Quantum error correction method, device and equipment, storage medium, program product

Abstract

The present disclosure relates to the technical field of computers, and particularly relates to a quantum error correction method, device and equipment, a storage medium and a program product. The quantum error correction method comprises: determining a measurement operator according to a gauge group in a quantum system, wherein the measurement operator corresponds to a plurality of generators in the gauge group, and the gauge group represents a gauge degree of freedom of the quantum system; determining a measurement result of a stabilizer in the quantum system according to the measurement operator, wherein the stabilizer represents an invariant in the quantum system; and correcting the quantum system according to the measurement result of the stabilizer.

Description

Technical Field

[0001] This disclosure relates to the field of computer technology, and in particular to a quantum error correction method, apparatus and device, storage medium and program product. Background Technology

[0002] As quantum computing technology continues to iterate and evolve, and its theory and hardware architecture continue to develop, quantum computing systems are gradually moving from laboratory prototypes to practical applications. The engineering implementation path is also giving rise to more and more new directions and feasible solutions, such as topological quantum computing platforms.

[0003] Currently, compared to the spin qubits used in most other quantum computing platforms, the basic unit of topological quantum platforms is mainly Majorana zero mode.

[0004] However, most of the quantum computing error correction codes proposed in related technologies can only be applied to one quantum computing platform, resulting in poor versatility of quantum error correction methods. Summary of the Invention

[0005] In view of this, the present disclosure provides a quantum error correction method, apparatus and device, storage medium and program product, which determines the measurement operator used for quantum computation error correction based on the generator in the gauge group of the quantum system. It can be applied equivalently to various types of quantum systems and can be used under various boundary conditions to perform quantum computation error correction based on the gauge subsystem in the quantum system, thereby improving the versatility of the quantum error correction method.

[0006] According to one aspect of this disclosure, a quantum error correction method is provided, comprising: determining a measurement operator based on a gauge group in a quantum system, wherein the measurement operator corresponds to a plurality of generators in the gauge group, the gauge group representing the gauge degrees of freedom of the quantum system; determining a measurement result of a stabilizer in the quantum system based on the measurement operator, wherein the stabilizer represents an invariant in the quantum system; and correcting the quantum system based on the measurement result of the stabilizer.

[0007] In some embodiments, determining a measurement operator based on the gauge group of a quantum system includes: determining a measurement operator corresponding to the size of the quantum system, the gauge group, and the size of the quantum system including the length of the qubit array in one of its linear dimensions.

[0008] In some embodiments, determining the measurement operator corresponding to the size of the quantum system and the gauge group includes: determining a candidate measurement period corresponding to the size based on the size; and determining the measurement operator based on a target measurement period among the candidate measurement periods and the gauge group.

[0009] In some embodiments, determining a candidate measurement period corresponding to the size includes: if the size is odd, determining a candidate measurement period as a first candidate measurement period; if the size is even, determining a candidate measurement period as a second candidate measurement period.

[0010] In some embodiments, the first candidate measurement period includes a single measurement period and three measurement periods, and the second candidate measurement period includes a single measurement period, two measurement periods, and three measurement periods.

[0011] In some embodiments, the canonical group includes triangle generators. Determining the measurement operator based on the target measurement period in the candidate measurement periods and the canonical group includes: when the target measurement period is a single measurement period, the measurement operator includes a measurement operator corresponding to a rectangular structure, where the rectangular structure corresponds to two triangle generators intersecting with its vertices; when the target measurement period is a double or triple measurement period, the measurement operator includes a measurement operator corresponding to a parallelogram structure, where the parallelogram structure corresponds to two triangle generators intersecting with its sides.

[0012] In some embodiments, determining the measurement operator based on the target measurement period in the candidate measurement periods and the specification group includes: when the target measurement period is a single measurement period, the measurement operator corresponding to the target measurement period includes a four-body joint measurement operator; when the target measurement period is a double measurement period, the measurement operator corresponding to the target measurement period includes a combination of two four-body joint measurement operators, or a combination of a four-body joint measurement operator and an eight-body joint measurement operator; when the target measurement period is a triple measurement period, the measurement operator corresponding to the target measurement period includes a combination of two four-body joint measurement operators and an eight-body joint measurement operator.

[0013] In some embodiments, determining the measurement results of stableons in the quantum system according to the measurement operator includes: determining a stable subgroup in the quantum system based on the size of the quantum system and the measurement period of the quantum system; and performing measurements on the quantum system according to the measurement operator to determine the measurement results of each stableon in the stable subgroup.

[0014] In some embodiments, measuring the quantum system according to the measurement operator and determining the measurement result of each stable element in the stable subgroup includes: when the measurement period is multiple measurement periods, updating the stable subgroup according to the measurement operator corresponding to each measurement period, and determining the measurement result of each stable element in the stable subgroup.

[0015] In some embodiments, error correction of the quantum system based on the measurement results of the stabilizers includes: determining an error set corresponding to the erroneous stabilizers based on the erroneous stabilizers whose measurement results do not match preset values, wherein each error in the error set, when applied to the quantum system, will cause the quantum system to include the erroneous stabilizer; and determining the location and type of physical errors in the quantum system based on the probabilities corresponding to each error in the error set.

[0016] In some embodiments, where the physical error includes a logical error, correcting the quantum system based on the measurement results of the stabilizer further includes: correcting the quantum system through an error correction operation to restore correct logical information in the quantum system; or recording the logical error and the current logical frame in the quantum system so as to compensate for the current logical frame during subsequent operations on the quantum system.

[0017] According to a second aspect of this disclosure, a quantum error correction device is provided, comprising: a measurement operator determination module configured to determine a measurement operator based on a gauge group in a quantum system, wherein the measurement operator corresponds to a plurality of generators in the gauge group, the gauge group representing the gauge degrees of freedom of the quantum system; a measurement module configured to determine a measurement result of a stabilizer in the quantum system based on the measurement operator, wherein the stabilizer represents an invariant in the quantum system; and an error correction module configured to perform error correction on the quantum system based on the measurement result of the stabilizer.

[0018] According to a third aspect of this disclosure, a quantum error correction device is provided, comprising: at least one memory; and at least one processor coupled to the at least one memory, the at least one processor being configured to execute a quantum error correction method as described in any embodiment of this disclosure based on instructions stored in the at least one memory.

[0019] According to a fourth aspect of this disclosure, a computer-readable storage medium is provided having computer instructions stored thereon that, when executed by a processor, implement the quantum error correction method as described in any embodiment of this disclosure.

[0020] According to a fifth aspect of this disclosure, a computer program product is provided that, when run on a computer, causes the computer to implement the quantum error correction method as described in any embodiment of this disclosure. Attached Figure Description

[0021] The accompanying drawings, which form part of this specification, illustrate embodiments of this disclosure and, together with the specification, serve to explain the principles of this disclosure.

[0022] This disclosure will become clearer with reference to the accompanying drawings and the following detailed description, wherein: Figure 1 A flowchart illustrating a quantum error correction method according to some embodiments of the present disclosure is shown; Figure 2 A schematic diagram of a Majorana fermion array according to some embodiments of the present disclosure is shown; Figure 3 A schematic diagram of equivalent qubits according to some embodiments of the present disclosure is shown; Figure 4 A schematic diagram of a modified lattice according to some embodiments of the present disclosure is shown; Figure 5 A schematic diagram of a triangular lattice according to some embodiments of the present disclosure is shown; Figure 6 A schematic diagram of a circle operator according to some embodiments of the present disclosure is shown; Figure 7 A schematic diagram of a double-loop operator according to some embodiments of the present disclosure is shown; Figure 8 A schematic diagram of a measurement operator according to some embodiments of the present disclosure is shown; Figure 9 A schematic diagram of a parallelogram operator generator according to some embodiments of the present disclosure is shown; Figure 10 A schematic diagram of a single-cycle measurement operator according to some embodiments of the present disclosure is shown; Figure 11 A schematic diagram of a single-cycle measurement operator according to other embodiments of the present disclosure is shown; Figure 12 A schematic diagram of a verification operator according to some embodiments of the present disclosure is shown; Figure 13 Schematic diagrams of transformation operators according to other embodiments of the present disclosure are shown; Figure 14 A schematic diagram of an open-boundary lattice according to some embodiments of the present disclosure is shown; Figure 15 A block diagram of a quantum error correction device according to some embodiments of the present disclosure is shown; Figure 16 A block diagram of a quantum error correction device according to some embodiments of the present disclosure is shown; Figure 17 A block diagram of a computer system for implementing some embodiments of the present disclosure is shown.

[0023] It should be understood that the dimensions of the various parts shown in the accompanying drawings are not drawn to actual scale. Furthermore, the same or similar reference numerals denote the same or similar components. Detailed Implementation

[0024] Various embodiments of the present disclosure will now be described in detail with reference to the accompanying drawings. The descriptions of the embodiments are merely illustrative and are in no way intended to limit the scope of the disclosure or its application or use. The present disclosure may be implemented in many different forms and is not limited to the embodiments described herein. These embodiments are provided to make the disclosure thorough and complete, and to fully express the scope of the disclosure to those skilled in the art. It should be noted that, unless otherwise specifically stated, the relative arrangement of components and steps set forth in these embodiments should be interpreted as merely illustrative and not as limiting.

[0025] The terms “first,” “second,” and similar words used in this disclosure do not indicate any order, quantity, or importance, but are merely used to distinguish different parts. Words such as “including” mean that the element preceding the word covers the element listed after the word, and do not exclude the possibility of covering other elements as well.

[0026] It should also be understood that any component, data or structure mentioned in the embodiments of this disclosure can generally be understood as one or more unless expressly defined or given to the contrary in the context.

[0027] All terms used in this disclosure (including technical or scientific terms) have the same meaning as understood by one of ordinary skill in the art to which this disclosure pertains, unless otherwise specifically defined. It should also be understood that terms defined in a general dictionary, such as a dictionary, should be interpreted as having a meaning consistent with their meaning in the context of the relevant art, and not as having an idealized or highly formalized meaning, unless expressly defined herein.

[0028] Techniques, methods, and equipment known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and equipment should be considered part of the specification.

[0029] Current quantum computing platforms can be categorized based on the specific implementation of physical qubits. For example, they can be divided into three categories: traditional equivalent spin qubit platforms composed of ordinary two-level systems; platforms that encode a single equivalent spin qubit using multiple Majorana zero modes; and platforms that directly form a lattice from Majorana zero modes. These quantum computing platforms will be described in detail below.

[0030] The first type of conventional equivalent spin qubit platform can be a platform that uses a common two-level system as the physical qubit, such as superconducting qubits, cold atom or neutral atom qubits, ion trap qubits, and semiconductor spin qubits.

[0031] In the first type of platform, each physical unit can be represented by an effective spin degree of freedom, and the few-volume detection operator required for error correction code is realized through single-qubit operations, entanglement operations between adjacent qubits, and auxiliary qubit measurements.

[0032] For example, spin qubit platforms constructed from superconducting quantum circuits have enabled multi-round quantum error correction measurements of stabilizers or subsystems. Spin qubit platforms constructed from reconfigurable arrays of neutral atoms can flexibly configure the two-dimensional arrangement and interaction connections of qubits, simulating cellular lattice models and implementing quantum error correction operations. These platforms are suitable for verifying the encoding, syndrome extraction, decoding, and fault-tolerant performance of fault codes.

[0033] Furthermore, for platforms lacking native few-body interactions, the detection operator can be decomposed into several single-qubit gates and two-qubit gates, and the corresponding joint eigenvalues ​​can be read with the help of auxiliary qubits.

[0034] The main advantages of the first type of platform are that the fabrication, manipulation, and measurement technologies are relatively mature, the system size and boundary conditions can be easily changed, and different types of detection operators can be implemented through software or pulse sequences. However, the physical qubits of the first type of platform do not depend on Majorana zero modes, so they are mainly used to realize and verify the equivalent spin model of error-correcting codes, and do not directly possess the nonlocal coding characteristics of Majorana qubits.

[0035] The second type of equivalent spin qubit platform based on Majorana zero modes uses multiple Majorana zero modes to jointly encode a two-level system, and uses this two-level system as the physical qubit in the error correction code.

[0036] The typical structure of the second type of platform consists of four Majorana zero modes forming a qubit. For example, four Majorana zero modes can be carried by a superconducting island with a fixed total fermion parity, and its low-energy space can form an equivalent spin qubit. The Pauli operator of this qubit can be represented by a bilinear parity operator between different Majorana zero modes.

[0037] Multiple four Majorana qubits or similar Majorana superconducting islands can be arranged in a triangular lattice or other equivalent two-dimensional structure. The joint fermion parity between multiple superconducting islands can be measured by quantum dots, tunneling junctions, floating superconducting islands, or auxiliary readout units, thereby enabling few-volume detection operators in error-correcting codes.

[0038] In addition, for devices that cannot directly measure few-volume operators, auxiliary quantum dots or auxiliary qubits can be used to decompose the corresponding measurement into several local parity measurements.

[0039] The second type of platform can be realized using semiconductor-superconductor nanowires, quantum dot-superconductor arrays, topological insulator nanostructures, and other heterostructures capable of forming spatially separated Majorana zero modes. Its main advantage is that it retains the description method of standard qubit error-correcting codes while enabling individual physical qubits to have a nonlocal coding structure determined by fermion parity, thus serving as a transitional solution between traditional qubit platforms and direct Majorana lattice platforms.

[0040] The third type of lattice platform, which is directly composed of Majorana zero modes, no longer pre-combines multiple Majorana zero modes into independent four Majorana qubits. Instead, it first prepares Majorana zero modes, for example, arranged in a honeycomb lattice, and then defines the equivalent physical qubits on the triangular lattice by the pairing relationship of adjacent Majorana zero modes.

[0041] In the aforementioned lattice platform, the canonical detection operator required for Majorana error-correcting codes can be further generated through local coupling between adjacent Majorana zero modes or joint parity measurements. This implementation is closest to the original microstructure of the error-correcting code.

[0042] The third type of lattice platform can be composed of vortex zero-energy modes in topological superconducting materials or superconducting heterostructures. A common approach is to generate ordinary magnetic flux vortices in topological superconducting candidate materials and adjust the position of the vortices through artificial defects, strain, localized magnetic structures, or patterned pinning arrays.

[0043] exist , In systems such as topological insulator-superconductor heterostructures, vortex nucleus zero energy states, Majorana zero mode candidate signals, or ordered vortex zero mode arrays have been observed, providing an experimental basis for further construction of artificial honeycomb vortex lattices.

[0044] Furthermore, systems such as paired density wave half-flux vortices can also serve as carrier units for Majorana zero modes. By designing the pinning positions of half-vortices or ordinary vortices as a honeycomb structure, a honeycomb Majorana zero-mode lattice can be formed. Subsequently, by adjusting the vortex spacing, local potential, and measurement electrodes, local coupling and parity detection between adjacent Majorana zero modes can be achieved.

[0045] The advantage of the third type of platform is that its spatial structure and local coupling relationship directly correspond to the microscopic model of Majorana error-correcting codes, which can reduce the additional encoding steps from Majorana degrees of freedom to ordinary qubits.

[0046] In related technologies, most current quantum error-correcting codes are designed based on spin qubits and cannot be directly applied to topological qubit platforms. For example, when applying the above error-correcting codes to a topological quantum computing platform, it is necessary to combine the corresponding Majorana zero-energy modes into equivalent spin qubits in order to realize the corresponding error-correcting codes.

[0047] Current error-correcting codes based on Majorana fermions are limited by Majorana fermions themselves and cannot be applied to spin qubit platforms.

[0048] As an example, error-correcting codes based on Majorana zero modes will transform into traditional error-correcting codes based on quantum spin bits under periodic boundary conditions, and can no longer be represented as local error-correcting codes based on Majorana fermions, which makes the design of error-correcting codes somewhat difficult.

[0049] Therefore, it is very important to construct error-correcting codes based on Majorana zero modes and to make them easily extend to spin quantum platforms.

[0050] In view of this, this disclosure proposes a quantum error correction method, apparatus and device, storage medium and program product, which determines the measurement operator used for quantum computation error correction based on the generator in the gauge group of the quantum system. It can be applied equivalently to various types of quantum systems and can be used under various boundary conditions to perform quantum computation error correction based on the gauge subsystem in the quantum system, thereby improving the versatility of the quantum error correction method.

[0051] First, combined Figure 1 The quantum error correction method in this disclosure is described. Figure 1 A flowchart illustrating a quantum error correction method according to some embodiments of the present disclosure is shown.

[0052] like Figure 1 As shown, the quantum error correction method may include: step S1, determining a measurement operator based on the gauge group in the quantum system, wherein the measurement operator corresponds to multiple generators in the gauge group, and the gauge group characterizes the gauge degrees of freedom of the quantum system; step S2, determining the measurement result of a stabilizer in the quantum system based on the measurement operator, wherein the stabilizer characterizes an invariant in the quantum system; and step S3, correcting the quantum system based on the measurement result of the stabilizer.

[0053] In some embodiments, the quantum system can be a spin qubit platform, an equivalent spin qubit platform, or a Majorana fermion lattice platform. In other words, the three types of quantum computing platform systems described above can all be used to implement the error-correcting codes proposed in the embodiments of this disclosure under appropriate lattice arrangements, local coupling, and parity measurement conditions.

[0054] Among the aforementioned quantum error correction methods, new error-correcting codes can be defined for quantum systems based on gauge groups, such as Majorana XYZ error-correcting codes. Below, using a two-dimensional Majorana fermion array with a honeycomb lattice structure as an example, we will first introduce the basic concept of XYZ error-correcting codes.

[0055] Figure 2 A schematic diagram of a Majorana fermion array according to some embodiments of the present disclosure is shown. Figure 2 As shown, each point on the honeycomb lattice has a Majorana fermion.

[0056] There are two types of four-body interactions in the aforementioned honeycomb lattice, such as Figure 2 Triangles 21 and 22 are shown in the diagram. Triangle 21 includes four-body interactions between four lattice points 211 to 214. Triangle 22 includes four-body interactions between four lattice points 221 to 224. .

[0057] Because there are two types of inequivalent lattice points in the crystal lattice, the aforementioned four-body interactions can be inequivalent. For example, lattice points 214 and 224 can be different types of Majorana fermions, leading to the aforementioned four-body interactions being inequivalent.

[0058] For the above lattice, the gauge group can be defined as: .

[0059] In the above formula, and Used to represent two non-equivalent triangular structures in a crystal lattice, where i represents the number of the triangular structure. and Used for two unequal four-body interactions, where j denotes the number of Majorana fermions in a triangular structure.

[0060] The above formula can be understood as Let G be the generators of a normalized group. A generator is a set of minimal basic operators of a group, and all elements of the group can be represented as a finite product of them.

[0061] According to the definition of the subsystem error-correcting code, the elements within the canonical group G do not always commute, but the logical bits of the error-correcting code are defined as the corresponding logical operators. satisfy ,and In other words, a logical operator O is an operator that commutes with all elements in the canonical group G, but does not belong to the canonical group itself; that is, a logical operator O can change logical information without being detected by canonical measurements.

[0062] For typical error-correcting codes, generators in a stable subgroup S commutate with each other. These generators can be defined, for example, as equivalent Pauli operators, such as... Where n is the number of independent generators in the stable subgroup S. In other words, there are n independent stable elements in the stable subgroup S, corresponding to... A check operator can be used to display errors that occur in the system, thereby enabling error correction in quantum computing.

[0063] The system has In the case of n qubits, the above n stable elements can immediately be considered as n constraints on the qubits, and the corresponding number of logical bits in the quantum system is . .

[0064] For the above-mentioned normative groups The generators of this group can be divided into two categories, for example, represented as: .

[0065] In the above formula, for the same canonical bit i, the generator and generator The opposition between them is easy, that is For different standard bits i and j, the generator... and generator The opposition between them is easy, that is .

[0066] By writing the generator in the above representation, a stable subgroup can be defined based on the canonical group as follows: In other words, after sorting the generators in the canonical group that have anti-commutative relations, there are no generators that can be used as generators of stable subgroups.

[0067] As can be seen from the above definition, each generator of this stable subgroup commutes with all other generators. The number of generators in the stable subgroup is... The number of stable logical bits is thus the corresponding number of logical bits. .

[0068] However, due to the canonical group The corresponding operators Connecting different logical states can be understood as additional constraints, thereby further reducing the number of logical bits. For example, operators in a canonical group. The anti-commutation relation between them will remove the original first logical bit. Therefore, the actual number of logical bits in the quantum system is... And parameters satisfy .

[0069] Below, we will return to the Majorana zero mode in a cellular lattice as a specific example, combined with... Figure 3 This paper describes the specific behavior of the gauge group and the stabilizer in this quantum system. Figure 3 A schematic diagram of equivalent qubits according to some embodiments of the present disclosure is shown.

[0070] As mentioned earlier, two types of Majorana fermions can exist in a honeycomb lattice, considering their corresponding operators. and ,in, It can correspond to a Type I Majoranag fermion, such as lattice point 311. It can correspond to a second type of Majorana fermion, such as lattice 312.

[0071] same subscript A Majorana operator can define a cell; for example, lattice points 311 and 312 can define cell 31. Each cell corresponds to a quantum spin, meaning a spin operator can be defined at the midpoint of the corresponding edge. , representing in Pauli operators on lattice points.

[0072] According to this definition, Majorana fermions can be represented using Pauli operators and satisfy the same commutation relations. For example, the operator corresponding to a Majorana fermion can be represented as: .

[0073] In the above formula, This can be understood as multiplying the Pauli z-operators on all lattice points with numbers less than i, for example... Figure 3 As shown by the dashed line, this part can be used to distinguish different grid points in the system.

[0074] Through the operator transformations described above, the gauge group of the Majorana lattice system can be represented using Pauli operators. , and the corresponding stable subgroups and logical operators for logical bits.

[0075] It is important to note that the above transformation relies entirely on a one-dimensional ordering of Majorana fermions on the lattice. This ordering can be defined by a preset path traversing all the lattice points of the cellular lattice, which can be understood as unfolding the high-dimensional array into a one-dimensional ordering, for example, along the dashed line 321.

[0076] Therefore, for a dashed line ordered from a certain lattice point, it is necessary to connect its ends. Consequently, at the connected region, the spin operator representation of the Majorana fermion operator exhibits nonlocality. This property will lead to different behaviors of the transformation under different boundary conditions.

[0077] Based on the above transformations, we can consider the transformations of the two generators in the canonical group mentioned above, namely... The first generator can be transformed, for example, into: The second generator can be transformed, for example, into .

[0078] Based on the transformation relationship between Majorana fermion operators and Pauli operators, if the generators mentioned above are represented using Pauli operators, then the first one is given. The second one gives Similarly, there are two types of triangular operators.

[0079] It can be seen that the above gauge group can be represented by spin operators on a triangular lattice. Figure 4 A schematic diagram of a modified lattice according to some embodiments of the present disclosure is shown.

[0080] like Figure 4 As shown in (a), Majorana fermions in a honeycomb lattice can be transformed into spin qubits in a triangular lattice, where each lattice point of the triangular lattice corresponds to an edge of the honeycomb lattice. According to the above transformation relationship, the two types of generators can correspond to two types of inequivalent surface elements in the triangular lattice, for example, triangle 41 and triangle 42 are inequivalent.

[0081] from Figure 4 In the structure shown in (a), selecting the area enclosed by the dashed line and rotating it, we can obtain... Figure 4 (b) shows the triangular lattice morphology. The following explanation will use this morphology as an example. It should be understood that this morphology, as a representative local structure of a triangular lattice, exhibits properties that are general. In other words, the error-correcting codes, stable subgroups, and logical operators determined based on this morphology can all be extended to complete triangular lattice structures.

[0082] Below, we will combine Figure 5 and Figure 6 First, we will introduce the various properties of quantum systems under periodic boundary conditions. Figure 5 A schematic diagram of a triangular lattice is shown according to some embodiments of the present disclosure.

[0083] Under periodic boundary conditions, starting with spin operator representation, we consider the gauge group defined on a triangular lattice: .

[0084] In the above formula, the inverted triangle operator, for example... Figure 5 The operator corresponding to structure 51 in the diagram, starting from the bottom vertex, corresponds to the operator in reverse time. And upper triangular operators, such as... Figure 5 The operator corresponding to structure 52 in the diagram is assigned sequentially starting from the lower right vertex. .

[0085] exist Figure 5 In the triangular lattice shown, considering the properties of the aforementioned gauge groups, the most important of which is the size of the gauge group. Below, it will be proven that the size of the aforementioned gauge group is... , where L represents the system size, that is, the length of the qubit array in one of the linear dimensions of the system.

[0086] For example, in Figure 5 When the qubit array shown is an equilateral parallelogram, its size is equal to the number of qubits on one side. The total number of qubits in the qubit array is... .

[0087] First, it can be proven that the above system For example, in the above structure, one can select... One generator is generated, and from this, all other generators are generated. Note that... Includes The upper triangle and You can select one of the lower triangular elements. The upper triangle and The lower triangular element.

[0088] As shown in the image, for example, you can select the leftmost side. The upper triangle of the column, and the bottom left corner A downward-pointing triangle. Notice the same column... Multiplying the two upper triangles yields two columns. Multiply.

[0089] To simplify the discussion, we can define a circle operator. ,in For the system dimensions as described above. The operator is defined as follows: .

[0090] In the above formula, the indicator The index corresponding to the Pauli operator, and the index This corresponds to row, column, or diagonal indicators. It should be understood that for periodic boundary conditions, the number of rows, columns, or diagonals corresponds to the system size L. Therefore, the value range of indicator i in the above formula is 1 to L.

[0091] Figure 6 A schematic diagram of a circle operator according to some embodiments of the present disclosure is shown. Accordingly, the circle operator... It represents the product of Pauli operators of the same type that belong to the same row, column, or diagonal.

[0092] Figure 6 Operator 61 in the diagram represents the product of all Pauli X operators along the i-th diagonal, corresponding to... Operators. Operator 62 is the product of all Pauli Y operators along the j-th column, corresponding to... Operators. Operator 63 is the product of all Pauli Z operators along the k-th row, corresponding to... Operators.

[0093] As an example, based on the relationship between Majorana fermions and Pauli operators in a quantum system, we can define... When it is a row indicator , When listing indicators , When it is a diagonal indicator This is so that the quantities in the triangular lattice can be represented by circle operators later.

[0094] According to the standard group elements in the triangular lattice, i.e. Figure 5 The triangular operator shown, due to A loop operator has an even number of overlapping lattice points with its corresponding triangular operator, and each lattice point is anticommutative; or it has an odd number of overlapping lattice points and is identical to the Pauli operator. Therefore, the loop operator satisfies the following condition: In other words, the circle operator commutes with all elements in the canonical group.

[0095] Back Figure 5 Based on the above operators, the same column The product of the upper triangles can be expressed as: Similar to the same column The product of the lower triangles can be expressed as: In other words, the product of the upper triangles and the product of the lower triangles in the same column are equal.

[0096] Therefore, the remaining lower triangle at the top of the first column can be accessed through all the triangles in the leftmost column. The upper triangle and A lower triangle is generated. Similarly, the previous triangle can be generated. Repeat this process for each column to obtain the top row. A lower triangle.

[0097] As for the triangle in the rightmost column, it can be determined by looking at the left side. Triangle generation. For example, considering the product of all the triangular operators in the dashed section, after mutual cancellation, we get the large triangular operator 501, whose upper vertex is... Clockwise in order and Due to the periodic boundary condition, this operator is equivalent to the lower triangular operator 502. By calculating each of the larger triangular operators, all the triangular operators in the rightmost column can be obtained.

[0098] In summary, through A set of triangular generators can generate all triangular operators in the system, thus yielding the canonical group. Therefore, the rank of the canonical group can be obtained. .

[0099] Regarding the above The existence of multiple triangular generators can be used to further verify that these generators are linearly independent, meaning that the product of any number of triangles will not yield a unit element.

[0100] For example, we can assume that the above... A subset of triangles The product of all elements in the array is the identity element. For the upper right lattice point 511 of these triangles, due to the boundary conditions, the triangle including this lattice point is only the first... The top and bottom two upper triangles of the column, and the product of any one or both of these triangles is not a unity operator at the top-right grid point P. Therefore, the above set... The middle does not include the first The top and bottom two triangles of the column.

[0101] Based on this, for grid point 512 to the left of grid point 511, similarly, the triangle including this grid point is only the first... The top and bottom two upper triangles of the column require the set to be... Excluding the The top and bottom two triangles of the column.

[0102] And so on, until the final set This excludes all the upper triangles in the first and last rows. At this point, we can further examine grid point 513 below grid point 511, and similarly, we can... Line 1 An upper triangle and a lower triangle of the column are excluded from the set. In addition, and similarly, the first All other triangles in the row are excluded.

[0103] In summary, the final set that satisfies the conditions is obtained. It is an empty set, meaning there is no partial triangle product that has an identity element. In other words, the above... The three triangular generators are the independent generators of the canonical group.

[0104] Based on the determination of the canonical group, we can further consider the corresponding stable subgroups. Based on the stable subgroup property described above, i.e. It can be achieved through standardized groups The triangular operators construct elements that commute with the canonical group.

[0105] As mentioned earlier, multiplying all upper triangles or all lower triangles in the same row, column, or diagonal will result in the product of two circle operators, i.e., a double circle operator. . Figure 7 A schematic diagram of a double-loop operator according to some embodiments of the present disclosure is shown.

[0106] Figure 7 Operator 71 in the diagram is the product of all Pauli Z operators along the i-th and (i+1)-th rows, corresponding to Operators. Operator 72 is the product of all Pauli Y operators along the j-th and (j+1)-th rows, corresponding to... Operators. Operator 73 is the product of all Pauli X operators along the k-th and (k+1)-th rows, corresponding to... Operators.

[0107] Due to the circle operator themselves and the norm group Since bicyclic operators commute, they also commute with the canonical group. Furthermore, bicyclic operators can be constructed using canonical group generators, thus they belong to a stable subgroup. .

[0108] Since there are L triangles in each row, column, and diagonal, each double-loop operator... There are also L such instances. In other words, a total of [number] can be constructed using the double-circle operator. One generator.

[0109] However, the same type indivual Generators are not independent; the product of L double-loop operators of the same type is the identity element. Therefore, after initial screening, there are a total of There are 1 generator. Next, we need to further determine the generator. Are the generators linearly independent?

[0110] In system size In the case of an odd number, the above can be proven. Each generator is linearly independent.

[0111] For example, we can assume that there is a subset of them. If the product of all elements in the integer is a unit, then since and The product obtained by constructing the subset is not necessarily equal to the identity element. The middle needs to include .

[0112] And when there exists When, if subset Only exists in Because only For a double-loop z-operator, the product is not equal to zero. Therefore, subsets The middle part will simultaneously contain double-loop x operators and double-loop y operators in order to cancel out the double-loop z operators.

[0113] For a certain row of circle operators In order to make the final product a unit element, its corresponding and They need to be respectively This ensures that the product at each vertex of the row is a unit element. However, since the product of a double-loop x operator and a double-loop y operator always contains an even number of elements... and an even number of Therefore, when When the integers are odd, there are no corresponding double-loop x-operators and double-loop y-operators, thus proving the subset... It must be an empty set, that is Each generator is linearly independent.

[0114] In system size If the number is even, then it can be obtained through the same type. indivual If all lattice points are covered, the product of these operators is the Pauli operator for all lattice points. Multiplication. Therefore, the products of the three types of double-loop operators can be multiplied together, corresponding to the multiplication of three different Pauli operators at each lattice point, to obtain the identity operator.

[0115] For the remaining You can refer to this. A method for proving independence when the number of elements is odd. First, based on the above properties, generators can be eliminated, for example. Assuming the rest There exists a subset of double-circle operators. If the product of all elements in a subset is identity, then the subset... Need to include .and This will include those located in the first The grid points of the row, for which the generator does not contain Therefore, a unit element cannot be formed at this lattice point. Thus, the subset... It is an empty set, indicating Each generator is linearly independent.

[0116] As can be seen, when the system size is even, the above construction actually provides Each is an independent generator.

[0117] Furthermore, an additional generator can be constructed for the aforementioned quantum system. For example, you can multiply the lower right half of the lower triangle to get... In other words, the above generators It belongs to the canonical group G. Furthermore, since the circle operator commutes with the operators in the canonical group G, this generator also commutes with each operator in the canonical group G.

[0118] For size In the case of an odd number, since the product of the double-loop operators in each type cannot fill the entire triangular lattice, there is exactly one row, one column, or one diagonal remaining. Therefore, the generator... It can be used as previously constructed We obtain linearly independent generators.

[0119] Regarding size If the number is even, then it can be proven that... Unable to use We obtain linearly independent generators. For example, since for any The identity operator needs to have an even number of elements. .and There is 1 of each Multiplying by a double-loop operator always results in an odd number, making it impossible to obtain a unity operator.

[0120] In summary, based on the above analysis, it can be proven that for any system size, a stable subgroup... Size satisfies .

[0121] As mentioned earlier, the number of logical bits in the system can be expressed as: It should be greater than or equal to 0. Therefore, we can obtain... .

[0122] Furthermore, due to the number of logical bits If the integer is an integer, under the above conditions Only and Two possible values. When hour , and when hour .

[0123] The following will continue to demonstrate the number of logical bits. Therefore, combining the above proof results, we can obtain the number of logical bits. Stable subgroup size .

[0124] For example, logical operators can be defined as single-line operators. This operator commutes with the canonical group, but it does not belong to the canonical group.

[0125] Specifically, operators can be defined. Parity Pauli operator along a certain diagonal direction The parity of the number of elements. The corresponding does not include The contribution to parity is 0, while and Each includes one ,contribute .

[0126] Then single-line operator The initial parity is And any triangular operator always corresponds to parity. Therefore, it is impossible to obtain an operator with parity of 1 by multiplying triangular operators. Based on this, the above operators Not belonging to the normative group And with the normative group Since it commutes, this operator belongs to the logical operators of logical bits.

[0127] In summary, it can be proven that the number of logical bits... And completed the proof that for any system size: .

[0128] Based on the above discussion, we can further define complete logical operators, for example, as follows: .

[0129] All of the above logical operators commute with the canonical group and satisfy the Pauli commutation relation among themselves.

[0130] Furthermore, the code distance of the error-correcting code given by the logical bits can be calculated. Since logical operators are not unique, any logical operator multiplied by any stabilizer will result in a new operator with the same effect in the code space as the original logical operator. Therefore, the code distance of the error-correcting code does not depend on any specific logical operator, but rather on the minimum weight of elements in the entire equivalence class of logical operators.

[0131] For logical operators, it can be based on a logical operator and stable subgroups To define its equivalence class. For example, all equivalence classes with stable subgroups. Each element in the set represents a commutative operator. Treating operators with similarly stable operators in this set as belonging to the same class yields the equivalence class of logical operators, denoted as . That is, the element that commutes with S is removed from S.

[0132] Furthermore, the equivalence classes of logical operators can also be represented as That is, to represent the element The leading group The aforementioned accompanying group The weight of the element with the smallest weight in the error correction code is the code distance.

[0133] Returning to the stable subgroup example above, note that the stable subgroups given by the canonical group include all bicyclic operators. , , The aforementioned double-loop operator can be applied to previously obtained logical operators. superior.

[0134] First consideration ,Will Double-circle operator Effect on From above, one can obtain Non-overlapping Now consider the rest. Operators, because they are all related to Non-overlapping Opposition is easy, therefore The size of the operator is at least 1 .

[0135] Similarly, it can be Double-circle operator Effect on From above, one can obtain Non-overlapping Considering this at this time Operator, because it is related to Non-overlapping Opposition is easy, therefore Operator size is at least .

[0136] In summary, it can be proven that the size of a logical operator is at least 1 / 2. The above logical operators The size is already satisfied. Therefore, the code distance of the error-correcting code corresponding to this logical operator is: .

[0137] The preceding text has elaborated on the illustrative definitions of gauge groups, stabilizers, and logical operators in quantum systems. Now, we will return to... Figure 1 Using the crystal lattice mentioned above as an example, we will continue to introduce the specific implementation of error-correcting codes.

[0138] In step S1, the measurement operator can be determined based on the gauge group of the quantum system. As mentioned earlier, the gauge group can be represented as: .

[0139] in, That is, the stable subgroup of the quantum system To measure the error-correcting code, further measurements are needed. The specific form.

[0140] Figure 8 A schematic diagram of a measurement operator according to some embodiments of the present disclosure is shown. For example... Figure 8 As shown, in a spin triangular lattice, the parallelogram obtained by multiplying an upper triangle and a lower triangle corresponds to the multiplication of four identical Pauli operators.

[0141] For example, Figure 8 The parallelogram structure 81 corresponds to the Pauli x operator multiplication at four lattice points. Figure 8 The parallelogram structure 82 corresponds to the Pauli y operator multiplication at four lattice points. Figure 8 The parallelogram structure 83 corresponds to the Pauli z-operator multiplication of four lattice points.

[0142] Therefore, parallelogram operators of the same type can be defined as follows: The aforementioned parallelogram operator can be used as a measurement operator in the error correction process. In other words, the measurement operator in this embodiment corresponds to multiple generators in the canonical group, rather than directly to the generators.

[0143] The above construction method enables the measurement operator to be adaptively applied to various boundary conditions, and the measurement operator can be applied to quantum computing platforms of different systems through the conversion between Majorana fermion operators and spin Pauli operators, thereby improving the versatility of error correction codes.

[0144] Next, we will continue to discuss the properties of the set consisting of all parallelogram operators of the same type.

[0145] First, consider the system size. For odd numbers, use the parallelogram operator. For example, the rank of the group formed by this set as generators is This can be understood as shared ownership. Parallelogram Operators .

[0146] Specifically, one can consider the linearly independent parallelogram operators in the aforementioned group. . Figure 9 A schematic diagram of a parallelogram operator generator according to some embodiments of the present disclosure is shown.

[0147] Because of all of any row or any column Multiplication is a unit element, which can be used indivual We get the last one, so we can use Parallelogram operator Get all There are several parallelogram operators. These parallelogram operators are linearly independent; therefore, the parallelogram operator... The rank of the group consisting of generators is ,For example Figure 9 The parallelogram operator included in structure 91, i.e., the front row, before Parallelogram operator within a column.

[0148] also, Parallelogram operator It can also be equivalent to the lower left. Parallelogram operator ,For example Figure 9 The parallelogram operator included in structure 92, i.e., the front row, before Parallelogram operators within a column, and Double-circle operator ,For example Figure 9 Operator 93 in the example. The two representations have the same rank.

[0149] Similarly, for the other two parallelogram operators and Similarly, it can be proven that the rank of the group formed by them is... .

[0150] Furthermore, it can be further proven that by Parallelogram operator The generated group is composed of the remaining two types of double-loop operators, namely indivual and indivual The resulting group intersection is empty.

[0151] For example, assuming the intersection above is not empty, since the parallelogram operator is Then the operators in the intersection can be represented by Pauli operators. This is indicated by... On the other hand, in order to represent this using a double-loop operator, it is required that... and Passing through the grid point occupied by the symbol.

[0152] Consider a grid point in the intersection, since To make the final product a Pauli operator, the following steps were taken... The product requires multiplying each cell in the row by . Therefore, the lattice points in the above intersection will require all cyclo operators. Multiplication, and then the circle operator for all rows is required. Multiplying these together yields the Pauli operator for all lattice points. Multiply.

[0153] However, because the product of circle operators includes an odd number of Pauli operators. However, parallelogram operators can only represent an even number of Pauli operators. Therefore, the product of the above circle operators cannot be used. Parallelogram operator Generate. In other words, the intersection of the groups formed by the above generators is empty.

[0154] The three generators mentioned above, namely Parallelogram operator Two types of double-loop operators indivual and indivual The total number is That is, the canonical group The largest mutually commutative subgroup that can be obtained.

[0155] In other words, if this can be measured If all qubits are determined, then eventually only one logical bit will remain.

[0156] Similarly, for the case where the system size L is even, a similar proof can be used. Parallelogram operator The generated group and the remaining indivual , indivual and a three-circle arithmetic operator The resulting group intersection is empty.

[0157] The total number of the above generators is That is, the canonical group The largest mutually commutative subgroup that can be obtained.

[0158] In other words, if this can be measured If all qubits are determined, then eventually only one logical bit will remain.

[0159] Based on the above analysis, it can be determined that the operator corresponding to the parallelogram structure can be used as a measurement operator to realize quantum computing error correction based on the gauge subsystem.

[0160] In other words, when constructing a measurement operator through a gauge group, a measurement operator can be constructed using multiple generators in the gauge group, thereby enabling the measurement operator to be applied to quantum systems of various sizes and boundary conditions. Furthermore, the aforementioned measurement operator can be applied to quantum computing platforms of different systems through the conversion between Majorana fermion operators and spin Pauli operators, thus improving the versatility of error-correcting codes.

[0161] The following section will further elaborate on the measurement process based on different measurement operators, so that the corresponding instantaneous stable subgroups during the measurement process correspond to the largest commutative subgroups mentioned above.

[0162] In some embodiments, determining a measurement operator based on the gauge group of a quantum system includes: determining a measurement operator corresponding to the size of the quantum system, the gauge group, and the size of the quantum system including the length of the qubit array in one of its linear dimensions.

[0163] As mentioned earlier, the properties of a quantum system differ depending on whether its size is odd or even. Therefore, to achieve error correction in quantum computing, different measurement operators can be used for odd and even sizes.

[0164] For example, determining the measurement operator corresponding to the size of the quantum system and the gauge group may include: determining a candidate measurement period corresponding to the size based on the size; and determining the measurement operator based on a target measurement period among the candidate measurement periods and the gauge group.

[0165] The measurement period of quantum error correction can be divided into two cases: single measurement period and multiple measurement period.

[0166] Single-cycle measurement can correspond to the basic unit of static error correction, that is, complete all local operator measurements and obtain a set of transient symptoms within one cycle, thereby performing a single error inference and correction.

[0167] Multiple measurement cycles can form the basis of dynamic error correction. By measuring multiple cycles, the stable quantum and logical bits in the quantum system can be dynamically updated, thereby realizing error correction in quantum computing.

[0168] Below, we will first introduce the measurement process with reference to the case where the size of the quantum system is odd.

[0169] When the size of the quantum system is odd, the candidate measurement period can be determined as the first candidate measurement period.

[0170] For example, the first candidate measurement period can include a single measurement period and three measurement periods. The measurement implementation of the error correction code in these two cases will be described below.

[0171] First, we introduce the three measurement periods, specifically the case where the measurement period T=3. At time T=0, all parallelogram operators can be measured. At time T=1, all parallelogram operators can be measured. At time T=2, all parallelogram operators can be measured. .

[0172] Through the above three-period measurements, starting from time T=3, we can obtain the instantaneously stable subgroup that varies with time, which is equivalent to the largest mutually commutative subgroup mentioned above.

[0173] Measure all parallelogram operators at T=0. The corresponding stable subgroups are: .

[0174] Measure all parallelogram operators at T=1. , with all Measurement is easy The generator of a subgroup is Therefore, a new stable subgroup can be obtained after measurement: .

[0175] Measure all parallelogram operators at T=2. , with all Measurement is easy The generator of a subgroup is Therefore, a new stable subgroup can be obtained after measurement: .

[0176] After that, starting from T=3, all parallelogram operators can be measured again. A new stable subgroup is obtained: .

[0177] As can be seen, in time Then, cyclic measurements were performed according to three measurement cycles, and the number of generators was: .

[0178] Through the steady-state subgroup, we can obtain the logical bits that commute with it and the corresponding logical operators. The number of corresponding logical bits is 1, and the corresponding logical operators are... .

[0179] Using a method similar to that used to determine the code distance of the error-correcting code above, the code distance of this three-cycle error-correcting code can also be determined to be... .

[0180] In summary, when the target measurement period is three measurement periods, the measurement operators used for measurement include: , , It can be determined that the measurement operator includes a measurement operator corresponding to a parallelogram structure, and the parallelogram structure corresponds to two triangle generators whose sides intersect.

[0181] Furthermore, since the measurement with a period of T=3 involves all three parallelogram operators, as mentioned earlier, each triangular operator corresponds to four Majorana fermions based on the transformation relationship between fermions and spin.

[0182] for , The operator, since the two Majorana fermions in each triangle will cancel each other out, therefore , The operator involves the measurement of four fermions. And for... The operator, the fermions between the two triangles, do not cancel each other out, therefore The operator involves the measurement of eight fermions.

[0183] In other words, in a measurement process with a period of T=3, in order to measure all three parallelogram operators, two four-fermion measurements and one eight-fermion measurement are required. That is, the measurement operators include a combination of two four-body joint measurement operators and one eight-body joint measurement operator.

[0184] The above text introduced the situation of three measurement cycles. The following will combine... Figure 10 Next, we will discuss the case of a single measurement cycle. Figure 10 A schematic diagram of a single-cycle measurement operator according to some embodiments of the present disclosure is shown.

[0185] In a single measurement cycle, the actual measurement obtained is the statically stable subcode. A single measurement can determine the corresponding stable subgroup. Furthermore, operators that commute with stable subgroups but do not belong to stable subgroups are identified as logical operators.

[0186] like Figure 10As shown, when the target measurement period is a single measurement period, the measurement operator may include a measurement operator corresponding to the rectangular structure, and the rectangular structure corresponds to the two triangle generators intersecting the vertex.

[0187] For example, multiplying two triangles 1011 and 1012 yields a rectangular operator 101, which can be represented as: Different rectangular operators constructed in the above manner can commute with each other.

[0188] In system size When it is an odd number, we can get There are _ ... Each of the linearly independent generators can be determined by simultaneously measuring all the rectangular operators. One stable element is selected, and one logical bit is left.

[0189] Based on the above stabilizers, the corresponding logical operators can be defined as follows: .

[0190] Two logical operators satisfy an anti-commutation relation, that is... Furthermore, all logical operators commute with stable subgroups; therefore, if one of the logical operators belongs to a stable subgroup, the condition cannot be satisfied. ,visible None of them belong to a stable subgroup; they are logical operators of a single logical bit.

[0191] For the code distance of this rectangular error-correcting code, based on the measured rectangular operator, the distance along the left boundary direction can be... Multiplying rectangular operators together yields a double-loop operator. Applying double-loop operators to logical operators Up, because Since it is an odd number, all of them can be obtained. Non-overlapping .

[0192] The above operators are all related to logical operators. Opposition is easy, therefore arbitrary Certainly with Non-overlapping Each has at least one overlapping grid point, indicating The size is at least Similarly, and Its size is at least L.

[0193] Following a similar approach as above, the error correction code distance corresponding to this measurement is also L.

[0194] In summary, when the target measurement period is a single measurement period, the measurement operators used for measurement include: It can be determined that the measurement operator includes a measurement operator corresponding to the rectangular structure, which corresponds to the two triangle generators intersecting the vertex of the rectangular structure.

[0195] In a measurement process with a period of T=1, the rectangular operator can be directly implemented through four fermion measurements, that is, the measurement operator includes a four-body joint measurement operator.

[0196] The above text elaborated on the case where the system size is odd. Below, we will continue with the case where the system size is even.

[0197] When the size is an even number, the candidate measurement period can be determined as the second candidate measurement period.

[0198] For example, the second candidate measurement period may include a single measurement period, a double measurement period, and a triple measurement period.

[0199] Furthermore, when the system size is an even number, it can be further divided into two types based on whether the size is a multiple of 4. Below, we will introduce the error correction code implementation methods under three measurement cycles, according to the two sizes.

[0200] First, consider a size of , For any integer, a three-period measurement is performed. The three-period measurement method can be similar to that described earlier, i.e., measuring all parallelogram operators at T=0. Measure all parallelogram operators at T=1, and measure all parallelogram operators at T=2. , and then repeat.

[0201] At T=0, measure all parallelogram operators. A stable subgroup can then be obtained: .

[0202] At T=1, measure all parallelogram operators. .

[0203] To represent the stable subgroup at time T=1, we can generalize the definition of the circle operator, for example, by defining the circle operator... Defined as representing along the number straight line All Pauli operators The product of, where, This represents a line parallel to a horizontal edge, a vertical edge, or a diagonal. Based on the updated definition, the previously defined circle operator can be updated to... .

[0204] At time T=1 and all Measurement is easy The generator of the subgroup is Therefore, a new stable subgroup can be obtained after measurement: .

[0205] Note that at this point, due to the measurement of all parallelogram operators... New generators will be introduced, containing linearly independent ,as well as indivual And the original generator It can be equivalent to .

[0206] In addition, at this time The number of generators is The original norm group Three-circle operator in Not in middle.

[0207] At T=2, measure all parallelogram operators. With all Measurement is easy The generator of a subgroup is Therefore, a new stable subgroup can be obtained after measurement: .

[0208] Starting from T=3, all parallelogram operators can be measured again. A new stable subgroup is obtained: .

[0209] As can be seen, after time T≥3, the measurement is performed cyclically according to three measurement cycles, and the number of generators is: .

[0210] The corresponding number of logical bits is 2. The specific representation of this logical operator is given below.

[0211] Since the instantaneously stable subgroup obtained by measuring the generators in the canonical group during the above measurement process remains a logic bit under the above measurement, the logic bit that commutes with the canonical group remains a logic bit. Furthermore, this logic bit maintains commutation with the stable subgroup throughout the above measurement process, and therefore can be called a static logic bit. The logical operator corresponding to this logic bit is... .

[0212] Furthermore, because the number of generators in the instantaneously stable subgroup is always 2 when T>2. Then there exists another logical bit. Based on the above discussion, a logical operator for this logical bit is the three-circle operator. This operator commutes with all stable subgroups, and also with the logical operator of the first logical bit. Commutation. This operator can be written as .

[0213] Based on this, the logical operator of this logical bit It can be defined as This refers to the product of even-numbered circle operators. Due to system size... ,so correspond indivual Operator multiplication. This operator is multiplied by all parallelogram operators. It commutes, therefore it belongs to the logical operators. Also, operators... satisfy: .

[0214] Therefore, logical operators and An equivalent commutation relation satisfying two independent spin operators is a logical operator consisting of two logical bits.

[0215] Note the logical operators and Always with all parallelogram operators Commutation. Therefore, both logical operators are static logical operators.

[0216] Following a similar method, by analyzing the minimum weights of the logical operators of the aforementioned logical bits at each time step, it can be determined that two logical bits always have a code distance of at any given time. .

[0217] In summary, compared to the odd-numbered case, the method for performing three-cycle measurements on the system when the system size is even is similar, namely, measuring all three parallelogram operators in three cycles. The main differences lie in the different stabilizers used in the measurement process, and the different number and representation of logic bits.

[0218] For the system dimensions described above, when performing two-cycle measurements, all parallelogram operators can be measured when T is an even number. When T is odd, all parallelogram operators can be measured. .

[0219] Similarly, the changes in the stabilizers during the aforementioned two-period measurement process can be determined. At time T=0, all parallelogram operators can be measured. A stable subgroup is obtained: .

[0220] At time T=1, all parallelogram operators can be measured. A stable subgroup is obtained: .

[0221] At T=2, all parallelogram operators can be measured again. A new stable subgroup is obtained: .

[0222] Furthermore, starting from T>0, the number of generators in the stable subgroup is always 0. Therefore, two logical bits are given. The corresponding logical operators are the same as before, that is... Furthermore, both logic bits are static logic bits.

[0223] Because the instantaneously stable subgroups at odd and even times correspond to the first two instantaneously stable subgroups in the previous three-cycle measurements, respectively. Therefore, the aforementioned logical bits also have a code distance of... .

[0224] In summary, when the target measurement period is a dual measurement period, the measurement operators used for measurement include: , Similarly, it can be determined that the measurement operator includes the measurement operator corresponding to the parallelogram structure.

[0225] Furthermore, the above measurement method is merely exemplary; a measurement with a period of T=2 only requires two parallelogram operators, for example, it can be achieved through... , Measurements can also be taken through... , Take measurements.

[0226] As mentioned above, , The operator involves four fermion measurements, while The operator involves eight-fermion measurements. In other words, in a measurement process with a period of T=2, the measurement operator includes a combination of two four-body joint measurement operators, or a combination of one four-body joint measurement operator and one eight-body joint measurement operator.

[0227] The above text introduced the cases of three measurement cycles and two measurement cycles. The following will combine... Figure 11 We will now continue to discuss the case of a single measurement cycle in even-numbered systems. Figure 11 A schematic diagram of a single-cycle measurement operator according to other embodiments of the present disclosure is shown.

[0228] In the case of single-cycle measurement, similar to the method used when the system size is odd, all rectangular operators can be measured. In this case, all rectangular operators commutate with each other and can be divided into two groups, such as... Figure 11 The solid-line rectangles and dashed-line rectangles are shown.

[0229] like Figure 11 As shown, the grid points occupied by the dashed and solid rectangles do not overlap, thus the stable subgroup can be divided into two independent parts. Each part is a text-square grid model, which is equivalent to a toroidal code. In this case, the system size of each equivalent model is odd.

[0230] It is important to note that due to the distorted periodic boundary conditions, each lattice model in this case is not equivalent to the model with an odd system size under the usual periodic boundary conditions. Specifically, there are two constraints for each independent part, meaning there are two unequal ways to obtain the identity element.

[0231] For example, multiplying all the shaded rectangle operators together yields a unity element under the twisted boundary condition. Similarly, multiplying all the remaining solid rectangle operators also yields a unity element. The dashed rectangles also have two constraints.

[0232] Therefore, after applying the four constraints, we can obtain... There are linearly independent generators, i.e., the number of logical bits is . .

[0233] Below, the corresponding logical operators will be defined. Since the rectangular algorithm can be decomposed into two independent parts, each part can be defined with two logical bits. Taking a lattice composed of solid-line rectangular operators as an example, the corresponding circle operator is defined as follows: for The portion confined to this lattice yields the following logical operators: .

[0234] In the above formula, , These refer to the circle operators with odd numbers. , The portion projected onto the solid-line rectangular lattice. , These refer to the even-numbered circle operators. , The portion projected onto the solid-line rectangular lattice.

[0235] The above and Each of these operators satisfies the anti-commutation relation, and the pairs commute with each other, thus they are equivalent to two sets of spin operators. Furthermore, all of the above operators commute with stable subgroups, and none of them belong to stable subgroups. Therefore, the above operators give logical operators for two logical bits.

[0236] The code distance of this error-correcting code will be discussed further below. Similar to the previous discussion, based on the measured rectangular operator, , and The size is at least ,and , and The size is at least Therefore, the code distances of the above error correction codes are all... .

[0237] Similarly, the original circle operator can be restricted to a lattice composed of dashed rectangular operators to obtain another two logic bits of logic operators, namely: .

[0238] In the above formula, , These refer to the circle operators with odd numbers. , The portion projected onto the dashed rectangular lattice. , These refer to the even-numbered circle operators. , The portion projected onto the dashed rectangular lattice.

[0239] Following a similar proof, it can be obtained that the error-correcting code distances of the above logical operators are all... .

[0240] In summary, the method for single-cycle measurement of the system is similar when the system size is even, compared to the odd-numbered case: simultaneously measuring the rectangular operators in the system. The main difference lies in the different constraints during the measurement process, leading to differences in the number and representation of logic bits.

[0241] The above text describes the system size as... The situation has been described in detail. Next, we will continue to introduce the system size. The situation.

[0242] for The three-period measurement method is similar, that is, measuring all parallelogram operators at T=0. Measure all parallelogram operators at T=1 Measure all parallelogram operators at T=2 , and then repeat.

[0243] exist and Under these conditions, the instantaneously stable subgroups have the same form, therefore the system also has two logical bits. However, due to The logical operator of the second logical bit is no longer used, and the logical bit will have a different form.

[0244] First, due to logical operators It commutes with all instantaneously stable subgroups, but has no additional logical operators. Since it commutes with all the transiently stable subgroups, it can be determined that the second logic bit should be a dynamic logic bit, or a semi-dynamic logic bit.

[0245] For example, consider all parallelogram operators The generated group, that is, the group generated by considering all measurement operators simultaneously, is denoted as Due to the above Belongs to a group consisting of generators of the canonical group, and it can be proven that in hour Then it can be proven that there are no static logical operators. .

[0246] Figure 12 A schematic diagram of a verification operator according to some embodiments of the present disclosure is shown. For example... Figure 12 As shown, the above three-circle operators This can be represented by the product of all the upper triangular operators in the lower right half of the system section 1201. The product of these upper triangular operators can then be used as a test operator to prove that no static logical operator exists. On the other hand, for two adjacent upper triangles, the product of two different parallelogram operators can be used. For example, the product of upper triangle 1211 and upper triangle 1212 can be used to represent the product of the corresponding parallelograms 1221 and 1222, and the repeated operators will cancel each other out.

[0247] In other words, the pairwise multiplication of the three different types of parallelogram operators in the diagram is equivalent to the product of two adjacent upper triangles. Therefore, for the lower right... The large triangle, when At this point, all the upper triangles can be covered by trapezoids as shown in the figure. In other words, the product of all the upper triangle operators in the lower right half can be constructed using three different parallelogram operators, i.e., the three-circle operator. That is to prove .

[0248] As mentioned earlier, the logical operator of the second logical bit can be determined. This changes over time, that is, with different measurements within the measurement period. The logical operators will be further explained below. The specific evolutionary form over time.

[0249] At T=1, the largest stable subgroup has been obtained. Now, consider the single-cycle operator. This operator is compatible with all parallelogram operators. and The commutative nature of this subgroup is therefore consistent with the current instantaneously stable subgroup. Easy.

[0250] In addition, single-cycle operators AND logical operators exist Time opposes ease, therefore single-circle operator Not belonging to the instantaneously stable subgroup .

[0251] The above single-cycle operators That is, the instantaneous logical operator at that moment. It commutes with stable subgroups, does not belong to stable subgroups, and is associated with logical operators. Oppose easy, and the logical operator with the first logical bit. Easy.

[0252] At T=2, the previous instantaneous logical operator Not with all parallelogram operators Since they are interchangeable, we need to consider equivalent logical operators. Figure 13 A schematic diagram of a transformation operator according to some embodiments of the present disclosure is shown.

[0253] Transformation operators are, for example, operators constructed after a measurement at the previous time T=1, used to convert a logical operator at the previous time into another equivalent representation, thereby making the logical operator commutative with the measurement at the current time.

[0254] like Figure 13 As shown, in At that time, the parallelogram operator in the lower right corner can be... For example, multiplying parallelograms 1301 to 1304 together yields a three-circle operator. The three-circle operator It can be used as a transformation operator to realize the evolution of logical operators.

[0255] After measurement at T=1 Then, the three-circle arithmetic operator can be used. With instantaneous logical operators Multiplying them together yields an instantaneous logical operator with T=2. The operator With all parallelogram operators Easy, therefore That is, the measurement is completed at T=2. The instantaneous logical operator after that.

[0256] Similarly, when T=3, the lower right parallelogram operator can be used. Multiplying them together yields a three-circle operator. The three-circle operator With instantaneous logical operators Multiplication yields a new instantaneous logical operator. .

[0257] Similarly, when T=4, the lower right parallelogram operator can be used. Multiplying them together yields three circles of operators. The three-circle operator With instantaneous logical operators Multiplication yields the previously defined instantaneous logical operator. .

[0258] Therefore, the above logical operators can be understood as dynamic logical operators that evolve according to a three-cycle pattern, that is: .

[0259] For the two logical bits mentioned above, another dynamic logical operator representation can also be defined, namely: at T=0 (measurement... ), , ; at T=1 (measurement) ), , ; at T=2 (measurement) ), , .

[0260] For different logical operators The instantaneously stable subgroup at this point can also be given. Given a set of non-overlapping equivalent logical operators, we can prove that the size of each logical operator is at least 1. Therefore, the code distance between the two logical bits mentioned above is... .

[0261] for Dual-cycle and single-cycle measurements under certain conditions, and their relationship with Measurements under similar conditions will not be repeated here.

[0262] The above are various implementations of measurements for quantum systems of different sizes under periodic conditions. Below, we will combine... Figure 14 Further discussion on error correction code design under open boundary conditions. Figure 14 A schematic diagram of an open-boundary lattice according to some embodiments of the present disclosure is shown.

[0263] Based on the previous transformation relationship between spin operators on triangular lattices and Majorana operators on honeycomb lattices, i.e. .

[0264] Under open boundary conditions, consider as follows Figure 14 The numbering relationship is shown by the dashed line, where the endpoints with the same number are connected in pairs, starting from 1 in the upper left and ending at 1 in the lower right.

[0265] The above numbering has been traversed All Majorana fermions on the crystal lattice. Based on the above numbering method, consider... Figure 14 The bolded area includes all of structure 1401. The three-spin operators corresponding to these triangles can still be derived from the four Majorana fermion operators in the local area according to the above transformation relationship and numbering order. In particular, the transformation still holds for the blue triangle operators on the boundary.

[0266] Conversely, for the rightmost column, namely the triangles included in structure 1402, the corresponding three-spin operators cannot be represented by four local Majorana fermion operators. In other words, spin error-correcting codes with periodic boundaries cannot be directly translated into error-correcting codes represented by Majorana fermions using the above transformation. However, for conditions with one side open, this can be addressed by considering the above... Transform a set of triangles, which can be represented by both spin operators and Majorana operators.

[0267] For the three different parallelogram operators previously considered As mentioned above, among which All of these can be represented using four Majorana fermion operators, and Therefore, it can only be represented using an eight-fermion operator. The previous rectangular operator, however, can be represented using a four-fermion operator.

[0268] Under open boundary conditions, the corresponding gauge group can be defined as: .

[0269] Based on previous proofs, at this time The upper triangle and The lower triangle can generate all of them. The upper triangle and A lower triangle is obtained, thus yielding the same gauge group under periodic boundary conditions. Based on this, all previous properties concerning gauge groups still apply, therefore it can be determined that: .

[0270] Based on the properties of static subsystem error-correcting codes, since we are now equivalently starting from a consistent gauge group, the measurement method under the aforementioned periodic boundary conditions can be completely repeated, and only the method derived from this can be considered. The upper triangle and The parallelogram operator and rectangle operator are generated from the lower triangle. Ultimately, the same structure as under periodic boundary conditions can be obtained.

[0271] The above describes the subsystem error correction code design proposed in this embodiment. Using the method described above, the measurement operator used in the quantum system can be determined for subsequent quantum computing error correction.

[0272] It can be seen that the measurement operator proposed in this embodiment can be applied equivalently to various types of quantum systems, various boundary conditions, and various measurement periods, thereby improving the versatility of quantum error correction methods.

[0273] Next, we will return to Figure 1 Combining steps S2 and S3, we will continue to introduce the actual implementation of quantum error correction.

[0274] In step S2, the stable quantum in the quantum system can be measured according to the measurement operator.

[0275] As mentioned earlier, the representation of the stabilizers in the system is also given for each measurement operator. In the above example, the measurement results of each stabilizer in the system can be determined based on the parallelogram operator or the rectangular operator, so as to perform quantum computing error correction.

[0276] In some embodiments, determining the measurement results of stableons in the quantum system according to the measurement operator includes: determining a stable subgroup in the quantum system based on the size of the quantum system and the measurement period of the quantum system; and performing measurements on the quantum system according to the measurement operator to determine the measurement results of each stableon in the stable subgroup.

[0277] In other words, since the measurement operators in the system are determined based on the size and measurement period, and the stabilizers should correspond to the measurement operators, the corresponding stabilizers can also be determined based on the size and measurement period of the system.

[0278] Furthermore, measuring the quantum system according to the measurement operator and determining the measurement results of each stable quantum in the stable subgroup may also include: when the measurement period is multiple measurement periods, updating the stable subgroup according to the measurement operator corresponding to each measurement period, and determining the measurement results of each stable quantum in the stable subgroup.

[0279] In other words, the stabilizer can be dynamic during multi-period measurements. For dynamic measurement schemes, it is also necessary to record the instantaneous stable subgroups and logical operator representations corresponding to different measurement moments. With each round of measurement, the aforementioned stable subgroups and logical operators can be updated as the measurement results of the transient stabilizer.

[0280] Based on the aforementioned Majorana XYZ error-correcting code lattice structure, gauge detection operator, stable subgroup, and logic operator, a complete experimental procedure can be established, from logic information input, error detection, decoding and correction to logic information reading. This procedure, in principle, does not depend on any specific hardware and can be applied to conventional equivalent spin qubit platforms, four Majorana qubit platforms, and lattice platforms directly composed of Majorana zero-energy modes. Specific measurement methods and readout devices can be adjusted according to the chosen experimental platform.

[0281] Before determining the results of the stabilizer measurement, each physical qubit, Majorana zero mode, or equivalent spin degree of freedom can first be initialized to a predetermined state. Then, a set of gauge detection operators or stabilizer operators are measured to project the system into the coding space defined by the Majorana XYZ error-correcting code.

[0282] After the encoding space is established, the quantum information to be protected can be written into the logical degrees of freedom by acting on corresponding logical operators, preparing logical states, injecting assisted qubits, or measuring projections. For example, a logical ground state can be prepared, or a general logical superposition state can be input. After the input is completed, the logical information is no longer stored in a single physical unit, but is represented by nonlocal logical operators throughout the entire encoding region.

[0283] After the logical information is input, the quantum system can be measured according to the measurement operators and measurement methods determined in the embodiments of this disclosure.

[0284] For example, in traditional qubit platforms, few-volume detection operators can be implemented using auxiliary qubits and combinations of single-qubit and two-qubit gates, while in four Majorana qubit or direct Majorana lattice platforms, they can be implemented using joint fermion parity measurement, controlled tunneling, quantum dot readout, or equivalent interference measurement.

[0285] The above measurement process only reads the canonical detection operator or stable sub-operator, without directly measuring the stored logical information, so as to keep the logical state as intact as possible while correcting errors in quantum computing.

[0286] After determining the measurement results of the stabilizer, step S3 can be continued to correct the error in the quantum system.

[0287] In some embodiments, correcting errors in the quantum system based on the measurement results of the stabilizers may include: determining an error set corresponding to the erroneous stabilizers based on the erroneous stabilizers whose measurement results do not match preset values, wherein each error in the error set, when applied to the quantum system, will cause the quantum system to include the erroneous stabilizer; and determining the location and type of physical errors in the quantum system based on the probabilities corresponding to each error in the error set.

[0288] For example, the raw signal output by the measuring device can first be amplified, thresholded, and calibrated to convert it into discrete measurement results corresponding to each detection operator, usually represented as +1 or -1.

[0289] The current measurement result can then be compared with the expected result, the measurement results of adjacent regions, or the results of previous and subsequent measurement rounds to construct an error syndrome. If certain detection operators indicate that the measurement result of the stabilizer has an anomalous flip, it indicates that a physical qubit error, fermion parity flip, quasi-particle poisoning, local coupling error, or measurement error may have occurred in the corresponding region.

[0290] To distinguish between data errors and measurement errors, multiple rounds of measurement can be performed continuously, and the measurement results can be saved uniformly according to spatial location and measurement time to form a syndrome history for decoding.

[0291] For example, after the measurement is completed, the obtained error syndrome can be input into a pre-set decoder. The decoder can determine the most likely error location and error type based on the prior probability of different errors, experimental noise model, lattice boundary conditions, and the spatial and temporal distribution of the syndrome.

[0292] Since error-correcting codes based on canonical groups are subsystem error-correcting codes, different physical errors may differ only in operators within the canonical group or stable subgroups and have the same effect on logical information. Therefore, during decoding, it is not necessary to uniquely determine every actual micro-error, but only to determine its equivalent error category and whether the error may be equivalent to a nontrivial logical operator.

[0293] If the inferred error corresponds only to a local canonical change or a stable sub-change, it does not alter the stored logical information. However, if the error chain spans the encoded region or is equivalent to a nontrivial logical operator, a logical error may occur. The decoder then provides corresponding correction operations and logical frame update information accordingly.

[0294] Errors in quantum systems can be corrected based on the decoding results.

[0295] In some embodiments, where the physical error includes a logical error, correcting the quantum system based on the measurement results of the stabilizer further includes: correcting the quantum system through an error correction operation to restore correct logical information in the quantum system; or recording the logical error and the current logical frame in the quantum system so as to compensate for the current logical frame during subsequent operations on the quantum system.

[0296] In other words, one approach is to perform a correction operation on the corresponding physical qubit, Majorana zero mode, or local coupling unit, so that the combination of the error and the correction operation belongs to the gauge group or a stable subgroup, thereby recovering the correct logical information.

[0297] Another approach is to avoid immediately applying actual operations to the quantum system. Instead, the inferred Pauli operation or parity change is recorded in the classical control system, updating the corresponding logic frame. This logic frame is then uniformly compensated for during subsequent quantum operations and the final readout. This method reduces noise introduced by additional control operations.

[0298] After correction, the corresponding detection operator can be measured again to confirm whether the error syndrome has disappeared. If the abnormal syndrome still exists, the measurement, decoding, and correction process continues.

[0299] After completing the predetermined quantum storage, quantum evolution, or logical operation, the logical operator is measured to read the final logical quantum state. Logic X, logic Z, or other desired logical operators can be measured directly, or the inverse operation of the encoding process can be performed first to remap the logical information onto one or more easily readable physical qubits before measurement.

[0300] For implementations employing periodic measurements, the specific representation of the logic operator at the final measurement time should be determined. The measurement results of the logic operator are then combined with the correction records, logic frame changes, and identified error types obtained during the decoding process to recover the original input logic information.

[0301] Therefore, even if local physical errors occur during the experiment, as long as the error rate is within the range that the error correction code can handle, and the decoding and correction process is successful, the original logical quantum information can still be read correctly in the end.

[0302] The above is a quantum computing error correction method proposed in the embodiments of this disclosure. It can determine the measurement operator used for quantum computing error correction based on the generator in the gauge group of the quantum system. It can be applied equivalently to various types of quantum systems and can be used under various boundary conditions to perform quantum computing error correction based on the gauge subsystem in the quantum system, thereby improving the versatility of the quantum error correction method.

[0303] The following is for reference. Figure 15 A quantum error correction apparatus according to embodiments of the present disclosure is described, which is used to perform any of the embodiments of the quantum error correction method described above. Figure 15 A block diagram of a quantum error correction device according to some embodiments of the present disclosure is shown.

[0304] like Figure 15 As shown, the quantum error correction device 15 includes: a measurement operator determination module 151, configured to determine a measurement operator based on the gauge group in the quantum system, wherein the measurement operator corresponds to a plurality of generators in the gauge group, and the gauge group characterizes the gauge degrees of freedom of the quantum system; a measurement module 152, configured to determine the measurement result of a stabilizer in the quantum system based on the measurement operator, wherein the stabilizer characterizes an invariant in the quantum system; and an error correction module 153, configured to correct errors in the quantum system based on the measurement result of the stabilizer.

[0305] The measurement operator determination module 151 of the quantum error correction device 15 can, for example, be used to perform... Figure 1 Step S1. The measurement module 152 of the quantum error correction device 15 can, for example, be used to perform... Figure 1 Step S2. The error correction module 153 of the quantum error correction device 15 can, for example, be used to perform... Figure 1 Step S3.

[0306] It should be understood that the quantum error correction device may also include other modules for performing other steps in the quantum error correction method of the embodiments of this disclosure, such as a logic information extraction module.

[0307] Figure 16 A block diagram of a quantum error correction device according to some embodiments of the present disclosure is shown.

[0308] like Figure 16 As shown, the quantum error correction device 16 includes: at least one first memory 161; and at least one first processor 162 coupled to the at least one first memory 161, the at least one first processor 162 being configured to execute the quantum error correction method as described in any of the foregoing embodiments based on instructions stored in the at least one first memory 161.

[0309] The first memory 161 is used to store one or more computer-readable instructions. The first memory 161 may include any combination of various forms of computer-readable storage media, such as volatile memory and / or non-volatile memory, including but not limited to random access memory, dynamic random access memory, static random access memory, read-only memory, and flash memory. The first memory 161 may, for example, store operating systems, application programs, boot loaders, databases, and other programs, as well as various application programs and various data.

[0310] The first processor 162 is configured to execute computer-readable instructions to implement the quantum error correction method described in any of the foregoing embodiments. Specific implementations of each step of the method can be found in the above embodiments, for example... Figure 1 The steps involved are repeated here, so the details will not be repeated.

[0311] The first processor 162 can be various processing devices, such as a central processing unit (CPU), a network processor, etc.; it can also be a digital signal processor, an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components. The CPU can be based on x86 or ARM architectures, etc.

[0312] The first processor 162 and the first memory 161 can communicate with each other directly or indirectly. For example, the first processor 162 and the first memory 161 can communicate via a network. The network can include a wireless network, a wired network, and / or any combination of wireless and wired networks. The first processor 162 and the first memory 161 can also communicate with each other via a system bus, which is not limited in this disclosure.

[0313] It should be noted that Figure 16 The components of the quantum error correction device 16 shown are merely exemplary and not limiting. The quantum error correction device 16 may also have other components depending on the specific application requirements. The first processor 162 can control other components in the quantum error correction device 16 to perform desired functions. The quantum error correction device 16 can be implemented in software, firmware, and / or hardware and can be integrated into a device with the relevant application installed.

[0314] Figure 17 A block diagram of a computer system for implementing some embodiments of the present disclosure is shown.

[0315] like Figure 17 As shown, the computer system 17 can be represented in the form of a general computing device. The computer system 17 includes a second memory 171, a second processor 172, and a bus 170 connecting different system components.

[0316] The second memory 171 can be various forms of computer-readable storage media, such as system memory, non-volatile storage media, etc. System memory may store, for example, an operating system, application programs, a bootloader, and other programs. System memory may include volatile storage media, such as random access memory (RAM) and / or cache memory. Non-volatile storage media may store, for example, instructions for performing corresponding embodiments of quantum error correction methods. Non-volatile storage media include, but are not limited to, disk storage, optical storage, flash memory, etc.

[0317] The second processor 172 can be implemented using a general-purpose processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA) or other programmable logic devices, discrete gates or transistors, or other discrete hardware components. Accordingly, each module can be implemented by executing instructions in the central processing unit (CPU) memory to perform the corresponding steps, or by implementing dedicated circuits to perform the corresponding steps.

[0318] Bus 170 can use any of the various bus architectures. For example, bus architectures include, but are not limited to, Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, and Peripheral Component Interconnect (PCI) bus.

[0319] The computer system 17 may also include an input / output interface 173, a network interface 174, a storage interface 175, etc. These interfaces 173, 174, 175, as well as the second memory 171 and the second processor 172, can be connected via a bus 170. The input / output interface 173 provides a connection interface for input / output devices such as a monitor, mouse, and keyboard. The network interface 174 provides a connection interface for various networked devices. The storage interface 175 provides a connection interface for external storage devices such as floppy disks, USB flash drives, and SD cards.

[0320] According to embodiments of this disclosure, the processes described above with reference to the flowcharts can be implemented as computer software programs. For example, some embodiments of this disclosure include a computer program product that, when run on a computer, causes the computer to implement the quantum error correction method described in any of the foregoing embodiments. The computer program product includes computer instructions carried on a computer-readable medium, the computer instructions including program code for performing the methods shown in the flowcharts.

[0321] Various embodiments of this disclosure have now been described in detail. To avoid obscuring the concept of this disclosure, some details known in the art have not been described. Those skilled in the art will fully understand how to implement the technical solutions disclosed herein based on the above description.

[0322] While specific embodiments of this disclosure have been described in detail by way of examples, those skilled in the art should understand that the examples are for illustrative purposes only and not intended to limit the scope of this disclosure. Those skilled in the art should understand that modifications can be made to the above embodiments or equivalent substitutions can be made to some technical features without departing from the scope and spirit of this disclosure. The scope of this disclosure is defined by the appended claims.

Claims

1. A quantum error correction method, comprising: Measurement operators are determined based on the gauge group in the quantum system, wherein the measurement operators correspond to multiple generators in the gauge group, and the gauge group characterizes the gauge degrees of freedom of the quantum system. The measurement result of the stable quantum in the quantum system is determined according to the measurement operator, wherein the stable quantum characterizes the invariant in the quantum system; The quantum system is corrected based on the measurement results of the stabilizer.

2. The quantum error correction method according to claim 1, wherein, Based on the gauge group of a quantum system, the measurement operators are determined to include: Based on the size of the quantum system and the gauge group, a measurement operator corresponding to the size is determined, wherein the size of the quantum system includes the length of the qubit array in one of its linear dimensions.

3. The quantum error correction method according to claim 2, wherein, Based on the size of the quantum system and the gauge group, the measurement operator corresponding to the size is determined as follows: Based on the dimensions, determine the candidate measurement period corresponding to the dimensions; The measurement operator is determined based on the target measurement period in the candidate measurement periods and the norm group.

4. The quantum error correction method according to claim 3, wherein, Determining the candidate measurement period corresponding to the size, based on the size, includes: When the size is an odd number, the candidate measurement period is determined as the first candidate measurement period; If the size is an even number, the candidate measurement period is determined as the second candidate measurement period.

5. The quantum error correction method according to claim 4, wherein, The first candidate measurement period includes a single measurement period and a three-measurement period, and the second candidate measurement period includes a single measurement period, a double measurement period, and a three-measurement period.

6. The quantum error correction method according to claim 5, wherein, The canonical group includes triangle generators. Based on the target measurement period in the candidate measurement periods and the canonical group, the measurement operators are determined to include: When the target measurement period is a single measurement period, the measurement operator includes a measurement operator corresponding to the rectangular structure, which corresponds to the two triangle generators intersecting the vertex of the rectangular structure; When the target measurement period is a dual measurement period or a triple measurement period, the measurement operator includes a measurement operator corresponding to a parallelogram structure, which corresponds to two triangle generators whose sides intersect the parallelogram structure.

7. The quantum error correction method according to claim 5, wherein, Based on the target measurement period in the candidate measurement periods and the norm group, the measurement operator is determined to include: When the target measurement period is a single measurement period, the measurement operator corresponding to the target measurement period includes a four-body joint measurement operator; When the target measurement period is a dual measurement period, the measurement operator corresponding to the target measurement period includes a combination of two four-body joint measurement operators, or a combination of one four-body joint measurement operator and one eight-body joint measurement operator. When the target measurement period is three measurement periods, the measurement operators corresponding to the target measurement period include a combination of two four-body joint measurement operators and one eight-body joint measurement operator.

8. The quantum error correction method according to claim 1, wherein, Determining the measurement results of the stable quantum in the quantum system based on the measurement operator includes: Based on the size of the quantum system and the measurement period of the quantum system, the stable subgroup in the quantum system is determined; The quantum system is measured according to the measurement operator to determine the measurement results of each stable component in the stable subgroup.

9. The quantum error correction method according to claim 8, wherein, The measurement results of the quantum system, determined by measuring the quantum system according to the measurement operator, and identifying each stable component in the stable subgroup include: When the measurement period is multiple measurement periods, the stable subgroup is updated according to the measurement operator corresponding to each measurement period to determine the measurement result of each stable subgroup.

10. The quantum error correction method according to claim 1, wherein, Error correction of the quantum system based on the measurement results of the stabilizer includes: Based on the erroneous stabilizers whose measurement results do not match the preset values, an error set corresponding to the erroneous stabilizers is determined. Each error in the error set, when applied to the quantum system, will cause the quantum system to include the erroneous stabilizer. The location and type of physical error in the quantum system are determined based on the probability corresponding to each error in the error set.

11. The quantum error correction method according to claim 10, wherein, In the case where the physical error includes a logical error, error correction of the quantum system based on the measurement results of the stabilizer further includes: Error correction is performed on the quantum system to restore the correct logical information; or Record the logic error and the current logic frame in the quantum system so that the current logic frame can be compensated during subsequent operations on the quantum system.

12. A quantum error correction device, comprising: A measurement operator determination module is configured to determine a measurement operator based on a gauge group in a quantum system, wherein the measurement operator corresponds to a plurality of generators in the gauge group, and the gauge group characterizes the gauge degrees of freedom of the quantum system. A measurement module is configured to determine the measurement result of a stable quantum in the quantum system based on the measurement operator, wherein the stable quantum characterizes an invariant in the quantum system; An error correction module is configured to correct errors in the quantum system based on the measurement results of the stabilizer.

13. A quantum error correction device, comprising: At least one memory; as well as At least one processor coupled to the at least one memory, the at least one processor being configured to execute the quantum error correction method as described in any one of claims 1 to 11 based on instructions stored in the at least one memory.

14. A computer-readable storage medium having stored thereon computer instructions that, when executed by a processor, implement the quantum error correction method as described in any one of claims 1 to 11.

15. A computer program product, when run on a computer, causes the computer to implement the quantum error correction method as described in any one of claims 1 to 11.

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