinearities associated with the Josephson effect and excellent quantum coherence. All superconducting qubit designs use at least one Josephson junction as a nonlinear non-dissipative element. Summary of the invention

[0010] According to some aspects, a quantum information processing (QIP) system is provided. The QIP system includes a data qubit and an ancilla qubit, the ancilla qubit having an asymmetric error channel. The data qubit is coupled to the ancilla qubit.

[0011] According to some aspects, a method of performing QIP in a system including a data qubit coupled to an auxiliary qubit is provided. The method includes driving the auxiliary qubit with a stable microwave field to create an asymmetric error channel.

[0012] The foregoing is a non-limiting summary of the invention, which is defined by the appended claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Various aspects and embodiments are described with reference to the following drawings. The drawings are not necessarily drawn to scale. For clarity, not every component may be labeled in every drawing. In the drawings:

[0014] Figure 1 is a block diagram of a quantum information processing system according to some embodiments.

[0015] Figure 2 According to some embodiments Figure 1 Diagram of the joint system of data qubits and auxiliary qubits.

[0016] Figure 3 According to some embodiments, a transmission Figure 2 Diagram of a superconducting circuit element.

[0017] Figure 4 According to some embodiments, a superconducting nonlinear asymmetric inductor element (SNAIL) is provided. Figure 2 Diagram of a superconducting circuit element.

[0018] Figure 5 is a block diagram of a quantum information system based on cavity quantum electrodynamics, according to some embodiments.

[0019] Figure 6 Depicted is a cat-state based Bloch sphere according to some embodiments.

[0020] Fig. 7A Depicted is an eigenspectrum of a pumped cat oscillator according to some embodiments.

[0021] Figure 7B The potential of a pumped cat oscillator (PCO) in the limit of large parametric drive according to some embodiments is depicted.

[0022] Figure 8 is a quantum circuit diagram for error syndrome detection according to some embodiments.

[0023] Fig.9A is a graph of the dynamics of a PCO and associated qubits during a stabilizer measurement for a four-qubit torus code, according to some embodiments.

[0024] Fig. 9B is a graph of the dynamics of a PCO and associated qubits during a stabilizer measurement for a four-qubit torus code, according to some embodiments.

[0025] Fig. 10A is a graph of the dynamics of a PCO and associated qubits during a stable sub-measurement for a cat code, according to some embodiments.

[0026] Fig. 10B is a graph of the dynamics of a PCO and associated qubits during a stable sub-measurement for a cat code, according to some embodiments.

[0027] Fig.11A is a diagram of a quantum circuit for performing adaptive phase estimation according to some embodiments.

[0028] Fig. 11B is a diagram of a quantum circuit for performing adaptive phase estimation according to some embodiments.

[0029] Fig.12 is a diagram illustrating the readout process with respect to a Bloch sphere according to some embodiments.

[0030] Fig.13 is a schematic diagram of a quantum information processing system according to some embodiments.

[0031] Fig.14A is a graph of the magnitude versus time of multiple drive fields used to implement a bias-holding CNOT gate according to some embodiments.

[0032] Fig. 14B is a graph of phase variation over time of multiple drive fields for implementing a bias-holding CNOT gate according to some embodiments.

[0033] Fig.15 is a quantum circuit diagram of a technique for detecting errors according to some embodiments.

[0034] Fig.16 is a flow chart of a quantum information processing method according to some embodiments.

[0035] Fig.17 is a flow chart of a readout method according to some embodiments. DETAILED DESCRIPTION

[0036] Conventional QIP schemes encode information in one or more two-level quantum systems (i.e., "qubits"). The state of a single qubit can be represented by a quantum state |ψ>, which can be any arbitrary superposition of two quantum states |0) and |1>, for example, |ψ>=a|0>+b|1>, where a and b are complex numbers representing the probability amplitude of the logical qubit being in the states |0) and |1>, respectively. Here, |0) and |1> are computational bases, which can be physically implemented using any physical system with two orthogonal states.

[0037] In order to perform useful quantum information processing, conventional quantum information systems initialize a set of qubits, called "data qubits," because these qubits are used to encode the information to be processed into a specific quantum state, implement a set of quantum gates on the qubits, and measure the final quantum state of the qubits after completing the quantum gates. The first type of conventional quantum gate is a single-qubit gate, which converts the quantum state of a single qubit from a first quantum state to a second quantum state. Examples of single-qubit quantum gates include a collection of rotations of qubits on a Bloch sphere. The second type of conventional quantum gate is a two-qubit gate, which converts the quantum state of a first qubit based on the quantum state of a second qubit. Examples of two-qubit gates include controlled NOT (CNOT) gates and controlled phase gates. Conventional single-qubit gates and two-qubit gates unitarily evolve the quantum state of a qubit from a first quantum state to a second quantum state.

[0038] In order to make large-scale quantum computing feasible, the quantum states used in QIP must be protected from errors caused by inevitable and uncontrolled interactions with the environment. Techniques for reducing such errors include quantum error correction (QEC) schemes. In some conventional QEC schemes, quantum information is protected by linking errors and undesired interactions with low-weight quantum operators. For example, quantum information can be encoded in logical qubits using the non-local degrees of freedom of high-dimensional systems, rather than encoding information only in two quantum states of physical qubits. In such encodings, high-weight operators mean, for example, multi-body operators generated in a system of several qubits, or operators involving many quantum states of a single high-dimensional physical system (e.g., a quantum mechanical oscillator). High-weight operators that characterize the code space of quantum information are called "stabilizers" and are designed to commute with logical qubit operators and to counter-commute with errors in the system. In the absence of errors, the system is in the +1 eigenspace of the stabilizer, and after an error occurs, the system moves to the -1 eigenspace. Therefore, the location and type of error can be determined from the results of measuring the stabilizers, which are also called "error syndromes." The measurement of these high-weight stabilizers uses highly engineered, highly unnatural many-body interactions between the components of the quantum system.

[0039] The inventors have realized and appreciated that the above types of QEC techniques are not desirable for practical implementations of QIP. Instead, the inventors have realized and appreciated that it is desirable to synthesize stabilizer measurements via naturally available couplings between auxiliary systems and the data qubits of the system. Coupling the data qubits of a QIP system exposes the data qubits to a different set of errors that may also be challenges to mitigate. For example, if the measurements of the auxiliary system are not intelligently designed, errors from the auxiliary system may propagate to the data qubits, thereby corrupting the encoded quantum information beyond repair. Recognizing this, the inventors have developed techniques for reducing and / or in some cases eliminating such catastrophic backlash from the auxiliary system.

[0040] To help explain some aspects of the present application, a stable quantum measurement technique is described herein. To synthesize a stable quantum measurement, a system representing a logical data qubit The quantum information is encoded in N subsystems implemented using physical quantum bits. The code is defined by multiple stabilizers, but for simplicity, only a single stabilizer is considered here. A set of low-weight operators With stabilizer commutative and can be used to synthesize by coupling with auxiliary qubits Where i = 1, 2, ... N. As an example, a four-qubit operator is the stabilizer for the surface code, where is the z-Pauli operator acting on the ith quantum bit, As a second example, the parity operator is the stabilizer for single-mode bosonic cat codes, where and are the photon creation operator and the annihilation operator, respectively. The auxiliary system may be, for example, an auxiliary qubit coupled to the data qubit via the following interaction Hamiltonian.

[0041]

[0042] Among them, σ z is the z-Pauli operator of the auxiliary qubit, and g i is the controllable interaction strength between the auxiliary qubit and each of the physical qubits used to form the logical data qubit. The evolution of the joint system of data qubits and auxiliary qubits is described by the unitary operator:

[0043]

[0044] The duration T of the interaction between the data qubit and the auxiliary qubit and the coupling strength can be chosen so that the unitary operator acting on the joint system (up to the local rotation) becomes:

[0045]

[0046] The result of the interaction with interaction time T is therefore a phase flip of the auxiliary qubit conditional on whether the stabilizer is + 1 or - 1. This phase flip in the auxiliary qubit is an error syndrome.

[0047] The inventors have realized and appreciated that during the interaction time, the data qubits and the auxiliary qubits become entangled, and that for a successful QEC scheme, it is desirable to engineer the joint system so that errors in the auxiliary qubits do not propagate to the data qubits as uncorrectable errors, which is referred to as "fault tolerance." To prevent uncorrectable errors from propagating to the data qubits and to achieve fault tolerance, all possible errors in the auxiliary qubits should always be compared to the unitary operator. In the above example, the phase reversal error This condition is satisfied. Therefore, if a phase-flip error occurs at any time τ during the interaction duration, then at time T the state of the system is described by the following unitary operator:

[0048]

[0049] Based on this unitary operation, it is clear that a phase flip in the auxiliary qubit only introduces an error in the measurement of the syndrome without any adverse effect on the data qubit. However, it is important to note that a bit-flip error in the auxiliary qubit (given by the Pauli matrix ) and amplitude attenuation error (represented by the Pauli matrix means) has no unitary operator In fact, a bit-flip error on the auxiliary qubit Propagates to the data qubits as high-weight errors.

[0050] Conventional methods for fault-tolerant extraction of error syndromes are based on: using multiple auxiliary qubits prepared in a complex quantum state, performing multiple bit-wise entanglement gates between data qubits and auxiliary qubits, and then measuring the auxiliary qubits. The inventors have realized and appreciated that these conventional methods lead to rapid growth in the overhead of computationally expensive entanglement gates and auxiliary hardware, thereby imposing more stringent requirements on error rates and making fault-tolerant quantum computing impractical (if not impossible, making fault-tolerant quantum computing on a large scale). In addition, the inventors have realized and appreciated that efficient fault-tolerant syndrome extraction schemes will enable large-scale quantum information processing. Therefore, some aspects of the present application are directed to efficient fault-tolerant syndrome extraction.

[0051] The inventors have realized and recognised that in the above-mentioned stable quantum measurement scheme, if the auxiliary quantum bit does not have a bit flip Error, then the unitary operator There will be no adverse effect on the data qubit. Therefore, some aspects of the present application are directed to using auxiliary qubits with asymmetric error channels in which bit flip errors are suppressed relative to phase flip errors. To commutate bit-flip errors, a physical unitary operation can be engineered that more or less commutes with the auxiliary qubit’s error channel and will therefore be effectively transparent to auxiliary qubit errors.

[0052] Aspects of the present disclosure include methods for making fault-tolerant measurements in quantum systems. The techniques described herein can be used in at least three possible applications. First, these techniques can be used in quantum error correction schemes by allowing fault-tolerant extraction of error syndromes. Second, these techniques can be used for new, more efficient error correction codes and processes. Third, these techniques can be used to create bias-hold gates, such as controller-not-(CNOT) gates.

[0053] The inventors have realized and appreciated that fault-tolerant extraction of error syndromes can be performed using only local operations on auxiliary qubits whose error channels are strongly biased (i.e., asymmetric). Some embodiments improve the overhead requirements of fault-tolerant syndrome measurements relative to conventional schemes. Some embodiments include hardware efficient implementations of such syndrome extraction schemes using two-component cat states in parametrically driven nonlinear oscillators that exhibit highly biased noisy channels.

[0054] The inventors have further realized and appreciated the flexibility of the above method. In some embodiments, different codes can be used. In some embodiments, the syndrome extraction process is used for various codes, such as qubit-based torus codes, Bose cat codes (and as an extension, binomial and paired cat codes), and Gottesman-Kitaev-Preskill (GKP) codes. However, other codes may also be used.

[0055] The challenge of error correction with bias noise is to be able to maintain bias when performing basic gate operations such as CNOT gates, which is an important component of many error correction codes and general computing. In conventional systems using physical qubits as data qubits and / or auxiliary qubits, even if the potential noise is biased, native bias-maintained CNOT is impossible. The inventors have realized and appreciated that the aforementioned techniques developed for fault-tolerant syndrome extraction can be utilized and extended to implement bias-maintained CNOT gates between two stable cat states. In some embodiments, the CNOT gate is based on the structure of cat states in phase space. In this case, stable cat states can be achieved in parameter-driven nonlinear cavities or via dissipative engineering. Some embodiments including bias-maintained CNOT gates can achieve gains in thresholds for topological error correction codes (e.g., torus codes and surface codes).

[0056] In some embodiments, when combined with a ZZ(θ) gate, the threshold for so-called "magic state preparation" (which is an important but (in terms of overhead cost) expensive component to achieve generality) can be reduced. In some embodiments, the ZZ(θ) gate inherently maintains a bias and can be implemented with a stable cat. The inventors have realized and appreciated that combining bias-maintaining CNOT gates, ZZ(θ) gates, and syndrome measurements provides the basis for a fault-tolerant architecture for large-scale quantum computing with ultra-high thresholds and greatly reduced overhead requirements. Such an architecture that utilizes the biasing of the noisy channel of a stable cat qubit has no equivalent in conventional systems based on physical qubits.

[0057] Figure 1 A QIP system according to some embodiments is shown. QIP system 100 includes at least data qubit 110 and auxiliary qubit 120. Some embodiments also include microwave field source 150 and / or measurement device 125. Measurement device 125 may include a readout cavity 130 and a cavity state detector 140. Although not shown as such, microwave field source 150 may be considered part of measurement device 125 because the microwave field emitted by microwave field source 150 plays a role in the measurement.

[0058] Data qubit 110 may be any physical or logical qubit that can be coupled to auxiliary qubit 120. In some embodiments, data qubit 110 may include a superconducting circuit component. For example, data qubit 110 may include at least one Josephson junction. In some embodiments, data qubit 110 may include a transmon. In some embodiments, data qubit 110 may include a superconducting nonlinear asymmetric inductor element (SNAIL), which is an example of a superconducting circuit component including multiple Josephson junctions. In other embodiments, data qubit 110 may include an oscillator. Examples of linear oscillators that can be used include electromagnetic fields supported by a cavity, such as microwave radiation. The cavity may include a three-dimensional (3D) cavity or a planar transmission line cavity. In some embodiments, the cavity may be driven to include a particular type of quantum state. For example, as described in more detail below, the cavity may be driven to include a cat state or a GKP state. In some embodiments, a superconducting circuit component may be coupled to the cavity to form a Kerr nonlinear cavity.

[0059] Auxiliary qubit 120 can be any physical or logical qubit that can be coupled to data qubit 110. In some embodiments, auxiliary qubit 120 may include superconducting circuit components. For example, auxiliary qubit 120 may include at least one Josephson junction. In some embodiments, auxiliary qubit 120 may include a transmon. In some embodiments, auxiliary qubit 120 may include a SNAIL. In other embodiments, auxiliary qubit 120 may include an oscillator. Examples of linear oscillators that can be used include electromagnetic fields supported by a cavity, such as microwave radiation. The cavity may include a three-dimensional cavity or a planar transmission line cavity. In some embodiments, the cavity may be driven to include a particular type of quantum state. For example, as described in more detail below, the cavity may be driven to include a cat state or a GKP state. In some embodiments, superconducting circuit components may be coupled to the cavity to form a Kerr nonlinear cavity.

[0060] One or more properties of data qubit 110 may be measured by measurement device 125 using auxiliary qubit 120. For example, the interaction between data qubit 110 and auxiliary qubit 120 may be engineered so that the state of auxiliary qubit 120 is based on a particular property of data qubit 110. In some embodiments, the measurement of data qubit 110 is a quantum non-destructive measurement, meaning that the state of data qubit 110 is not affected by the measurement process. In some embodiments, quantum non-destructive measurement may be performed by measuring the state of auxiliary qubit 120 using measurement device 125 after data qubit 110 and auxiliary qubit 120 interact to determine the property of auxiliary qubit 120. In some embodiments, the interaction between data qubit 110 and auxiliary qubit 120 may be switched on by driving data qubit 110 and / or auxiliary qubit 120 with one or more microwave fields using microwave field source 150.

[0061] The readout cavity 130 is a cavity coupled to the auxiliary qubit 120 and is configured to support multiple electromagnetic radiation (e.g., microwave radiation) states based on the properties of the auxiliary qubit 120. In some embodiments, the interaction between the readout cavity 130 and the auxiliary qubit 120 is engineered so that the state of the readout cavity 130 depends on a specific property of the auxiliary qubit 120, which itself can be based on the properties of the data qubit 110. For example, if the property of the auxiliary qubit 120 is a first value, the interaction causes the readout cavity 130 to be in a first state; if the property of the auxiliary qubit 120 is a second value, the interaction causes the readout cavity to be in a second state. In some embodiments, the two states of the readout cavity 130 can be two different quasi-orthogonal coherent states. In other words, depending on the value of the property of the auxiliary qubit 120, the readout cavity 130 can be shifted in different ways. In some embodiments, the process can be performed using a technique referred to herein as "Q switching", which uses a frequency conversion technique to conditionally shift the readout cavity 130 based on the properties of the auxiliary qubit 130. In some embodiments, the interaction between readout cavity 130 and ancillary qubit 120 may be switched on by driving readout cavity 130 and / or ancillary qubit 120 with one or more microwave fields using microwave field source 150 .

[0062] The cavity state detector 140 may be, for example, a microwave radiation detector capable of distinguishing possible states of the readout cavity 130 caused by the interaction between the readout cavity 130 and the auxiliary qubit 120. In some embodiments, the cavity state detector may be a phase-sensitive detector capable of measuring not only the amplitude of the electromagnetic field of the readout cavity 130 but also the phase of the electromagnetic field of the readout cavity 130. For example, the cavity state detector 140 may be a homodyne detector or a heterodyne detector. In some embodiments, the detection result is directly related to the error syndrome.

[0063] In some embodiments, the auxiliary qubit 130 includes a cavity, and the state of the cavity can be directly measured using a homodyne detector. However, if the cavity of the auxiliary qubit 130 is a high-Q cavity, homodyne detection will be slow. Therefore, the readout cavity can be a low-Q cavity that can be read out quickly.

[0064] Figure 2is a diagram of a particular embodiment of a joint system 200 including an example of a data qubit 110 and an example of an auxiliary qubit 120, according to some embodiments. Data qubit 110 includes a data cavity 210 and a data superconducting circuit 212. Ancillary qubit 120 includes an auxiliary cavity 220 and an auxiliary superconducting circuit 222. The two cavities are coupled together via an interface 230, which may include, for example, a microwave waveguide and / or a plug connector.

[0065] The data cavity 210 may be a three-dimensional cavity and include at least one microwave port 214 for receiving a microwave field 216 from the microwave field source 150. The auxiliary cavity 220 may be a three-dimensional cavity including at least one microwave port 224 for receiving a microwave field 226 from the microwave field source 150. In some embodiments, the microwave port may include a plug connector and / or a microwave waveguide. Figure 2 Only a single port is shown for each cavity, but each cavity may include more than one port for receiving and / or transmitting microwave fields. Figure 2 Not shown is a port for coupling the auxiliary cavity 220 to the readout cavity 130 .

[0066] In some embodiments, the data superconducting circuit element 212 and the auxiliary superconducting circuit element 222 may include nonlinear circuit elements. For example, the superconducting circuit element may be a transmon or a SNAIL. Figure 3 An example of a superconducting circuit element 300 that can be used as a data superconducting circuit element 212 and / or an auxiliary superconducting circuit element 222 is shown. The superconducting circuit element 300 includes an antenna and a transmon 301 including a single Josephson junction, the antenna including a first antenna portion 303 and a second antenna portion 305. The two antenna portions together form a dipole antenna, through which the transmon 301 is coupled to a three-dimensional cavity in which the superconducting circuit element 300 is located.

[0067] Figure 4 An example of a superconducting circuit element 400 that can be used as a data superconducting circuit element 212 and / or an auxiliary superconducting circuit element 222 is shown. The superconducting circuit element 400 includes an antenna and a SNAIL 401 including a single Josephson junction, the antenna including a first antenna portion 403 and a second antenna portion 405. The two antenna portions together form a dipole antenna through which the SNAIL 401 is coupled to a three-dimensional cavity in which the superconducting circuit element 400 is located.

[0068] SNAIL 401 is a nonlinear superconducting circuit element that has additional tenability relative to a transmon. Figure 5is a schematic diagram of a SNAIL 500 according to some embodiments. The SNAIL 500 includes a superconducting loop 501 having two nodes 511 and 512. There are two paths along two different portions of the superconducting loop 501 connecting the first node 511 and the second node 512.

[0069] The first loop portion includes a plurality of Josephson junctions 505-507 connected in series. In some embodiments, there are no other circuit elements between one Josephson junction and the next Josephson junction. For example, a Josephson junction is a dipole circuit element (i.e., it has two nodes). The first node of the first Josephson junction 505 is directly connected to the first node 511 of the SNAIL, and the first node 511 can lead to some other external circuit elements (not shown). The second node of the first Josephson junction 505 is directly connected to the first node of the second Josephson junction 506. The second node of the second Josephson junction 506 is directly connected to the first node of the third Josephson junction 507. The second node of the third Josephson junction 507 is directly connected to the second node 512 of the SNAIL, and the second node 512 can lead to some other external circuit elements (not shown), such as a part of an antenna.

[0070] Although Figure 5 A first ring portion including three Josephson junctions is shown, but any suitable number of Josephson junctions greater than one may be used. For example, three, four, five, six, or seven Josephson junctions may be used. For the example shown, three Josephson junctions were selected because three Josephson junctions are the minimum number of Josephson junctions (other than zero or one) that can be formed using a Dolan bridge fabrication process, which may be used in some embodiments.

[0071] In some embodiments, the Josephson junctions 505 to 507 are formed to be identical. For example, the tunneling energy, critical current, and size of the Josephson junctions 505 to 507 are all identical.

[0072] The second loop portion of the SNAIL 500 includes a single Josephson junction 508. In some embodiments, no other circuit elements are present in the second loop portion. The first node of the single Josephson junction 508 is directly connected to the first node 511 of the SNAIL, which may lead to some other external circuit element (not shown), such as a portion of an antenna. The second node of the single Josephson junction 508 is directly connected to the second node 512 of the SNAIL, which may lead to some other external circuit element (not shown), such as a portion of an antenna.

[0073] The single Josephson junction 508 has a tunneling energy that is smaller than the tunneling energy of each of the Josephson junctions 505 to 507. For this reason, the single Josephson junction 508 may be referred to as a "small" Josephson junction, and the Josephson junctions 505 to 507 may be referred to as "large" Josephson junctions. The terms "large" and "small" are relative terms that are used only to mark the relative size of the Josephson junction 508 compared to the Josephson junctions 505 to 507. The Josephson energy and the Josephson junction size in the large Josephson junction are larger than the Josephson energy and the Josephson junction size in the small Josephson junction. The parameter α is introduced to represent the ratio of the small Josephson energy to the large Josephson energy. Therefore, the Josephson energy of the large Josephson junctions 505 to 507 is E J , and the Josephson energy of the small Josephson junction 508 is αE J , where 0<α<1.

[0074] Figure 5 The right side shows the Figure 4 The circuit element symbol of SNAIL 500 used in superconducting circuit element 400. The parameter characterizing SNAIL 500 is the Josephson energy E J Superconducting phase difference with small Josephson junction 508 Of note is the fact that SNAIL 500 has only two nodes 511 and 512 that can connect corresponding parts of the antenna.

[0075] Although in Figure 2 Two separate 3D cavities (one for data qubit 110 and one for auxiliary qubit 120) are shown, but other cavity arrangements may be used. In some embodiments, data superconducting circuit element 212 and auxiliary superconducting circuit element 222 may be located within a single shared 3D cavity. In other embodiments, data superconducting circuit element 212 and auxiliary superconducting circuit element 222 may be coupled to corresponding two-dimensional (2D) transmission line cavities.

[0076] When using Figure 3 The embodiment of transmon shown or Figure 4 In the illustrated embodiment of the SNAIL, the superconducting circuit elements coupled to the cavity form a Kerr nonlinear oscillator, which can be used as a data qubit and / or an auxiliary qubit. In some embodiments, the two-photon pump received from the microwave field generator 150 can be used to generate a two-component cat state:

[0077]

[0078] in, and β is the complex amplitude associated with the coherent state |β> associated with the cat state. And cat status are degenerate orthogonal eigenstates of a Kerr nonlinear oscillator pumped by a two-photon driving field. These cat states can be used to form a Bloch sphere, the orientation of which is arbitrary with respect to the cat state ground state. Figure 6 The Bloch sphere 600 used in this application is shown. The basis of the logical qubit formed using these cat states is such that: the +Z axis and -Z axis of the Bloch sphere 600 correspond to the superposition state where for large values ​​of β, it is very similar to the coherent state |±β>; the +X and –X axes of the Bloch sphere 600 correspond to the cat states And cat status And the +Y axis and -Y axis of the Bloch sphere 600 correspond to the superposition state Figure 6 Also shown is a simplified phase space diagram of each state associated with the axes of the Bloch sphere 600.

[0079] The above-described type of cat state has the following property: natural coupling only causes rotations about the Z axis of the Bloch sphere 600, because the pump used to generate the cat state generates a large energy barrier that prevents phase rotations (i.e., rotations from the coherent state |+β> to the coherent state |-β>, and rotations from the coherent state |-β> to the coherent state |+β>). Therefore, according to some embodiments, using the Bloch sphere 600, the noise channel associated with photon losses corresponds to phase flip errors, and in some embodiments, phase flip errors dominate the error channel for the logical qubit, while bit flip errors are suppressed to generate an asymmetric error channel. Phase flip errors, for example, increase with the |β| 2 The size of the determined cat state increases linearly with the increase in the size of the cat state, or equivalently increases with the increase in the intensity of the microwave field used to pump the cat state. On the other hand, based on the size of the cat state |β| 2 Or the intensity of the microwave field used to pump the cat state exponentially suppresses bit flip errors and amplitude decay errors. Therefore, in some embodiments, when the pumped cat state of the Kerr nonlinear cavity is used as the physical realization of the auxiliary qubit, fault-tolerant syndrome measurement can be performed.

[0080] Some embodiments extract error syndromes based on conditional rotations of cat states around the Z axis. In some examples, this can be achieved using only low-weight local interactions. In some embodiments, the fault-tolerant technique can be used with various error correction codes such as stable subcodes. Examples of stable subcodes include, but are not limited to, torus codes, Bose cat codes, and GKP codes. Some embodiments can use error correction codes based on non-stable subcodes, such as non-additive quantum codes. In addition, some embodiments can use asymmetric error channels of auxiliary qubits to perform fault-tolerant quantum gates.

[0081] In some embodiments, the interaction between the data qubit and the auxiliary qubit is achieved using the inherent nonlinearity of the auxiliary qubit implemented using cat states in a Kerr nonlinear cavity. In this way, some embodiments do not require additional coupling elements. Therefore, by utilizing the techniques described herein, a hardware-efficient quantum information processing scheme can be achieved.

[0082] Error check sub-detection

[0083] In some embodiments, a Kerr nonlinear oscillator implemented, for example, using the hardware described above, can be driven by a two-photon driver with a frequency equal to twice the resonant frequency of the oscillator. When driven by such a microwave field, the oscillator is called a pumped-cat oscillator (PCO), and the Hamiltonian in the rotating wave approximation is:

[0084]

[0085] in, and are the photon production and annihilation operators of PCO, K is the strength of the Kerr nonlinearity, and P is the strength of the two-photon driving field. Rewriting the PCO Hamiltonian gives:

[0086]

[0087] coherent state |±β> and (equivalently) cat state are the degenerate eigenstates of the Hamiltonian, where the eigenenergy is Kβ 4 =P 2 / K. For simplicity, it will be assumed that the driving field strength (P) will be a positive real number, causing β to be a real number as well. The coherent state |±β> is quasi-orthogonal (<β|-β> = exp(-2β 2 )) and cat status is completely orthogonal. It is also the photon number parity operator ±1 eigenstates of the photon number. Since the PCO Hamiltonian commutes with the photon number parity operator, the eigenstates of the Hamiltonian are also the eigenstates of the parity operator. As a result, we can Fig. 7A The eigenspace 700 of the PCO Hamiltonian is shown to be divided into an even parity subspace 701 and an odd parity subspace 702, denoted by superscripts ±, respectively. The cat subspace of the eigenspace 700 is represented by Represented by the energy gap ω gap ∝4Kβ 2 Separated from the rest of the states of eigenspace 700.

[0088] In the rotating frame, the PCO Hamiltonian is described by quasi-energy eigenstates that exhibit negative energy. When applied to the PCO Hamiltonian, the shifted Hamiltonian becomes:

[0089]

[0090] Among them, because have and The term disappears. The constant term representing energy transfer E = P 2 / K is also discarded. Vacuum state|0> is eigenstates of , so in the original frame, the coherent state |±β> or equivalently its superposition is the degenerate eigenstate of the original PCO Hamiltonian. In the limit of large β (i.e., large pump value), β 2 >>β, which leads to pass A good approximation, is the Hamiltonian of the inverse harmonic oscillator. The first excited state is the Fock state In=1>, whose energy is 4Kβ 2 is lower than that of the vacuum state |0). Therefore, the shifted Fock state D(±β)|n=1> is two degenerate excited states in the original unshifted frame. Since the eigenstates of the PCO Hamiltonian are also the eigenstates of the parity operator, it is convenient to represent the excited states as two orthogonal states The two orthogonal states are the even parity state and the odd parity state, where is a normalizing constant. Therefore, the cat subspace is The energy gap between gap ∝4Kβ 2 .

[0091] Figure 7B The potential of a PCO in the limit of large parameter drive according to some embodiments is shown. When the driving microwave field is large (e.g., large β, or equivalently, large P), the PCO behaves like two harmonic oscillators shifted at ±β. Tunneling between the two harmonic oscillators is exponentially suppressed with β, since tunnel splitting can be achieved by overlapping Approximately, where f(β 2 ) is β 2The eigenspectrum of the PCO Hamiltonian is thus reduced to a superposition of pairs of degenerate shifted Fock states [D(β)±D(-β)]|n). For a fixed value of β, this approximation becomes less valid for higher values ​​of n and decreases to 1 in the range n~β. 2 When the drive is zero (β = P = 0), the Hamiltonian becomes the Hamiltonian of an undriven nonlinear oscillator with degenerate Fock states |n = 0> and |n = 1> and nondegenerate next two excited states |n = 2> and |n = 3>. In this case, ω gap becomes equal to the gap between the Fock state |n=0> and the Fock state |n=2>, which is equal to 2 K. The eigenspectrum is described in a frame rotating at the frequency of the oscillator ω PCO , which means that the energy gap in the laboratory frame is ω PCO -ω gap Therefore, an external drive (such as a microwave field) or a disturbance at this frequency can cause transition between excited states.

[0092] In some embodiments, the PCO is connected to a system The data qubits represented by interact with each other so that the interaction Hamiltonian in the rotating frame is:

[0093]

[0094] To understand the effect of this interaction / coupling on PCO, note that the cat state undergoes a bit flip under the action of the photon annihilation operator in, because Therefore for large β, p→1. Although the annihilation operator Cat Space A state within is converted into Another state in the inner, but the photon produces the operator It is possible to move the PCO out of the cat subspace. However, for small coupling Due to the energy gap between the cat subspace and other states of the eigenspectrum, these spurious excitations outside the subspace are suppressed. In this restricted subspace, Therefore, the interaction Hamiltonian can be approximated as:

[0095]

[0096] Among them, χ′ i (t) = x i (t)(p+p -1 ) / 2~χ i (t) and is the Pauli operator in the cat subspace. Therefore, the interaction Hamiltonian is an entangled interaction that is identical to the interaction Hamiltonian described above in the example stable submeasurement technique is the same as above, and thus gives rise to the unitary operator Equivalent unitary evolution. Therefore, the coupling χi(t) and the interaction time can be chosen so that the evolution of the system after time t = T is given by:

[0097]

[0098] Based on the above, the auxiliary cat state of PCO is based on the stable or In some embodiments, the error syndrome can be extracted by measuring the state of PCO at time T.

[0099] In some embodiments, a An alternative coupling of For example, when using GKP codes, such coupling can be used to extract error syndromes.

[0100] Figure 8 is a quantum circuit diagram of error syndrome detection 800 according to some embodiments. The three horizontal lines represent the readout cavity 801, the auxiliary qubit 802, and the data qubit 803. Time increases from left to right, so that the operations appearing on the left side of the figure are performed before the operations shown on the right side of the figure. The readout cavity 801 is initialized to be in the vacuum state |0>, and the auxiliary qubit 802 is initialized to be in the even cat state And data qubit 803 is initialized to be in any state |ψ that the data qubit is in based on other operations that may be performed on data qubit 803 before error syndrome detection 800 M >. In some embodiments, as described above, auxiliary qubit 802 includes a PCO.

[0101] The first action of error syndrome detection 800 is to map the error syndrome onto the state of the PCO. This is called syndrome measurement 810. For example, the auxiliary qubit 802 can be kept in the cat state based on at least one property of the data qubit 803. Or be transformed into cat state In some embodiments, a syndrome measurement may be implemented using controlled Z rotation 811, where the state of the PCO is conditionally rotated about the Z axis of the sphere based on the state of the data qubit 803. In some embodiments, syndrome measurement 810 does not change the state of the data qubit 803. As such, syndrome measurement 810 may be a quantum non-destructive measurement.

[0102] After performing syndrome measurement 810, error syndrome detection 800 includes a readout operation 820. Readout operation 820 is performed, for example, by determining whether auxiliary qubit 802 is in a cat state. Still in cat form to determine the state of auxiliary qubit 802. In some embodiments, the readout of auxiliary qubit 802 may include mapping the state of auxiliary qubit 802 onto readout cavity 801. In some embodiments, readout operation 820 may include two separate operations. The first operation may be a rotation operation 821 on auxiliary qubit 802. For example, rotation operation 821 may convert the cat state Rotate to a nearly coherent state |±β>. The second operation of the readout operation 820 includes a "Q-switching" operation 823, in which the single-photon exchange coupling between the PCO and the readout cavity 801 is turned on by applying an appropriate microwave field from the microwave field generator 150. The result of the Q-switching operation 823 is to conditionally shift the readout cavity 801 based on the state of the PCO. Finally, after the readout operation 820 is completed, the readout cavity 801 is measured using, for example, a homodyne detection scheme to obtain an error syndrome.

[0103] False channeling due to single photon loss

[0104] The error channel of a PCO (sometimes called the noise channel) is dominated by single-photon losses in the oscillator, which are caused by single-photon exchange coupling with the bath. As mentioned above, if the coupling with the bath is smaller than the energy gap between the cat-state subspace C and the other states of the eigenspectrum, the dynamics of the PCO is confined to the cat-state subspace. In this confined subspace, assuming that there are no thermal excitations in the bath, that is, the PCO can only lose photons but not gain photons, then single-photon exchange coupling with the bath causes phase-flip errors to dominate over bit-flip errors, which are exponentially small with respect to the strength of the pump field, β. The bath confines the cat-state subspace to The double degeneracy of β 2 This is because Given the odd cat state The number of photons under and is given by The even cat state given The number of photons under the condition differs by an exponentially smaller amount. Photons are more likely to be lost to the environment than to is lost to the environment. This asymmetry increases the degeneracy between the two cat states. However, since the difference in the number of photons decreases exponentially with β, even for moderate pump intensities, such as β~2 (for which exp(-2β 2 )=3.3×10-4 ), Status Cat Status Almost degenerate.

[0105] The preservation of the degenerate cat subspace in some embodiments makes the PCO a good candidate for instrumentation for syndrome detection, since the coupling to the slot commutates with the interaction Hamiltonian and does not affect the data qubits. However, a single photon loss to the slot may give rise to two cat states This causes random flipping between the auxiliary qubits, which reduces the measurement accuracy of the auxiliary qubit. However, since the backlash is exponentially suppressed, the accuracy can be restored by repeating the measurement multiple times. Therefore, in some embodiments, multiple measurements of the auxiliary qubits are performed, and a majority vote is used to determine the error syndrome.

[0106] Other noise sources

[0107] In some embodiments, there are other noise sources, such as photon gain, pure-dephasing, two-photon loss. Single-photon gain and pure-dephasing can lead to leakage out of the cat-state subspace. However, by ensuring that the spectral density of these noise sources is narrower than the energy gap between the cat-state subspace and other states of the eigenspectrum, leakage can be suppressed. Therefore, some embodiments are engineered so that the PCO has single-photon gain and a pure-dephasing spectral density that is smaller than the energy gap. In such embodiments, regardless of the underlying noise cause, the error channel of the PCO is dominated by phase-flip errors, while bit-flip errors are exponentially suppressed.

[0108] Additionally, in some embodiments, it is possible that stray excitations or sudden non-perturbative effects overcome the energy barrier and cause excitations to states outside the cat state subspace. It is noteworthy that the error tolerance of the syndrome measurement is still maintained under these errors.

[0109] In the example of single-photon gain, As a cat state subspace The effect of the cat state on a portion of the cat state leads to both leakage out of the cat state subspace and phase flip errors. As mentioned above, in the limit of large β, the first excited state is approximately equivalent to a shifted single-photon Fock state (see, for example, Figure 7B ). Under this approximation, and single-photon gain excitation of the first excitation subspace. In the first excitation subspace, in, Because the two states of the first excitation subspace is approximately degenerate, so the term can induce transitions between these two states, but cannot induce transitions out of the first excitation subspace. Recall from above that the coupling between the PCO and the data qubit is related to proportional to, and in the cat-state subspace middle, Thus, the excited state forms another two-level auxiliary qubit that is coupled to the data qubit in the same way as the auxiliary qubit in the cat state subspace is coupled to the data qubit. As a result, the data qubit does not gain any information about whether the PCO is in the cat subspace (e.g., in the first excited subspace). Equivalently, the data qubit is transparent to leakage errors in the PCO. If the PCO experiences n photon gain events, then the PCO is excited to As long as the nth excitation subspace is doubly degenerate, it will behave as a two-level auxiliary qubit that has the same coupling to the data qubit For high excited states (e.g., large n), this approximation becomes less valid. Therefore, it may be beneficial to reduce such excitations by dissipative processes (e.g., single-photon loss or two-photon loss). This is because photon loss events transfer the population from the nth excited subspace to the (n-1)th excited subspace. In summary, the backlash due to out-of-subspace excitations in the PCO depends on the presence of paired degenerate eigenstates in the spectrum of the PCO. Due to the paired eigenstates The energy difference decreases exponentially with the magnitude of the cat state, β, and so the backlash decreases in the same way.

[0110] In the example of pure phase loss error, the jump operator causes leakage. In the limit of large β, Using arguments similar to those described above in connection with single-photon gain, it can be shown that the data qubit pair is not a cat-state subspace The excitation in the state of a part of remains transparent and the adverse effects due to leakage errors are suppressed.

[0111] Example Stabilizer Measurement: Toroidal Codes

[0112] In some embodiments, n qubits Stabilizers are used in conjunction with torus codes, which are examples of topological quantum error correction codes. In some embodiments, two-dimensional torus codes may be used. Four-qubit stabilizers may be measured using, for example, direct eigenspace-preserving measurements. Stabilizer The Hilbert space can be divided into an even eigenspace ε and an odd eigenspace In some embodiments, the eightfold degenerate even (odd) subspace includes The +1(-1) eigenstate of the even eigenspace ε and the odd eigenspace can be defined as code subspace and error subspace respectively, so that The measurement of produces -1 or +1 depending on whether there is an error. Therefore, the measurement indicates an error syndrome.

[0113] In some embodiments, the stabilizer Direct measurement of would require a five-body interaction between the data qubit and the ancillary qubit, which is challenging to implement experimentally. Instead, some embodiments perform syndrome measurements using only two-body interactions. This can be accomplished by using Substituting the above interaction Hamiltonian The resulting interaction Hamiltonian is:

[0114]

[0115] in, This has the form of longitudinal qubit oscillator coupling. For simplicity, all interaction strengths are assumed to be equal, although they do not need to be. As long as the interaction strengths are known, the duration of the interaction with each qubit can be adjusted to perform syndrome measurements. An alternative approach is to keep the duration of the interaction fixed, but use paired bit-flip drive field pulses for each qubit appropriately separated in time.

[0116] After analyzing the example stabilizer measurement technique above, the unitary operator corresponding to this interaction Hamiltonian is:

[0117]

[0118] In some embodiments, the PCO can be initialized to the CAT state by first Then, in an embodiment where the interaction strength is the same for all qubits, the system is z Internal evolution More generally, for implementations where the interaction strength is different for each qubit, the interaction duration for the ith qubit is T i，z , where, for each qubit, At the end of the interaction duration, the unitary operator simplifies to:

[0119]

[0120] In this expression for the unitary operator after the interaction duration, the initial exponential term is the local phase rotation of the qubit. In some embodiments, the local phase rotation can be kept track of in classical software while performing subsequent operations on the qubit and is described later. In other embodiments, the local phase rotation can be set during or after the syndrome measurement. Gates are applied to the qubits to compensate for these phase rotations. or code Starting qubit, PCO at time T z The status after that is or

[0121] In some embodiments, the time-dependent qubit-oscillator interaction is achieved by switching on and then off the coupling between the qubit and the oscillator. In the maximum entangled state |ψ o > Initialize the four qubits:

[0122]

[0123] In some embodiments, the PCO is initialized to an even state.

[0124] Fig.9A and Fig. 9B is a graph of the dynamics of the PCO and qubit during the following stabilizer measurement for: single photon loss rate, κ = 0 (solid line) and κ = K / 200 (dashed line), the value P = 4K (equivalent to β = 2), χ0=K / 20 and T z =π / (8χ0β). Fig.9A The following graph 900 is shown: The graph 900 shows that when the PCO is initialized to be in state and the qubit is initialized to be in an odd-parity state |ψ o >, PCO is in the state of The probability that the qubit is in state |ψ o >903. During the interaction duration T z Afterwards, in the case of no photon loss (solid line), the density matrix The probability that the qubit represented by is in the maximum entangled odd-parity state is And the density matrix The probability that the PCO represented by is in the odd cat state is When photon loss (dashed line) is introduced, the PCO is z After that, the probability of being in the odd cat state decreases to Relative to the case without photon loss, the quantum bit has an interaction duration T z The probability of being in the odd-parity state afterwards remains unchanged: If the photon loss increases to κ = K / 10, the PCO will z After that, the probability of being in the odd cat state decreases to The fidelity of mapping the syndrome onto the auxiliary qubit is therefore reduced to 52%, which is close to the 50% point where the majority vote fails, but the backlash on the data qubit is still suppressed.

[0125] Fig. 9B The following graph 910 is shown: Graph 910 shows that when the PCO is initialized to be in state and the qubit is initialized to be in an even parity state |ψ e >, PCO is in the state of The probability that the qubit is in state |ψ e >913, where

[0126]

[0127] During the interaction duration T z Afterwards, in the case of no photon loss (solid line), the density matrix The probability that the qubit represented by is in an even parity state is And the density matrix The probability that the PCO represented by is in the odd cat state is Therefore, the data qubit is transparent to errors in the PCO. Although single photon losses in the PCO reduce the fidelity of syndrome extraction, the fidelity can be restored by repeating the protocol multiple times and taking a majority vote. For example, in the case of κ = K / 200, the fidelity of the controlled Z rotation is reduced to 93% (dashed line), but by repeating the process 5 times, the probability of accurately mapping the syndrome to the PCO is increased to 99.7%. As can be seen from the dashed line associated with curve 913, the state of the data qubit after the interaction duration is not affected by photon losses.

[0128] To highlight the fault tolerance of measurements using PCOs, the case where the measurements are performed using conventional two-level physical qubits with the same relaxation rate γ = K / 200 is shown. With conventional physical qubits, the probability that the data qubit remains in its original state is significantly reduced to (and when γ=K / 10, )and (and when γ=K / 10, ). In other words, it takes more than 100 repetitions of the measurement with the PCO before the data qubit is corrupted as it would be with a single measurement using only a conventional physical qubit. This clearly demonstrates the exponential suppression of backaction when performing measurements using the PCO as an auxiliary qubit.

[0129] When photon loss (dashed line) is introduced, the PCO is z After that, the probability of being in the odd cat state decreases to Relative to the case without photon loss, the quantum bit has an interaction duration T z The probability of being in the odd-parity state afterwards remains unchanged: If the photon loss increases to κ = K / 10, the PCO will z After that, the probability of being in the odd cat state decreases to The fidelity of mapping the syndrome onto the auxiliary qubit is therefore reduced to 52%, which is close to the 50% point where the majority vote fails, but the backlash on the data qubit is still suppressed.

[0130] Example Stabilizer Measurement: Cat Code

[0131] In some embodiments, a cat code stabilizer is used, which is a class of Bose error correction codes in which information is encoded in a superposition of coherent states. The stabilizer for cat codes is the photon number parity operator in, and is the photon creation and annihilation operator for a data qubit (sometimes referred to herein as a storage qubit or storage cat or storage oscillator). When measured, the photon number parity operator indicates whether the state of the data qubit has an even or odd number of photons.

[0132] In some embodiments, the doubly degenerate code subspace consists of cat states with even photon numbers and Definition, cat state and yes The eigenstates of the case where the eigenvalue is +1. The error subspace consists of cat states with odd photon numbers and Definition, cat state and yes The eigenstate when the eigenvalue is -1.

[0133] In some embodiments for performing cat syndrome measurements, a storage oscillator encoding a cat codeword is dispersively coupled to an auxiliary qubit. The dispersive coupling between the storage oscillator and the auxiliary qubit can be used to map the parity of the storage cat, which is a property of the data qubit, onto the auxiliary qubit. However, random relaxation of the auxiliary qubit during measurement can cause random phase rotations of the cat codeword, making the scheme intolerant. The inventors have realized and appreciated that this can be achieved by using a photon number operator Replacing the above interaction Hamiltonian Operators in To engineer a fault-tolerant syndrome detection scheme. In some embodiments, the interaction Hamiltonian of the storage oscillator and the auxiliary PCO is then given by:

[0134]

[0135] This interaction is equivalent to the longitudinal interaction between the storage oscillator and the auxiliary PCO. In some embodiments, this interaction can be generated in a tunable manner.

[0136] The unitary operator corresponding to this interaction Hamiltonian is:

[0137]

[0138] In some embodiments, the PCO can be initialized to the CAT state by first Then, the interaction between the storage oscillator and the auxiliary PCO is switched on for the interaction duration T p , so that During the interaction duration T p At the end, the unitary operator simplifies to:

[0139]

[0140] In this expression for the unitary operator after the interaction duration, the initial exponential term is the deterministic rotation of the reference frame of the storage cat. In some embodiments, the deterministic rotation can be kept track of in classical software while performing subsequent operations on the qubits and explained later. If the storage oscillator is in the code subspace The auxiliary PCO and the storage oscillator interact for a duration of T p The subsequent states are and (Ignore deterministic frame rotations). On the other hand, if the storage oscillator is in the error subspace In the case of PCO at time Tp Evolved to state While the storage cat is still in state Therefore, the status of the auxiliary PCO indicates an error syndrome In some embodiments, the PCO measures only the parity of the stored cat, without revealing information about the actual photon statistics, as long as χ is small and the dynamics of the PCO can be restricted to a stable cat subspace. 2 ,leave The probability of excitation of the subspace is small, which may cause phase diffusion in the storage cat. In some embodiments, the PCO is driven in the opposite direction to cancel the average departure The excitation of the subspace makes the corrected Hamiltonian It is possible to partially correct for this diffusion.

[0141] In some embodiments, the time-dependent qubit-oscillator interaction is achieved by switching on and then off the coupling between the storage cavity and the auxiliary PCO. In some embodiments, the storage cavity is initialized to an odd parity state And the auxiliary PCO is initialized to be in the cat state

[0142] Fig. 10A and Fig. 10B is a graph of the dynamics of the PCO and storage cavity during the following stabilizer measurements for: single photon loss rate, κ = 0 (solid line) and κ = K / 200 (dashed line), the value P = 4K (equivalent to β = 2), χ0=K / 15 and T P =π / 4χ0β). Fig. 10A The following graph 1000 is shown: The graph 1000 shows that when the PCO is initialized to be in state and the storage cavity is initialized to be in an odd parity state |ψ o >, PCO is in the state of The probability that the storage cavity is in the state |ψ o >1003. During the interaction duration T p Afterwards, in the case of no photon loss (solid line), the density matrix The probability that the storage cavity represented by is in the state of maximum entangled odd-parity is And the density matrix The probability that the PCO represented by is in the odd cat state is When photon loss (dashed line) is introduced, the PCO is zAfter that, the probability of being in the odd cat state decreases to Relative to the case without photon loss, the storage cavity has an interaction duration T P The probability of being in the odd-parity state afterwards remains unchanged:

[0143] Fig. 10B is the following graph 1010: The graph 1010 shows that when the PCO is initialized to an even parity state And the auxiliary PCO is initialized to be in the cat state When PCO changes with time, it is in the state The probability that the storage cavity is in the state |ψ e >1013. For no photon loss (κ=0), at time T P The probability that the storage cavity is in an even parity state is And the probability that PCO is in an odd parity state is When photon loss (κ = K / 200) is introduced, the probability of the auxiliary PCO being in an odd-parity state decreases to But the probability of the storage cavity being in an even parity state remains

[0144] To highlight the fault tolerance of measurements using PCOs, the case where measurements are performed with conventional two-level physical qubits having the same relaxation rate γ = K / 200 is shown. With conventional physical qubits, the backaction on the storage cavity is increased by about two orders of magnitude.

[0145] Example stable submeasurement: Gottesman-Kitaev-Preskill (GKP) code

[0146] In some embodiments, a GKP code is used, which is a class of Bose error correction codes designed to correct random displacement errors in phase space. In some embodiments, the codeword for the GKP code is the phase space displacement of the storage cavity. and At the same time +1 eigenstate, where and are the position operator and momentum operator respectively. According to the photon annihilation and generation operators of the storage cavity, and Defined as and and and is a displacement operator, where

[0147] The two ideal GKP codewords are Position operator at even and odd integer multiples of These GKP states are the sum of an infinite number of infinitely squeezed states and are non-physical (non-standardizable) because they have an unlimited number of photons. The infinitely squeezed states can be replaced by squeezed Gaussian states. And the uniform superposition of these states is replaced by an overall envelope function such as Gaussian, binomial, etc., to achieve a more realistic codeword that can be used in some embodiments. The GKP code provides protection against low rate errors, which can be represented by and The small phase-space displacement of the oscillator is given by . The shifted GKP states are also respectively about the eigenvalues and The stabilizer and The measurement of the stabilizer will produce the eigenvalues ​​and thus uniquely determine the displacement errors u and v. In some embodiments, when When the displacement error is smaller than the translation distance between codewords This is possible when the

[0148] In some embodiments, a method for measuring and The eigenvalue of and A simple method is based on an adaptive phase estimation protocol (APE). In such an embodiment, a shift operation is repeatedly performed on the storage cavity, which is conditioned on the state of the auxiliary qubit. Therefore, some embodiments are directed to a fault-tolerant protocol for APE of stabilizers of GKP codes using a stable cat in a PCO.

[0149] In some embodiments, to achieve controlled displacement for use in APE, the storage cavity is coupled to the PCO via a tunable single-photon exchange interaction (also known as a beam splitter operation) defined by the following Hamiltonian:

[0150]

[0151] where g(t) is the dynamic coupling strength between the storage cavity and the PCO. In some embodiments, the tunable beam splitter operation can be achieved using the three-wave or four-wave mixing capability of the PCO and an external microwave drive of an appropriate frequency received from the microwave field generator 150. For small values ​​of |g|, the beam splitter Hamiltonian can be approximated as:

[0152]

[0153] For large amplitude β, the Hamiltonian The second term of becomes negligibly small, and the evolution under the Hamiltonian results in a controlled displacement along the position or moment of momentum, depending on the phase chosen for the coupling g(t). In this limit, when the phase and amplitude of the coupling g(t) are chosen so that g(t) = g * (t) = |g(t)| and When , the unitary operator corresponding to the above beam splitter interaction Hamiltonian simplifies to:

[0154]

[0155] The above unitary operator According to some embodiments, Conditional transfer of the APE storage cavity.

[0156] Similarly, when the phase and amplitude of the coupling g(t) are chosen so that g(t)=i|g(t)|, g * (t) = i|g(t)| and When , the unitary operator corresponding to the above beam splitter interaction Hamiltonian simplifies to:

[0157]

[0158] The above unitary operator According to some embodiments, Conditional transfer of the APE storage cavity.

[0159] Fig.11A and Fig. 11B A diagram of a quantum circuit for executing the APE protocol according to some embodiments is shown. Fig.11A Shown is the estimate Protocol 1100, and Fig. 11B Shown is the estimate APE protocol 1150. The three horizontal lines represent the readout cavity 1101, the auxiliary qubit 1102, and the data qubit 1103. Time increases from left to right, so that the operations appearing on the left side of the figure are performed before the operations shown on the right side of the figure. The readout cavity 1101 is initialized to be in the vacuum state |0>, and the auxiliary qubit 1102 is initialized to be in the even cat state And data qubit 1103 is initialized to be in any state |ψ that the data qubit is in based on other operations that may be performed on data qubit 1103 before the APE protocol N-1 >. In some embodiments, as described above, auxiliary qubit 1102 includes a PCO.

[0160] For estimation The protocol 1100 includes: performing a first joint unitary operation 1110 on a data qubit 1102 and an auxiliary qubit 1103 so as to achieve In some embodiments, the first joint unitary operation 1110 includes two separate actions. First, the data qubit 1103 is shifted Then, the data qubit 1103 is shifted based on the state of the auxiliary qubit 1102. Conditional bit shift operation 1113.

[0161] Then, to estimate The protocol 1100 includes: performing a rotation angle about the Z axis on the auxiliary quantum bit 1102 In some embodiments, the rotation operation 1120 is performed by driving an auxiliary qubit 1102 with a microwave field, which can be a PCO. In some embodiments, the rotation operation 1120 can be performed by driving an auxiliary qubit 1102 with a microwave field, which can be a PCO. The previous iteration of protocol 1100 determined The value of .

[0162] Then, to estimate The protocol 1100 includes a readout operation 1130 for determining the state of the auxiliary quantum bit 1102. The readout operation 1130 is performed, for example, by determining whether the auxiliary quantum bit 1102 is in the cat state. Still in cat form to determine the state of auxiliary qubit 1102. In some embodiments, the readout of auxiliary qubit 1102 may include mapping the state of auxiliary qubit 1102 onto readout cavity 1101. In some embodiments, readout operation 1130 may include two separate operations. The first operation may be a rotation operation 1131 on auxiliary qubit 1102. For example, rotation operation 1131 may convert the cat state The second operation of the readout operation 1130 includes a "Q-switching" operation 1133, in which the single-photon exchange coupling between the PCO 1102 and the readout cavity 1101 is turned on by applying an appropriate microwave field from the microwave field generator 150. The result of the Q-switching operation 1133 is to conditionally shift the readout cavity 1101 based on the state of the PCO 1102. Finally, after the readout operation 1130 is completed, the readout cavity 1101 is measured using, for example, a homodyne detection scheme.

[0163] In the implementation for estimating After the protocol 1100, the readout cavity 1101 and the auxiliary qubit 1103 can be reset to their respective initialization states (|0> and ).

[0164] For estimation The protocol 1150 includes: performing a second joint unitary operation 1160 on the data qubit 1102 and the auxiliary qubit 1103 so as to achieve In some embodiments, second joint unitary operation 1160 includes two separate actions. First, a shift is performed on data qubit 1103. Then, the data qubit 1103 is shifted based on the state of the auxiliary qubit 1102. Conditional bit shift operation 1163.

[0165] Then, to estimate The protocol 1100 includes: performing a rotation operation 1170 on an auxiliary qubit 1102 about the Z axis by an angle v. In some embodiments, the rotation operation 1170 is performed by driving the auxiliary qubit 1102 with a microwave field, and the auxiliary qubit 1102 can be a PCO. In some embodiments, the rotation operation 1170 can be performed by using a microwave field to drive the auxiliary qubit 1102, which can be a PCO. The value of v is determined by a previous iteration of protocol 1100.

[0166] Then, to estimate The protocol 1100 includes a readout operation 1180 for determining the state of the auxiliary qubit 1102. The readout operation 1180 determines, for example, whether the auxiliary qubit 1102 is in the cat state. Still in cat form to determine the state of auxiliary qubit 1102. In some embodiments, the readout of auxiliary qubit 1102 may include mapping the state of auxiliary qubit 1102 onto readout cavity 1101. In some embodiments, readout operation 1180 may include two separate operations. The first operation may be a rotation operation 1181 on auxiliary qubit 1102. For example, rotation operation 1181 may convert the cat state Rotate to a nearly coherent state |±β>. The second operation of the readout operation 1180 includes a "Q-switching" operation 1183, in which the single-photon exchange coupling between the PCO 1102 and the readout cavity 1101 is turned on by applying an appropriate microwave field from the microwave field generator 150. The result of the Q-switching operation 1183 is to conditionally shift the readout cavity 1101 based on the state of the PCO 1102. Finally, after the readout operation 1180 is completed, the readout cavity 1101 is measured using, for example, a homodyne detection scheme.

[0167] In the implementation for estimating After the protocol 1100, the readout cavity 1101 and the auxiliary qubit 1103 can be reset to their respective initialization states (|0> and ).

[0168] As described above, the amounts of rotation performed in the above rotation operations 1120 and 1160 are respectively and φ, and can be determined based on previous iterations of the corresponding estimation protocol. In this way, the measurement results are fed back into subsequent iterations of the APE protocol. To understand how these feedback phases are determined, consider that data qubit 1103 is in a stable About the eigenvalue The eigenstate of After the first joint unitary operation 1110, the state of the auxiliary qubit becomes If the auxiliary qubit 1102 is further rotated around the Z axis by an angle φ / 2 by a rotation operation 1120, the state of the auxiliary qubit 1102 becomes Therefore, after a single iteration of phase estimation, auxiliary qubit 1102 remains at The probability of the state is Therefore, in order to accurately predict the value of u, the probability distribution should be In some embodiments, this is achieved in the APE by: based on the auxiliary qubit 1102 being measured in a previous iteration of the protocol Status is still state to select the feedback phase v. Similar analysis applies to The eigenvalue and feedback phase of Execute APE protocol 1150.

[0169] Based on the above, in some embodiments, the APE protocol 1100 and the APE protocol 1150 may be iterated to estimate the stable sub-eigenvalues. As the number of iterations of the phase estimation increases, the accuracy of the estimation of u,v also increases, and therefore, the eigenvalues and The uncertainty is reduced.

[0170] Assisted readout

[0171] In some of the above embodiments (for example, see Figure 8 and FIG. 11A to FIG. 11B), a readout operation is performed to measure the auxiliary qubit. Although the state of the auxiliary qubit can be directly measured by directly measuring the state of the cavity using homodyne detection when the auxiliary qubit is a PCO, such a measurement will be slow due to the high Q of the PCO cavity. Therefore, in some embodiments, after mapping the state of the auxiliary qubit to the state of the readout cavity with a Q value less than the Q value of the auxiliary cavity, the readout cavity is measured using homodyne detection. In some embodiments, the readout of the PCO can be a quantum non-destructive (QND) measurement, although it does not have to be so (for example, the readout may introduce bit flip errors or other errors in the state of the auxiliary qubit). Such non-QND measurements are possible because the interaction between the auxiliary PCO and the data qubit can be turned off when measuring the PCO, so that auxiliary errors do not propagate to the data qubit. Such direct measurements of the auxiliary qubit can be performed using a superconducting transmon.

[0172] In some embodiments, the readout of the auxiliary PCO includes measurements along the Z-axis of the Bloch sphere, and does not introduce any other nonlinearities into the system. As described above, the state along the Z-axis of the Bloch sphere is a nearly coherent state and can be measured using homodyne detection of the field at the output of the PCO. To overcome the slow speed of direct homodyne detection of the PCO cavity, a Q-switching operation is performed, whereby the PCO state is switched to a low-Q readout cavity via frequency conversion. In some embodiments, the Q-switching operation conditionally shifts the readout cavity based on the state of the PCO along the Z-axis.

[0173] As described above, the readout operation may include a first operation in which the cat state of the PCO is rotated into a coherent state. Then, the coherent state Q of the PCO is switched to the readout cavity. Finally, the readout cavity is measured. Fig.12 A readout process 1200 according to the Bloch sphere is shown.

[0174] In some embodiments, the cat state of the PCO is performed using a microwave drive field from the microwave field generator 150. The Bloch sphere 1201 of the readout process 1200 shows the cat state of the PCO positioned along the X-axis of the Bloch sphere 1201 and then rotated about the Z-axis of the Bloch sphere 1201. In some embodiments, rotation about the Z axis is performed using single photon drive using the following Hamiltonian:

[0175]

[0176] The result of the single photon driving the Hamiltonian is that the cat state rotates from the Z axis to the Rotate to These are called parity-free cat states and correspond to states along the Y axis of the Bloch sphere (see Bloch sphere 1203 of readout process 1200). To map these parity-free cat states to coherent states, the two-photon pump that generates the cat states is turned off for a time T = π / 2K (see Bloch sphere 1203), thereby allowing the state of the PCO to be in the Kerr nonlinear Hamiltonian. As shown in the Bloch sphere 1205 of the readout process 1200, the free evolution under the Kerr nonlinear Hamiltonian leads to the state Transformed into a nearly coherent state Once the free evolution of the PCO state is complete, the two-photon cat pump is reapplied, thereby again stabilizing the cat subspace to prevent bit flips. As a result, the PCO remains in a coherent state as shown by the Bloch sphere 1205 of the readout process 1200.

[0177] After converting the PCO from the cat state to the coherent state via the above rotation, the state of the PCO is in place along the Z-axis of the Bloch sphere. In some embodiments, the PCO is then coupled to a non-resonant readout cavity. In the absence of an external microwave drive field, the coupling between the PCO and the readout cavity is negligible due to the large detuning between the PCO and the readout cavity. In some embodiments, single-photon exchange coupling (beam splitter coupling) is turned on by applying at least one microwave drive field from a microwave field generator to compensate for the frequency difference between the PCO and the readout cavity. Three-wave or four-wave mixing between the driver, PCO, and readout cavity results in an interaction between the PCO and the readout cavity, which causes a resonant single-photon exchange between the PCO and the readout cavity. This controllable coupling is called Q-switching. As shown in the phase space diagram 1207 of the readout process 1200, the result of the Q-switching operation is a transfer of the readout cavity conditioned on the state of the PCO. The Q-switching Hamiltonian for this process is given by Given, where and are the production and annihilation operators of the readout cavity, and g is the tunable coupling strength between the PCO and the readout cavity. For small values ​​of g, the Q-switching Hamiltonian can be approximated as:

[0178]

[0179] In the case of moderately large β, the final term becomes negligibly small, and the result is a displacement of the readout oscillator conditioned on the state of the PCO, where the amplitude of the field of the readout cavity is

[0180]

[0181] Among them, κ r is the linewidth of the field.

[0182] After conditionally shifting the sense cavity, a homodyne detector is used to determine the state of the sense cavity, thereby determining the state of the PCO, which is equivalent to extracting the error syndrome.

[0183] Bias-maintained quantum gate

[0184] The inventors have realized and appreciated that the above technique of using an auxiliary qubit's asymmetric error channel to detect error syndromes can be extended to implement bias-preserving quantum gates. For qubits with biased noisy channels (i.e., asymmetric error channels), operations that do not commute with the dominant error type can unbias or depolarize the qubit's noisy channel, thereby reducing the benefit of the biased noisy channel.

[0185] To understand how a non-commutative operation can debias the noisy channel of a qubit that has a biased noisy channel, consider a system that maintains the noisy bias. For example, consider the following two-qubit gate:

[0186]

[0187] in, is the Z-Pauli operator of the ith qubit, and θ is the adjustable phase angle. When θ = π / 2, the ZZ(θ) gate becomes a controlled phase gate (also called a CZ gate), which depends on the local Pauli rotation and the overall phase. The unitary evolution is given as The ZZ(θ) gate is realized by the interaction Hamiltonian of . Under this unitary evolution, the ZZ(θ) gate is realized after the interaction duration T = θ / 2V. If a phase flip occurs in either of the two qubits at time τ during the interaction time, the evolution is modified as follows:

[0188]

[0189] Therefore, an erroneous gate operation is equivalent to an error-free gate followed by a phase flip. Therefore, the ZZ(θ) gate maintains the erroneous bias of the qubit.

[0190] On the other hand, a controlled-not (CNOT) gate (also called a CX gate) between two qubits can be implemented using the following CX Hamiltonian:

[0191]

[0192] Among them, the unitary evolution is given by Given, where the control qubit and target qubit of the CNOT gate are labeled by 1 and 2, respectively. Under this evolution, the CNOT gate is realized after the interaction duration T = π / 2V, such that

[0193]

[0194] where the overall phase is neglected. In the case of this CNOT gate, the phase flip error in the target qubit at time τ during the interaction time modifies the unitary evolution as follows:

[0195]

[0196] Therefore, depending on when the phase error in the target qubit occurs, a phase flip error in the target qubit introduces a phase flip error in the control qubit. Importantly, the phase flip of the target qubit during the CNOT gate is propagated as a combination of a phase flip error and a bit flip error in the same qubit. Therefore, the CNOT gate reduces the bias of the noisy channel by introducing a bit flip in the target qubit. Similarly, coherent errors in the gate operation caused by the uncertainty of V and T will also cause bit flip errors in the target qubit. As a result, it is impossible to achieve a native bias-preserving CNOT gate.

[0197] The inventors have realized and appreciated that in the absence of bias-holding CNOT gates, alternative circuits are needed to extract error syndromes. These alternative circuits increase complexity and limit the gain in the fault-tolerance threshold for error correction resulting from the use of qubits with bias noise. Therefore, the inventors have developed a novel solution to this problem by engineering a bias-holding CNOT gate using the same two-component cat state implemented in the above-described parameter-driven nonlinear oscillator.

[0198] As above Figure 6 As shown, the Bloch sphere is oriented so that the superposition state is oriented along the Z axis of the Bloch sphere. Furthermore, for the purpose of the CNOT gate, the Z axis is chosen as the computational basis such that:

[0199]

[0200] where α is the complex amplitude of the coherent state associated with the cat state.

[0201] The cat states and their superpositions - |0> and |1> - are degenerate eigenstates of a parameter-driven Kerr nonlinear oscillator. As described above, the PCO exhibits a strong noise bias, so that bit flips are exponentially suppressed. Some embodiments use the PCO to implement a native CNOT gate while maintaining the error bias, thereby overcoming the problems of the example CNOT gate described above. In some embodiments, the CNOT gate is based on a topological phase that results from a rotation of the cat state around the Bloch sphere, which is generated by changing the phase of the parameter drive. The topological properties of some embodiments allow the CNOT gate to maintain the error bias in the qubit. The ability to maintain a noise bias presents only one advantage: using a continuously variable physical system (such as a PCO) to implement a logical qubit, rather than using a two-level physical system as the basis for the qubit.

[0202] In some embodiments, the time-dependent unitary evolution of a qubit undergoing a CNOT gate does not include an explicit operator (i.e., the X-Pauli operator), because as described in the above example of the CNOT gate, The operator does not preserve the noise bias of the qubit. In some embodiments, an alternative technique that does preserve the noise bias is used to engineer operator-equivalent evolution. To see how this is achieved, note that the cat state is The operator makes In addition, the orientation of the cat state on the Bloch sphere is defined by the phase φ of the two-photon driving field that produces the cat state in the PCO, where the Hamiltonian of the PCO is given by:

[0203]

[0204] This Hamiltonian is the same as the PCO Hamiltonian discussed previously, but the drive field is no longer considered real and positive, resulting in the inclusion of a phase φ. In some embodiments, this phase of the two-photon pump is varied to implement a CNOT gate. For example, if the phase is varied adiabatically from 0 to π, the cat state changes from Convert to Therefore, rotating the phase of the two-photon pump field by π is equivalent to achieving Operator.

[0205] In some embodiments, a two-qubit bias-hold CNOT gate is based on a conditional phase-space rotation of the target qubit according to the state of the control qubit. To show how conditional rotation leads to a CNOT gate, consider two PCOs, each stabilized / pumped with its own two-photon microwave pump field. The initial state of the two-qubit system is:

[0206]

[0207] where the first and second terms in the tensor product refer to the control qubit and the target qubit, respectively, and the term c i and item d i is simply the probability amplitude of each component of the superposition, and can be arbitrarily chosen to be any initial state. If the phase of the two-photon drive applied to the target PCO is conditioned on the state of the controlling PCO, the state of the system will evolve as follows, so that for any given time t, the state is:

[0208]

[0209] If the time-varying phase φ(t) is such that φ(0) = 0 and φ(T) = π, then at time T, the state becomes:

[0210]

[0211] The above results show that the CNOT gate is realized by rotating the phase of the cat in the target PCO by π, conditional on the state of the control PCO. The CNOT operation is realized because: during this rotation, Status relative to The state acquires a π phase. The acquired phase difference between the two cat states is the topological phase, which originates from the state According to the phase of α, it is 2π periodic, and the state The phase is π-periodic according to α. The topological phase does not depend on energy as the dynamical phase does. The topological phase also does not depend on the geometry of the path as is the case with the geometric phase. The phase will rise as long as the state |±α> moves along a loop in phase space that is not too close to the origin (e.g., the cat size α should be large enough so that the geometric phase difference between two cat states is exponentially small and topology is the only source of phase difference between the two cat states). If the state |±α> orbits around the origin to The degree of is given by u, and the phase obtained by |±α> is exp(iuπ). In other words, u is the winding number.

[0212] It can be shown that, as previously described, Unlike the CNOT gate of the operator, the topological phase-based CNOT gate maintains the bias of the noise channel of the qubit. In particular, the occurrence of a phase flip error in the control PCO during the evolution of the CNOT gate is equivalent to the occurrence of a phase flip on the control qubit after the ideal CNOT gate is realized. Similarly, during the evolution of the CNOT gate, the phase flip error on the target PCO is equivalent to the phase flip error on the control qubit and the target qubit that occurs after the ideal CNOT gate. Therefore, the CNOT gate according to some embodiments does not unbias the noise channel. This result contrasts with the aforementioned CNOT gate implementation between two strictly two-level qubits, and shows an advantage of using a larger Hilbert space (e.g., an oscillator) to perform quantum information processing.

[0213] In some embodiments, a specific Hamiltonian is used to implement the time evolution of the above state |ψ(t)>. In general, it is assumed that the amplitude α of the cat in the control PCO is different from the amplitude β of the cat in the target PCO. The following is the time-dependent interaction Hamiltonian for implementing a bias-hold CNOT gate between two PCOs according to some embodiments:

[0214]

[0215] In the Hamiltonian In the first row, a parametrically driven nonlinear oscillator stabilizes the control cat qubit. The phase of the drive for this oscillator is fixed at φ = 0. To understand the other two rows, recall that Thus, if the control qubit is in state |0〉 in the computation basis (which is approximately equal to |β〉 for large β), and from The exponentially smaller contribution of the term is neglected, and the CNOT Hamiltonian simplifies to:

[0216]

[0217] As a result, when the control qubit is in state |0>, the state of the target oscillator remains unchanged.

[0218] On the other hand, if the control qubit is in the state |1〉 in the computational basis (which is approximately equal to |−β〉 for large β), then the CNOT Hamiltonian simplifies to:

[0219]

[0220] The second line of the above expression shows the cat state is the instantaneous eigenstate of the target PCO. As a result, if the constraint The phase φ(t) is changed adiabatically, and the directional follower of the target PCO state φ(t) and α evolves in time to αeiφ(t) During the rotation in phase space, the target PCO also acquires the geometric phase The geometric phase is proportional to the area under the phase space path and depends on the state of the target PCO, where Geometric phase obtained from two cat states The difference reflects the fact that for the two cat states, the average number of photons is different and the phase space is represented by The area of ​​the path followed is larger than that given by The area of ​​the path followed by . In the limit of large α, the difference between the two geometric phases is α 2 decreases exponentially, making Therefore, for large α, the two geometric phases are approximately equal and do not result in a phase difference between the two states. Instead, it is the overall phase shift that results in the additional Z c (Φ g ) rotation. This overall phase shift can be taken into account classically in software, or by applying an additional rotation Z c (-Φ g ) to eliminate the additional rotation to account for the overall phase shift. Alternatively, the method given above for The additional interaction given by the last term in the expression of , cancels this extra rotation by the operation of the CNOT gate itself. The projection of this last term on the cat state basis is given by:

[0221]

[0222] This term leads to a dynamical phase that exactly cancels the geometric phase. Hence, the CNOT Hamiltonian leads to the two-qubit evolution of a bias-preserving CNOT gate.

[0223] In some embodiments, a bias-holding CNOT gate is physically implemented using three-wave mixing between two oscillators. The natural coupling between the two oscillators is a beam splitter coupling. Therefore, in some embodiments, the oscillators themselves are fourth-order, Kerr nonlinear. Thus, the bias-holding CNOT gate can be physically implemented using three-wave mixing between two oscillators. The natural coupling between the two oscillators is a beam splitter coupling. Therefore, in some embodiments, the oscillators themselves d =2ω t -ω c The parameters drive the target oscillator to generate a three-wave mixture, where ω t and ω care the frequencies of the target oscillator and the control oscillator, respectively. Under such a driving field, the fourth-order nonlinearity converts a photon in the driving field and a photon in the control oscillator into two photons in the target oscillator. Thus, effective three-wave mixing is achieved between the control oscillator and the target oscillator. In some embodiments, the Kerr nonlinearity of the oscillator itself is sufficient to realize the CNOT interaction Hamiltonian, and no additional coupling elements are required. In addition, due to the parametric nature of the interaction, the coupling is controllable.

[0224] Fig.13 is a schematic diagram of a quantum information processing device 1300 configured to implement a bias-hold CNOT gate according to some embodiments. Figure 1 and Figure 2 Compared to the details about the drive field provided in the block diagram of Fig.13 Additional details about the drive field are provided. Fig.13 The schematic diagram is the circuit diagram equivalent of the quantum information processing device 1300. In some embodiments, the physical system is as described above in combination with Figures 1 to 5 It was achieved as discussed.

[0225] Quantum information processing device 1300 includes control qubit 1301 and target qubit 1303. In some embodiments, qubits 1301 and 1303 are Kerr nonlinear cavities. As described above, superconducting circuit elements such as transmon or SNAIL can be used to control the nonlinearity of the cavity. Fig.13 In the example shown, both the control qubit 1301 and the target qubit 1303 include SNAILs. The resonant frequency of the SNAIL of the control qubit 1301 is ω c , the resonant frequency of the SNAIL of the target qubit 1303 is ω t In some embodiments, the SNAIL is biased with an external magnetic field to engineer three-wave and / or four-wave mixing interactions between the control qubit 1301 and the target qubit 1303. Using this engineered interaction, a two-photon driven Kerr nonlinear oscillator is generated and can be used to generate a PCO with a bias noise channel.

[0226] As shown by capacitor 1309, control qubit 1301 and target qubit 1303 are capacitively coupled to each other. A microwave field can be coupled to control qubit 1301 via input port 1305, and a microwave field can be coupled to target qubit 1303 via input port 1307. Figure 1 The microwave field generator 150 discussed receives the microwave field. In some embodiments, microwave fields of more than one frequency may be applied to a given input port at one time.

[0227] In order to conditionally rotate the state of the target PCO in phase space based on the state of the control PCO, the CNOT Hamiltonian described above is implemented The terms of the extended CNOT Hamiltonian can help understand what driving fields are needed to realize this Hamiltonian. The extended CNOT Hamiltonian can be written as:

[0228]

[0229] This expression can be further simplified by converting the Hamiltonian into a rotating frame where the frequencies of both PCOs are zero:

[0230]

[0231] in, In this form, it becomes clear how the Hamiltonian can be parameter engineered using four-wave mixing based on the Kerr nonlinearity and the drive field. For example, in an embodiment using SNAIL to implement PCO, and The proportional terms are implemented using three-wave mixing (where i = c, t denotes the control (c) qubit and the target (t) qubit) and are related to and The proportional terms are implemented using four-wave mixing. and The proportional term does not require nonlinearity and is achieved by applying only the drive field to the control qubit. In addition, φ(t) is a phase shift that varies with time and increases adiabatically from 0 to π in time T. Using all the above information, the CNOT Hamiltonian can be expressed in terms of the microwave field amplitude and phase as:

[0232]

[0233] In which, it is assumed that the field amplitude A i The driving microwave fields corresponding to amplitudes A1, A2, A3, A4 and A5 are respectively at frequencies 2ω c , 2ω t , 2ω t -ω c ,ω c and ω c was imposed.

[0234] In some embodiments, during an interaction duration T, a specific field sequence is applied to the control qubit 1301 and / or the target qubit 1303. During this interaction time, the execution of the CNOT gate is performed. In some embodiments, during the time outside of this interaction duration, the amplitude and phase take the following fixed values: A1 = Kβ2 ,Φ1=0,A2=Kα 2 , Φ2=0, A3=A4=A5=Φ3=Φ4=Φ5=0. During the CNOT interaction time, the phase Φ i (t) is time-varying and changes from a value of 0 to π. The phase value between 0 and T can be varied in any manner as long as the variation is adiabatic. In some embodiments, the phase varies linearly. For example, Φ i (t) = πt / T.

[0235] Fig.14A is a graph of the amplitude versus time of five drive fields used to implement a CNOT gate according to some embodiments, and Fig. 14B is a graph of the time-dependent phase variation of the five drive fields used to implement the CNOT gate.

[0236] First, at frequency 2ω c A first microwave field is applied to the control cavity with a fixed amplitude A1 and a time-dependent phase Φ1(t). The first microwave field provides a two-photon term to drive the control cavity via three-wave mixing. The fixed amplitude A1 is applied to the control cavity with a fixed amplitude A1 and a time-dependent phase Φ1(t). Fig.14A The line 1401 in FIG. 1400 shows that the time-dependent phase Φ1(t) is Fig. 14B The line 1411 in FIG. 14 is shown. In the example shown, the phase decreases linearly.

[0237] Next, at frequency 2ω t , time-dependent amplitude A2 and time-dependent phase Φ2(t) apply a second microwave field to the target cavity. The second microwave field provides a two-photon term to drive the target cavity via three-wave mixing. The changing amplitude A2 is achieved by Fig.14A The time-dependent phase Φ2(t) is shown by line 1402 in FIG. Fig. 14B 14. The amplitude varies sinusoidally with time. In the example shown, the phase is constant at a first phase value during a first portion of the gate duration, and is constant at a second phase value during a second portion of the gate duration, wherein the first phase value is less than the second phase value. This is because the amplitude always takes a positive value. When the amplitude A2 as a sine function crosses the zero amplitude point (instead of becoming negative), the amplitude begins to increase again, and the phase takes a different value instead.

[0238] The frequency 2ω is converted to the time-dependent amplitude A3 and the time-dependent phase Φ3(t). t -ω c The third microwave field is applied to the target cavity. This third microwave field realizes the same The amplitude of the change A3 is proportional to the coupling term. Fig.14AThe line 1403 in FIG. 14 shows that the time-dependent phase Φ3(t) is Fig. 14B The line 1413 in FIG. 14 shows that the amplitude varies as a cosine with time. In the example shown, the phase increases linearly with time.

[0239] The frequency ω is converted to the time-dependent amplitude A4 and the time-dependent phase Φ4(t). c The fourth microwave field is applied to the control cavity. The fourth microwave field realizes the single photon driving of the control cavity. The amplitude A4 of the change is Fig.14A The line 1404 in FIG. 14 shows that the time-dependent phase Φ4(t) is Fig. 14B 14 is shown by line 1414 in . The amplitude varies as a cosine over time. In the example shown, the phase decreases linearly during the first part of the gate duration, and decreases linearly during the second part of the gate duration. The linear decrease has the same slope in both parts of the gate duration, but halfway through the gate duration, there is a jump in phase. This is because the amplitude is always positive. When the amplitude A4 as a cosine crosses the zero amplitude point (instead of becoming negative), the amplitude begins to increase again, and the phase jumps to a different value instead.

[0240] Finally, with frequency ω c A fifth microwave field is applied to the target cavity with a fixed amplitude A5 and a time-dependent phase Φ5(t). The fifth microwave field provides the last term in the realization of the CNOT Hamiltonian. The fixed amplitude A5 is obtained by Fig.14A The line 1405 in FIG. 1404 shows that the time-dependent phase Φ5(t) is Fig. 14B The line 1415 in FIG. 14 is shown. In the example shown, the phase decreases linearly.

[0241] Error correction code tailored for bias noise

[0242] The inventors have realized and appreciated that aspects of the above-described stabilizer measurement scheme can be used to efficiently implement error correction codes that are customized for bias noise because the measurement scheme maintains the noise bias. Above, the preparation of cat states in data qubits and ancillary qubits is described. Quantum gates (such as Z-axis rotations and ZZ(θ) gates) are also described above. In addition, measurements along the Z axis can be performed, for example, by homodyne detection using the above techniques. Measurements along the X axis can be performed using additional gates and ancillary qubits. The inventors have realized and appreciated that these state preparation techniques, quantum gates, and detections can be combined with a bias-maintained CNOT gate between two cat qubits to achieve universal fault-tolerant quantum computing. Therefore, some embodiments use a bias-maintained set of operations. to realize an efficient and compact cascade-based circuit for fault-tolerant error correction, where It's cat-like Preparation, is the measurement along the X-axis, and is measured along the Z axis.

[0243] In some embodiments, the biased noise qubit is encoded as a repetitive code And the dominant error type (e.g., phase flip error) is corrected. A repetition code with n qubits can correct for (n-1) / 2 phase flip errors. In some embodiments, the codeword is and in, and Wherein there are n cat states for each codeword state. The result of this first encoding is a more symmetrical noise channel with reduced noise intensity. In some embodiments, the repetition code with errors below the threshold can then be cascaded to the CSS code To further reduce errors.

[0244] The n-1 stable subgenerator for the repetitive code is In some embodiments, the most naive way to detect errors is to use auxiliary qubits to measure each stabilizer generator. Such a technique is described by Fig.15 A quantum circuit diagram 1500 is shown. In the example of n=3, horizontal lines 1501-1503 represent code qubits, and horizontal lines 1504-1505 represent auxiliary qubits.

[0245] As shown by triangles 1517 and 1527, each auxiliary qubit 1504-1505 is initialized to be in the state Then, two CNOT gates 1510 and 1515 are implemented between the first auxiliary qubit 1504 and the first two code qubits 1501 and 1502, and two CNOT gates 1520 and 1525 are implemented between the second auxiliary qubit 1505 and the second two code qubits 1502 and 1503. Finally, as shown by triangles 1519 and 1529, the auxiliary qubits 1504-1505 are measured along the X-axis. For fault tolerance, some embodiments measure each stabilizer generator r times and determine the syndrome bit by a majority vote on the measurement results. If the measured m≥(r+1) / 2 is wrong, the syndrome bit is incorrect.

[0246] This decoding scheme is equivalent to constructing an r-bit repetition code for each of the (n-1) stabilizer generators of the repetition code. Therefore, each bit of the syndrome from the inner code is itself encoded in the [r,1,r] repetition code, so that it can be decoded by first decoding the syndrome bits and then decoding the resulting syndrome. This naive way of decoding the syndromes produces a simple analytical expression for the logical error rate. However, the inventors have realized and appreciated that this naive approach may not be the preferred method of decoding, and in some embodiments, it may be replaced by a decoder that directly infers the most likely error on the n-qubit repetition code given s measured syndrome bits. Fig.15 Therefore, in some embodiments, the concept of measurement codes is introduced, which uses the above insights to improve the naive scheme by constructing a block code that can directly correct bit-flip errors on n data qubits in a single decoding step.

[0247] In some embodiments, to construct the measurement code, the syndrome measurement process measures a total of s elements of the stable subgroup (not necessarily the designated generator) through coupling to the auxiliary qubits, and corrects any t=(d-1) / 2 phase-flip errors on the n qubits. Therefore, there are classical codes with parameters [n+2, n, d]. However, not every classical code with these parameters is allowed, because the classical parity check should be compatible with the stabilizers of the original quantum code (in this example, the repetition code). In particular, each parity check in the measurement code should have equal weight when restricted to the data qubits so that it is consistent with the logic of the quantum phase-flip code. Operator commutation. In some embodiments, consistency with a stable subset of the underlying quantum code is the only restriction on the measurement code.

[0248] The matrix H can be checked by M The general form of the measurement code is specified by . This general form is in turn based on the (usually redundant) parity check H of the quantum repetition code z and another set of s auxiliary bits marking the measurement result. Z , the parity check matrix of the measurement code is the block matrix H M =(H Z I S ), where I S is an s×s identity matrix. Since there are s auxiliary bits for readout, H M is an s×(n+s) matrix. In some embodiments, H ZThe rows of have equal weight because they come from the stabilizer of the quantum repetition code. The rows are linearly independent so that for some d ≤ n, the associated code has parameters [n+s, n, d]. The distance can never be greater than n because the distance on the data qubit corresponding to a 1 that happens to be on the first n bits is The string of operators is always located in H M in the kernel.

[0249] The measurement of the jth parity check in the measurement code can be done by a standard choice of circuits. In some embodiments, if there is a 1 in the i-th column, a CNOT gate is applied to the i-th qubit and the ancillary qubit labeled in the n+j column is targeted. Note that by construction, H M There is a 1 in position (j; n+j) of . The effective error rate of this bare-assisted measurement tool depends on the number of CNOT gates used and, therefore, on the weights of the stabilizers being measured. Therefore, when designing measurement codes, lower weight rows are preferred when all other things (e.g., code distance) are equal. The two examples considered here are based on the following for H z The selection generated by , shown here transposed to save space:

[0250]

[0251] These example codes saturate the distance bounds so that for each code, d=n (e.g., d=3 and d=5, respectively). In contrast, for n=r=3, the measurement code associated with repeating the measurement of the standard generator r times is:

[0252]

[0253] As measurement codes, both this choice and the above 3×3 choice have a distance d=3. However, the above 3×3 choice corresponds to a [6, 3, 3] measurement code, while the naive repetition generator method produces a [12, 3, 3] measurement code. In general, the naive scheme produces [n ( n-1) r, n, d(n, r)] code, and for smaller r, the distance has not yet saturated to n. For the case of n=5, r must be equal to 2 before the measuring code has a distance of 3, and r=4 before the distance saturates at d=5. Therefore, the naive scheme produces a [13, 5, 3] code or a [21, 5, 5] code, which are inferior to the [14, 5, 5] code obtained according to the selection in terms of distance or ratio, respectively. According to some embodiments, these examples also illustrate the counter-intuitive characteristics of the measuring code. Consider again the naive repetition generator method with n=5 and r=2 or 4. If the decoder works by first decoding the syndrome bits separately, the data is only protected against at most (r-1) / 2=0 or 1 arbitrary errors, respectively. However, a decoder using the structure of the associated measuring code can correct 1 or 2 arbitrary data errors using these corresponding parameters, which then reduces the preamble behavior of the code failure probability.

[0254] Both example codes above are small enough that the exact probability of decoding failure can be calculated via an exhaustive lookup table. To demonstrate the advantage of the measured code over naive encoding and decoding, the measured code in the second example above is used to estimate the probability of logical errors in the CNOT tool for n=5. The corresponding threshold is ~6×10 -3 . On the other hand, to achieve similar thresholds using a naive decoder requires n=11 and r=5. Therefore, in some embodiments, the decoder requires fewer resources than the naive decoder. Typically, an optimal (maximum likelihood) decoder is not feasible to implement because it requires exponential resources in n and s to perform the computation, so substantially larger codes will require decoding heuristics, such as a message passing algorithm, to achieve peak decoding performance. In some embodiments, the decoder declares failure whenever a data error is not completely guessed correctly, even though this is not necessary. When repeated rounds of error correction occur, it is sufficient to define success as reducing the weight of any correctable errors.

[0255] Methods for implementing QIP

[0256] Various methods of performing QIP are discussed above in conjunction with measuring error syndromes and performing bias hold gates. Fig.16 is a flow chart of a method 1600 for performing QIP, which is generally applicable to most of the above-described embodiments using data qubits coupled to auxiliary qubits. In some embodiments, the physical implementation of the data qubits and auxiliary qubits can be any of the above-described physical systems.

[0257] At action 1602, method 1600 includes driving an auxiliary qubit with a stabilizing field. In some embodiments, the stabilizing field generates asymmetry in an error channel of the auxiliary qubit, which is employed to measure error syndromes and perform bias-preserving quantum gates. The stabilizing field can be applied to the auxiliary qubit using a microwave field generator 160.

[0258] At act 1604, method 1600 includes generating a Kerr nonlinearity in the auxiliary qubit using at least one Josephson junction of the auxiliary qubit. In some embodiments, coupling a superconducting circuit element to the cavity generates the Kerr nonlinearity. For example, a transmon or SNAIL can be located in a 3D cavity to generate a Kerr nonlinear cavity.

[0259] At action 1605, method 1600 includes applying a plurality of microwave fields to the auxiliary qubits and the data qubits. In some embodiments, these microwave fields may be applied to produce pumped cat states in a Kerr nonlinear cavity. In some embodiments, microwave fields may be applied to perform a rotation on the state of a data qubit or an auxiliary qubit. In some embodiments, microwave fields may be applied to implement a conditional gate, for example, a conditional rotation of one qubit based on the state of another qubit. In some embodiments, microwave fields may be applied to couple an auxiliary qubit to a data qubit or to couple an auxiliary qubit to a readout cavity. Alternatively, as described above, any number of operations may be performed by applying microwave fields to data qubits and / or auxiliary qubits.

[0260] At action 1608, the method includes measuring the auxiliary qubit. As discussed above, the auxiliary qubit can be measured directly by, for example, performing homodyne detection on the cavity of the auxiliary qubit. Alternatively, the auxiliary qubit can be measured by coupling the auxiliary qubit to a readout cavity, conditionally shifting the state of the readout cavity based on the state of the auxiliary qubit, and then measuring the state of the readout cavity. In some embodiments, the measurement of the auxiliary qubit is a QND measurement.

[0261] Fig.17 is a flow chart of a method 1700 for performing readout of an auxiliary qubit according to some embodiments. In some embodiments, the method 1700 can be used to measure a property of a data qubit coupled to an auxiliary qubit. In some embodiments, the method 1700 can implement a QND measurement.

[0262] At act 1702, method 1700 includes applying at least one rotating microwave field to an auxiliary qubit. In some embodiments, the rotation may be about a Z axis of a Bloch sphere associated with the auxiliary qubit. In some embodiments, the rotation may be about a Z axis of a Bloch sphere associated with the auxiliary qubit. Rotate to

[0263] At act 1704, method 1700 includes turning off the stabilizing microwave field for a certain amount of time. In some embodiments, this allows the auxiliary qubit to evolve freely. In some embodiments, the auxiliary qubit may include a Kerr nonlinear cavity, and the state of the auxiliary qubit may evolve freely under the Kerr nonlinear Hamiltonian. In some embodiments, the free evolution of the auxiliary qubit results in a rotation of the state of the auxiliary qubit that cannot be performed if the stabilizing field is still applied to the auxiliary qubit.

[0264] At act 1706, method 1700 includes reapplying a stabilizing microwave field to the auxiliary qubit. In some embodiments, reapplying the stabilizing microwave field stops the free evolution of the state of the auxiliary qubit. In some embodiments, reapplying the stabilizing microwave field maintains the state of the auxiliary qubit in one of two coherent states. In some embodiments, reapplying the stabilizing field suppresses certain types of errors, such that the error channel of the auxiliary qubit is asymmetric. For example, the stabilizing field can suppress bit flip errors.

[0265] At act 1708, method 1700 includes applying an exchange microwave field to the auxiliary qubit. In some embodiments, the exchange microwave field generates an interaction between the auxiliary qubit and the readout cavity. In some embodiments, applying the exchange microwave field generates a three-wave or four-wave mixing interaction. In some embodiments, applying the exchange microwave field causes a Q-switching operation.

[0266] Other considerations

[0267] Having thus described several aspects of at least one embodiment of the present invention, it will be appreciated that various changes, modifications and improvements will be readily conceived by those skilled in the art. Such changes, modifications and improvements are intended to be a part of the present disclosure and are intended to fall within the spirit and scope of the present invention. In addition, although the advantages of the present invention have been pointed out, it will be appreciated that not every embodiment of the present invention will include each advantage described. Some embodiments may not implement any of the features described as being advantageous in this article and in some examples. Therefore, the foregoing description and accompanying drawings are exemplary only.

[0268] The various aspects of the present invention can be used alone, in combination, or in a variety of arrangements not specifically discussed in the above-described embodiments, and are therefore not limited in their application to the details and arrangements of the components set forth in the above description or shown in the accompanying drawings. For example, the various aspects described in one embodiment can be combined in any manner with the various aspects described in other embodiments.

[0269] The use of ordinal terms such as "first", "second", "third", etc. in the claims to modify claim elements does not itself imply any priority, precedence or order of one claim element relative to another claim element or the temporal order of the actions of performing the method, but is merely used as a mark to distinguish one claim element with a certain name from another element with the same name (except for the use of ordinal terms) to distinguish the claim elements.

[0270] As defined and used herein, all definitions should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0271] As used herein in the specification and claims, the indefinite articles "a" and "an" should be understood to mean "at least one", unless explicitly indicated to the contrary.

[0272] As used herein in the specification and claims, when referring to a list of one or more elements, the phrase "at least one" should be understood to mean at least one element selected from any one or more elements in the list of elements, but not necessarily including at least one of each and every element specifically listed in the list of elements, and not excluding any combination of elements in the list of elements. This definition also allows that elements may optionally be present instead of the elements specifically identified in the list of elements to which the phrase "at least one" refers, whether related or unrelated to those elements specifically identified.

[0273] As used herein in this specification and claims, the phrase "equal" or "same" when referring to two values ​​(e.g., distance, width, etc.) means that the two values ​​are the same within manufacturing tolerances. Thus, two values ​​being equal or the same can mean that the two values ​​differ from each other by ±5%.

[0274] The phrase "and / or" as used herein in the specification and claims should be understood to refer to "either or both" of the elements so connected (i.e., elements that are present in conjunction in some cases and separately in other cases). Multiple elements listed with "and / or" should be interpreted in the same manner, i.e., "one or more" elements so combined. In addition to the elements specifically identified by the "and / or" clause, other elements may optionally be present, whether related or unrelated to the specifically identified elements. Thus, as a non-limiting example, when used in conjunction with open-ended language such as "comprising", in one embodiment, reference to "A and / or B" may refer to only A (optionally including elements other than B); in another embodiment, reference to "A and / or B" may refer to only B (optionally including elements other than A); in yet another embodiment, reference to "A and / or B" may refer to both A and B (optionally including other elements); etc.

[0275] As used herein in the specification and claims, "or" should be understood to have the same meaning as "and / or" as defined above. For example, when the items in the list are separated, "or" or "and / or" should be interpreted as inclusive, i.e., including at least one of a plurality of elements or a list of elements, but also including more than one, and (optionally) other unlisted items. Only terms such as "only one" or "just one" or when used in the claims that are clearly indicated to the contrary, "consisting of..." will refer to including exactly one element in a plurality of elements or a list of elements. In general, the term "or" used herein, when in front of an exclusive term (such as "one", "one of", "only one of them" or "just one of them"), should only be interpreted as indicating exclusive alternatives (i.e., "one or the other but not both"). When used in the claims, "consisting essentially of..." should have the common meaning used in the field of patent law.

[0276] In addition, the words and terms used herein are for descriptive purposes and should not be considered limiting. The use of "includes," "comprising," or "having," "containing," "involving," and variations thereof herein is meant to encompass the items listed thereafter and their equivalents as well as additional items.

Claims

1. A quantum information processing system, comprising: Data qubits; as well as An auxiliary qubit is coupled to the data qubit and is configured to have an asymmetric error channel when the auxiliary qubit is driven by a pump microwave field generated by a microwave field source, the pump microwave field being configured to generate a cat state in the auxiliary qubit, the cat state being associated with the asymmetric error channel.

2. The quantum information processing system according to claim 1, further comprising: A measurement device is configured to measure a property of the data qubit using the auxiliary qubit.

3. The quantum information processing system according to claim 2, wherein: The measurement device is configured to measure a property of the data qubit using quantum non-destructive measurement.

4. The quantum information processing system according to claim 3, wherein: The measuring device is configured to perform the quantum non-destructive measurement by: causing the data qubit and the ancillary qubit to interact so that a state of the ancillary qubit is based on a property of the data qubit; as well as The state of the auxiliary qubit is measured to determine the state of the data qubit.

5. The quantum information processing system according to claim 4, wherein: The measurement device is configured to suppress a first type of error in the auxiliary qubit by repeatedly performing the quantum non-destructive measurement.

6. The quantum information processing system according to claim 4, wherein: The measuring device comprises: a readout cavity coupled to the auxiliary qubit; and A cavity state detector is configured to measure a state of the readout cavity.

7. The quantum information processing system according to claim 6, wherein: The measuring device is configured to measure the state of the auxiliary qubit by: causing the auxiliary qubit and the readout cavity to interact so that a state of the readout cavity is based on a state of the auxiliary qubit; The state of the readout cavity is measured to determine the state of the auxiliary qubit.

8. The quantum information processing system according to claim 7, wherein: The cavity state detector comprises a phase sensitive detector.

9. The quantum information processing system according to claim 8, wherein: The cavity state detector comprises a homodyne detector.

10. The quantum information processing system of claim 2, further comprising the microwave field source, the microwave field source being configured to cause the data qubit and the auxiliary qubit to interact so that a z-component of a state of the auxiliary qubit is based on a property of the data qubit.

11. The quantum information processing system according to claim 10, wherein: The auxiliary quantum bit is an auxiliary superconducting nonlinear asymmetric inductive element having a resonant frequency; and The microwave field source is configured to apply a pump microwave field to the auxiliary superconducting nonlinear asymmetric inductive element at a pump frequency that is twice a resonant frequency of the auxiliary superconducting nonlinear asymmetric inductive element.

12. The quantum information processing system according to claim 11, wherein: The microwave field source is configured to: Applying at least one initialization microwave field to the auxiliary quantum bit to initialize the cat state of the auxiliary quantum bit along the x-axis of the Bloch sphere; as well as At least one driving microwave field is applied to at least one of the data qubit and the auxiliary qubit to generate an interaction between the data qubit and the auxiliary qubit.

13. The quantum information processing system according to claim 12, wherein: The microwave field source is configured to apply the at least one initialization microwave field to the auxiliary qubit by: adiabatically increasing the amplitude of the pump microwave field of the auxiliary superconducting nonlinear asymmetric inductive element prepared in a vacuum state from 0 to a value equal to K|α| 2 The pump amplitude value of , wherein K is the strength of the Kerr nonlinearity of the auxiliary superconducting nonlinear asymmetric inductive element, and α is the coherent state amplitude associated with the cat state.

14. The quantum information processing system according to claim 11, wherein: The data qubit is a transporter; The property of the data qubit is the σ of the transporter Z ;and The microwave field source is further configured to apply a driving microwave field to the transmission sub-element at a resonant frequency of the auxiliary superconducting nonlinear asymmetric inductive element and in phase with the pump microwave field, so as to cause the data qubit and the auxiliary qubit to interact, so that the z component of the state of the auxiliary qubit depends on the σ of the data qubit. Z .

15. The quantum information processing system according to claim 11, wherein: The data qubit comprises a cat state in a linear oscillator; The property of the data qubit is the photon number parity of the cat state; and The microwave field source is also configured to apply a driving microwave field to the cat state in the linear oscillator at the resonant frequency of the auxiliary superconducting nonlinear asymmetric inductive element and in phase with the pump microwave field, so as to cause the data quantum bit and the auxiliary quantum bit to interact so that the z component of the state of the auxiliary quantum bit depends on the photon number parity of the data quantum bit.

16. The quantum information processing system according to claim 11, wherein: The data qubit comprises a Gottesman-Kitayev-Preskill state in a linear oscillator; The property of the data qubit is a stabilizer of the Gottesmann-Kitayev-Preskill state; and The microwave field source is further configured to apply a driving microwave field to the auxiliary superconducting nonlinear asymmetric induction element at a driving frequency equal to the difference between the resonant frequency of the auxiliary superconducting nonlinear asymmetric induction element and the resonant frequency of the linear oscillator, so as to cause the data qubit and the auxiliary qubit to interact so that the z component of the state of the auxiliary qubit depends on the stabilizer of the Gottesman-Kitayev-Preskill state of the data qubit.

17. The quantum information processing system according to claim 16, wherein: The microwave field source is configured to apply the drive microwave field to the Gottesman-Kitayev-Preskill state in phase with the pump microwave field; and The stabilizer of the Gottesmann-Kitayev-Preskill state of the data qubit is S q Stabilizer.

18. The quantum information processing system according to claim 16, wherein: The microwave field source is configured to apply the drive microwave field to the Gottesman-Kitayev-Preskill state 90° out of phase with the pump microwave field; and The stabilizer of the Gottesmann-Kitayev-Preskill state of the data qubit is S p Stabilizer.

19. The quantum information processing system according to claim 11, wherein: The data qubit includes a cat state in a data superconducting nonlinear asymmetric sensing element; The property of the data quantum bit is the σ of the data superconducting nonlinear asymmetric induction element. Z ;and The microwave field source is further configured to apply a plurality of microwave fields to the data qubit and the auxiliary qubit within an interaction duration to cause the data qubit and the auxiliary qubit to interact so that the z component of the state of the auxiliary qubit depends on the σ of the data qubit. Z , wherein applying the plurality of microwave fields comprises: Apply a frequency 2ω to the auxiliary quantum bit. c The first microwave field, where ω c is the resonant frequency of the auxiliary qubit; A frequency of 2ω is applied to the data qubit. t The second microwave field, where ω t is the resonant frequency of the data qubit; A frequency of 2ω is applied to the data qubit. t -ω c The third microwave field; Applying a frequency ω to the auxiliary quantum bit c A fourth microwave field; and Applying a frequency ω to the data qubit c The fifth microwave field.

20. The quantum information processing system according to claim 19, wherein: The amplitude of the first microwave field is constant during the interaction duration; The amplitude of the second microwave field is time-varying during the interaction duration; The amplitude of the third microwave field is time-varying during the interaction duration; The amplitude of the fourth microwave field is time-varying during the interaction duration; and The amplitude of the fifth microwave field is constant during the interaction duration.

21. A quantum information processing system according to claim 20, wherein: The phase of the first microwave field decreases linearly over the interaction duration; the phase of the second microwave field being constant at a first phase value during a first portion of the interaction duration and being constant at a second phase value during a second portion of the interaction duration, wherein the first phase value is less than the second phase value; The phase of the third microwave field increases linearly during the interaction duration; The phase of the fourth microwave field decreases linearly during a first portion of the interaction duration and decreases linearly during a second portion of the interaction duration; and The phase of the fifth microwave field decreases linearly over the interaction duration.

22. The quantum information processing system according to claim 10, wherein: The microwave field source is configured to make the data qubit and the auxiliary qubit interact with each other so that a controlled NOT gate is implemented between the data qubit and the auxiliary qubit.

23. The quantum information processing system according to claim 1, wherein: The auxiliary quantum bit includes a Kerr nonlinear cavity.

24. The quantum information processing system according to claim 23, wherein: The Kerr nonlinearity of the Kerr nonlinear cavity is greater than 1 MHz.

25. The quantum information processing system according to claim 24, wherein: The intensity of the parameter driving of the auxiliary quantum bit is greater than the Kerr nonlinearity of the Kerr nonlinear cavity.

26. The quantum information processing system according to claim 23, wherein: The Kerr nonlinearity of the Kerr nonlinear cavity is greater than the phase flip rate in the auxiliary quantum bit.

27. The quantum information processing system according to claim 23, wherein: The auxiliary quantum bit includes a cat state in the Kerr nonlinear cavity.

28. The quantum information processing system according to claim 27, wherein: The auxiliary qubit includes at least one Josephson junction configured to generate Kerr nonlinearity in the Kerr nonlinear cavity.

29. The quantum information processing system according to claim 28, wherein: The auxiliary quantum bit includes a transporter in the Kerr nonlinear cavity.

30. The quantum information processing system according to claim 28, wherein: The auxiliary qubit includes a superconducting nonlinear asymmetric inductor element in the Kerr nonlinear cavity.

31. The quantum information processing system according to claim 1, wherein: An error syndrome is determined based on a conditional rotation of the cat state.

32. A quantum information processing system according to claim 31, wherein: The conditional rotation is achieved using only local interactions.

33. The quantum information processing system according to claim 1, wherein: The error channel of the auxiliary qubit is dominated by phase flip errors rather than bit flip errors.

34. A quantum information processing system according to claim 33, wherein: The phase flip error is caused by photon loss.

35. A quantum information processing system according to claim 34, wherein: When a phase flip error occurs in the auxiliary qubit, the state of the data qubit is not affected.

36. A quantum information processing system according to claim 33, wherein: The bit flip errors are suppressed exponentially depending on the power of the pump used to generate the cat state in the auxiliary qubit.

37. A quantum information processing system according to claim 1, wherein: A quantum data gate is implemented between the data qubit and the auxiliary qubit.

38. A quantum information processing system according to claim 37, wherein: The quantum data gate is a controlled NOT gate, the data qubit is a target qubit of the controlled NOT gate, and the auxiliary qubit is a control qubit of the controlled NOT gate.

39. A quantum information processing system according to claim 38, wherein: The control qubit includes a first cat state, and the target qubit includes a second cat state.

40. A quantum information processing system according to claim 39, wherein: The controlled NOT gate is realized by rotating the first cat state in the phase space.

41. The quantum information processing system of claim 40, further comprising the microwave field source, the microwave field source being configured to apply a plurality of microwave fields to the data qubit and the auxiliary qubit within a gate duration to implement the controlled NOT gate.

42. A quantum information processing system according to claim 41, wherein: The microwave field source is configured to: A frequency of 2ω is applied to the control qubit. c The first microwave field, where ω c is the resonant frequency of the control qubit; Apply a frequency 2ω to the target quantum bit. t The second microwave field, where ω t is the resonant frequency of the target qubit; Applying a frequency 2ω to the target quantum bit t -ω c The third microwave field; Applying a frequency ω to the control qubit c A fourth microwave field; and Applying a frequency ω to the target quantum bit c The fifth microwave field.

43. A quantum information processing system according to claim 42, wherein: The amplitude of the first microwave field is constant during the gate duration; The amplitude of the second microwave field is time-varying during the gate duration; The amplitude of the third microwave field is time-varying during the gate duration; The amplitude of the fourth microwave field is time-varying during the gate duration; and The amplitude of the fifth microwave field is constant during the gate duration.

44. A quantum information processing system according to claim 43, wherein: The phase of the first microwave field decreases linearly during the gate duration; a phase of the second microwave field being constant at a first phase value during a first portion of the gate duration and being constant at a second phase value during a second portion of the gate duration, wherein the first phase value is less than the second phase value; The phase of the third microwave field increases linearly during the gate duration; The phase of the fourth microwave field decreases linearly during a first portion of the gate duration and decreases linearly during a second portion of the gate duration; and The phase of the fifth microwave field decreases linearly within the gate duration.

45. The quantum information processing system of claim 1, further comprising the microwave field source, the microwave field source being configured to apply a stable microwave field to the auxiliary qubit to create the asymmetric error channel.

46. ​​A quantum information processing system according to claim 45, wherein: The stabilizing microwave field stabilizes the state of the auxiliary qubit to prevent rotation about at least one axis of the auxiliary Bloch sphere.

47. A quantum information processing system according to claim 46, further comprising a readout cavity coupled to the auxiliary qubit.

48. The quantum information processing system according to claim 47, further comprising: A cavity state detector configured to determine a state of the readout cavity, the cavity state detector configured to measure a state of the auxiliary qubit.

49. A quantum information processing system according to claim 47, wherein: The readout cavity is configured to conditionally transfer based on a state of the auxiliary qubit.

50. A quantum information processing system according to claim 49, wherein: The microwave field source is configured to conditionally shift the readout cavity by: applying at least one rotating microwave field to rotate the state of the auxiliary qubit about an axis of the Bloch sphere; Turning off the stabilizing microwave field for an amount of time to allow the state of the auxiliary qubit to evolve based on the Kerr nonlinearity of the auxiliary qubit; reapplying the stable microwave field to stabilize the state of the auxiliary quantum bit again; as well as A switching microwave field is applied to the auxiliary qubit to switch on the exchange coupling between the auxiliary qubit and the readout cavity.

51. A method of performing quantum information processing in a system comprising a data qubit coupled to an auxiliary qubit, the method comprising: The auxiliary qubit is driven with a stable pump microwave field using a microwave field source, the stable pump microwave field being configured to create a first cat state and an associated asymmetric error channel in the auxiliary qubit.

52. The method of claim 51 further comprising: The auxiliary qubit is used to measure a property of the data qubit.

53. The method of claim 52, wherein: Measuring the property of the data qubit includes performing a quantum non-destructive measurement.

54. The method of claim 53, wherein: Performing the quantum non-destructive measurement includes: causing the data qubit and the ancillary qubit to interact so that a state of the ancillary qubit is based on a property of the data qubit; and The state of the auxiliary qubit is measured to determine the state of the data qubit.

55. The method of claim 54, further comprising: A first type of error in the auxiliary qubit is suppressed by repeatedly performing the quantum non-destructive measurement.

56. The method of claim 54, wherein: The quantum non-destructive measurement is performed using a measuring device, the measuring device comprising: a readout cavity coupled to the auxiliary qubit; and A cavity state detector is configured to measure a state of the readout cavity.

57. The method of claim 56, wherein: Performing the quantum non-destructive measurement includes: causing the auxiliary qubit and the readout cavity to interact so that a state of the readout cavity is based on a state of the auxiliary qubit; The state of the readout cavity is measured to determine the state of the auxiliary qubit.

58. The method of claim 57, wherein: The cavity state detector comprises a phase sensitive detector.

59. The method of claim 58, wherein: The cavity state detector comprises a homodyne detector.

60. The method of claim 52, further comprising: The data qubit and the ancillary qubit are caused to interact such that a z-component of a state of the ancillary qubit is based on a property of the data qubit.

61. The method of claim 60, wherein: The auxiliary quantum bit is an auxiliary superconducting nonlinear asymmetric induction element with a resonant frequency; and Causing the data qubit and the auxiliary qubit to interact with each other includes applying a pumping microwave field to the auxiliary superconducting nonlinear asymmetric inductive element at a pumping frequency that is twice a resonant frequency of the auxiliary superconducting nonlinear asymmetric inductive element.

62. The method of claim 61 further comprising: Applying at least one initialization microwave field to the auxiliary quantum bit to initialize the auxiliary quantum bit to be in a cat state along the x-axis of the Bloch sphere; as well as At least one driving microwave field is applied to at least one of the data qubit and the auxiliary qubit to generate an interaction between the data qubit and the auxiliary qubit.

63. The method of claim 62, wherein: Applying the at least one initialization microwave field to the auxiliary quantum bit comprises: adiabatically increasing the amplitude of the pump microwave field of the auxiliary superconducting nonlinear asymmetric inductive element prepared in a vacuum state from 0 to a value equal to K|α| 2 The pump amplitude value of , wherein K is the strength of the Kerr nonlinearity of the auxiliary superconducting nonlinear asymmetric inductive element, and α is the coherent state amplitude associated with the cat state.

64. The method of claim 61, wherein: The data qubit is a transporter; The property of the data qubit is the σ of the transporter Z ;and The method further includes: applying a driving microwave field to the transmission sub-element at the resonant frequency of the auxiliary superconducting nonlinear asymmetric induction element and in phase with the pump microwave field, so as to make the data qubit and the auxiliary qubit interact with each other, so that the z component of the state of the auxiliary qubit depends on the σ of the data qubit. Z .

65. The method of claim 61, wherein: The data qubit comprises a cat state in a linear oscillator; The property of the data qubit is the photon number parity of the cat state; and The method also includes: applying a driving microwave field to the cat state in the linear oscillator at the resonant frequency of the auxiliary superconducting nonlinear asymmetric inductive element and in phase with the pump microwave field to cause the data quantum bit and the auxiliary quantum bit to interact so that the z component of the state of the auxiliary quantum bit depends on the photon number parity of the data quantum bit.

66. The method of claim 61, wherein: The data qubit comprises a Gottesman-Kitayev-Preskill state in a linear oscillator; The property of the data qubit is a stabilizer of the Gottesmann-Kitayev-Preskill state; and The method further includes applying a driving microwave field to the auxiliary superconducting nonlinear asymmetric induction element at a driving frequency equal to a difference between a resonant frequency of the auxiliary superconducting nonlinear asymmetric induction element and a resonant frequency of the linear oscillator, so as to cause the data qubit and the auxiliary qubit to interact with each other, so that a z-component of a state of the auxiliary qubit depends on a stabilizer of the Gottesmann-Kitayev-Preskill state of the data qubit.

67. The method of claim 66, wherein: applying the drive microwave field to the Gottesman-Kitayev-Preskill state is accomplished in phase with the pump microwave field; and The stabilizer of the Gottesmann-Kitayev-Preskill state of the data qubit is S q Stabilizer.

68. The method of claim 66, wherein: Applying the drive microwave field to the Gottesman-Kitayev-Preskill state is accomplished 90° out of phase with the pump microwave field; and The stabilizer of the Gottesmann-Kitayev-Preskill state of the data qubit is S p Stabilizer.

69. The method of claim 61, wherein: The data qubit includes a cat state in a data superconducting nonlinear asymmetric sensing element; The property of the data quantum bit is the σ of the data superconducting nonlinear asymmetric induction element. Z ; and The method further includes applying a plurality of microwave fields to the data qubit and the auxiliary qubit during an interaction duration to cause the data qubit and the auxiliary qubit to interact with each other so that the z component of the state of the auxiliary qubit depends on the σ of the data qubit. Z , wherein applying the plurality of microwave fields comprises: Apply a frequency 2ω to the auxiliary quantum bit. c The first microwave field, where ω c is the resonant frequency of the auxiliary qubit; A frequency of 2ω is applied to the data qubit. t The second microwave field, where ω t is the resonant frequency of the data qubit; A frequency of 2ω is applied to the data qubit. t -ω c The third microwave field; Applying a frequency ω to the auxiliary quantum bit c A fourth microwave field; and Applying a frequency ω to the data qubit c The fifth microwave field.

70. The method of claim 69, wherein: The amplitude of the first microwave field is constant during the interaction duration; The amplitude of the second microwave field is time-varying during the interaction duration; The amplitude of the third microwave field is time-varying during the interaction duration; The amplitude of the fourth microwave field is time-varying during the interaction duration; and The amplitude of the fifth microwave field is constant during the interaction duration.

71. The method of claim 70, wherein: The phase of the first microwave field decreases linearly over the interaction duration; the phase of the second microwave field being constant at a first phase value during a first portion of the interaction duration and being constant at a second phase value during a second portion of the interaction duration, wherein the first phase value is less than the second phase value; The phase of the third microwave field increases linearly during the interaction duration; The phase of the fourth microwave field decreases linearly during a first portion of the interaction duration and decreases linearly during a second portion of the interaction duration; and The phase of the fifth microwave field decreases linearly over the interaction duration.

72. The method of claim 60, wherein: Allowing the data qubit and the ancillary qubit to interact results in a controlled NOT gate being implemented between the data qubit and the ancillary qubit.

73. The method of claim 51, wherein: The auxiliary quantum bit includes a Kerr nonlinear cavity.

74. The method of claim 73, wherein: The Kerr nonlinearity of the Kerr nonlinear cavity is greater than 1 MHz.

75. The method of claim 74, wherein: The intensity of the parameter driving of the auxiliary quantum bit is greater than the Kerr nonlinearity of the Kerr nonlinear cavity.

76. The method of claim 73, wherein: The Kerr nonlinearity of the Kerr nonlinear cavity is greater than the phase flip rate in the auxiliary quantum bit.

77. The method of claim 73, wherein: The auxiliary quantum bit includes a cat state in the Kerr nonlinear cavity.

78. The method of claim 77, further comprising: Kerr nonlinearity is generated in the Kerr nonlinear cavity using at least one Josephson junction of the auxiliary qubit.

79. The method of claim 78, wherein: The auxiliary qubit includes a transporter in the Kerr nonlinear cavity.

80. The method of claim 78, wherein: The auxiliary qubit includes a superconducting nonlinear asymmetric inductor element in the Kerr nonlinear cavity.

81. The method of claim 77, further comprising: An error syndrome is determined based on a conditional rotation of the cat state.

82. The method of claim 81 further comprising: Only local interactions are performed to achieve the conditional rotation.

83. The method of claim 51, wherein: The error channel of the auxiliary qubit is dominated by phase flip errors rather than bit flip errors.

84. The method of claim 83, wherein: The phase flip error is the result of photon loss.

85. The method of claim 84, wherein: When a phase flip error occurs in the auxiliary qubit, the state of the data qubit is not affected.

86. The method of claim 83, further comprising: A pump microwave field is applied to the auxiliary quantum bit to generate a cat state, so that the bit flip error is suppressed exponentially according to the power of the pump microwave field.

87. The method of claim 51, further comprising: A quantum data gate is implemented between the data qubit and the auxiliary qubit.

88. The method of claim 87, wherein: The quantum data gate is a controlled NOT gate, the data qubit is a target qubit of the controlled NOT gate, and the auxiliary qubit is a control qubit of the controlled NOT gate.

89. The method of claim 88, wherein: The control qubit includes a first cat state, and the target qubit includes a second cat state.

90. The method of claim 89, further comprising: The first cat state is rotated in phase space to realize the controlled NOT gate.

91. The method of claim 88, further comprising: A plurality of microwave fields are applied to the data qubit and the auxiliary qubit within a gate duration to implement the controlled NOT gate.

92. The method of claim 91, further comprising: A frequency of 2ω is applied to the control qubit. c The first microwave field, where ω c is the resonant frequency of the control qubit; Applying a frequency 2ω to the target quantum bit t The second microwave field, where ω t is the resonant frequency of the target qubit; Apply a frequency 2ω to the target quantum bit. t -ω c The third microwave field; Applying a frequency ω to the control qubit c A fourth microwave field; and Applying a frequency ω to the target quantum bit c The fifth microwave field.

93. The method of claim 92, wherein: The amplitude of the first microwave field is constant during the gate duration; The amplitude of the second microwave field is time-varying during the gate duration; The amplitude of the third microwave field is time-varying during the gate duration; The amplitude of the fourth microwave field is time-varying during the gate duration; and The amplitude of the fifth microwave field is constant during the gate duration.

94. The method of claim 93, wherein: The phase of the first microwave field decreases linearly during the gate duration; a phase of the second microwave field being constant at a first phase value during a first portion of the gate duration and being constant at a second phase value during a second portion of the gate duration, wherein the first phase value is less than the second phase value; The phase of the third microwave field increases linearly during the gate duration; The phase of the fourth microwave field decreases linearly during a first portion of the gate duration and decreases linearly during a second portion of the gate duration; and The phase of the fifth microwave field decreases linearly within the gate duration.

95. The method of claim 51, wherein: The stable pump microwave field stabilizes the state of the auxiliary qubit to prevent rotation about at least one axis of the auxiliary Bloch sphere.

96. The method of claim 95, further comprising: A readout cavity is coupled to the auxiliary qubit.

97. The method of claim 96, further comprising: The readout cavity is conditionally shifted based on a state of the auxiliary qubit.

98. The method of claim 97, wherein: Conditionally shifting the read cavity comprises: applying at least one rotating microwave field to rotate the state of the auxiliary qubit about an axis of the Bloch sphere; Turning off the stable pump microwave field for a certain amount of time to allow the state of the auxiliary qubit to evolve based on the Kerr nonlinearity of the auxiliary qubit; reapplying the stable pump microwave field to re-stabilize the state of the auxiliary quantum bit; and A switching microwave field is applied to the auxiliary qubit to switch on the exchange coupling between the auxiliary qubit and the readout cavity.

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