Magic state factory construction for generating CCZ and T states

Abstract

Methods, systems, and apparatus for generating a CCZ state and a T state. In one aspect, a method for purifying a CCZ state includes preparing a plurality of target qubits, auxiliary qubits, and stabilizer qubits in a zero state, performing, for each stabilizer qubit, an X-gate on the plurality of auxiliary qubits or one of the plurality of auxiliary qubits and the target qubits using the stabilizer qubits as control, measuring stabilizer qubits, executing a Z1 / 4 gate and a Hadamard gate on each auxiliary qubit, measuring each auxiliary qubit, executing a non-operation on the selected stabilizer qubit or executing a non-operation on the selected stabilizer qubit and executing a Z gate on one or more corresponding target qubits by taking the state of each measured auxiliary qubit as a condition, a Z-gate is performed on the target qubits and an X-gate is performed on each target qubit for each target qubit and on the condition of the measured state of the corresponding stabilizer qubit.

Description

[0001] This application is a divisional application of the invention patent application with an international application date of November 27, 2019, Chinese application number 201980090034.5, and invention name “Magic state factory structure for generating CCZ and T states”. Technical Field

[0002] This specification relates to quantum computing. Background Art

[0003] In order to perform quantum computation, techniques are needed so that a universal set of quantum gates can be implemented fault-tolerantly. Most error-correcting codes allow fault-tolerant implementation of gates from the Clifford group, a set of non-universal gates. By adding Toffoli or π / 8 phase gates to the Clifford group, fully functional quantum computation can be obtained. Executing these gates involves preparing high-fidelity magic states for injecting the gates into the main computation. These magic states are usually required in large quantities, and their preparation requires a large part of the equipment to operate as a dedicated magic state factory. Summary of the invention

[0004] This specification describes magic state factory methods and constructs for distilling CCZ states and T states.

[0005] One innovative aspect of the subject matter described in this specification can be implemented in a method for purifying a CCZ state, the method comprising preparing a qubit register in a zero state, the register comprising a plurality of target qubits, a plurality of auxiliary qubits, and a plurality of stabilizer qubits; for each stabilizer qubit, performing an X gate on one of i) the plurality of auxiliary qubits or ii) the plurality of auxiliary qubits and the target qubit using the stabilizer qubit as a control; measuring the stabilizer qubit to determine a corresponding stabilizer qubit state; performing a Z^(1 / 4) quantum gate and a Hadamard quantum gate on each auxiliary qubit; measuring each auxiliary qubit to determine a corresponding auxiliary qubit state; conditioned on each determined auxiliary qubit state, i) performing a NOT operation on a selected stabilizer qubit, or ii) performing a NOT operation on a selected stabilizer qubit and performing a Z gate on one or more corresponding target qubits; for each target qubit and conditioned on the determined state of the corresponding stabilizer qubit, performing a Z gate on the target qubit; and performing an X gate on each target qubit to obtain a CCZ state.

[0006] Other embodiments of this aspect include corresponding devices, computer systems, means, or computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more classical and / or quantum computers may be configured to perform specific operations or actions by having software, firmware, hardware, or a combination thereof installed on the system that causes the system to perform actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that cause the device to perform actions when executed by a data processing device. For example, a quantum computing device includes: a qubit register, the qubit register including a plurality of target qubits, a plurality of auxiliary qubits, and a plurality of stabilizer qubits, each qubit being prepared in a corresponding initial state; a plurality of control lines coupled to the qubit register; a plurality of control devices coupled to the plurality of control lines; and one or more classical processors may be configured to perform the actions of the method.

[0007] The foregoing and other embodiments can each optionally include one or more of the following features, alone or in combination. In some embodiments, the plurality of target qubits includes three target qubits, the plurality of auxiliary qubits includes eight auxiliary qubits, and the plurality of stabilizer qubits includes four stabilizer qubits.

[0008] In some embodiments, for each stabilizer qubit, performing an X-gate on one of i) a plurality of ancillary qubits or ii) a plurality of ancillary qubits and a target qubit using the stabilizer qubit as a control and measuring the stabilizer qubit to determine a corresponding stabilizer qubit state comprises: performing a sequence of operations from a first set of operations, the first set of operations comprising: performing an X-gate on a first target qubit using a first stabilizer qubit as a control; performing an X-gate on a first, second, third, and fourth ancillary qubit using the first stabilizer qubit as a control; performing an X-gate on a first, second, third, and fourth ancillary qubit using a second ... The method further comprises performing an X gate on the first, second, fifth, and sixth ancillary qubits; performing an X gate on the third target qubit using the third stabilizer qubit as control; performing an X gate on the first, second, fifth, and seventh ancillary qubits using the third stabilizer qubit as control; performing an X gate on the second target qubit using the fourth stabilizer qubit as control; performing an X gate on the first, second, fifth, and sixth ancillary qubits using the fourth stabilizer qubit as control; measuring the first stabilizer qubit; measuring the second stabilizer qubit; measuring the third stabilizer qubit; and measuring the fourth stabilizer qubit.

[0009] In some embodiments, for each auxiliary qubit and conditioned on the determined state of the corresponding auxiliary qubit, i) performing a NOT operation on the second stabilizer qubit or ii) performing a NOT operation on the second stabilizer qubit and performing a Z-gate on the one or more target qubits comprises: performing a sequence of operations from a second set of operations comprising: performing a Z-gate on each of the three target qubits and performing a NOT operation on the second stabilizer qubit, conditioned on the state of the first auxiliary qubit; performing a Z-gate on the first and second target qubits and performing a NOT operation on the second stabilizer qubit, conditioned on the state of the second auxiliary qubit; performing a Z-gate on the first and third target qubits and performing a NOT operation on the second stabilizer qubit, conditioned on the state of the third auxiliary qubit; performing a Z-gate on the first and third target qubits and performing a Z-gate on the first and third target qubits .... performing a Z gate on the third target qubit and performing a NOT operation on the second stabilizer qubit; conditioned on the state of the fourth ancillary qubit, performing a Z gate on the first target qubit and performing a NOT operation on the second stabilizer qubit; conditioned on the state of the fifth ancillary qubit, performing a Z gate on the second and third target qubits and performing a NOT operation on the second stabilizer qubit; conditioned on the state of the sixth ancillary qubit, performing a Z gate on the second target qubit and performing a NOT operation on the second stabilizer qubit; conditioned on the state of the seventh ancillary qubit, performing a Z gate on the third target qubit and performing a NOT operation on the second stabilizer qubit; and conditioned on the state of the eighth ancillary qubit, performing a NOT operation on the second stabilizer qubit.

[0010] In some embodiments, for each target qubit and conditioned on the determined state of the stabilizer qubit, performing a Z-gate on the target qubit includes: performing a Z-gate on the first target qubit conditioned on the state of the first stabilizer qubit; performing a Z-gate on the third target qubit conditioned on the state of the third stabilizer qubit; and performing a Z-gate on the second target qubit conditioned on the state of the fourth stabilizer qubit.

[0011] In some embodiments, for each stabilizer qubit, performing an X-gate on one of i) the plurality of auxiliary qubits or ii) the plurality of auxiliary qubits and a target qubit using the stabilizer qubit as a control comprises: using the stabilizer qubit as an X-axis control by applying a Hadamard gate to the stabilizer qubit before and after using the stabilizer qubit as a control.

[0012] In some embodiments, measuring the stabilizer qubit includes performing a Pauli product measurement on the stabilizer qubit.

[0013] In some embodiments, the method further includes performing a post-select operation, including: determining whether an error occurs using a parity calculation; and in response to determining that an error occurs, discarding the obtained CCZ state.

[0014] In some embodiments, parity calculations are performed on selected stabilizer qubits.

[0015] In some embodiments, using post-select operation includes implementing classical control software.

[0016] In some embodiments, the method is accomplished using lattice surgery techniques.

[0017] In some embodiments, the method is implemented to purify a first CCZ state and the method further includes simultaneously purifying a second CCZ state by partially overlapping execution of operations corresponding to the purifying of the first CCZ state and the purifying of the second CCZ state.

[0018] In some implementations, execution of operations corresponding to purification of the first CCZ state and purification of the second CCZ state are reordered before overlapping.

[0019] In some embodiments, the method further comprises providing the CCZ state for subsequent calculations.

[0020] Another innovative aspect of the subject matter described in this specification can be implemented in a method for transforming a CCZ state into three output T states, the method comprising: obtaining i) three target qubits prepared in a CCZ state, ii) a first auxiliary qubit in a T state, iii) a plurality of additional auxiliary qubits prepared in a zero state, and iv) a plurality of stabilizer qubits prepared in a zero state; performing an X-gate on the target qubits and the additional auxiliary qubits using the first stabilizer qubit as a control and measuring the first stabilizer qubit to determine the state of the first stabilizer qubit; performing an X-gate on the plurality of target qubits and the additional auxiliary qubits using the second stabilizer qubit as a control and measuring the second stabilizer qubit to determine the state of the second stabilizer qubit; performing an X-gate on the additional auxiliary qubits 1 / 2 performing a Z gate on the plurality of target qubits and the first auxiliary qubit using the third stabilizer qubit as a control; performing a Hadamard gate on the first auxiliary qubit and measuring the first auxiliary qubit, the additional auxiliary qubit, and the third stabilizer qubit to determine the corresponding qubit states; performing an X gate on the additional auxiliary qubit 1 / 2gate; performing a Z gate on the target qubit and the additional auxiliary qubit using a fourth stabilizer qubit as a control; performing a Hadamard gate on the additional auxiliary qubit and measuring the additional auxiliary qubit and the fourth stabilizer qubit to determine corresponding qubit states; performing a NOT operation on the stabilizer qubit using the additional auxiliary qubit as a control, and performing two NOT operations on the additional auxiliary qubit and the stabilizer qubit using different stabilizer qubits as controls; performing a Z gate on a plurality of target qubits and the additional auxiliary qubit using two stabilizer qubits as controls; performing a Hadamard gate on the additional auxiliary qubit and measuring the additional auxiliary qubit and the stabilizer qubit to determine corresponding qubit states; and performing a plurality of classically controlled Pauli operators on the three target qubits conditioned on the determined states of the auxiliary qubits.

[0021] Other embodiments of this aspect include corresponding devices, computer systems, means, or computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more classical and / or quantum computers may be configured to perform specific operations or actions by having software, firmware, hardware, or a combination thereof installed on the system that causes the system to perform actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that cause the device to perform actions when executed by a data processing device. For example, a quantum computing device includes: a qubit register, the qubit register including a plurality of target qubits, a plurality of auxiliary qubits, and a plurality of stabilizer qubits, each qubit being prepared in a corresponding initial state; a plurality of control lines coupled to the qubit register; a plurality of control devices coupled to the plurality of control lines; and one or more classical processors may be configured to perform the actions of the method.

[0022] The foregoing and other embodiments can each optionally include one or more of the following features, alone or in combination. In some embodiments, the three target qubits include first, second, and third target qubits, the plurality of additional auxiliary qubits include second, third, and fourth auxiliary qubits, and the plurality of stabilizer qubits include first, second, third, fourth, and fifth stabilizer qubits.

[0023] In some embodiments, performing an X-gate on the target qubit and the additional ancillary qubit using the first stabilizer qubit as a control includes: performing an X-gate on the third target qubit using the first stabilizer qubit as a control; and performing an X-gate on the third ancillary qubit using the first stabilizer qubit as a control.

[0024] In some embodiments, X is performed on the additional auxiliary qubits. 1 / 2The gate consists of performing X on the third auxiliary qubit 1 / 2 Door.

[0025] In some embodiments, performing an X-gate on the plurality of target qubits and the additional ancillary qubit using the second stabilizer qubit as a control includes: performing an X-gate on the first and second target qubits using the second stabilizer qubit as a control; and performing an X-gate on a fourth ancillary qubit using the second stabilizer qubit as a control.

[0026] In some embodiments, performing a Z-gate on the plurality of target qubits and the ancillary qubit using the third stabilizer qubit as a control includes performing a Z-gate on the plurality of target qubits and the first ancillary qubit using the third stabilizer qubit as a control.

[0027] In some embodiments, performing a Z-gate on the plurality of target qubits and the first auxiliary qubit using the third stabilizer qubit as an X-axis control includes: performing a Z-gate on each of the three target qubits using the third stabilizer qubit as a control; and performing a Z-gate on the first auxiliary qubit using the third stabilizer qubit as a control.

[0028] In some embodiments, measuring the first ancillary qubit, the additional ancillary qubit, and the stabilizer qubit to determine corresponding qubit states includes measuring the first ancillary qubit, the third ancillary qubit, and the third stabilizer qubit to determine corresponding qubit states.

[0029] In some embodiments, X is performed on the additional auxiliary qubits. 1 / 2 The gate consists of performing X on the second auxiliary qubit 1 / 2 Door.

[0030] In some embodiments, performing a Z-gate on the target qubit and the additional ancillary qubit using the fourth stabilizer qubit as a control includes: performing a Z-gate on the third target qubit using the fourth stabilizer qubit as a control; and performing a Z-gate on the fourth ancillary qubit using the fourth stabilizer qubit as a control.

[0031] In some embodiments, performing a Hadamard gate on the additional auxiliary qubit and measuring the additional auxiliary qubit includes: performing a Hadamard gate on a fourth auxiliary qubit; and measuring the fourth auxiliary qubit and a fourth stabilizer qubit.

[0032] In some embodiments, performing a NOT operation on the stabilizer qubit using the additional auxiliary qubit as a control and performing two NOT operations on the additional auxiliary qubit and the stabilizer qubit using different stabilizer qubits as controls includes: performing a NOT operation on the third stabilizer qubit using the third auxiliary qubit as a control; performing a NOT operation on the third auxiliary qubit using the first stabilizer qubit as a control; and performing a NOT operation on the third stabilizer qubit using the first stabilizer qubit as a control.

[0033] In some embodiments, performing a Z-gate on the plurality of target qubits and the additional ancillary qubits using the two stabilizer qubits as controls includes performing a Z-gate on each of the three target qubits and the second ancillary qubit using the third and fifth stabilizer qubits as controls.

[0034] In some embodiments, performing a Hadamard gate on the additional auxiliary qubit includes performing a Hadamard gate on the second auxiliary qubit.

[0035] In some embodiments, measuring the additional ancillary qubits and the stabilizer qubit includes measuring a second ancillary qubit and a fifth stabilizer qubit.

[0036] In some embodiments, multiple classically controlled Pauli operators are performed on three target qubits conditioned on the determined states of the auxiliary qubits to reduce decoherence.

[0037] In some embodiments, performing a plurality of classically controlled Pauli operators on three target qubits conditioned on the determined states of the ancillary qubits includes: performing a Z-gate on one or more of the three target qubits using i) at least one of the one or more stabilizer qubits and ii) at least one of the one or more ancillary qubits as controls; performing an X-gate on one or more of the three target qubits using the stabilizer qubit or the ancillary qubit as controls; and performing an X-gate on the target qubits.

[0038] In some embodiments, performing a Z-gate on a target qubit using as control at least one of i) one or more stabilizer qubits and ii) one or more ancillary qubits comprises: performing a sequence of operations from a third set of operations, the third set of operations comprising: performing a Z-gate on a first, second, and third target qubit using the third and fifth stabilizer qubits as control; performing a Z-gate on a first, second, and third target qubit using the third ancillary qubit and the third stabilizer qubit as control; performing a Z-gate on a first, second, and third target qubit using the second ancillary qubit and the third stabilizer qubit as control; performing a Z-gate on a third target qubit using the second stabilizer qubit as control; performing a Z-gate on a third target qubit using the fourth ancillary qubit as control; and performing a Z-gate on a first, second, and third target qubit using the first ancillary qubit as control.

[0039] In some embodiments, performing an X-gate on one or more of the three target qubits using a stabilizer qubit or an ancillary qubit as a control includes: performing an X-gate on the first and second target qubits using a fourth stabilizer qubit as a control; and performing an X-gate on the first, second, and third target qubits using a third ancillary qubit as a control.

[0040] In some embodiments, performing an X-gate on the target qubit includes performing an X-gate on a third target qubit.

[0041] In some embodiments, using the stabilizer qubit as a control includes using the stabilizer as an X-axis control by applying a Hadamard gate to the qubit before and after using the qubit as a control.

[0042] In some embodiments, the obtained first auxiliary qubit in the T state includes a catalyst T-state output from a previous transition of the CCZ state to three T states.

[0043] In some implementations, the method further includes providing two of the three output T-states for subsequent calculations.

[0044] In some embodiments, the method also includes determining whether the third output T state has accumulated an amount of error greater than a predetermined acceptable threshold; in response to determining that the third output T state has accumulated an amount of error less than a predetermined acceptable threshold, providing the third output T state as a catalyst T state for a subsequent transition to the CCZ state; in response to determining that the third output T state has accumulated an amount of error greater than or equal to a predetermined acceptable threshold, providing the stored T state for a subsequent transition to the CCZ state.

[0045] In some embodiments, the obtained CCZ state is purified using the above-described method for purifying a CCZ state.

[0046] In some implementations, the three output T-states are less noisy than the input T-states.

[0047] In some implementations, different groups of operations are performed in parallel.

[0048] Another innovative aspect of the subject matter described in this specification can be implemented in a method of transforming a CCZ state into three T states, the method comprising: obtaining a first target qubit, a second target qubit, and a third target qubit in a CCZ state; performing X on the third target qubit -1 / 2 gate; use the third target qubit as control to perform an X gate on the first target qubit and the second target qubit; use the third qubit as the X-axis control to perform a Z gate on the first target qubit and the second target qubit; perform a Z gate on the third target qubit -1 / 4 and performing a Z gate on the first target qubit and the second target qubit using the third qubit as an X-axis control to obtain three T states. The method may also include providing one of the three T states obtained as a catalyst for subsequent transition of the CCZ state to the three T states.

[0049] Other embodiments of this aspect include corresponding devices, computer systems, means, or computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more classical and / or quantum computers may be configured to perform specific operations or actions by having software, firmware, hardware, or a combination thereof installed on the system that causes the system to perform actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that cause the device to perform actions when executed by a data processing device. For example, a quantum computing device includes: a qubit register, the qubit register including a plurality of target qubits, a plurality of auxiliary qubits, and a plurality of stabilizer qubits, each qubit being prepared in a corresponding initial state; a plurality of control lines coupled to the qubit register; a plurality of control devices coupled to the plurality of control lines; and one or more classical processors may be configured to perform the actions of the method.

[0050] The disclosed subject matter can be implemented in a specific manner to realize one or more of the following advantages.

[0051] In fault-tolerant quantum computing based on surface codes—a possible component of future error-corrected quantum computers due to the relatively high threshold and planar connectivity requirements of surface codes—the cost of a quantum algorithm can be well approximated by the number of non-Clifford operations included in the algorithm. This is because non-Clifford operations are performed via magic state purification, and state purification is expensive. For example, the spacetime volume (qubit-seconds) of some existing T-state factories is two orders of magnitude larger than the volume of CNOT operations between adjacent qubits. Non-Clifford operation counts will likely be particularly important for the earliest error-corrected quantum computers, which may not have enough space to purify magic states in parallel.

[0052] The presently described method and factory configuration for generating CCZ states (referred to herein as CCZ state factory) has a footprint of 12d×6d, where d represents the surface code distance, generates one CCZ state every 5.5d surface code period, and can generate 10 CCZ states before an error occurs. 13 states (on average, assuming a physical gate error rate of 10 -3 The CCZ state factory described herein can speed up algorithms that are affected by the cost of applying Tofelice gates (such as some quantum chemistry algorithms) by a factor of four compared to using known T-state factories (e.g., 12d×8d×6.5d T-state factories). Furthermore, reducing the footprint of the methods and factory constructions used to generate CCZ states can make them more suitable for implementation on space-constrained early quantum computers.

[0053] The currently described methods and plant configurations for producing T-states (referred to herein as T-state plants) use CCZ states (e.g., the CCZ states of the output of the currently described CCZ state plants) and catalyst T-states to precisely transform a single CCZ state into two T-states. The T-state plant has a footprint 25% smaller than other known T-plants and outputs T-states twice as fast. The currently described CCZ state plants and T-state plants can be combined to produce highly efficient T-state plants.

[0054] The generalization of the catalytic circuit to arbitrary phase angles is also described. The generalization can be used to generate states - for example, five T states can be used to generate two state.

[0055] The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description and the drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 An example quantum computing system is shown.

[0057] Figure 2 is a flow chart of an example process for purifying the CCZ state.

[0058] Figure 3 is a diagram of an example quantum circuit used to purify the CCZ state.

[0059] Figure 4 An example time slice of lattice surgery activity when applying an example process for purifying a CCZ state to purify one CCZ state is shown.

[0060] Figure 5 An example time slice of lattice surgery activity is shown when applying an example process for purifying CCZ states to purify two CCZ states.

[0061] Figure 6 is a flow chart of a first example process for transitioning a CCZ state to three T states.

[0062] Figure 7 is a diagram of a first example quantum circuit for generating three T states using CCZ states.

[0063] Figure 8 is a flow chart of a second example process for generating three T-states using a CCZ state.

[0064] Fig. 9 is a diagram of a second example quantum circuit for generating three T states using CCZ states.

[0065] Fig.10 Shows the use of Fig. 9 An example time slice of lattice surgery activity during an example quantum circuit producing three T states.

[0066] Fig.11 is used to use the phased plus state Z θ The third qubit under |+> performs the target phase operation Z on the first and second qubits θ Figure 2. Example of a generalized phase catalytic circuit.

[0067] Fig.12 is intended for use in The third qubit in the state performs the operation on the first qubit prepared in the first input state and the second qubit prepared in the second input state. Diagram of an example quantum circuit of gates.

[0068] Fig.13An example time slice of lattice surgery activities during parallel operation of a T-catalyst plant and a CCZ plant is shown.

[0069] Fig.14 Diagram of the spatial layout and data flow of the known T-factory configuration, the presently described CCZ-factory configuration, and the presently described T-catalyzed T-factory. DETAILED DESCRIPTION

[0070] This specification describes a method for generating a T state with improved footprint and space-time volume. and novel magic state factory construction and state purification techniques for CCZ states (defined as the output of a CCZ gate applied to an input state |++0>).

[0071] Example Hardware

[0072] Figure 1 An example quantum computing system 100 is shown. System 100 is an example of a system implemented as a quantum and / or classical computer program on one or more quantum computing devices and / or classical computers located at one or more locations in which the systems, components, and techniques described below may be implemented.

[0073] System 100 includes a quantum computing device 102 in data communication with one or more classical processors 104. For convenience, quantum computing device 102 and classical processors 104 are shown as separate entities, however, in some implementations, one or more classical processors may be included in quantum computing device 102.

[0074] The quantum computing device 102 includes components for performing quantum computing. For example, the quantum computing device 102 includes at least a quantum circuit 106 and a control device 108.

[0075] The quantum circuit 106 includes components for performing quantum computing, such as components for implementing various quantum circuits and operations described in this specification. For example, a quantum circuit may include a quantum system that includes one or more multi-level quantum subsystems, such as a qubit register 114. A qubit is a physical qubit that can be used to perform an algorithmic operation or quantum computing. The specific implementation of one or more qubits and their interactions may depend on a variety of factors, including the type of quantum computing to be performed by the quantum computing device 102. For example, a qubit may include a qubit implemented by an atomic, molecular, or solid-state quantum system. In other examples, a qubit may include, but is not limited to, a superconducting qubit, such as a Gmon or Xmon qubit; or a semiconductor qubit. In some cases, it may be convenient to attach one or more resonators to one or more superconducting qubits. In other cases, an ion trap, a photonic device, or a superconducting cavity may be used (in which case a state can be prepared without the need for a qubit). Other examples of implementing a multi-level quantum subsystem include fluxmon qubits, silicon quantum dots, or phosphorus impurity qubits.

[0076] Quantum circuits including different quantum logic operations (e.g., the single qubit gate, two qubit gate, and three qubit gate described in this specification) can be constructed using quantum circuit 106. The constructed quantum circuit can be operated / implemented using control device 108.

[0077] The type of control device 108 included in the quantum system depends on the type of qubit included in the quantum computing device. For example, in some cases, multiple qubits may be frequency-adjustable. That is, each qubit may have an associated operating frequency that can be adjusted using one or more control devices. Example operating frequencies include qubit idle frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that qubits can perform. For example, setting the operating frequency to a corresponding idle frequency can place the qubit in a state that does not strongly interact with other qubits and can be used to perform a single-qubit gate. In these examples, the control device 108 may include a device for controlling the frequency of the qubits included in the quantum circuit 106, an excitation pulse generator, and a control line that couples the qubit to the excitation pulse generator. The control device can then adjust the frequency of each qubit toward or away from the quantum gate frequency of the excitation pulse on the corresponding control drive line.

[0078] The control device 108 may also include a measurement device, such as a readout resonator. The measurement results obtained by the measurement device may be provided to the classical processor 104 for processing and analysis. The measurement device directly or indirectly performs physical measurements on the properties of the qubit, from which the state(s) of the qubit may be inferred.

[0079] Quantum circuit 106 may also include one or more magic state factories, such as CCZ state factory 112 and T state factory 110. CCZ state factory 112 generates CCZ states that can be consumed by quantum circuit 106 when performing quantum computations. For example, CCZ state factory 112 may be Factory, which is based on the following reference Figure 2-5 The technology described generates the CCZ state. (In this specification, the factory refers to the use of the symbol The left side |In> represents the state of the input factory, the right side |Out> represents the state of the factory output, and the function f(∈) above the arrow represents the amount of error suppression to the leading term, that is, f(∈) is shorthand for the true suppression f(∈)+O(∈f(∈)). For example, T-state purification based on the known 15-qubit Reed-Muller code is called factory.)

[0080] The T-state factory 110 generates T-states that can be consumed by the quantum circuit 106 when performing quantum computations. For example, the T-state factory 110 can be based on the following reference Figure 6-10 The described technique generates a T-state |T>-catalyzed |CCZ>→2|T> factory.

[0081] The CCZ state factory 112 and the T state factory 110 can also be combined to produce The factory can refer to the following Figure 2-13 The described technique generates a T-state.

[0082] Operating hardware: Example method implemented by the factory

[0083] Figure 2 is a flow chart of an example process for purifying a CCZ state. For convenience, process 200 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to the present specification, such as Figure 1 The system 100 may perform the process 200. In addition, for convenience, the operations described below are described with reference to an ordered list of steps, but the order of some operations may be changed, and different groups of operations may be performed in parallel.

[0084] The system prepares a qubit register in a zero state (step 202). In other words, the system prepares each qubit in the qubit register in a zero state. The qubit register includes a plurality of target qubits, a plurality of auxiliary qubits, and a plurality of stabilizer qubits. The target qubit is the qubit that will encode the purified CCZ state. The plurality of auxiliary qubits and the stabilizer qubit are auxiliary qubits (in this specification, the term "stabilizer qubit" is used to distinguish between different groups of qubits in the qubit register and to describe a second plurality of auxiliary qubits). Figure 3 An example qubit register is illustrated and described.

[0085] For each stabilizer qubit, the system performs an X-gate on i) one of the plurality of ancillary qubits, or ii) the plurality of ancillary qubits and a target qubit using the stabilizer qubit as a control (step 204). In some embodiments, the stabilizer qubit can be used as an X-axis control by applying a Hadamard gate to the stabilizer qubit before and after using the stabilizer qubit as a control.

[0086] The system measures the stabilizer qubit to determine the corresponding stabilizer qubit state (step 206). For example, the system can perform a corresponding Pauli product measurement on the stabilizer qubit to determine the corresponding stabilizer qubit state. In some embodiments, whenever an X-axis control is directly preceded by a measurement operation, the measurement operation can be a Pauli product measurement.

[0087] The system performs Z on each auxiliary qubit. 1 / 4 quantum gate, followed by a Hadamard quantum gate (step 208).

[0088] The system measures each ancillary qubit to determine the corresponding ancillary qubit state (step 210).

[0089] The system conditions each determined auxiliary qubit state on either i) performing a NOT operation on a selected stabilizer qubit, or ii) performing a NOT operation on the selected stabilizer qubit and performing a Z-gate on one or more corresponding target qubits (step 212).

[0090] The system performs a Z-gate on the target qubit for each target qubit and conditioned on the determined state of the corresponding stabilizer qubit (step 214).

[0091] The system performs an X gate on each target qubit to obtain a CCZ state (step 216). The system can provide the obtained CCZ state for subsequent calculations, for example, as input to the example process 800.

[0092] In some embodiments, the system may also perform a classical post-selection operation implemented by classical control software to determine whether to retain or discard the obtained CCZ state. For example, the system may use a parity calculation performed on a selected stabilizer qubit to determine whether an error has occurred. If an error has occurred, the system may discard the obtained CCZ state. If no error has occurred, the system may retain the obtained CCZ state, e.g., as described above, to provide the CCZ state for subsequent calculations.

[0093] In some embodiments, the system may perform the example process 200 twice to purify the first CCZ state and the second CCZ state. In these embodiments, as described below with reference to Figure 5 As shown, the system may perform both processes simultaneously and partially overlap the execution of operations corresponding to purification of the first CCZ state and purification of the second CCZ state (which may require some reordering of the steps of example process 200 prior to execution of the overlapping operations).

[0094] Figure 3 is a diagram of an example quantum circuit 300 for purifying a CCZ state. The example quantum circuit 300 operates on a qubit register, Figure 3 The qubit register includes three target qubits labeled 1, 2, and 3. The qubit register also includes eight auxiliary qubits labeled ah and one qubit labeled X1X. abcd ,X abcdefgh ,X3X aceg ,X2X abef Each qubit in the qubit register is initialized to a zero state 302, as described above with reference to Figure 2 As described in step 202.

[0095] The example quantum circuit 300 includes a first set of operations 304. The first set of operations 304 is related to Figure 2 The operations in the first set of operations 304 include a plurality of X-gates, such as X-gate 306. Each X-gate in the first set of operations 304 is applied to an auxiliary qubit or a target qubit and controlled by a corresponding stabilizer qubit. In the example quantum circuit 300, the control is an X-axis control, such as X-axis control 308, which represents applying a Hadamard gate to a stabilizer qubit before and after the stabilizer qubit is used as a control.

[0096] For example, the X-gate may include a first X-gate applied to the first target qubit 1, a second X-gate applied to the first auxiliary qubit a, a third X-gate applied to the second auxiliary qubit b, a fourth X-gate applied to the third auxiliary qubit c, and a fifth X-gate applied to the fourth auxiliary qubit d. Each of the first, second, third, fourth, and fifth X-gates uses the first stabilizer qubit X1X abcd The X-gate also includes a sixth X-gate applied to the first auxiliary qubit a, a seventh X-gate applied to the second auxiliary qubit b, an eighth X-gate applied to the third auxiliary qubit c, a ninth X-gate applied to the fourth auxiliary qubit d, a tenth X-gate applied to the fifth auxiliary qubit e, an eleventh X-gate applied to the sixth auxiliary qubit f, a twelfth X-gate applied to the seventh auxiliary qubit g, and a thirteenth X-gate applied to the eighth auxiliary qubit h. Each of the sixth to thirteenth X-gates uses the second stabilizer qubit X abcdefgh As control. The X gate also includes a fourteenth X gate applied to the third target qubit 3, a fifteenth X gate applied to the first auxiliary qubit a, a sixteenth X gate applied to the third auxiliary qubit c, a seventeenth X gate applied to the fifth auxiliary qubit e, and an eighteenth X gate applied to the seventh auxiliary qubit g. Each of the fourteenth to eighteenth X gates uses the third stabilizer qubit X3X aceg The X gates also include a nineteenth X gate applied to the second target qubit 2, a twentieth X gate applied to the first auxiliary qubit a, a twenty-first X gate applied to the second auxiliary qubit b, a twenty-second X gate applied to the fifth auxiliary qubit e, and a twenty-third X gate applied to the sixth auxiliary qubit f. Each of the nineteenth to twenty-third X gates uses the fourth stabilizer qubit X2X abef As a control.

[0097] The first set of operations 304 also includes four measurement operations, such as measurement operation 310. Each measurement operation is applied to a corresponding stabilizer qubit. In some embodiments, the measurement operation can be a Pauli product measurement.

[0098] The example quantum circuit 300 also includes a plurality of Z 1 / 4 Gate (T gate), for example, Z 1 / 4 Gate 312, and multiple Hadamard gates, such as Hadamard gate 314. Multiple Z 1 / 4 Gates and multiple Hadamard gates are applied to the auxiliary qubits ah. For example, each auxiliary qubit ah consists of a Z 1 / 4 The example quantum circuit 300 also includes a method for applying Z 1 / 4 The gates and Hadamard gates are then applied to multiple measurement operations on the auxiliary qubit ah.

[0099] The example quantum circuit 300 also includes a second set of operations 316. The second set of operations 316 is related to Figure 2 The operations in the second set of operations 316 include a plurality of Z gates, for example, Z gate 318. Each Z gate in the second set of operations 316 is applied to a target qubit conditioned on the measurement state of the corresponding auxiliary qubit, as shown by a solid circle (e.g., circle 320). For example, the Z gates may include a first Z gate applied to a first target qubit 1, a second Z gate applied to a second target qubit 2, and a third Z gate applied to a third target qubit 3. The first, second, and third Z gates are executed with the measurement state of the first auxiliary qubit a, for example, if the measurement state of the first auxiliary qubit is 1, then executed. The Z gates may also include a fourth Z gate applied to the first target qubit 1 and a fifth Z gate applied to the second target qubit 2. The fourth and fifth Z gates are executed with the measurement state of the second auxiliary qubit b. The Z gates may also include a sixth Z gate applied to the first target qubit 1 and a seventh Z gate applied to the third target qubit 3. The sixth and seventh Z gates are executed with the measurement state of the third auxiliary qubit c. The Z gates also include an eighth Z gate applied to the first target qubit 1. The eighth Z-gate is executed conditioned on the measurement state of the fourth auxiliary qubit d. The Z-gates also include a ninth Z-gate applied to the second target qubit 2 and a tenth Z-gate applied to the third target qubit 3. The ninth and tenth Z-gates are executed conditioned on the measurement state of the fifth auxiliary qubit e. The Z-gates also include an eleventh Z-gate applied to the second target qubit 2. The eleventh Z-gate is executed conditioned on the measurement state of the sixth auxiliary qubit f. The Z-gates also include a twelfth Z-gate applied to the third target qubit 3. The twelfth Z-gate is executed conditioned on the measurement state of the seventh auxiliary qubit g.

[0100] The second set of operations 316 also includes a plurality of non-operations, such as non-operation 322. Each non-operation is performed on the same stabilizer qubit - the second stabilizer qubit X. abcdefgh - performed and conditioned on the measured state of the corresponding ancillary qubit. The total number of NOT operations equals the total number of ancillary qubits. For example, a NOT operation may include performing a NOT operation on a second stabilizer qubit X conditional on the measured state of the first ancillary qubit a. abcdefgh The first negation operation performed, conditional on the measured state of the second auxiliary qubit b, is performed on the second stabilizer qubit x. abcdefgh The second negation operation performed and the stabilizer qubit X conditioned on the measured state of the third auxiliary qubit c abcdefgh The third NOT operation performed is like that.

[0101] Example quantum circuit 300 also includes three Z-gates 322a-c, which are applied to each of target qubits 1-3. Each of the three Z-gates 322a-c is executed conditioned on the measured state of the corresponding stabilizer qubit. For example, Z-gate 322a is conditioned on the measured state of the first stabilizer qubit X1X abcd The Z gate 322b is executed on the first target qubit 1 with the measurement state of the third stabilizer qubit X3X aceg The Z gate 322c is executed on the third target qubit 3 with the measurement state of the fourth stabilizer qubit X2X abef The measurement state of is performed on the second target quantum bit 3 as a condition.

[0102] The example quantum circuit 300 also includes three X-gates, for example, X-gate 324, each of which is applied to a corresponding target qubit 1-3. The three target qubits 1-3 are then output as |CCZ> states.

[0103] The example quantum circuit 300 also includes a post-selection operation 328. The post-selection operation 328 is executed by classical control software and is implemented to determine whether an error is detected during the state purification process. If the post-selection operation indicates that an error is present and thus "failed", the output CCZ state 326 can be discarded.

[0104] The operations included in the example quantum circuit 300 can be converted into lattice surgery operations. Figure 4 Shown in use Figure 3 Example quantum circuit 300 of generates an example time slice 400 of lattice surgery activity during a CCZ state. Example time slice 400 corresponds to one of many possible transformations of example quantum circuit 300 to a lattice surgery operation, and is therefore one of many possible time slices (with matching qubit labels and operation labels). Figure 4 In the figure, the squares correspond to different qubits (identified by the labels inside the squares), while the shaded rectangles correspond to X-stabilizer measurements between groups of qubits. Each arrow-marked T corresponds to a noisy T state entering the system.

[0105] Time slice 400a corresponds to a first column of X-gates included in example quantum circuit 300 (applied to the first target qubit, the first ancillary qubit, the second ancillary qubit, the third ancillary qubit, and the fourth ancillary qubit, controlled by the first stabilizer qubit) and a measurement of the first stabilizer qubit.

[0106] Time slice 400b corresponds to the measurement of the second column of X-gates (applied to each auxiliary qubit ah and controlled by the second stabilizer qubit) and the second stabilizer qubit included in example quantum circuit 300. In time slice 400b, no operation is performed on the first target qubit 1.

[0107] Time slice 400c corresponds to the third column of X gates (applied to auxiliary qubits a, c, e, g, controlled by the third stabilizer qubit) and the measurement of the third stabilizer qubit included in example quantum circuit 300. In time slice 400c, no operations are performed on auxiliary qubits b, d, f, h and target qubit 1.

[0108] Time slice 400d corresponds to the fourth column of X-gates (applied to the second target qubit 2 and auxiliary qubits a, b, e, f, controlled by the fourth stabilizer qubit) and the measurement of the fourth stabilizer qubit included in example quantum circuit 300. Target qubits 1 and 3 are moved, and no other operations are performed on them. Time slice 400d also corresponds to the measurement of the fourth stabilizer qubit in the fourth column of X-gates (applied to the second target qubit 2 and auxiliary qubits a, b, e, f, controlled by the fourth stabilizer qubit). Figure 2 The T-gate (Z^1 / 4 gate), Hadamard gate, and measurement operations applied to auxiliary qubits c, g, d, and h at steps 208 and 210 correspond.

[0109] Time Slice 400e and Figure 2 The remaining T-gates (Z^1 / 4 gates), Hadamard gates, and measurement operations applied to auxiliary qubits a, b, e, and f at steps 208 and 210 of FIG. 400e also correspond to the time slice 400e at step 408 of FIG. Figure 3 The operations performed on target qubits 1 and 3 during second set of operations 316 (e.g., the third, fourth, seventh, and eighth columns in second set of operations 316) correspond.

[0110] Time slice 400f and Figure 3 The remaining operations included in the second group of operations 316 (eg, the first, second, fifth, and sixth columns in the second group of operations 316) correspond to the remaining operations included in the second group of operations 316.

[0111] Time slice 400g corresponds to the remaining operations in example quantum circuit 300 (eg, the last four columns of operations in example quantum circuit 300).

[0112] Time slice 400h corresponds to outputting the CCZ state after verification.

[0113] Figure 5 Shows the use of Example time slice of lattice surgery activity during a factory-generated |CCZ> state and corresponding to two crosslinking transformations of the example quantum circuit 300 for improved utilization.

[0114] Time slice 502 and Figure 4 That is, the first process for purifying the CCZ state is at time slice 400f, at which time the second process for purifying the CCZ state starts and is at time slice 400a. Figure 4The time slices 400g and 400b correspond to each other. The time slice 506 corresponds to Figure 4 The first process is then completed. Time slice 508 corresponds to time slices 400h and 400c. Figure 4 The time slice 400d corresponds to the time slice 510. Figure 4 Time slice 512 corresponds to the start of a third process that is between time slice 400a and time slice 400f for the second process. Time slice 514 corresponds to Figure 4 The time slices 400b and 400g correspond to each other. The time slice 516 corresponds to Figure 4 The second process is then completed. Time slices 518 and 520 correspond to 400d and 400e. The loop can then continue to complete the third process and start the fourth process.

[0115] The currently described process for purifying CCZ states has a naive depth of 4 (stabilizer measurements) + 1.5 (T state injection) + 1 (X or Y basis measurement, depending on the T injection measurement) + 2 (detection errors) = 8.5. However, the process can be performed in a parallel example where the execution of the operations can be partially overlapped, resulting in an effective depth of 5.5.

[0116] The error rate of the CCZ state produced using the currently described procedure can be calculated as follows: It is reasonable to assume that the physical gate error rate is 10 -3 , and a post-selection state injection technique can be used to create a T0 state with approximately this error probability. The known Reed-Muller code-based The factory generates the T1 state, and the error rate is approximately equal to 35(10 -3 ) 3 ≈10 -7 . Then, the currently described factory, resulting in a final error rate of 28(10 -7 ) 2 ≈10 -13 . 10 -13 The error rate is sufficient to run classically intractable algorithms. For example, if we conservatively assume that Shor's algorithm performs 2n 3 Tophilimon, the currently described The factory can support factoring numbers of 10,000 digits with a failure rate of less than 50%.

[0117] Operating hardware: |T>-catalyzed|CCZ>→2| Example method implemented by T> factory

[0118] Figure 6is a flow chart of a first example process 600 for converting a CCZ state into three output T states. For convenience, process 600 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to the present specification, such as Figure 1 The system 100 may perform the process 600. In addition, for convenience, the operations described below are described with reference to an ordered list of steps, but the order of some operations may be changed, and different groups of operations may be performed in parallel.

[0119] The system obtains a first target qubit, a second target qubit, and a third target qubit in a CCZ state (step 602). In some embodiments, the CCZ state may be a CCZ state output using the example process 200 and the example quantum circuit 300.

[0120] The system performs X on the third target qubit -1 / 2 The system performs an X gate on the first target qubit and the second target qubit using the third target qubit as a control (step 606). The system performs a Z gate on the first target qubit and the second target qubit using the third qubit as an X-axis control (step 608). The system performs a Z gate on the third target qubit. -1 / 4 The system performs a Z gate on the first target qubit and the second target qubit using the third qubit as an X-axis control to obtain three T states (step 612).

[0121] Because the |T> state can be used to perform a T-gate, in some embodiments, one of the three T-states obtained can be used as a catalyst for subsequently transforming the CCZ state into three T-states. For example, a T-gate used to transform |CCZ> into three |T> states can be powered by a |T> state output from a previous iteration of circuit 300. If the |T> state output from iteration k is fed into iteration k+1, a circuit that adopts a |CCZ> state and outputs two |T> states is effectively obtained. Under this interpretation of the circuit, the third |T> state is an auxiliary state required to make the transformation possible, but is not consumed by the transformation. Therefore, the third |T> can be referred to as a catalyst. Therefore, circuit 300 as a whole is referred to as |T>-catalyzed 8|T>→2|T>, or simply as a "C2T factory."

[0122] Although the catalyst |T> state is not consumed by the C2T factory, it accumulates noise from incoming |CCZ> states. If a catalyst |T> has cycled through n iterations of the C2T factory, and each |CCZ> contains an error with probability ∈, then there is a probability Θ(N∈) that the catalyst is contaminated and will cause the factory to produce bad output. However, because every error in the catalyst is ultimately traced back to an error in the |CCZ> state, the probability of any error occurring grows as Θ(N∈), rather than Θ(N∈) as expected from a naive calculation assuming uncorrelated errors. 2 ) growth. Purification protocols often require inputs with uncorrelated errors, so in certain embodiments it may be beneficial to include a C2T plant as the last step in the purification chain. This use means that correlation between errors is a benefit rather than a cost.

[0123] Figure 7 7 is a diagram of a first example quantum circuit 700 for generating three T states using a CCZ state. The example quantum circuit 700 operates on a qubit register 702 including three qubits prepared in a CCZ state. The example quantum circuit 700 includes an X applied to the third qubit in the qubit register 702. -1 / 2 Example quantum circuit 700 also includes a first X-gate 706 applied to a first qubit in qubit register 702 and a second X-gate 708 applied to a second qubit in qubit register 702. X-gates 706 and 708 are controlled by a third qubit. Example quantum circuit 700 also includes a first Z-gate 710 applied to the first qubit and a second Z-gate 712 applied to the second qubit. The application of first Z-gate 710 and second Z-gate 712 is controlled by a third qubit (X axis). ... -1 / 4 The example quantum circuit 700 also includes a third Z-gate 716 applied to the first qubit and a fourth Z-gate 718 applied to the second qubit. The application of the third Z-gate 716 and the fourth Z-gate 718 is controlled by the third qubit (X axis). The example quantum circuit 700 outputs the first qubit, the second qubit, and the third qubit in T states 720a-c.

[0124] Z -1 / 4 Application of gate 714 requires a T state, and thus, in some cases, the T state output by the first embodiment of example quantum circuit 700 may be provided as a catalytic input to the second embodiment of example quantum circuit 700 , as indicated by arrow 722 .

[0125] The example quantum circuit 700 is compact, but is not in an ideal form for embedding into a lattice. Figure 8 and Fig. 9 An equivalent circuit that can be converted to lattice surgery is shown. The result of the conversion is Fig.10 It is shown in Fig.10 An example time slice of a lattice surgery operation occurring while the plant is operating is shown.

[0126] Figure 8 is a flow chart of a second example process 800 for generating three T states using CCZ states. For convenience, process 800 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to the present specification, such as Figure 1 The system 100 may perform the process 800. In addition, for convenience, the operations described below are described with reference to an ordered list of steps, but the order of some operations may be changed, and different groups of operations may be performed in parallel.

[0127] The system obtains a qubit register (step 802). The qubit register includes a first target qubit, a second target qubit, and a third target qubit that are prepared in a positive CCZ state. For example, the CCZ state may be Figure 2 The example process 200 of purifying CCZ. The qubit register also includes a first auxiliary qubit prepared in the T state. The qubit register also includes a plurality of additional auxiliary qubits, each of which is prepared in the zero state, for example, a second, third, and fourth auxiliary qubit, each of which is prepared in the zero state. The qubit register also includes a plurality of stabilizer qubits prepared in the zero state, for example, a first, second, third, fourth, and fifth stabilizer qubit, each of which is prepared in the zero state. Figure 7 An example qubit register is illustrated and described.

[0128] The system performs an X-gate on the target qubit and the attached ancillary qubit using the first stabilizer qubit as a control (step 804). For example, continuing the above example, the system may perform an X-gate on the third target qubit using the first stabilizer qubit as a control, and perform an X-gate on the third ancillary qubit using the first stabilizer qubit as a control. In this description, when the stabilizer qubit is used as a control, the control may be an X-axis control that includes applying a Hadamard gate to the stabilizer qubit before and after the stabilizer qubit is used as a control.

[0129] The system then measures the first stabilizer qubit to determine the state of the first stabilizer qubit.

[0130] The system performs an X-gate on the plurality of target qubits and the additional ancillary qubits using the second stabilizer qubit as a control (step 806). The system then measures the second stabilizer qubit to determine the state of the second stabilizer qubit. For example, continuing the example above, the system may: perform an X-gate on the first target qubit using the second stabilizer qubit as a control, perform an X-gate on the second target qubit using the second stabilizer qubit as a control, and perform an X-gate on the fourth ancillary qubit using the second stabilizer qubit as a control. The system then measures the second stabilizer qubit to determine the state of the second stabilizer qubit.

[0131] The system performs X on the attached auxiliary qubit. 1 / 2 Gate (step 808). For example, the system can perform X on the third auxiliary qubit 1 / 2 Door.

[0132] The system performs a Z-gate on each of the plurality of target qubits and the first ancillary qubit using the third stabilizer qubit as a control for each Z-gate (step 810). For example, continuing the example above, the system may perform a Z-gate on the first, second, and third target qubits and the first ancillary qubit using the third stabilizer qubit as a control in each case.

[0133] The system performs a Hadamard gate on the first auxiliary qubit and measures the first auxiliary qubit, the additional auxiliary qubits, and the third stabilizer qubit to determine corresponding qubit states (step 812). For example, after performing the Hadamard gate, the system may measure the first auxiliary qubit, the third auxiliary qubit, and the third stabilizer qubit to determine corresponding states of the first auxiliary qubit, the third auxiliary qubit, and the third stabilizer qubit.

[0134] The system performs X on the attached auxiliary qubit. 1 / 2 Gate (step 814). For example, the system can perform X on the second auxiliary qubit 1 / 2 Door.

[0135] The system performs a Z-gate on the target qubit and the attached ancillary qubits using the fourth stabilizer qubit as a control for each Z-gate (step 816). For example, the system may perform a Z-gate on the third target qubit using the fourth stabilizer qubit as a control and perform a Z-gate on the fourth ancillary qubit using the fourth stabilizer qubit as a control.

[0136] The system performs a Hadamard gate on the additional ancillary qubit and measures the additional ancillary qubit and the fourth stabilizer qubit to determine the corresponding qubit states (step 818). For example, the system may perform a Hadamard gate on the fourth ancillary qubit and measure the fourth ancillary qubit and the fourth stabilizer qubit.

[0137] The system performs one NOT operation on the stabilizer qubit using the additional auxiliary qubit as a control, and performs two NOT operations on the additional auxiliary qubit and the stabilizer qubit using different stabilizer qubits as controls (step 820). For example, the system may perform a NOT operation on the third stabilizer qubit using the third auxiliary qubit as a control, perform a NOT operation on the third stabilizer qubit using the first stabilizer qubit as a control, and perform a NOT operation on the third auxiliary qubit using the first stabilizer qubit as a control.

[0138] The system performs corresponding Z-gates on the plurality of target qubits and the additional ancillary qubits using the two stabilizer qubits as controls (step 822). For example, the system can perform a Z-gate on each of the three target qubits and the second ancillary qubit using the third and fifth stabilizer qubits as controls.

[0139] The system performs a Hadamard gate on the additional auxiliary qubit (step 824). For example, the system may perform a Hadamard operation on the second auxiliary qubit. The system then measures the additional auxiliary qubit (e.g., the second auxiliary qubit) and the fifth stabilizer qubit to determine the corresponding qubit states.

[0140] The system performs a plurality of classically controlled Pauli operators on the three target qubits conditioned on the determined state of the auxiliary qubit (step 826). Performing a plurality of classically controlled Pauli operators on the three target qubits conditioned on the determined state of the auxiliary qubit can reduce decoherence.

[0141] Performing a plurality of classically controlled Pauli operators on three target qubits conditioned on the determined state of the auxiliary qubits includes: performing a plurality of Z-gates on one or more of the target qubits using i) the one or more stabilizer qubits and ii) at least one of the one or more auxiliary qubits as controls. For example, the system may perform a sequence of operations from a set of operations including: applying Z-gates to the first, second, and third target qubits using the third and fifth stabilizer qubits as controls; applying Z-gates to the first, second, and third target qubits using the third auxiliary qubit and the third stabilizer qubit as controls; applying Z-gates to the first, second, and third target qubits using the second auxiliary qubit and the third stabilizer qubit as controls; applying Z-gates to the third target qubit using the second stabilizer qubit as controls; applying Z-gates to the third target qubit using the fourth auxiliary qubit as controls; applying Z-gates to the first, second, and third target qubits using the first auxiliary qubit as controls. The system can then perform an X-gate on one or more of the target qubits using the stabilizer qubit or the ancillary qubit as control, e.g., perform an X-gate on the first and second target qubits using the fourth stabilizer qubit as control and perform an X-gate on the first, second, and third target qubits using the third ancillary qubit as control, and perform an X-gate on a target qubit (e.g., the third target qubit).

[0142] The system can provide three target states, all of which are T states after step 826, for subsequent calculations. For example, in some embodiments, the system can provide one or more of the T states as input to a subsequent implementation of the example process 800. The remaining T states can then be used in some subsequent calculations performed by the quantum computing device. In some cases, if the output T state does not accumulate an amount of error above a predetermined acceptable threshold, the system can only provide the output T state for subsequent calculations or as input to a subsequent implementation of the example process 800. If the output T state accumulates an amount of error above a predetermined acceptable threshold, the system can discard the state and / or provide a different T state, such as a stored T state, to a subsequent calculation / implementation.

[0143] Fig. 9is a diagram of a second example quantum circuit 900 for generating three T states using CCZ states. Example quantum circuit 900 operates on a qubit register 902, where each qubit is represented by a corresponding horizontal line. Qubit register 902 includes three target qubits labeled 1, 2, and 3. Target qubits 1, 2, and 3 are prepared in the CCZ state. Qubit register 902 also includes a first auxiliary qubit labeled T that is prepared in the T state. Qubit register 902 also includes a second, third, and fourth auxiliary qubits labeled S, B, and A, respectively. The second, third, and fourth auxiliary qubits are each prepared in the zero state. Qubit register 902 also includes a second auxiliary qubit labeled X3X B ,X 12 X A ,Z 123 Z T ,Z3Z A ,Z 123 Z S Each of the five stabilizer qubits is prepared in the zero state.

[0144] The example quantum circuit 900 includes a plurality of Pauli product measurements 904-912. The first Pauli product measurement 904 is Figure 8 The first Pauli product measurement 904 includes a first X-gate (e.g., X-gate 918) applied to the third target qubit and a second X-gate applied to the third auxiliary qubit B. The first stabilizer qubit X3X B Serves as the X-axis control for the two X-gates included in the first Pauli product measurement 904. The first Pauli product measurement 904 also includes an X-axis control applied to the first stabilizer qubit X3X B measurement operation.

[0145] The second Pauli product measurement 906 and Figure 8 The second Pauli product measurement 906 includes a first X-gate applied to the first target qubit, a second X-gate applied to the second target qubit, and a third X-gate applied to the fourth auxiliary qubit A. The second stabilizer qubit X 12 X A Serves as an X-axis control for each X-gate included in the second Pauli product measurement 906. The second Pauli product measurement 906 also includes an X-axis control applied to the second stabilizer qubit X 12 X A measurement operation.

[0146] After the second Pauli product measurement 906, X 1 / 2 Gate 922 is applied to the third auxiliary qubit B. 1 / 2 Door 922 and Figure 8 Corresponding to step 808.

[0147] The third Pauli product measurement 908 and Figure 8 The third Pauli product measurement 908 includes a first Z-gate (e.g., Z-gate 924) applied to the first target qubit, a second Z-gate applied to the second target qubit, a third Z-gate applied to the third target qubit, and a fourth Z-gate applied to the first auxiliary qubit. The third stabilizer qubit Z 123 Z T Serves as the X-axis control for each Z gate included in the third Pauli product measurement 908. The third Pauli product measurement 908 also includes a Hadamard gate 926 and a measurement operation applied to the first auxiliary qubit T. The third Pauli product measurement 908 also includes a measurement operation applied to the third auxiliary qubit B and a measurement operation applied to the third stabilizer qubit Z. 123 Z T measurement operation.

[0148] After the third Pauli product measurement 908, X 1 / 2 Gate 928 is applied to the second auxiliary qubit S. 1 / 2 Door 928 and Figure 8 Corresponding to step 814.

[0149] The fourth Pauli product measurement 910 and Figure 8 The fourth Pauli product measurement 910 includes a first Z-gate applied to the third target qubit 3 and a second Z-gate applied to the fourth auxiliary qubit. A Serves as the X-axis control for each Z gate included in the fourth Pauli product measurement 910. The fourth Pauli product measurement 910 also includes a Hadamard gate and measurement operation applied to the fourth auxiliary qubit A. The fourth Pauli product measurement 910 also includes an operation applied to the fourth stabilizer qubit Z3Z A measurement operation.

[0150] After the fourth Pauli product measurement 910, the third stabilizer qubit Z is measured using the third auxiliary qubit B as control. 123 Z T A first NOT operation is performed, for example, NOT operation 930. Using the first stabilizer qubit X3X B As a control, the third stabilizer qubit Z 123 Z T Perform the second NOT operation and use the first stabilizer qubit X3X B As a control, a third NOT operation is performed on the third auxiliary qubit B. The first and second NOT operations are applied in the same manner as Figure 8 Corresponding to step 820.

[0151] The fifth Pauli product measurement 912 and Figure 8The fifth Pauli product measurement 912 includes a first Z-gate applied to the first target qubit 1, a second Z-gate applied to the second target qubit 2, a third Z-gate applied to the third target qubit 3, and a fourth Z-gate applied to the second auxiliary qubit S. The third stabilizer qubit Z 123 Z T and the fifth stabilizer qubit Z 123 Z S Serves as a control for each Z gate included in the fifth Pauli product measurement 912. The fifth Pauli product measurement 912 also includes a Hadamard gate applied to the second auxiliary qubit S, a measurement operation applied to the second auxiliary qubit S, and a measurement operation applied to the fifth stabilizer qubit Z. 123 Z S measurement operation.

[0152] The example quantum circuit 900 also includes a set 914 of multiple classically controlled Pauli operators. The set 914 of multiple classically controlled Pauli operators is Figure 8 Corresponding to step 826.

[0153] Several classically controlled Pauli operators including the use of the third Z 123 Z T and the fifth Z 123 Z S The stabilizer qubit acts as a control, applying a Z gate to the first, second, and third target qubits 1-3; using the third auxiliary qubit B and the third stabilizer qubit Z 123 Z T As control, apply Z gates to the first, second, and third target qubits 1-3; use the second auxiliary qubit S and the third stabilizer qubit Z 123 Z T As control, apply Z gates to the first, second, and third target qubits 1-3; use the second stabilizer qubit X 12 X A As control, a Z gate is applied to the third target qubit 3; a Z gate is applied to the third target qubit 3 using the fourth auxiliary qubit A as control; and a Z gate is applied to the first, second, and third target qubits 1-3 using the first auxiliary qubit T as control.

[0154] Multiple classically controlled Pauli operators also include the use of a fourth stabilizer qubit Z3Z A As control, X gates are applied to the first and second target qubits 1, 2; X gates are applied to the first, second, and third target qubits 1-3 using the third auxiliary qubit B as control; and X gates are applied to the third target qubit 3. Then, the three target states 1, 2, 3 are each output at a corresponding T state 916.

[0155] As described above, the operations included in the example quantum circuit 900 can be converted into lattice surgery operations. Fig.10 Shown in use Fig. 9 The example quantum circuit 900 generates three T states during the example time slice 1000 of the lattice operation activity. Each time slice can be related to Fig. 9 The steps in match, and Fig.10 The qubit labels used in Fig. 9 The qubit labels used in correspond to the stabilizer measurements. The black and shaded bars correspond to the stabilizer measurements. The auxiliary qubits are shown in the filled boxes. When the single-qubit Clifford operation is applied, the code distance of the auxiliary qubits is doubled to ensure adequate error suppression. The CCZ box will be used by the CCZ factory to generate the CCZ states to be transformed.

[0156] Time slice 1002 and Fig. 9 The time slice 1004 corresponds to the first Pauli product measurement 904. Fig. 9 The time slice 1006 corresponds to the second Pauli product measurement 906. Fig. 9 The time slice 1008 corresponds to the third Pauli product measurement 908 of Fig. 9 The fourth Pauli product measurement 910 corresponds to the time slice 1010. Fig. 9 The fifth Pauli product measurement 912 corresponds to the time slice 1012. Fig. 9 914 corresponds to a set of multiple classically controlled Pauli operators in .

[0157] Operating hardware: arbitrary angle phase catalysis

[0158] The catalytic technique used in the C2T plant described here can be generalized to phase angles other than 45° for the T-gate. Fig.11 Shows Figure 7 Generalization to any angle θ. Fig.11 is used to use the phased plus state Z θ The third qubit under |+> performs the target phase operation Z on the first and second qubits θ FIG. 1 is a diagram of an example generalized phase catalytic quantum circuit 1100. In the example quantum circuit 1100, a first qubit, a second qubit, and a third qubit are represented by horizontal lines 1102a-c. Qubits 1102a and 1102b represent qubits on which a target phase operation Z is to be performed. θThe first and second qubits of the example quantum circuit 1100 are provided with qubits 1102a and 1102b in input states |ψ0> and |ψ1>, respectively. The input states |ψ0> and |ψ1> may be the initial states of qubits 1102a or 1102b, i.e., qubits 1102a or 1102b may have been prepared in an arbitrary initial state, or may be a state representing the output of a previous computation. Qubit 1102c represents a qubit prepared in the directed-add state Z. θ The third qubit under |+>. Qubit 1102c can be prepared in a phased-add state using any of a variety of conventional techniques.

[0159] Example quantum circuit 1100 includes a sequence of gates applied to qubits 1102a-1102c. The gate sequence includes a first NOT operation 1104 applied to a third qubit 1102c. Then, a first set of operations is applied to qubits 1102a-1102c to compute a controlled adder operation on qubits 1102a-1102c. The first set of operations includes a multi-target CNOT gate 1106 applied to three qubits 1102a-1102c, wherein the first qubit acts as a control for the multi-target CNOT gate. The first set of operations also includes a logical AND operation 1108 performed between the second and third qubits. The result of the local AND operation is encoded in the fourth qubit 1110. The first set of operations also includes a CNOT gate 1112 applied to the first qubit 302a and the fourth qubit 1110, wherein the first qubit 1102a acts as a control for the CNOT gate.

[0160] The gate sequence also includes the square of the target phase operation 1114 applied to fourth qubit 1110. A second set of operations is then applied to qubits 1102a-1102d to uncompute the controlled adder operation performed by the first set of operations. The second set of operations includes a CNOT operation 1116 applied to first qubit 1102a and fourth qubit 1110, where first qubit 1102a acts as a control for CNOT operation 1116. The second set of operations also includes uncompute a logical AND operation between second qubit 1102b and third qubit 1102c.

[0161] The gate sequence also includes a CNOT operation 1118 applied to second qubit 1102b and third qubit 1102c, where second qubit 1102b acts as a control for the CNOT operation. The gate sequence also includes a multi-target CNOT operation 1120 applied to first qubit 1102a, second qubit 1102b, and third qubit 1102c, where first qubit 1102a acts as a control for multi-target CNOT operation 1120. The gate sequence includes a NOT operation 1122 applied to third qubit 1102c. NOT operation 1122 returns the qubits to the original phased-add state Z. θ |+>.

[0162] After the example quantum circuit is applied to three qubits 1102a-c (and a fourth qubit 1110), the target phase operation Z θ has been applied to first qubit 1102a and second qubit 1102b. That is, example quantum circuit 1100 operates on a resource state (i.e., a |+> state phased by an angle equal to the angle of the target phase operation) and two input states to be phased by that angle, phases the two input states in the target manner, and returns the resource state.

[0163] Fig.12 is a diagram of an example quantum circuit 1200 that converts Fig.11 The generalized phase catalytic circuit 1100 is specialized to θ = 22.5°, that is, to This specialization creates two Assume that the physical gate error rate is 10-3 and the target error rate is 10 -10 , the example quantum circuit 1200 can produce a space-time volume that is an order of magnitude smaller than previous technologies For example, some repeat-until-success circuits use ≈45 T gates to achieve ∈=10 -10 implement gates, while some include direct synthesis The approach to synthesizing T states uses ≈25 times more volume than directly synthesizing T states (although this ratio increases with the error rate of the physical gates).

[0164] Hardware operation: By Example method implemented by the factory

[0165] Reference above Figure 2-5 The CCZ plant described can be compared with the one referenced above Figure 6-12The described C2T factory combination, for example, example process 600 or 800 can be appended to example process 200. Combining the two factories / processes results in a T-catalyzed T-factory that transforms eight noisy T-states into two T-states with quadratic noise reduction, thereby achieving a 4:1 ratio of input T-states to output T-states. This is competitive with the 3:1 ratio of block code state purification and is particularly advantageous because conventionally a large number of T-states must be worked to achieve such a good ratio.

[0166] In some embodiments, a CCZ factory (or example process 200) can be modified when combined with a T-catalyzed T factory (or example processes 600 / 800). For example, some or all stabilizer measurements can be reordered and the output qubits can be located in different positions so that the CCZ factory fits tightly into the T-catalyzed T factory. This means that some steps of example processes 200 and 600 (or 800) can be combined and the order can be changed. In addition, the factory no longer needs to be cross-linked to itself.

[0167] Fig.13 An example time slice of lattice surgery activity during combined operation of a T catalytic plant and a CCZ plant is shown. The qubit tags can be used with Figure 4 match to verify that the correct stabilizers are being measured (although in a different order).

[0168] Time slice 1302 and Figure 4 The time slice 400c and Fig.10 1008 corresponds to the time slice 1304. Figure 4 The time slice 400d and Fig.10 The time slice 1306 corresponds to 1010. Figure 4 The time slice 400a and Fig.10 The time slice 1308 corresponds to 1012. Figure 4 The time slice 400b and Fig.10 1002 corresponds to. Time slice 1310 corresponds to Figure 4 The time slices 400d and 400e and Fig.10 The time slice 1312 corresponds to the time slice 1004. Figure 4 Time slice 400e-h and Fig.10 Corresponding to time slice 1006.

[0169] Fig.14 For known Factory Construction 1402, Current Description Factory Construction 1404 and the currently described |T>-Catalysis Diagram of the spatial layout and data flow of the factory 1406. Error rates shown assume a physical gate error rate of 10-3, and assume that the code distance is large enough so that the dominant source of errors in the output are purification errors. The first-level T factories 1408a-f are implemented at half the code distance to balance the contribution from purification errors and code errors. Fig.14 It is shown how the presently described CCZ-factory and T-catalyzed T-factory configurations have a smaller footprint, faster output, and sufficient inhibition to run the proposed algorithm beyond the classical simulation scenario.

[0170] The digital and / or quantum subject matter and implementations of digital function operations and quantum operations described in this specification may be implemented in digital electronic circuits, suitable quantum circuits or more generally quantum computing systems, tangibly embodied digital and / or quantum computer software or firmware, digital and / or quantum computer hardware, including the constructions disclosed in this specification and their structural equivalents, or in a combination of one or more of them. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptographic system, or a quantum simulator.

[0171] The embodiments of the digital and / or quantum subject matter described in this specification may be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by a data processing device or for controlling the operation of a data processing device. The digital and / or quantum computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or additionally, the program instructions may be encoded on an artificially generated propagation signal capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, which is generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by a data processing device.

[0172] The terms quantum information and quantum data refer to information or data carried, held or stored by a quantum system, wherein the smallest non-trivial system is a qubit, i.e., a system that defines a unit of quantum information. It should be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, for example, systems with two or more levels. For example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the base state and the first excited state are used to identify the reference state of the calculation, however, it should be understood that other arrangements where higher-level excited states are used to identify the state of the calculation are also possible.

[0173] The term "data processing device" refers to digital and / or quantum data processing hardware, and covers all kinds of devices, equipment and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The device may also be or may also include a dedicated logic circuit, such as an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a specific quantum system. In particular, a quantum simulator is a special-purpose quantum computer that does not have the ability to perform general-purpose quantum computations. In addition to the hardware, the device may optionally include code that creates an execution environment for a digital and / or quantum computer program, such as code constituting processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0174] A digital computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and may be deployed in any form, including as a stand-alone program or as a module, component, subroutine or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or may be written in a quantum programming language, such as QCL or Quipper.

[0175] A digital and / or quantum computer program may, but need not correspond to a file in a file system. A program may be stored in a portion of a file that stores other programs or data, such as one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, such as files storing one or more modules, subroutines, or portions of code. A digital and / or quantum computer program may be deployed to be executed on a digital or quantum computer, or on multiple digital and / or quantum computers located in one location or distributed in multiple locations and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can transmit quantum data using a quantum system (e.g., qubit). Typically, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0176] The processes and logic flows described in this specification may be performed by one or more programmable digital and / or quantum computers, operating in conjunction with one or more digital and / or quantum processors where appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating outputs. The processes and logic flows may also be performed by a dedicated logic circuit (e.g., FPGA or ASIC) or a quantum simulator, and the apparatus may also be implemented as a dedicated logic circuit (e.g., FPGA or ASIC) or a quantum simulator, or by a combination of a dedicated logic circuit or a quantum simulator and one or more programmed digital and / or quantum computers.

[0177] For a system of one or more digital and / or quantum computers, being "configured to" perform a particular operation or action means that the system has software, firmware, hardware, or a combination thereof installed thereon that, when run, causes the system to perform the operation or action. For one or more digital and / or quantum computer programs configured to perform a particular operation or action, it means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer can receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform the operation or action.

[0178] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general or special purpose digital and / or quantum processor or both, or any other type of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory or a quantum system suitable for transmitting quantum data (e.g. photons or a combination thereof).

[0179] Elements of a digital and / or quantum computer include a central processing unit for executing or running instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory may be supplemented or incorporated by a dedicated logic circuit or a quantum simulator. Typically, a digital and / or quantum computer also includes one or more mass storage devices (e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information) for storing digital and / or quantum data, or is operably coupled to receive digital and / or quantum data from the one or more mass storage devices or to transmit digital and / or quantum data to the one or more mass storage devices or both. However, a digital and / or quantum computer may not require such a device.

[0180] Digital and / or quantum computer readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including, by way of example, semiconductor memory devices such as EPROM, EEPROM and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; CD-ROMs and DVD-ROMs; and quantum systems, such as trapped atoms or electrons. It should be understood that quantum memory is a device capable of storing quantum data for a long time with high fidelity and efficiency, such as a light-matter interface, where light is used for transmission and matter is used for storage and protection of quantum features such as superposition or quantum coherence of quantum data.

[0181] The control of the various systems described in this specification or portions thereof may be implemented in a digital and / or quantum computer program product that includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems described in this specification or portions thereof may each be implemented as an apparatus, method, or system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0182] Although this specification contains many specific implementation details, these details should not be interpreted as limitations on the scope of the claimed protection, but rather as descriptions of features for specific embodiments. Certain features described in this specification in the context of separate embodiments may also be implemented in combination in a single embodiment. On the contrary, the various features described in the context of a single embodiment may also be implemented in multiple embodiments separately or in any suitable sub-combination. In addition, although features may be described above as working in certain combinations, and even initially claimed as such, in some cases, one or more features from the claimed combination may be deleted from the combination, and the claimed combination may point to a variant of a sub-combination or a sub-combination.

[0183] Similarly, although operations are depicted in a particular order in the accompanying drawings, this should not be understood as requiring that the operations be performed in the particular order shown or in a sequential order, or should not be understood as requiring that all of the operations shown be performed to obtain the desired results. In some cases, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the above-described embodiments should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0184] Specific implementations of the subject matter have been described. For example, the actions listed in this disclosure can be performed in a different order and still obtain the desired results. As an example, the processes depicted in the accompanying drawings do not necessarily require the specific order or sequential order shown to obtain the desired results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for converting a CCZ state into three output T states, the method comprising: obtaining i) three target qubits prepared in the CCZ state, ii) a first auxiliary qubit in the T state, iii) a plurality of additional auxiliary qubits prepared in the zero state, and iv) a plurality of stabilizer qubits prepared in the zero state; performing an X-gate on a target qubit and an attached ancillary qubit using a first stabilizer qubit as a control and measuring the first stabilizer qubit to determine a state of the first stabilizer qubit; performing an X-gate on the plurality of target qubits and the additional ancillary qubit using a second stabilizer qubit as a control and measuring the second stabilizer qubit to determine a state of the second stabilizer qubit; Perform X on the attached auxiliary qubit 1 / 2 Door; performing a Z-gate on a plurality of target qubits and the first ancillary qubit using a third stabilizer qubit as a control; performing a Hadamard gate on the first ancillary qubit and measuring the first ancillary qubit, an additional ancillary qubit, and a third stabilizer qubit to determine corresponding qubit states; Perform X on the attached auxiliary qubit 1 / 2 Door; performing a Z-gate on the target qubit and the attached ancillary qubit using the fourth stabilizer qubit as a control; performing a Hadamard gate on the additional ancillary qubit and measuring the additional ancillary qubit and the fourth stabilizer qubit to determine corresponding qubit states; performing a NOT operation on a stabilizer qubit using an attached auxiliary qubit as a control, and performing two NOT operations on the attached auxiliary qubit and the stabilizer qubit using different stabilizer qubits as controls; performing Z-gates on multiple target qubits and attached auxiliary qubits using two stabilizer qubits as controls; performing a Hadamard gate on the additional ancillary qubit and measuring the additional ancillary qubit and the stabilizer qubit to determine corresponding qubit states; and A plurality of classically controlled Pauli operators are performed on the three target qubits conditioned on the determined states of the auxiliary qubits.

2. The method according to claim 1, wherein: The three target qubits include a first target qubit, a second target qubit, and a third target qubit, the plurality of additional auxiliary qubits include a second auxiliary qubit, a third auxiliary qubit, and a fourth auxiliary qubit, and the plurality of stabilizer qubits include a first stabilizer qubit, a second stabilizer qubit, a third stabilizer qubit, a fourth stabilizer qubit, and a fifth stabilizer qubit.

3. The method according to claim 2, wherein: Performing an X-gate on a target qubit and an attached ancillary qubit using the first stabilizer qubit as a control includes: performing an X-gate on the third target qubit using the first stabilizer qubit as a control; and An X-gate is performed on a third auxiliary qubit using the first stabilizer qubit as a control.

4. The method according to claim 3, wherein: Perform X on the attached auxiliary qubit 1 / 2 The gate comprises performing X on the third auxiliary qubit 1 / 2 Door.

5. The method according to any one of claims 2 to 4, wherein: Performing an X-gate on multiple target qubits and additional ancillary qubits using a second stabilizer qubit as a control includes: performing an X-gate on the first target qubit and the second target qubit using the second stabilizer qubit as a control; and An X-gate is performed on a fourth auxiliary qubit using the second stabilizer qubit as a control.

6. The method according to any one of claims 2 to 4, wherein: Performing a Z-gate on the plurality of target qubits and the ancillary qubit using the third stabilizer qubit as a control includes performing a Z-gate on the plurality of target qubits and the first ancillary qubit using the third stabilizer qubit as a control.

7. The method according to claim 4, wherein: Performing a Z-gate on a plurality of target qubits and the first ancillary qubit using the third stabilizer qubit as a control includes: performing a Z-gate on each of the three target qubits using the third stabilizer qubit as a control; and A Z-gate is performed on the first auxiliary qubit using the third stabilizer qubit as a control.

8. The method according to any one of claims 2 to 4, wherein: Measuring the first ancillary qubit, the additional ancillary qubit, and the stabilizer qubit to determine corresponding qubit states includes measuring the first ancillary qubit, the third ancillary qubit, and the third stabilizer qubit to determine corresponding qubit states.

9. The method according to any one of claims 2 to 4, wherein: Perform X on the attached auxiliary qubit 1 / 2 The gate consists of performing X on the second ancillary qubit 1 / 2 Door.

10. The method according to any one of claims 2 to 4, wherein: Performing a Z-gate on a target qubit and an attached ancillary qubit using a fourth stabilizer qubit as a control includes: performing a Z-gate on the third target qubit using the fourth stabilizer qubit as a control; and A Z-gate is performed on the fourth auxiliary qubit using the fourth stabilizer qubit as a control.

11. The method according to any one of claims 2 to 4, wherein: Performing a Hadamard gate on the attached ancillary qubit and measuring the attached ancillary qubit involves: performing a Hadamard gate on the fourth auxiliary qubit; and The fourth ancillary qubit and the fourth stabilizer qubit are measured.

12. The method according to any one of claims 2 to 4, wherein: Performing a NOT operation on a stabilizer qubit using an additional auxiliary qubit as a control and performing two NOT operations on the additional auxiliary qubit and the stabilizer qubit using different stabilizer qubits as controls includes: performing a NOT operation on the third stabilizer qubit using the third auxiliary qubit as a control; performing a NOT operation on the third auxiliary qubit using the first stabilizer qubit as a control; and A NOT operation is performed on the third stabilizer qubit using the first stabilizer qubit as a control.

13. The method according to any one of claims 2 to 4, wherein: Performing a Z-gate on the plurality of target qubits and the additional ancillary qubit using the two stabilizer qubits as controls includes performing a Z-gate on each of the three target qubits and the second ancillary qubit using the third stabilizer qubit and the fifth stabilizer qubit as controls.

14. The method according to any one of claims 2 to 4, wherein: Performing a Hadamard gate on the additional ancillary qubit includes performing a Hadamard gate on the second ancillary qubit.

15. The method according to any one of claims 2 to 4, wherein: Measuring the additional ancillary qubits and the stabilizer qubit includes measuring the second ancillary qubit and the fifth stabilizer qubit.

16. The method according to any one of claims 1 to 4, wherein: A plurality of classically controlled Pauli operators are performed on the three target qubits conditioned on the determined states of the auxiliary qubits to reduce decoherence.

17. The method according to any one of claims 2 to 4, wherein: Executing a plurality of classically controlled Pauli operators on the three target qubits conditioned on the determined state of the auxiliary qubit includes: performing a Z-gate on one or more of the three target qubits using as a control at least one of i) the one or more stabilizer qubits and ii) the one or more ancillary qubits; performing an X-gate on one or more of the three target qubits using the stabilizer qubit or the ancillary qubit as a control; and Perform an X gate on the target qubit.

18. The method according to claim 17, wherein: Performing a Z-gate on the target qubit using at least one of i) one or more stabilizer qubits and ii) one or more ancillary qubits as a control comprises: performing a sequence of operations from a third set of operations, the third set of operations comprising: performing a Z-gate on the first target qubit, the second target qubit, and the third target qubit using the third stabilizer qubit and the fifth stabilizer qubit as controls; performing a Z-gate on the first target qubit, the second target qubit, and the third target qubit using the third auxiliary qubit and the third stabilizer qubit as controls; performing a Z-gate on the first target qubit, the second target qubit, and the third target qubit using the second ancillary qubit and the third stabilizer qubit as controls; performing a Z-gate on the third target qubit using the second stabilizer qubit as a control; performing a Z-gate on the third target qubit using the fourth auxiliary qubit as a control; A Z-gate is performed on the first target qubit, the second target qubit, and the third target qubit using the first auxiliary qubit as a control.

19. The method according to claim 17, wherein: Performing an X-gate on one or more of the three target qubits using a stabilizer qubit or an ancillary qubit as a control includes: performing an X-gate on the first target qubit and the second target qubit using the fourth stabilizer qubit as a control; and An X gate is performed on the first target qubit, the second target qubit, and the third target qubit using the third auxiliary qubit as a control.

20. The method according to claim 17, wherein: Performing an X-gate on the target qubit includes performing an X-gate on the third target qubit.

21. The method according to any one of claims 1-4, 7 and 18-20, wherein: Using the stabilizer qubit as a control includes using the stabilizer qubit as an X-axis control by applying a Hadamard gate to the qubit before and after using the qubit as a control.

22. The method according to any one of claims 1-4, 7 and 18-20, wherein: The obtained first auxiliary qubit in the T state includes the catalyst T state output from the previous transition of the CCZ state to the three T states.

23. The method of any one of claims 1-4, 7 and 18-20, further comprising providing two of the three output T-states for subsequent calculations.

24. The method according to claim 23, further comprising: determining whether the third output T state has accumulated an amount of error above a predetermined acceptable threshold; In response to determining that the third output T-state has accumulated an amount of error below the predetermined acceptable threshold, providing the third output T-state as a catalyst T-state for a subsequent transition to a CCZ state; In response to determining that the third output T-state has accumulated an amount of errors that is greater than or equal to the predetermined acceptable threshold, providing the stored T-state for a subsequent transition to a CCZ state.

25. The method according to any one of claims 1-4, 7, 18-20 and 24, wherein: The CCZ state obtained is purified using a method according to any one of claims 1 to 14.

26. The method according to claim 25, wherein: The three output T-states are less noisy than the input T-states.

27. The method according to any one of claims 1-4, 7, 18-20, 24 and 26, wherein: Execute different groups of operations in parallel.

28. A method for converting a CCZ state into three T states, the method comprising: Obtaining a first target qubit, a second target qubit, and a third target qubit in a CCZ state; Perform X on the third target qubit -1 / 2 Door; performing an X gate on the first target qubit and the second target qubit using the third target qubit as a control; Using the third qubit as an X-axis control, performing a Z gate on the first target qubit and the second target qubit; Perform Z on the third target qubit -1 / 4 Door; and Using the third qubit as an X-axis control, a Z gate is performed on the first target qubit and the second target qubit to obtain the three T states.

29. The method of claim 28, further comprising providing one of the obtained three T states as a catalyst for subsequent conversion of the CCZ state to the three T states.

30. A quantum computing device comprising: a qubit register, the qubit register comprising a plurality of target qubits, a plurality of auxiliary qubits, and a plurality of stabilizer qubits, each qubit being prepared in a respective initial state; a plurality of control lines coupled to the qubit register; a plurality of control devices coupled to the plurality of control lines; and One or more classical processors; Wherein, the quantum computing device is configured to perform classical and quantum operations according to the method of any one of claims 1-29.

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