Regarner loop stabilizer code for error correction of fermi quantum simulation

Abstract

Methods, systems, and apparatus for error correction for Fermi quantum simulation. In one aspect, a method includes representing a Fermi subsystem as a graph of vertices and edges, where each vertex represents a Fermi sub-pattern of the Fermi subsystem and each edge represents an interaction between two respective Fermi sub-patterns; assigning a qubit to each edge in the graph to form a qubit system; quantum bit operators meeting a set of Fermi pair easy and dependency relationship are determined, and the quantum bit operators are non-uniform relative to graph vertexes; determining a stable sub-operator corresponding to the product of the secondary Margarner operators on the corresponding loop in the graph, and defining a code sub-space for encoding the state of the Fermi subsystem to be simulated by a public intrinsic space of the defined stable sub-operator; and emulating a Fermi subsystem by evolving a qubit system under a qubit Hamiltonian, the qubit Hamiltonian including the determined qubit operator and stabilizer operator.

Description

[0001] This application is a divisional application of the invention patent application with application date of December 17, 2019, application number 201980083716.3, and invention name “Majorana loop stable subcode for error correction of fermion quantum simulation”. Background Art

[0002] This specification relates to quantum computing.

[0003] Applications of quantum computing include quantum simulation. The simulation of quantum systems (e.g., systems of electrons) has applications in a variety of different fields, from drug synthesis to the design of new catalysts and materials. However, simulating complex quantum systems using classical techniques may be unsustainable due to the exponential scaling of the required resources as a function of the size N of the system. Quantum computers and quantum simulation techniques could provide a computationally feasible solution to this task. Summary of the invention

[0004] This specification describes techniques for error detection and correction of fermionic quantum simulations. In particular, this specification describes techniques for error detection and correction that are particularly suitable for implementing fermionic quantum simulations on near-term quantum computing hardware with only short-range interactions between qubits.

[0005] In general, one innovative aspect of the subject matter described in this specification can be implemented in a method for quantum simulation of a fermionic system, wherein the fermionic system is characterized by a fermionic Hamiltonian, the fermionic Hamiltonian comprising elements from a group of quadratic Majorana operators of even weight, wherein the quadratic Majorana operators satisfy a set of commutation relations and dependency relations implying that the quadratic Majorana operators are not independent, the method comprising: representing the fermionic system as an interaction graph of vertices and edges, wherein each vertex represents a fermionic mode of the fermionic system and each edge represents an interaction between two corresponding fermionic modes; Assigning a qubit to each edge in the interaction graph to form a qubit system; determining qubit operators that satisfy the set of commutation relations and dependency relations, wherein the qubit operators are non-uniform with respect to the interaction graph vertices; determining stabilizer operators corresponding to products of quadratic Majorana operators on corresponding loops in the interaction graph, wherein a common eigenspace of the defined stabilizer operators defines a code subspace encoding a state of the fermionic system; and simulating the fermionic system by evolving the qubit system under a qubit Hamiltonian, wherein the qubit Hamiltonian includes the determined qubit operators and the determined stabilizer operators.

[0006] Other embodiments of this aspect include corresponding computer systems, devices, and computer programs recorded on one or more computer storage devices, each of which is configured to perform the actions of the method. A system of one or more classical and / or quantum computers may be configured to perform specific operations or actions by installing hardware, software, firmware, or a combination thereof on the system, which software, firmware, hardware, or a combination thereof causes the system to perform actions in operation. One or more computer programs may be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform an action.

[0007] The foregoing and other embodiments can each optionally include one or more of the following features, alone or in combination. In some embodiments, the qubit operator comprises a Pauli-Y operator.

[0008] In some embodiments, evolving the qubit system under the qubit Hamiltonian includes correcting all single qubit errors.

[0009] In some embodiments, the determined qubit operator has a weight greater than or equal to three. In some embodiments, the determined qubit operator has a weight less than or equal to four.

[0010] In some embodiments, a fermionic system includes a two-dimensional square lattice of fermions.

[0011] In some embodiments, the lattice comprises periodic boundary conditions. In some embodiments, the lattice comprises open boundary conditions.

[0012] In some embodiments, the qubit operators include edge operators and vertex operators, and wherein the method further comprises: adding a first number of dangling edges to the interaction graph, wherein the first number is equal to the number of boundary vertices in the interaction graph plus four; determining a corresponding boundary stabilizer operator for each dangling edge, comprising, for each dangling edge: determining a corresponding dangling edge operator, wherein the dangling edge operator anti-commutes with the incident vertex and edge operator and commutes with the determined stabilizer operator; determining an edge operator in the corresponding boundary lattice; where the product of the edge operators in the corresponding boundary element lattice commutes with the determined qubit operator and anticommutes with each adjacent product of edge operators in adjacent boundary element lattices; determining a conjugate operator where the conjugate operator anticommutes with the dangling edge operator of the dangling edge, commutes with the determined qubit operator, commutes with the dangling edge operator of the other dangling edges, and commutes with the other conjugate operators; defining the boundary stabilizer operator of the dangling edge to be equal to i times the product of the edge operators in the corresponding boundary element lattice times the conjugate operator of the dangling edge.

[0013] In some embodiments, determined boundary stabilizer operators commute with determined qubit operators, determined stabilizer operators, and with each other.

[0014] In some embodiments, the method further comprises assigning a qubit to each dangling edge.

[0015] In some embodiments, the method also includes applying a transformation to the quadratic Majorana operators to generate transformed quadratic Majorana operators, including: representing each quadratic Majorana operator as a 2n-dimensional vector on a Galois field of two elements; constructing a matrix whose rows correspond to the transformed quadratic Majorana operators; and generating a set of transformed quadratic Majorana operators by applying the constructed matrix to each of the quadratic Majorana operators.

[0016] In some embodiments, the quadratic Majorana operator of the transformation satisfies a set of commutation relations.

[0017] In some embodiments, the method further includes modifying the determined qubit operator and updating the determined stabilizer operator based on the modified determined qubit operator, including repeatedly until each qubit operator and stabilizer operator has been updated: identifying a Pauli operator that anti-commutes with at least one determined stabilizer operator; selecting a determined qubit operator and updating the selected qubit operator by multiplying the selected determined qubit operator by the identified Pauli operator; identifying a set of determined qubit operators, wherein the set of determined qubit operators excludes the determined qubit operator and anti-commutes with the identified Pauli operator; updating the determined qubit operators in the identified set of determined qubit operators by multiplying each qubit operator by one of the at least one determined stabilizer operator; and updating at least one determined stabilizer operator using the updated qubit operator.

[0018] In some embodiments, during the evolution of the qubit system, a defined stabilizing sub-operator constrains the state of the qubit system to a code subspace.

[0019] In some embodiments, the determined qubit operators include qubit operators corresponding to corresponding edges in the interaction graph, and wherein defining a stabilizer operator corresponding to a product of quadratic Majorana operators on corresponding loops in the interaction graph includes defining a stabilizer operator equal to a product of qubit edge operators on corresponding loops in the interaction graph.

[0020] In some embodiments, the defined stabilizer operators commute with each of the determined qubit operators, and with each other.

[0021] In some embodiments, the qubit system preserves the locality of a fermion system.

[0022] In some embodiments, determining the qubit operator includes: for each vertex in the interaction graph, selecting an order of all incident edges to the vertex; and mapping the quadratic Majorana operator to the corresponding qubit operator using the selected order.

[0023] In some implementations, evolving the qubit system includes: evolving the qubit system under the qubit Hamiltonian for a predetermined time to generate an evolved qubit system; and experimentally probing the evolved qubit system to determine a property of the fermionic system.

[0024] In some embodiments, the fermion system is represented by a Hubbard model, and wherein evolving the qubit system using the quantum computing device to emulate the fermion system includes evolving the qubit system using the quantum computing device to emulate properties of the fermion system.

[0025] In some embodiments, the fermionic system represents a material, and wherein the properties of the fermionic system include superconductivity, magnetism, the strength of interaction-driven metal-insulator transitions, optical conductivity, or the behavior of spin correlations under doping.

[0026] In some embodiments, the material includes a cuprate material.

[0027] In some embodiments, the fermionic system comprises a copper oxide superconductor, and wherein the characteristic of the fermionic system comprises the presence of phase separation.

[0028] The subject matter described in this specification can be implemented in a specific way to realize one or more of the following advantages.

[0029] Fermion-to-qubit mappings that preserve geometric locality are particularly useful for simulating lattice fermion models (e.g., the Hubbard model) on quantum computers. They avoid the overhead associated with geometrically nonlocal parity terms in mappings such as the Jordan-Wigner and Bravyi-Kitaev transforms. As a result, they typically provide quantum circuits with lower depth and gate complexity. In such encodings, fermion states are encoded in a common +1 eigenspace of a set of stabilizers, similar to stabilizer quantum error correction codes.

[0030] The technique described here introduces a geometry-locality-preserving Majorana circuit stabilizer code whose stabilizers correspond to the product of Majorana operators on closed paths of a fermion hopping graph. The Majorana circuit stabilizer code can correct all single-qubit errors on a two-dimensional square lattice - whereas previous geometry-locality-preserving codes could only detect single-qubit errors on the same lattice. Compared to existing codes, the Majorana circuit stabilizer code maps the relevant fermion operators to lower-weight qubit operators, albeit with a larger code distance.

[0031] The Majorana loop stabilization subcode can also be applied to models other than lattice models. For example, the Majorana loop stabilization subcode is compatible with state-of-the-art algorithms for simulating quantum chemistry and can provide the same error correction properties for these simulations without the additional asymptotic overhead.

[0032] The above properties make Majorana loop-stable subcodes particularly suitable for error-mitigated quantum simulations of fermions on close-range devices, which typically have a finite number of qubits arranged in a lattice and / or are limited to short-range (e.g., nearest neighbor) interactions between qubits. However, Majorana loop-stable subcodes can also be used for error-mitigated quantum simulations of fermions on devices including a large number of qubits with short-range and / or relatively long-range interactions.

[0033] Furthermore, the currently described Majorana loop stabilizer code could be applied to a wide range of technological applications, including the efficient design or testing of materials, the prediction of the rates of certain chemical reactions, or the efficient synthesis of drugs.

[0034] Details of one or more implementations of the subject matter of this specification are set forth in the specification and appendices. Other features, aspects, and advantages of the subject matter will become apparent from the specification, appendices, and claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 is a diagram of an example system for implementing quantum simulation of error correction for a fermionic system.

[0036] Figure 2 is a flow chart of an example process for error correction of a Majorana loopstabilizer code for quantum simulation of fermionic systems.

[0037] Figure 3 An example interaction diagram representing a fermion system is shown.

[0038] Figure 4A An example structure of an MLSC code for a 2D square lattice is shown.

[0039] Figure 4B -E shows the Figure 4A Logical operators and stabilizer operators for the plaquette shown in .

[0040] Figure 5 An example MLSC on a 4 by 4 lattice with open boundary conditions is shown.

[0041] Like reference numbers and designations in the various drawings indicate like elements. DETAILED DESCRIPTION

[0042] Overview

[0043] Before a fermionic system can be simulated on a quantum computer, it must be mapped to a qubit system. Existing techniques for performing such mapping include the Jordan-Wigner transform or the Bravyi-Kitaev transform. Both techniques have disadvantages - the Jordan-Wigner transform introduces non-local parity terms that act on a large number of qubits. These parity terms add considerable overhead to the quantum simulation of local fermionic systems. The Bravyi-Kitaev transform reduces these overheads, however, the qubit operators produced by the BK transform are still geometrically non-local. This problem becomes prominent on near-term quantum devices where only short-range qubit-qubit interactions are available. In order to realize the simulation of fermionic systems on such near-term quantum devices, it is desirable to map the fermionic system to a qubit system with short-range (e.g., nearest neighbor) interactions.

[0044] The Bravyi-Kitaev ultrafast (BKSF) transformation is an example fermion-to-qubit mapping that preserves geometric locality. However, the no-go theorem shows that if the vertex degree of the interaction graph representing the fermion system is greater than or equal to 6, then the BKSF transformation cannot correct all single-qubit errors without including additional vertices and edges in the interaction graph (i.e., without increasing the complexity of the simulation). BKSF can only detect single-qubit errors.

[0045] This specification describes a geometric locality preserving fermion-qubit mapping that can correct all single-qubit errors on a graph with vertex degree equal to 4 - without introducing any additional vertices or edges. As with BKSF, the logic (qubit) operators in the currently described code satisfy the standard commutation relation for the corresponding Majorana operators. The stabilizer of the code is defined as the product of Majorana operators on the loop. Therefore, the code is described herein as a Majorana loop stabilizer code (MLSC). The fermion operators corresponding to the edges in the interaction graph representation of the fermion system can be non-uniform and / or include Pauli-Y operators, which allows the mapping to circumvent the infeasibility theorem. For example, by using logic operators that are non-uniform or include Pauli-Y operators, the weights of the logic operators can be balanced. Since the minimum weight of the logic operator defines the code distance, balancing the weights of the logic operators can increase the code distance while reducing the simulation complexity.

[0046] Majorana Operator

[0047] The fermion system to be simulated is characterized by a fermion Hamiltonian. The fermion Hamiltonian can be expressed by a single-mode Majorana operator. A single-mode Majorana operator can be defined as a linear combination of fermion creation and annihilation operators:

[0048] , (1)

[0049] in, , They represent the fermion creation and annihilation operators respectively. The fermion ladder operator satisfies the relationship:

[0050] (2)

[0051] in, Express opposition to the exchange of children, and is the Kronecker delta function.

[0052] Single-mode Majorana operator Together with the phase factor i, we generate the group , among which, group Any element in takes the following form:

[0053] (3)

[0054] in is the overall phase factor. The set is the support of Majorana operators. The product The order of the operators in is important and follows the convention that the index k increases monotonically from left to right. The weight of a Majorana operator is equal to the number of fermion modes it acts non-trivially, i.e. The weight of is |A|. Two Majorana operators and Either for or against:

[0055] (4)

[0056] The commutation relation is determined by the parity of the overlap |A∩B| when |A| or |B| is even.

[0057] In terms of Majorana operators, the fermion occupation operator is:

[0058]

[0059] in, represents the unit operator. The fermion hopping operator term takes the following form:

[0060]

[0061] Majorana operators include Majorana operators corresponding to corresponding vertices in the interaction graph representation of the fermion system and Majorana operators corresponding to corresponding edges in the interaction graph. For convenience, in this specification, fermions can be represented in an undirected graph with vertices V and edges E. Jump between adjacent vertices. For the set of quadratic Majorana operators:

[0062] , for each vertex

[0063] , for each edge

[0064] The occupation term in equation (5) and the jump term in equation (6) can be expressed as:

[0065]

[0066]

[0067] The operator generating group in (7a, 7b) , which consists of Majorana operators of even weight. Any physical fermion Hamiltonian can be written as a group The sum of the elements in .

[0068] The quadratic Majorana generator in (7a, 7b) is both Hermitian and unitary. It satisfies the commutation relation:

[0069]

[0070] In addition to equations (9a-c), the following conditional closed algebra on any closed path in graph G:

[0071]

[0072] where l is the length of the path. This relationship implies that the generators in (9a-c) are not independent. The total number of independent closed paths is .

[0073] in, Represents the total number of vertices (edges) of the graph G. As shown below Figure 2 As mentioned above, when fermion operators are mapped to qubit operators, the condition in equation (10) is not automatically satisfied.

[0074] Example Hardware

[0075] Figure 1An example system 100 for implementing error-corrected quantum simulation of fermionic systems is depicted. Example system 100 is an example of a system implemented as a classical or quantum computer program on one or more classical computers or quantum computing devices at one or more locations in which the systems, components, and techniques described below may be implemented.

[0076] System 100 includes quantum hardware 102 in data communication with a classical processor 104. For convenience, classical processor 104 and quantum computing hardware 102 are shown as separate entities, however, in some embodiments, classical processor 104 may be included in quantum computing hardware 102, e.g., quantum computing hardware 102 may include one or more components for performing classical computing operations.

[0077] The system 100 may receive, as input, data that may include data representing a physical system of interest, e.g., input data 106. The received data representing a physical system of interest may include data representing a physical system to be modeled or simulated. The physical system may be a fermionic system characterized by a fermionic Hamiltonian, as described above with reference to equations (1)-(10). The physical system may be a periodic or aperiodic system.

[0078] The system can generate data representing the results of simulating a physical system of interest, e.g., output data 108, as output, and can include data that can be used to determine properties of the physical system. For example, in some embodiments, the physical system can be a material, such as a semiconductor. In these cases, the data representing the simulation results can be used to determine properties of the material, such as electrical conductivity, magnetism, the strength of an interaction-driven metal-insulator transition, optical conductivity, or the behavior of spin correlations under doping.

[0079] System 100 is configured to perform classical computations in conjunction with quantum computations using classical processor 104 and quantum hardware 102 .

[0080] The quantum hardware 102 includes a plurality of physical qubits 110 for performing algorithmic operations or quantum computations. The specific implementation of the qubits included in the quantum computing hardware 102 and how they interact with each other depends on a variety of factors, including the type of quantum computation being performed by the quantum computing hardware. For example, a qubit may include a qubit physically implemented via an atomic, molecular, or solid-state quantum system. In other examples, a qubit may include, but is not limited to, a superconducting qubit or a semiconductor qubit. Near-term quantum computing devices are typically limited to planar geometric structures of qubits (e.g., a finite 2D lattice), with only short-range (e.g., nearest neighbor) interactions between qubits.

[0081] In some embodiments, qubits 110 may be frequency-adjustable. For example, each qubit may have an associated operating frequency, which may be adjusted, for example, using one or more control devices 112 by applying voltage pulses via one or more drive lines coupled to the qubit. Example operating frequencies include qubit idle frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that a qubit may perform. For example, setting the operating frequency to a corresponding idle frequency may cause the qubit to enter a state in which it does not strongly interact with other qubits, and a state in which it may be used to perform a single-qubit gate (e.g., Pauli X, Y, and Z gates). As another example, in the case where qubits interact via a coupler with a fixed coupling, the qubits may be configured to interact with each other by setting their respective operating frequencies to a gate-dependent frequency that is detuned from their common interaction frequency. In other cases, for example, when qubits interact via an adjustable coupler, the qubits may be configured to interact with each other by setting the parameters of their respective couplers to enable interaction between qubits and then by setting the respective operating frequencies of the qubits to a gate-dependent frequency that is detuned from their common interaction frequency. Such interactions can be performed in order to execute multi-qubit gates.

[0082] The control device 112 may also include a measurement device, for example, a readout resonator. The measurement results (measurement data) obtained via the measurement device may be provided to a classical processor included in the quantum computing hardware 102 or to the classical processor 104 for processing and analysis.

[0083] The classical processor 104 may include various components for performing the classical operations described in this specification. For example, the classical processor 104 includes an interaction graph generator 116 configured to represent the fermion system specified by the input data 106 as an interaction graph of vertices and edges, where each vertex represents a fermion mode of the fermion system and each edge represents an interaction between two corresponding fermion modes. The interaction graph generator 116 is also configured to assign qubits to each edge in the generated interaction graph to form a qubit system, and provide data representing the qubit system to the quantum computing hardware 102. Additional operations performed by the interaction graph generator 116 are described below with reference to Figure 2 Give a description.

[0084] The classical processor 104 also includes a qubit and stabilizer operator generator 114. The qubit and stabilizer operator generator 114 is configured to determine qubit operators that satisfy fermion commutation relations and dependency relations for the qubit system defined by the interaction graph generator 116. The qubit operators generated by the qubit and stabilizer operator generator 114 may be non-uniform with respect to the interaction graph vertices. The qubit and stabilizer operator generator 114 is configured to determine the stabilizer operator corresponding to the product of the quadratic Majorana operators on the corresponding loops in the interaction graph, wherein the common eigenspace of the defined stabilizer operators defines a code subspace that encodes the state of the fermion system. Referring to Figure 2 Additional operations performed by qubit and stabilizer operator generator 114 are described.

[0085] Classical processor 104 may provide data 118 representing the generated interaction graph / qubit system and the determined qubit and stabilizer operators to quantum computing hardware 102. Quantum computing hardware 102 may then perform a quantum simulation of the physical qubit system specified by input data 106 by evolving the physical qubit system under a qubit Hamiltonian (e.g., a qubit Hamiltonian corresponding to a fermionic Hamiltonian and including the determined qubit operator and the determined stabilizer operator) specified by data 118. Additional operations that classical processor 104 and quantum computing hardware 102 are configured to perform include the following references Figure 2-5 Describes the operation.

[0086] Operating Hardware: Example Implementation of MLSC

[0087] Figure 2 is a flow chart of an example process 200 for Majorana loop stable subcode (MLSC) error correction for quantum simulation of fermionic systems. In the example process 200, the fermionic system is characterized by a fermionic Hamiltonian comprising elements from a group of even-weighted quadratic Majorana operators, where the quadratic Majorana operators satisfy a set of commutation relations and dependency relations that imply that the quadratic Majorana operators are not independent. For convenience, the process 200 will be described as being performed by a quantum computing system. For example, a system appropriately programmed according to the present specification Figure 1 The system 100 may perform the process 200 .

[0088] The system maps the fermionic system to a corresponding locality preserving qubit system. To map the fermionic system to a corresponding locality preserving qubit system, the system represents the fermionic system as an interaction graph of vertices and edges (step 202). Each vertex in the interaction graph represents a fermionic mode of the fermionic system. Each edge in the interaction graph represents an interaction between two corresponding fermionic modes. The number of vertices and edges depends on a particular form of the fermionic system, such as the form of a Hamiltonian that characterizes the fermionic system. In some embodiments, the vertex degree of the interaction graph can be equal to 4. The system then assigns a qubit to each edge in the interaction graph to form a qubit system that preserves the locality of the fermionic system (step 204).

[0089] Figure 3 An example interaction graph 300 representing a fermionic system is shown in FIG. In the example interaction graph 300, the vertices F0-F5 of the graph represent fermionic modes. The edges between the vertices represent the interactions between the fermionic modes, for example, edge 302 represents the interaction between vertices F4 and F5. Fermions can jump between adjacent vertices. Qubits Q01, Q03, Q05, Q12, ... Q45 are introduced to the corresponding edges of the graph 300, for example, qubit Q45 is introduced to edge 302. The fermionic state of the fermionic system is then encoded in the subspace of qubits.

[0090] return Figure 2 , the system determines a qubit operator that shares the same algebra as the fermion operator. That is, the system determines a qubit operator that satisfies the quadratic Majorana operator commutation relation and dependency relation given in equations (9a-c) and 10 (step 206). Determining the qubit operator includes, for each vertex in the interaction graph: selecting an order of all incident edges of the vertex, and mapping the single-mode Majorana operator to the corresponding qubit operator using the selected order. For example, the system may select an ordering of the incident edges of each vertex of G, such as, , which means that at the vertex Among all the incident edges of Placed in Before. The quadratic Majorana operators in (9a-c) can then be mapped to the corresponding qubit operators in the following way:

[0091]

[0092] Among them, for each edge Arbitrary choice of antisymmetric coefficients . Qubit operators (also called logical operators) and The same commutation relations as those for fermion operators in (9a-c) are satisfied. Unlike other fermion-to-qubit mappings that preserve geometric locality, the logical operators can be allowed to be more general, e.g., non-uniform with respect to interaction graph vertices, and can involve Pauli-Y operators. For example, in the Bravyi-Kitaev ultrafast transformation, the logical operators are translationally invariant across a lattice site. However, in the MLSC described in the present paper, this translational invariance is broken.

[0093] The mapping given by equation (11) artificially introduces an additional conservation quantity:

[0094]

[0095] It corresponds to the even-parity subspace of the original fermion system. To simulate the odd-parity subspace, the sign in the mapping given by Eq. (11) can be changed for a specific value of k (e.g., k = 0),

[0096]

[0097] The system determines a stabilizer operator corresponding to the product of quadratic Majorana operators on the corresponding loop in the interaction graph (step 208). The stabilizer operator is defined to be equal to the product of qubit edge operators on the corresponding loop in the interaction graph. For example, the stabilizer operator corresponding to the conserved quantity given in equation (10) above is given by

[0098]

[0099] Its index These loop operators form an Abelian group, and the fermionic state is encoded in their common +1 eigenspace. These operators are called stabilizer operators (or gauge operators, not to be confused with the non-commutative metric operators in subsystem quantum error correction codes). As a consequence of the commutation relations given in (9a-c), stabilizer operators are identical to all logical operators. and They also commute with each other, and for all closed paths in the graph G , their common +1 eigenspace defines the code subspace :

[0100] , if and only if (16).

[0101] Therefore, the restriction of the closed-loop stabilizer to the code space acts as an identity, just as in the case of Majorana operators. Applying the stabilizer operator to the qubit system corresponding to the fermion system restricts the state of the qubit system to the code subspace , which corresponds to restricting the state of the original fermion system to the corresponding Fock space.

[0102] The above description for determining qubit operators that satisfy the quadratic Majorana operator commutation and dependency relations given in equations (9a-c) and 10 and determining the stabilizer operator corresponding to the product of the quadratic Majorana operators on the corresponding loop in the interaction graph (steps 206 and 208) can be further generalized to construct arbitrary MLSCs.

[0103] One approach for generalizing MLSC consists in introducing a new set of qubit operators that belong to the same centralizer of the stable subgroup (the set of elements in the Pauli group that commutes with all elements of the stable subgroup), i.e., the stable subgroup generated by the above stabilizer operators remains the same. This is achieved by applying a transformation to the vector representation of the Majorana operator. This transformation preserves the anti-commutation relation of the Majorana operator - similar to applying unitary in the Clifford group to the Pauli operator.

[0104] More specifically, the system represents each multimode Majorana operator acting on n fermionic modes (up to the sign) by the corresponding 2n-dimensional vector over GF(2) (a two-element Galois field). For example, the Majorana operator acting on two fermionic modes is can be represented by the vector (1 1 1 0) (up to the total phase factor i), where 1 (0) indicates the presence (absence) of the corresponding single-mode Majorana operator in the product. The system then introduces a matrix M whose rows correspond to the new Majorana operators in terms of the old Majorana operators. For the specific example of M with n = 2, it takes the following form:

[0105]

[0106] The new Majorana operator satisfies the standard anti-commutation relation in equation (2). Then, the matrix M can be applied to the vector representation of each Majorana operator to generate a new Majorana operator. The new Majorana operator can be used to replace the original Majorana operator in MLSC.

[0107] Since total parity is conserved by any quadratic Majorana operator, the Hamming weights of all row vectors in M ​​are either odd or even. It plays a similar role to the two-qubit gate in the Clifford group, and the permutation of Majorana modes is similar to the single-qubit Clifford operation. The general transformation matrix M for 2n Majorana modes can be decomposed into four Majorana transformations and a sequence of row permutation matrices. Due to the anti-commutation relation of the Majorana operator given in (2), any two rows of M are orthogonal to each other. In order to maintain geometric locality, the transformation can be restricted to transformations between adjacent Majorana modes.

[0108] Another method includes modifying a qubit operator and updating a stabilizer group based on the modified qubit operator. The system identifies a Pauli operator Δ that anti-commutes with at least one stabilizer operator (e.g., S). The system selects a qubit operator and updates the stabilizer group via Update the selected qubit operator. The system identifies a set of qubit operators , excluding L, which anti-commutes with the Pauli operator Δ. The system updates these qubit operators by multiplying them with the stabilizer operator, i.e. . The system uses the updated qubit operator and updates the stabilizer operator based on equation (10), and repeats the process for the remaining qubit operators until each qubit operator is updated. This process produces a new MLSC with modified qubit operators and stabilizers. In order to make the generated MLSC code geometrically local, the choice of Δ can also be geometrically local. By combining the two methods, arbitrary MLSCs can be constructed.

[0109] To simulate the fermionic system, the system evolves the qubit system under the qubit Hamiltonian, e.g., for a predetermined time t (step 210). The system generates the qubit Hamiltonian by mapping terms of the fermionic Hamiltonian including quadratic Majorana operators to corresponding terms of the qubit Hamiltonian including qubit operators using the qubit operators determined in step 206.

[0110] During the evolution, the system applies a defined stabilizer operator to the qubit system and performs multiple stabilizer measurements. The qubit Hamiltonian commutes with the stabilizer operator, so under a perfect (error-free) qubit Hamiltonian evolution, the measured value of the stabilizer operator remains unchanged. However, the error operator (e.g., all single-qubit Pauli operators) anti-commutes with at least one of the stabilizer operators. Therefore, single-qubit errors can be detected by measuring the value of the stabilizer operator. In the currently described MLSC, the measured value of the stabilizer operator (also called syndrome) is different for different single-qubit errors. Therefore, all single-qubit errors on the interaction graph can be corrected during the evolution.

[0111] In order to simulate the lattice fermion problem directly (i.e., without mapping to the corresponding qubit system), it is necessary to physically implement the jump terms given in Eqs. (8a-b) above and occupied items . In the implementation of the BKSF on a 2D lattice, both vertical and horizontal hopping terms are mapped to qubit operators of weight 6. Using the currently described technique, the hopping terms are mapped to qubit operators of weight no more than 4 (sometimes 3). The occupation terms can also be implemented with qubit operators of lower weight. The ability to implement logic operations with low-weight Pauli operators while correcting all single-qubit errors is an important feature of near-term quantum error mitigation schemes. The weights of the stabilizers in the currently described technique range from 4 to 10. Measurement is often the bottleneck for error-corrected quantum computations implemented on near-term devices. For example, reading out the state of a superconducting qubit with a fidelity of ~0.95 may take up to ~1μs, while a typical two-qubit gate time is ~40ns with a fidelity of ~0.995. Symptom readout should only be performed sparsely on these near-term devices, e.g., each syndrome readout cycle is used for at least tens of logic gate cycles. Therefore, the total number of gates is dominated by logic operations rather than syndrome readout.

[0112] Table I below shows the differences between the currently described technique and BKSF on a 2D lattice in terms of code distance, weights of occupied terms, and weights of jump terms. The currently described technique exhibits improved distance and weight (logic) properties. The weights of stabilizers in the currently described code ("our code") range from 4 to 10, which is generally higher than BKSF.

[0113]

[0114] Table I

[0115] The system can use the evolved qubit system to determine the characteristics of the fermion system. For example, the system can perform a measurement operation on the evolved qubit system to determine the corresponding measurement result. The measurement result can then be used to infer the characteristics of the fermion system. Performing the measurement operation on the evolved qubit system can include applying a decoding circuit that maps the stabilizer and fermion states to different registers, and measuring the different registers separately. Alternatively, an ancilla qubit can be used to measure the correlation function (corresponding to the characteristic of interest) and the stabilizer. The choice of measurement scheme depends on the type of device used to perform the methods described herein.

[0116] In some embodiments, the fermion system can be a Hubbard model. The fermion system can represent a material, for example, a cuprate or copper oxide superconductor. In these examples, the methods described herein can be used to determine the properties of the material, such as superconductivity, magnetism, the strength of the interaction-driven metal-insulator transition, optical conductivity, the behavior of spin correlations under doping, or the presence of phase separation.

[0117] Figure 4A An example structure 400 of an MLSC code for a 2D square lattice is shown. Each circle represents a vertex (fermion mode) in the 2D lattice, where the shading of the vertices represents different types of vertices. One stabilizer operator S0-S15 is associated with each element, where the shading of the element represents different types of corresponding stabilizer operators.

[0118] Figure 4B Shown with Figure 4A The logical operators and stabilizer operators associated with the lower right lattice (i.e., the lattice consisting of vertices m, n, q, p) of the lattice. A physical qubit is located on each edge of the lattice. The Pauli operators X, Y, and Z act on the qubits on the corresponding edges. Each quadratic Majorana operator in (7a-b) is mapped to the product of single-qubit Pauli operators connected by lines. Vertex operator is mapped to the product of three Pauli-Z operators linked by a curved dash-dot line around the point m, and similarly for the other vertex operators. The Pauli operator mapped onto the upper dotted line , where only the Pauli-X operator acts on the edge (m, n). The same rules apply to the remaining edge operators , where the Pauli-X operator always acts on These operators satisfy the commutation relation from (9a-c) (which can be verified numerically). The lattice has a 4th-order rotational symmetry. Figure 4B The stabilizer operator associated with the element lattice is plotted in part (b) of

[0119]

[0120] ,

[0121] Until the overall minus sign, where, using the definition and convention of formula (15), if point k is to the right or below point j, then .

[0122] For naming purposes, the item is called a generalized edge operator. In some embodiments, Figure 4BThe weight of any generalized edge operator in the lattice of part (b) can be greater than or equal to three, for example, when all single-qubit errors will be corrected. Other transformations, such as BKSF, violate this condition on 2D lattices regardless of the ordering of the incident edges.

[0123] You can do this by Figure 4B The four types of vertices shown in are put together to construct the entire code, where each Pauli X operator is shared by two adjacent vertices. Figure 4B similar, Figure 4C Shown with Figure 4A The logical operators and stabilizer operators associated with the lower left lattice (i.e., the lattice consisting of vertices l, m, o, p) of . This lattice has second-order rotational symmetry. Figure 4D Shown with Figure 4A The logical operators and stabilizer operators associated with the upper-left cell (i.e., the cell consisting of vertices i, j, l, and m). Figure 4E Shown with Figure 4A The logical operators and stabilizer operators associated with the upper right lattice of (i.e., the lattice consisting of vertices j, k, m, n). Through π rotation, these logical operators are related to Figure 4D The logical operator in part (a) is the same as that in part (b). By π rotation, the corresponding stabilizer shown in part (b) is also the same as Figure 4D The stabilizer is the same as in part (b).

[0124] exist Figure 4C -E, any generalized operator in the corresponding element may have a weight greater than or equal to three. In some embodiments, any generalized operator in the corresponding element will have a weight greater than or equal to three. For example, up to the total symbol, such a generalized operator M can be specified as:

[0125]

[0126] in, and represents the set of edges and vertices that determine M. Without loss of generality, we can assume that for any vertex , for a vertex k, there must be an edge Otherwise, you can Remove j from the Figure 4E In Remove Increase its weight from 5 to 6). If exactly one incident edge of vertex j is in In the above example, the vertex j can be called a single pairing vertex. Whether or not, at least one of the Pauli-Z operators on the other three incident edges of j contributes to M. If If there is more than one single-pairing vertex in M, then the weight of M must be greater than or equal to three. This is because in addition to the at least two Pauli-Z operators associated with the single-pairing vertices, there must be some Pauli-X or Y components in M ​​to make it anti-commutative with the vertex operators on the single-pairing vertices. When two edges are involved, the only case where there are fewer than two single-pair vertices is when two parallel edges are placed next to each other. Figure 4B -E shows that the weight of such a logical operator is always greater than or equal to three. Figure 4A The specific choice of mid-vertex layout is important to prevent the opposite from happening.

[0127] The syndrome of a single-qubit error can be described by a stabilizer operator that anticommutates with it. As an example, Figure 4A The syndrome of single-qubit errors on all incident edges of vertex m is:

[0128]

[0129] These single-qubit errors (Pauli-Y errors are not listed) have different syndromes. Under periodic boundary conditions, all single-qubit errors have different syndromes on an 8 by 8 lattice (128 qubits).

[0130] Operational Hardware: Example Implementation of MLSC for Open Boundary Conditions

[0131] As above reference Figure 2-4E As shown, MLSC can be defined on a lattice with periodic boundary conditions. However, in practical experimental setups, periodic boundary conditions are often unavailable or very expensive to implement. Boundary effects are important for lattices of moderate size, such as those available for near-term quantum computing devices. For example, most vertices are located at the boundaries of a 4 by 4 square lattice. Errors on boundary qubits are more difficult to correct because they involve a smaller number of stabilizers, i.e., fewer check codons are available to correct / detect errors. This problem is particularly prominent for qubits around lattice corners where two boundary edges meet. Therefore, using the above MLSC with open boundary conditions may lead to uncorrectable single-qubit errors.

[0132] To overcome this problem, dangling boundary edges are added during the example process 200. A dangling edge is a horizontal or vertical edge on a square lattice that extends from a boundary vertex in the interaction graph and does not connect the boundary vertex to another vertex. For convenience, the following description of adding dangling boundary edges is described with reference to a 4 by 4 square lattice. However, larger sized open boundary MLSCs can be constructed using the same process.

[0133] Figure 5An example MLSC on a 4 by 4 lattice with open boundary conditions is shown. Each vertex represents a fermion mode (16 total), and each edge in the code bulk (solid black) is associated with a logical edge operator (24 total). Each added dangling edge (dash-dot line) introduces an additional physical qubit at the boundary (16 total). The logical operators—including all vertex operators and edge operators—are constructed using the same techniques as in the previous reference. Figure 2 -4. This is possible because any vertex (edge) operator of the MLSC involves only Pauli operators on the edges incident on that vertex (edge), i.e., the logical operators in the code block have no support beyond the dangling edges. Therefore, the commutation relation given in (9a-c) is satisfied, and the stable sub-operators in the code block The stabilizers remain the same as in the periodic boundary case. The introduction of dangling edges adds an extra degree of freedom, which allows the construction of additional stabilizers To correct errors at the boundaries. Since the number of boundary stabilizers is equal to the number of dangling edges, this eliminates the extra degree of freedom. Because each qubit at (or near) the boundary involves more stabilizers, errors on them are more likely to be correctable.

[0134] To define the boundary stabilizer, the edge operator associated with the dangling edge is defined ,in, corresponds to the vertical dangling edge (labeled 502) of vertex i, and the remaining edge operators Follow clockwise along the border. These operators use the same Figure 2 -4, but ignoring the Pauli operators on edges other than dangling edges. Each dangling edge operator is defined to be anti-commutative with the vertex and edge operators incident on it. Thus, the dangling edge operator is identical to all the stable operators in the code block. Easy to make.

[0135] Then the operator is defined as the product of all edge operators in the boundary element grid. For example, are the two dangling edge operators incident on vertex i ( and ), and is the edge operator on (i; j) and the two vertical dangling edge operators incident on vertices i and j ( and ). These operators commute with all logical operators, since every logical operator The two edge operators in are anti-commutative. However, the adjacent operators and Oppose each other. For example, and Oppose Yi, and Therefore, and Oppose Yi.

[0136] In order to construct boundary stabilizers that commutate with each other, conjugation operators for each dangling edge are introduced. These are denoted as , starting with the vertical dangling edge of vertex i and counting clockwise along the boundary. Conjugation operator and Opposition is commutative, but with all logical operators and For easy ). Conjugate operators also commute with each other, for all When all edge operators incident on a dangling edge have only Pauli-Z components on them, the conjugate operator is the single-qubit Pauli-Z operator on the corresponding dangling edge. For example, is equal to the Pauli-Z operator on the horizontal dangling edge incident on vertex i. The situation is more complicated when some edge operators have Pauli-Y components on dangling edges. For example, is the product of the Pauli-Z operators on the two perpendicular dangling edges incident on i and j and on the edge (i; j). In general, these operators are defined as the product of the Pauli-Z operators on the dangling edge, the second edge whose edge operator has a Pauli-Y component on the dangling edge, and the adjacent dangling edge (if any) whose edge operator has a Pauli-Y component on the second edge.

[0137] The boundary stabilizer can then be defined as:

[0138]

[0139] Boundary stabilizers commute with all logical operators and stabilizers in the bulk. They also commute with each other, i.e., Any Pauli operator of weight one or two anticommutes with at least one stabilizer, so the code distance is three and all single-qubit errors are correctable.

[0140] Operational Hardware: Extension of MLSC for General Fermion Models

[0141] The MLSC described so far can be applied to models other than the lattice models described above. For example, MLSC can be applied to quantum simulation tasks with interacting fermion models with a high degree of connectivity, e.g., a molecular electronic structure Hamiltonian given by:

[0142]

[0143] in, and represents the annihilation and creation operator of spin-orbit p, and represents the scalar coefficients, which can be given by known integrals over basis functions depending on the particular discretization scheme. This Hamiltonian has terms, where N is the number of orbitals (basis functions) into which the system is discretized. Therefore, applying MLSC directly to this Hamiltonian would require qubits, which is an unacceptable overhead in most environments.

[0144] However, instead of encoding the Hamiltonian of the system to be simulated directly, it is possible to encode the algorithm used to perform the simulation. That is, if the Hamiltonian (or more generally, any algorithm) is simulated using only a set of gates defined in terms of the evolution under fermionic operators on the lattice, then the algorithm can be encoded into an MLSC representation with only a constant factor increase in the number of qubits, making the error correction properties of these codes practical to use in a more general setting.

[0145] As an example, a tensor factorization of the Coulomb operator can be performed to express the electronic structure Hamiltonian as:

[0146]

[0147] in, and is a scalar quantity, and for chemical applications ,and represents a unitary rotation of the single-particle orbital basis. The time-evolving Trotter step can then be implemented using known techniques, e.g., and Each algorithm uses only the nearest neighbor Fermi term The nearest neighbor gates in the generated group All of these Fermi terms are represented within the MLSC applied to the planar lattice. Therefore, the entire Trotter step requires only gates (converted to gates, matching the lowest known gate complexity of the Trotter step for this Hamiltonian, and requiring only 2N qubits. Then, one can use MLSC encoding on a 2D lattice, and then 1D paths can wind through this lattice, over which these nearest neighbor algorithms are implemented in the code.

[0148] By encoding the lattice fermionic Hamiltonian whose individual terms generate all the gates in these algorithms, the error correction properties of MLSC can be exploited in a more general setting than the lattice model.

[0149] The digital and / or quantum subject matter and implementations of digital functional operations and quantum operations described in this specification may be implemented in digital electronic circuits, suitable quantum circuits, or more generally, quantum computing systems, in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in a combination of one or more of them. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptographic system, or a quantum simulator.

[0150] The embodiments of the digital and / or quantum subject matter described in this specification may be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by a data processing device or to control the operation of a data processing device. The digital and / or quantum computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or additionally, the program instructions may be encoded on an artificially generated propagation signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information, which is generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by a data processing device.

[0151] The terms quantum information and quantum data refer to information or data carried, held or stored by a quantum system, wherein the smallest non-trivial system is a qubit, i.e., a system defining a unit of quantum information. It is understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding environment. Such quantum systems may include, for example, multi-level systems having two or more levels. For example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the computational basis states are identified with the ground state and the first excited state, however it is understood that other arrangements are possible in which the computational states are identified with higher order excited states.

[0152] The term "data processing device" refers to digital and / or quantum data processing hardware, and covers all kinds of devices, equipment and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers and combinations thereof. The device may also be or further include a dedicated logic circuit, such as an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit) or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a specific quantum system. Specifically, a quantum simulator is a dedicated quantum computer that does not have the ability to perform universal quantum computing. In addition to the hardware, the device may optionally include code that creates an execution environment for a digital and / or quantum computer program, for example, code that constitutes a processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0153] A digital computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages; and it may be deployed in any form, including as a stand-alone program or as a module, component, subroutine or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, software application, module, software module, script or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages; and translated into a suitable quantum programming language, or may be written in a quantum programming language (e.g., QCL or Quipper).

[0154] A digital and / or quantum computer program may but need not correspond to a file in a file system. A program may be stored in a portion of a file that stores other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordination files (e.g., a file storing a portion of one or more modules, subroutines, or codes). A digital and / or quantum computer program may be deployed on a digital or quantum computer, or on multiple digital and / or quantum computers located at a site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can use a quantum system (e.g., qubit) to transmit quantum data. Typically, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0155] The processes and logic flows described in this specification may be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or quantum processors, where appropriate, running one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows may also be performed by a dedicated logic circuit (e.g., FPGA or ASIC), or a quantum simulator, or a combination of a dedicated logic circuit or quantum simulator and one or more programmed digital and / or quantum computers, and the apparatus may also be implemented as a dedicated logic circuit (e.g., FPGA or ASIC), or a quantum simulator, or a combination of a dedicated logic circuit or quantum simulator and one or more programmed digital and / or quantum computers.

[0156] For a system of one or more digital and / or quantum computers to be "configured to" perform a particular operation or action means that the system has installed thereon software, firmware, hardware, or a combination thereof, which, in operation, causes the system to perform the operation or action. For one or more digital and / or quantum computer programs to be configured to perform a particular operation or action means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer may receive instructions from a digital computer that, when executed by a quantum computing device, causes the device to perform the operation or action.

[0157] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general or special purpose digital and / or quantum processor or both, or any other kind of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory or a quantum system suitable for transmitting quantum data (e.g., photons), or a combination thereof.

[0158] Elements of a digital and / or quantum computer may include a central processing unit for executing or running instructions, and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and memory may be supplemented or incorporated by a dedicated logic circuit or quantum simulator. Typically, a digital and / or quantum computer will also include one or more large-capacity storage devices (e.g., magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information) for storing digital and / or quantum data, or operably coupled to receive digital and / or quantum data from one or more large-capacity storage devices (e.g., magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information) for storing digital and / or quantum data or to transmit digital and / or quantum data or both thereto. However, a digital and / or quantum computer need not have such a device.

[0159] Digital and / or quantum computer readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including, for example: semiconductor memory devices, such as EPROM, EEPROM and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems, such as trapped atoms or electrons. It is understood that quantum memory is a device that can store quantum data with high fidelity and efficiency for a long time, such as a light-matter interface, where light is used for transmission and matter is used to store and retain quantum features of quantum data, such as superposition or quantum coherence.

[0160] Control of the various systems or portions thereof described in this specification may be implemented in a digital and / or quantum computer program product, which includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions thereof described in this specification may each be implemented as an apparatus, method, or system, which may include one or more digital and / or quantum processing devices and a memory for storing executable instructions to perform the operations described in this specification.

[0161] Although this specification contains many specific implementation details, these details should not be interpreted as limitations on the scope of possible claims, but rather as descriptions of features that may be specific to a particular implementation. Certain features described in this specification in the context of separate implementations may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented in multiple implementations individually or in any suitable sub-combination. In addition, although features may be described above as functioning in certain combinations and even initially claimed as such, in some cases one or more features in the claimed combination may be cut out of the combination, and the claimed combination may be directed to a sub-combination or a variant of the sub-combination.

[0162] Similarly, although operations are depicted in a particular order in the accompanying drawings, this should not be interpreted as requiring that such operations be performed in the particular order shown or in a sequential order or requiring that all operations shown be performed in order to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. In addition, the separation of various modules and components in the embodiments described above should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0163] Specific embodiments of the subject matter have been described. Other embodiments are within the scope of the appended claims. For example, the actions recited in the claims may be performed in a different order and still achieve the desired results. As an example, the processes depicted in the accompanying drawings do not necessarily require the particular order shown or sequential order to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method performed by a quantum computing device, the method comprising: executing a quantum algorithm comprising evolving the qubit system for a predetermined time under a qubit Hamiltonian, wherein each qubit in the qubit system corresponds to a respective interaction between two fermion modes in the fermion system and terms of the qubit Hamiltonian correspond to respective terms of the fermion Hamiltonian; Determine if errors occur when executing quantum algorithms, including: applying a stabilizer operator to the qubit system during the evolving qubit system, wherein the stabilizer operator corresponds to a product of quadratic Majorana operators on corresponding loops in an interaction graph of vertices and edges, wherein each vertex represents a corresponding fermionic mode in the fermionic system and each edge represents a corresponding interaction between two fermionic modes; and performing a plurality of stable sub-measurements during the evolution of the qubit system to detect the occurrence of one or more errors; and The one or more errors are corrected.

2. The method according to claim 1, wherein: The quantum algorithm includes an algorithm for quantum simulation of the fermion system.

3. The method according to claim 1, wherein: Evolving the qubit system under the qubit Hamiltonian for a predetermined time includes applying quantum logic gates from a set of gates defined in terms of fermion operators on a lattice.

4. The method according to claim 3, wherein: The fermion system is characterized by a Hamiltonian, whose multiple monomer operator terms are in the leading order N 2 Scaled, where N represents the number of tracks.

5. The method according to claim 1, wherein: The common eigenspace of the stabilizer operators defines a code subspace encoding the state of the fermionic system.

6. The method according to claim 1, wherein: The one or more errors include single qubit errors.

7. The method according to claim 1, wherein: One or more of the following: The qubit operators include inhomogeneous or Pauli-Y operators, The qubit operator has a weight greater than or equal to three, or The qubit operator has a weight less than or equal to four.

8. The method according to claim 1, wherein: The stabilizer operators commute with each qubit operator and with each other.

9. The method according to claim 1, wherein: The qubit operators include edge operators and vertex operators, and wherein the method further includes determining the qubit operators, including: adding a first number of dangling edges to the interaction graph, wherein the first number is equal to the number of boundary vertices in the interaction graph plus four, and each dangling edge corresponds to a qubit in the qubit system; Determine the corresponding boundary stabilizer operator for each dangling edge, including for each dangling edge: Determine a corresponding dangling edge operator, wherein the dangling edge operator anti-commutes with the incident vertex operator and the edge operator, and commutes with the determined stabilizer operator; Determining a product of edge operators in a corresponding boundary element lattice, wherein the product of edge operators in the corresponding boundary element lattice commutes with the determined qubit operator and anti-commutes with each adjacent product of edge operators in adjacent boundary element lattices; determining a conjugate operator, wherein the conjugate operator anti-commutes with the dangling edge operator of the dangling edge, commutes with a determined qubit operator, commutes with the dangling edge operator of the other dangling edges, and commutes with other conjugate operators; A boundary stabilizer operator for the dangling edge is defined as being equal to i times the product of edge operators in the corresponding boundary element lattice times the conjugate operator of the dangling edge, wherein the boundary stabilizer operator commutes with qubit operators, stabilizer operators, and each other.

10. The method according to claim 1, wherein: The qubit system preserves the locality of the fermion system.

11. A method performed by a quantum computing device, the method comprising: Represent a fermion system as an interaction graph of vertices and edges, where Each vertex represents a fermion mode of a fermion system, and each edge represents an interaction between two corresponding fermion modes; adding a first number of dangling edges to the interaction graph, wherein the first number is equal to the number of boundary vertices in the interaction graph plus four; evolving the qubit system for a predetermined time under a qubit Hamiltonian, wherein each qubit in the qubit system corresponds to a respective edge in the interaction graph and qubit operators included in the qubit Hamiltonian correspond to respective terms of the fermionic Hamiltonian; Determining whether an error occurs during evolving the qubit system under the qubit Hamiltonian for a predetermined time comprises: applying stabilizer operators to the qubit system during evolving the qubit system, wherein the stabilizer operators correspond to products of Majorana operators on corresponding loops of the interaction graph, wherein the stabilizer operators include boundary stabilizer operators for each dangling edge; and performing a plurality of stable sub-measurements during the evolution of the qubit system to detect the occurrence of one or more errors; and The one or more errors are corrected.

12. The method according to claim 11 further comprises determining a corresponding boundary stabilizer operator for each dangling edge, comprising for each dangling edge: Determine the corresponding dangling edge operator, where The dangling edge operator anti-commutes with the incident vertex operator and the edge operator, and commutes with the determined stabilizer operator; determining a product of edge operators in a corresponding boundary element lattice, wherein the product of edge operators in the corresponding boundary element lattice commutes with the qubit operator and anti-commutes with each adjacent product of edge operators in an adjacent boundary element lattice; determining a conjugate operator, wherein the conjugate operator anti-commutes with the dangling edge operator of the dangling edge, commutes with a qubit operator, commutes with the dangling edge operator of other dangling edges, and commutes with other conjugate operators; and The boundary stabilizer operator of the dangling edge is defined as being equal to the product of i times the edge operator in the corresponding boundary element lattice times the conjugate operator of the dangling edge.

13. The method according to claim 11, wherein: Simulation of the fermion system is performed by evolving the qubit system under the qubit Hamiltonian for a predetermined time.

14. The method according to claim 11, wherein: Evolving the qubit system under the qubit Hamiltonian for a predetermined time includes applying quantum logic gates from a set of gates defined in terms of fermion operators on a lattice.

15. The method according to claim 14, wherein: The lattice includes open boundary conditions.

16. A method for executing a quantum algorithm, the method being executed by a quantum computing device, the method comprising: Modifying a qubit operator in a qubit Hamiltonian representing a fermion system, the modification comprising: identifying a Pauli operator that is anticommutative with at least one stabilizer operator for a quantum algorithm, wherein the stabilizer operator for the quantum algorithm corresponds to a product of Majorana operators on corresponding loops in an interaction graph of the fermion system; selecting a qubit operator in the qubit Hamiltonian and multiplying the selected qubit operator by the identified Pauli operator; identifying a set of qubit operators, wherein the qubit operators in the set of qubit operators exclude the selected qubit operator and anti-commutate with the identified Pauli operator; and multiplying each qubit operator in the identified set by one of the at least one stabilizer operator for the quantum algorithm; evolving the qubit system under the qubit Hamiltonian for a predetermined time, including applying a stabilizer operator to the qubit system during the evolving qubit system, and performing a plurality of stabilizer measurements to detect occurrence of one or more errors during the evolving qubit system; and The one or more errors are corrected.

17. A method performed by a quantum computing device, the method comprising: executing a quantum algorithm comprising evolving the qubit system for a predetermined time under a qubit Hamiltonian, wherein each qubit in the qubit system corresponds to a respective interaction between two fermion modes in the fermion system and terms of the qubit Hamiltonian correspond to respective terms of the fermion Hamiltonian; Determine if errors occur when executing quantum algorithms, including: applying a stabilizer operator to the qubit system during the evolution of the qubit system, wherein the stabilizer operator corresponds to a product of multi-mode Majorana operators on corresponding loops in an interaction graph of the fermion system, each multi-mode Majorana operator comprising a product of a transformation matrix and a corresponding quadratic Majorana operator, the transformation matrix comprising rows representing the multi-mode Majorana operator as vectors on a Galois field of two elements; and performing a plurality of stable sub-measurements during the evolution of the qubit system to detect the occurrence of one or more errors; and The one or more errors are corrected.

18. The method according to claim 16 or 17, wherein: The quantum algorithms include algorithms for quantum simulation of fermion systems.

19. The method according to claim 16 or 17, wherein: Evolving the qubit system under the qubit Hamiltonian for a predetermined time includes applying quantum logic gates from a set of gates defined in terms of fermion operators on a lattice.

20. The method according to claim 16 or 17, wherein: The fermion system is characterized by a Hamiltonian, whose multiple monomer operator terms are in the leading order N 2 Scaled, where N represents the number of tracks.

21. The method according to claim 16 or 17, wherein: The common eigenspace of stabilizer operators defines a code subspace that encodes the state of the fermion system.

22. The method according to claim 16 or 17, wherein: The one or more errors include single qubit errors.

23. The method according to claim 16 or 17, wherein: One or more of the following: The qubit operators include inhomogeneous or Pauli-Y operators, The qubit operator has a weight greater than or equal to three, or The qubit operator has a weight less than or equal to four.

24. The method according to claim 16 or 17, wherein: The stabilizer operators commute with each qubit operator and with each other.

25. An apparatus comprising: One or more classical processors; as well as Quantum hardware; The device is configured to perform the operations of the method according to any one of claims 1-24.

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