COMPUTER-IMPLEMENTED METHOD FOR DETERMINING A CONTROL SEQUENCE FOR PERFORMING A SEQUENCE OF QUOBIT INTERACTIONS, COMPUTER PROGRAM PRODUCT, QUANTUM CIRCUIT, AND METHOD FOR DETERMINING SYSTEM PROPERTIES - Patent application

Abstract

1. A computer-implemented method for determining a control sequence for executing a sequence of steps operating on qubits on a plurality of qubits on a quantum device, the method comprising: obtaining a plurality of qubit entities; decomposing at least one qubit entity; combining at least two qubit entities; and compressing the determined combination of at least one of the at least two qubit entities.

Description

[Technical field]

[0001] The present invention relates generally to quantum devices and, more particularly, to determining a set of qubit interactions performed on a quantum device by utilizing decomposition and compression of qubit entities such as qubit operators. [Background technology]

[0002] Quantum computers can provide a powerful tool for studying various types of problems, such as many-body problems, and using Hamiltonian equations, the properties of quantum many-body systems can be studied, for example, where the electronic structure of a molecule can be determined.

[0003] Quantum computers are suitable for simulating quantum many-body systems, but current quantum computers are limited by errors such as noise, imperfections, and loss of quantum coherence. The accuracy of quantum computing results can degrade rapidly as the number of gate operations and the depth of the circuit increase.

[0004] Furthermore, quantum error correction can be used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise, but quantum error correction requires additional qubits, the number of which is limited by the total number of available qubits. A problem associated with some of the known methods for simulating many-body interactions is the use of auxiliary spins, which further reduces the number of remaining qubits available.

[0005] In particular, the study of fermionic quantum systems, which can be characterized by a fermionic Hamiltonian, is important in many areas of technology, such as the study of superconductivity, the design of batteries, the optimization of chemical reactions, fertilizers, and novel drugs.

[0006] Prior to simulation of quantum many-body systems, the interactions to be performed must be converted into qubit interactions. For example, fermionic interactions can be represented as qubit interactions involving multiple qubits. However, to be able to perform computations on quantum devices, these multi-qubit interactions may need to be represented as one-qubit and / or two-qubit interactions.

[0007] Some prior art methods also utilize multiple CNOT gates to solve many-body problems. One problem with such approaches in digital quantum computing is that CNOT gates are not always native for all qubit pairs of currently available quantum systems. Gates that are not native to the system must be decomposed into a sequence of native gates, thus increasing the number of gates used.

[0008] When a computation is performed on a quantum device, a quantum circuit is implemented, which includes a sequence of quantum gates that may correspond to qubit interactions or qubit operators that describe the interactions between qubits that are being performed. Since the accuracy of the result of a computation is affected by the number of gates implemented and / or the depth (number of layers) in the quantum circuit, it is advantageous to try to find ways to provide quantum circuits with as little circuit depth and / or number of quantum gates as possible and / or to provide ways to reduce the circuit depth and / or number of quantum gates associated with a particular problem. Summary of the Invention [Problem to be solved by the invention]

[0009] It is an object of the present invention to alleviate at least some of the problems in the prior art. [Means for solving the problem]

[0010] According to one aspect of the invention, there is provided a computer-implemented method for determining a control sequence for performing a series of steps operating on a plurality of qubits on a quantum device, the method comprising: - obtaining a plurality of qubit entities including at least a first qubit entity and a second qubit entity, the qubit entities including a sequence of steps acting on qubits, the at least second qubit entity including a plurality of qubit entities including one multi-qubit interaction term; - decomposing the multi-qubit interaction terms of the second qubit entity such that the multi-qubit interaction terms are decomposed into a sequence of one-qubit and / or two-qubit interaction terms and determining a decomposed sequence of steps acting on the qubits; - combining a first sequence of steps acting on qubits of a first qubit entity with a decomposed sequence of steps acting on qubits of a second qubit entity to obtain a combined sequence of steps acting on qubits; - compressing the combined sequence of steps operating on qubits to provide a compressed sequence of steps operating on qubits including only one-qubit and / or two-qubit interactions, wherein the number of steps operating on qubits in the compressed sequence of steps operating on qubits is less than or equal to the number of steps operating on qubits in the combined sequence of steps operating on qubits; providing a control sequence for performing a sequence of steps operating on quantum bits, the control sequence including at least a condensed sequence of steps operating on quantum bits.

[0011] In this section, the terms "operator," "interaction," and "gate" may be used interchangeably. The terms "control sequence" and "quantum circuit" or "circuit" may be used interchangeably. A "series" or "sequence" of "steps acting on qubits" may also refer to a quantum circuit herein, where such a series or sequence includes steps, each step being defined as a time interval during which a qubit interaction is performed between the qubits specified by the series. Thus, a series of steps acting on qubits may include, for example, a two-qubit gate, and specify the qubits that act and the temporal order in which the gate is performed. A circuit may also refer to an empty circuit that includes one or more steps in which no qubit interaction is performed.

[0012] In the present invention, a solution may be provided for providing many-body or multi-qubit gates by using native two-qubit gates (TQGs) (or possibly native one-qubit gates (SQGs) and TQGs), which may be advantageous compared to solutions utilizing CNOT gates.

[0013] In an embodiment of the method, the final control sequence may include both SQG and TQG.

[0014] The number of gates, and in particular the number of TQGs, required to perform a multi-qubit interaction may be reduced by at least 1 compared to prior art approaches. Additionally or alternatively, the circuit depth of the quantum circuit provided through the control sequences may be reduced compared to prior art methods of solving the same many-body problem.

[0015] The depth of a quantum circuit may refer to the number of time steps required for its completion, and therefore, since quantum computations may experience increased errors the longer the computation takes, reducing the depth of the circuit may result in reduced error in addition to reduced computation time.

[0016] The present invention may provide a method (or quantum compiler) for determining control sequences that can be used to implement multi-qubit interactions on a quantum device such that the resulting circuit depth is less than using other prior art methods. Decomposing at least one qubit entity corresponding to one multi-qubit operator or interaction into a series of two-qubit interactions or combinations of SQGs and TQGs by iterative decomposition may result in control sequences that lead to less circuit depth than prior art methods.

[0017] Furthermore, when a decomposed qubit entity, which was previously, i.e., a single multi-qubit entity, is combined with another qubit entity, the circuit depth may be further reduced, and the combination compressed to provide a compressed circuit. The circuit depth of the compressed circuit may be less than the circuit depth of a circuit obtained by simply decomposing and combining the qubit entities. The method may provide an advantageous way of providing quantum circuits that implement multiple multi-qubit operators or combine decomposed multi-qubit operators into a compressed quantum circuit, such that a reduced circuit depth is provided.

[0018] The first qubit entity may include a first sequence of steps operating on qubits that include one-qubit and / or two-qubit interaction terms and / or include steps that do not include interactions between qubits. If the sequence of steps operating on qubits of the first qubit entity only includes steps without interactions between qubits, the first qubit entity is considered an empty quantum circuit. However, alternatively, the first qubit entity may include only one-qubit interaction terms, or only two-qubit interaction terms, or a combination of one-qubit and two-qubit interaction terms.

[0019] In the simplest use case scenario, the first qubit entity may be an empty quantum circuit and the second qubit entity may be a multi-qubit operator, which for example corresponds to a Hamiltonian operator containing only one term, with no further qubit entities being considered.

[0020] In some embodiments, a first qubit entity that is a quantum circuit including one qubit and / or two qubit interaction terms is not obtained, but for example if all inputs include one multi-qubit interaction term, a first qubit entity can be generated by decomposing one of the multi-qubit interaction terms to determine a first sequence of steps acting on a qubit that is a quantum circuit including one qubit and / or two qubit interaction terms, which is then used to combine and compress together with the second qubit entity.

[0021] According to one embodiment, the first qubit entity may be considered as an "empty circuit" generated by the first computing device that is combined with the obtained second qubit entity that was first decomposed to provide a decomposed quantum circuit, and the combination is compressed. Any further obtained one multi-qubit operator may be considered as a subsequent or further qubit entity.

[0022] Thus, the multiple qubit entities may include at least one subsequent qubit entity, which subsequent qubit entity includes one multiple qubit interaction term, and the method includes decomposing the subsequent qubit entity to determine a subsequent decomposed sequence of steps acting on the qubits of the subsequent qubit entity, which is combined with the compressed sequence of steps acting on the qubits determined in the previous compression to determine a final sequence of steps acting on the qubits as the sequence of compressed steps acting on the qubits determined in the last performed compression.

[0023] Subsequent qubit entities may be decomposed using the same decomposition as the compressed sequence of steps acting on the qubits determined in the previous compression.

[0024] Subsequent qubit entities may be decomposed, resulting in a decomposition that differs from the compressed sequence of steps acting on the qubits determined in the previous compression.

[0025] The decomposition may include iterative decomposition steps of multi-qubit interaction terms, with each decomposition step including at least one interaction term that includes fewer qubits than the interaction term of the previous decomposition step.

[0026] The final decomposition step of the decomposition of the multi-qubit interaction terms may include a final decomposition sequence of steps including at least one sequence of three two-qubit interaction terms, which is the last decomposition step performed before combining.

[0027] The method may additionally include obtaining information indicative of a native interaction of the quantum device, the method including applying the one-qubit interaction in conjunction with the two-qubit interaction that does not correspond to the native interaction of the quantum device to obtain one-qubit and / or two-qubit interaction terms corresponding to the native interaction of the quantum device.

[0028] The compression may include performing at least one action selected from the group of swapping through a two-qubit operator, canceling a two-qubit operator, shifting a two-qubit operator, restructuring a two-qubit operator, or merging into a two-qubit operator, where the action is performed on at least one pair of adjacent one-qubit or two-qubit operators adjacent to a two-qubit operator acting on at least one same qubit and / or on adjacent steps acting on qubits.

[0029] The compression can be performed recursively: it can be any combination of different types of actions, and any type of action can be repeated until the compressed sequence cannot be compressed any further, which means that the different types of actions no longer affect the interaction terms in the sequence of steps of the compressed sequence, and the depth of the compressed sequence cannot be reduced any further.

[0030] The compression may include performing at least an operation of shifting operators to adjacent steps.

[0031] The compression may include at least one of the effects of cancelling a two-qubit operator and / or merging and / or cancelling into a two-qubit operator, optionally combined with at least one further effect of cancelling or shifting a two-qubit operator, or merging into a two-qubit operator.

[0032] The compression can include at least two two-qubit gates operating on the same qubit being compressed into one two-qubit gate.

[0033] The compression may include compressing at least two two-qubit gates operating on at least one common qubit into one one-qubit gate and one two-qubit gate.

[0034] Compression may include the act of exchanging at least two gates.

[0035] The compression may include the act of reconstructing at least one two-qubit gate and at least one one-qubit gate.

[0036] The decomposition may include determining a decomposed sequence of steps acting on the qubits that can be combined with the first sequence of steps acting on the qubits to obtain multiple commutative pairs of two-qubit interaction terms.

[0037] The decomposition is to separate the multi-qubit interaction terms into unitary e iuO where u is the coupling strength coefficient of O between the qubits on which the primary operator O acts, and the unitary e iγH The second interaction term is written as and the negative-O unitary e of the main operator -iuO and a third interaction term written as:

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[0038] Next, the method also - selecting at least one of the qubits identified as a central qubit; and - The first auxiliary operator H is H qubits, each of which acts on a different qubit, and this qubit is represented by the operator H d and a selection range including at least one central qubit, M H is less than the number M of Pauli matrices of the multi-qubit operator being decomposed; and -M O selecting a first principal operator O as a tensor product of Pauli matrices,O is less than the number of Pauli matrices of the multi-qubit operator being decomposed, each Pauli matrix in the tensor product acts on a different qubit, and the Pauli matrices of this qubit and the first principal operator O are selected such that at least one qubit involved in the first principal operator O is one of at least one central qubit, and H d is proportional to the commutator of the primary operator O and the auxiliary operator H; - selecting a coupling strength coefficient of the main operator O as u=π / 4+a*π (where a is an integer) to separate one M-body term; - the primary operator O and the auxiliary operator H are chosen to be anti-commutative, the square of the primary operator O is equal to the identity matrix, The iterative decomposition involves iteratively selecting subsequent principal and ancillary operators until a final principal and ancillary operator is obtained that is a tensor product of two Pauli matrices and thus corresponds to a two-qubit interaction, and the decomposition of a previously determined principal operator O involves re-selecting a central qubit before selecting subsequent principal and ancillary operators, the central qubit being selected from the qubits of the operator being decomposed.

[0039] The primary operator in each decomposition step acts on a qubit on a first side of a central qubit in the considered multi-qubit operator and may include a tensor product of Pauli matrices with a Pauli matrix acting on the central qubit, and the ancillary operator in each decomposition step acts on a qubit on a second side of the central qubit in the considered multi-qubit operator and may include a tensor product of Pauli matrices with a Pauli matrix acting on the central qubit, where the Pauli matrices acting on the central qubit in the considered multi-qubit operator are selected to be of a first type, the Pauli matrices acting on the central qubit in the primary operator are selected to be of a second type, and the Pauli matrices acting on the central qubit in the ancillary operator are selected to be of a third type.

[0040] The decomposition of at least one of the multi-qubit interaction terms may be performed multiple times with multiple alternative decompositions to obtain multiple alternative decomposition sequences of steps operating on the qubits, and the method additionally includes determining multiple alternative combined sequences of steps operating on the qubits and determining multiple alternative compressed sequences of steps operating on the qubits. The method may then include selecting the compressed sequence of steps operating on the qubits that provides the least circuit depth of the multiple alternative compressed sequences of steps operating on the qubits as the compressed sequence of steps operating on the qubits used in a subsequent join procedure or as the compressed sequence of steps operating on the qubits used as the final sequence of steps operating on the qubits.

[0041] The alternative decomposition may be accomplished by selecting a different qubit as the central qubit and / or selecting different types of Pauli matrices for the primary and auxiliary operators in at least one of the iterations in the alternative decomposition.

[0042] Using the decomposition as above may eliminate the need to add auxiliary spins and therefore provide the best possible solution within the limits.

[0043] The method may be used to determine a control sequence for simulating a Hamiltonian operator that includes multiple interaction terms corresponding to multiple qubit operators, where the multiple qubit entity includes at least a multiple qubit operator that corresponds to the interaction terms of the Hamiltonian operator.

[0044] The method can be used to determine a control sequence for simulating a fermion Hamiltonian, the method comprising: - obtaining parameters of a simulated fermion Hamiltonian, the parameters being at least Multiple fermion lattice points L, Multiple fermion modes m in the fermion lattice, and fermionic operator m corresponding to the interaction between fermionic modes m, - projecting the fermionic lattice onto a qubit layout of a quantum device such that all fermionic modes are assigned to a qubit of the quantum device, said qubit referred to as qubit P, where the projection between the fermionic modes and the physical qubit is one-to-one, and a plurality of further qubits of the quantum device, referred to as ancilla qubits A, where ancilla qubits A are not assigned to any fermionic modes; The physical qubits P and the ancilla qubits A are arranged on single horizontal lines of a two-dimensional square lattice, each single horizontal line including at least one column P′ and at least one ancilla qubit A, each column P′ including one or more physical qubits P; the placement of the physical qubits P and the ancilla qubits A in each single horizontal line of the qubit layout is the same; and associating each physical qubit P with at least one edge operator E and one vertex operator V; Each physical qubit P is treated as a vertex operator V p It is to associate with V p is a first type of Pauli operator selected from Pauli operator types X, Y, and Z acting on the physical qubit p; In the horizontal dimension, for any pair p and q of physical qubits that are directly adjacent with no physical qubits or ancilla qubits between them or that are separated by one or two ancilla qubits, a horizontal edge operator associated with the qubits

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[0045] In one aspect of the invention there may be provided a computer program product as claimed in independent claim 24. In yet further aspects of the invention there may be provided a quantum circuit as claimed in claim 25 and a method for determining at least one property of a system as claimed in claim 26.

[0046] The novel features which are regarded as characteristic of the invention are set forth with particularity in the appended claims, but the invention itself, both as to its structure and its method of operation, together with additional objects and advantages thereof, will best be understood from the following description of certain illustrative embodiments when read in connection with the accompanying drawings.

[0047] The invention will now be explained in more detail with reference to exemplary embodiments according to the accompanying drawings, in which: [Brief description of the drawings]

[0048] [Figure 1A] 1 illustrates a flow chart according to one embodiment of the present invention. [Figure 1B] 1 illustrates a flow chart according to one embodiment of the present invention. [Diagram 2] 1 illustrates a schematic decomposition according to one embodiment of the present invention. [Diagram 3] 2 illustrates a flow chart of at least a portion of a decomposition method according to one embodiment of the present invention. [Figure 4] 1 illustrates a flow chart of a decomposition method according to one embodiment of the present invention. [Diagram 5] 1 illustrates a quantum circuit that may result from the decomposition. [Figure 6] FIG. 1 shows a diagrammatic notation that may be used in connection with the present invention, the diagram showing options for two-qubit operators that may be obtained by decomposition. [Figure 7] We show possible decompositions of three-qubit operators. [Figure 8] We show possible decompositions of four-qubit operators. [Figure 9] We show possible decompositions of five-qubit operators. [Figure 10]1 illustrates a sequence of steps that operate on a qubit. [Figure 11] 1 shows a sequence of steps operating on qubits, illustrating shifting, swapping, merging, and canceling. [Figure 12] We present the reconstructions for one-qubit and two-qubit operators. [Figure 13] Demonstrates the two-qubit operators exchange, cancellation, and merging. [Figure 14] Illustrates combining anticommutative two-qubit operators into one two-qubit operator. [Figure 15] We present results for decomposing, combining, and compressing two multi-qubit operators. [Figure 16] Illustrates the results of decomposing, combining, and compressing four multi-qubit operators. [Figure 17] Some examples of fermion lattice geometries are given. [Figure 18] Some examples of vertex and edge operators are illustrated below. [Figure 19] An example of a fermion lattice is given, along with the connectivity and interaction terms considered. [Figure 20] Illustrates the layout of qubit hardware. [Figure 21] 1 illustrates an exemplary hopping operator. [Figure 22] Quantum circuits are shown. [Figure 23-1] 1 illustrates examples of edge and vertex operators. [Figure 23-2] 1 illustrates examples of edge and vertex operators. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0049] The computer-implemented method of the present invention may be at least partially performed by a first computing device, typically a classical computer, such that the method may additionally include providing a control sequence obtained through the method as a computer-readable output deliverable for execution on a second computing device, which is a quantum device. The quantum device and the computing device on which the method of the present invention is performed may be completely separate devices or they may be combined devices, and the control sequence may be directly deliverable to or directly executable on the quantum device. The control sequence may include or correspond to a sequence of quantum gates.

[0050] A quantum device may include at least one quantum processor. A quantum processor may include multiple qubits or other quantum elements. In embodiments of the invention, for example, the connectivity of qubits may be arbitrary.

[0051] In one embodiment, the multiple quantum elements may be arranged in a square lattice layout. Each quantum element may be connectable / coupleable with up to four other neighboring quantum elements on the square lattice. The quantum processor may also include multiple qubits arranged in a multiple square lattice qubit layout. The square lattice may include rows or lines of qubits arranged in two dimensions, e.g., a horizontal dimension and a vertical dimension.

[0052] In one embodiment, the method may include information indicative of native interactions or gates of the quantum device. This information may be used in the method to convert non-native SQGs and / or TQGs to native gates, if desired. Information characterizing the native gates of the quantum device (or second computing device) may be obtained, for example, as an input, or may be known.

[0053] 1A, a method may include obtaining 002 at least a first qubit entity and a second qubit entity, the qubit entity including a sequence of steps operating on a qubit. The first qubit entity may include a first sequence of steps operating on a qubit. The second qubit entity may include one multi-qubit interaction term.

[0054] A multi-qubit interaction term (also called an operator) involves M qubits and can be expressed as a tensor product of Pauli matrices, identifying the qubits it involves and the type of corresponding Pauli matrix.

[0055] According to a first embodiment of the invention, at least the second qubit entity is decomposed to determine a decomposed sequence of steps acting on the qubit 004, the decomposition being performed in a manner described hereinafter. The decomposition may include an iterative decomposition of multi-qubit interaction terms.

[0056] The decomposed sequence of steps acting on qubits of the second qubit entity is then combined with the other qubit entity that is the first qubit entity in a first step of the combination, or is a first sequence of steps acting on qubits 006. In this way, a combined sequence of steps acting on qubits is determined.

[0057] The combined sequence of steps acting on the qubits is then compressed 008 to determine a compressed sequence of steps acting on the qubits. The compression may include iterative or recursive steps of compression, where one step of compression may include a combination of one or simultaneous actions of compression, where the action may be swapping through one two-qubit operator, canceling one two-qubit operator, shifting one two-qubit operator, restructuring one two-qubit operator, or merging into one two-qubit operator. Subsequent compression steps may include one or more compression actions, which may be of the same type as the previous step or different. The compression may be any combination of the different types of actions mentioned above, and any type of action may be repeated until the compressed sequence cannot be further compressed, which means that the different types of actions no longer affect the interaction terms in the sequence of steps of the compressed sequence, and the depth of the compressed sequence cannot be further reduced.

[0058] The decomposition may be performed separately on any of the obtained qubit entities that comprise the multi-qubit operator.

[0059] Combining with another qubit entity may include combining the decomposed sequence of steps acting on qubits of the second qubit entity with the compressed sequence of steps acting on qubits determined as a result of a previously performed decomposition, combination and compression procedure to obtain at least one further or subsequent combined sequence of steps acting on qubits. The further or subsequent combined sequence of steps acting on qubits may then be compressed again to determine a further or subsequent compressed sequence of steps acting on qubits.

[0060] When it is determined that the multi-qubit entity being considered does not include any further multi-qubit operators, the compressed sequence of steps operating on qubits determined in the previous step of compression may be provided as a final sequence of steps operating on qubits 010.

[0061] A first join procedure performed in the method may be performed in a first sequence of steps operating on qubits, including a series of one-qubit and / or two-qubit operators and a second qubit entity that is a multi-qubit operator, where only at least the second qubit entity (or the second qubit entity and any further qubit entities) may be received as input, e.g. from a user, while the method may include obtaining or generating a first qubit entity for the first join procedure where, e.g., only one multi-qubit operator is obtained as the qubit entity.

[0062] 1B illustrates a further embodiment of a method in which a plurality of qubit entities are obtained 002, including at least a first qubit entity and a second qubit entity, the second qubit entity including one multi-qubit interaction term, The obtained qubit entities also include at least a subsequent qubit entity including a subsequent multi-qubit interaction term.

[0063] The steps of 004a, 006a and 008a may essentially correspond to the steps performed in the embodiment of Figure 1A. After obtaining a first combined sequence of steps acting on qubits in 008a, subsequent qubit entities may be (iteratively) decomposed in 004b to determine a subsequent decomposed sequence of steps acting on qubits. In 006b, the subsequent decomposed sequence of steps acting on qubits may be combined with the compressed sequence of steps determined in 008a as the initially compressed sequence of steps (or the compressed sequence of steps obtained in a previously performed compression step) to determine a subsequent combined sequence of steps.

[0064] At 008b, the determined subsequent combined sequence of steps operating on the qubits is compressed to determine a compressed sequence of subsequent steps.

[0065] The decomposition, combination, and compression may be repeated for each further or subsequent qubit entity that is a multi-qubit interaction term. The final sequence of steps operating on qubits may be determined as the last determined compressed sequence of steps operating on qubits.

[0066] In the following, the decomposition is first disclosed in more detail, with the multi-qubit operator H d A decomposition of a qubit entity (which may refer to a qubit entity if it is a multi-qubit operator) into a two-qubit operator or two-qubit interaction is discussed. The result of the decomposition may be a determined two-qubit operator. The determined two-qubit operator may then be further manipulated to provide a circuit or control sequence or a sequence of steps acting on qubits that includes only one-qubit and / or two-qubit operators, which may correspond to native interactions of a quantum device.

[0067] Related to many-body problems, one challenge to consider in quantum computing is the problem of solving multi-qubit operators H, such as many-body Hamiltonian operators that contain a set of Pauli terms acting on multiple distinct qubits. d The objective of quantum computing is to optimally implement the Hamiltonian, which describes the behavior of many-body systems, and to implement the dynamics generated by the Hamiltonian. The unitary implementation is

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[0068] One may carry out a non-commutative unitary transformation such that the primary and auxiliary operators do not commute, but involve a series of Pauli interactions, where H d =e iO He -iO where O is a primal operator and H is an auxiliary operator. The primal operator O and the auxiliary operator H may be anti-commutative.

[0069] The task at hand is to find a suitable scheme to take the H and O terms and decompose them into TQGs, preferably such that the number of TQGs and / or the circuit depth are optimal. We can make use of the identities for matrices P and R given below, where (1) is the Baker-Campbell-Hausdorff expansion, and the second identity ((2)) is derived from the fact that P is invertible, and P -1 This leads to the third identity (equation (3)) if and only if =-P.

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[0070] In light of the above,

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[0071] With the O,H Hermitian operators [O,H] ≠ 0 and {O,H} = 0, 2 and H 2 must satisfy the following identity: e iαO =cos(α)I+isin(α)O (6) Combining the above properties,

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[0072] Using the above, a sequence of terms can be generated to appropriately set the coupling strength of the principal operator O to generate isolated many-body terms. The terms can then also be rotated individually to obtain the required combination of Pauli matrix type X, Y and Z manipulations of the many-body terms. Such a protocol can be implemented on a quantum computer, without auxiliary spins, using digital algorithms where the unitary transformations are represented by gates.

[0073] In some embodiments, after resolution, the first and / or second control sequences of the interaction are H d =e iO’ H'e -iO’where H' is the central final auxiliary operator H, O' is the sum of all final primal operators O and final auxiliary operators H except for the central final auxiliary operator, H' and O' do not commute, and any summation of O' commutes with any other summation of O'.

[0074] As illustrated diagrammatically in Figure 2, the method may involve decomposing a multi-qubit interaction term 110 into a sequence of three interaction terms 111, 112, 113. The first interaction term 111 is a unitary e iuO where u is the coupling strength coefficient of O between the qubits on which the primary operator O acts. The second interaction term 112 is the unitary e iγH The third interaction term 113 is written as the unitary e of the negative -O of the main operator. -iuO where H and O are each the tensor product of at least two Pauli matrices. The first (111), second (112), and third (113) interaction terms produced by decomposition 104 contain fewer qubits than the original multi-qubit interaction term 110.

[0075] A desired multi-qubit operator H with M qubits d may be known or may be determined. d may be expressible as a tensor product of M Pauli matrices, and identifies the type of qubit and corresponding Pauli matrix to be involved. d may refer to qubit entities. For example, at least one of the received or acquired qubit entities may be decomposed into a multi-qubit operator H d The obtained number of qubit entities can be decomposed into a multi-qubit entity H d It could be.

[0076] With reference to FIG. 3, for one embodiment of the present invention, a multi-qubit operator H dmay be obtained as an input by a first computing device 102 that includes at least one processor utilized to perform the method for determining a control sequence. d may be known or may be determined based on the problem to be solved involving a system that includes M-bodies, where certain properties of the system are determined. d may be a Hamiltonian operator, or may be, for example, one term of a Hamiltonian operator. When considering a Hamiltonian operator with multiple terms, each term that is a multi-qubit operator may be a multi-qubit operator H d and the decomposition may be performed separately for each such term. The terms may then be combined and advantageously condensed to provide a final quantum circuit implementation, as described further below.

[0077] Any of the input(s) considered may be obtained by the first computing device, e.g., provided by a user of the first computing device, or by another computing program that determines the input(s) based on additional information provided by the user, where the additional information may be, e.g., properties describing the quantum device in which a particular series of qubit interactions is implemented, and / or, e.g., desired properties of the Hamiltonian to be simulated via the series of qubit interactions. At least some of the methods contemplated herein can be performed without all possible information contemplated herein being provided as possible inputs.

[0078] The term "qubit number" may refer to a number used to identify a particular qubit contained in a particular quantum device. Of course, other identifiers may be used instead of numbers.

[0079] The decomposition may include an iterative decomposition of interaction terms with primary and auxiliary operators until the first and / or second multi-qubit interaction terms 110 are decomposed into a sequence of two-qubit interaction terms 120, each of which is described by an operator that includes a tensor product of two Pauli matrices. In the decomposed sequence, any two-qubit interaction term that corresponds to a non-native interaction may be converted into a combination of a one-qubit interaction term and a two-qubit interaction term.

[0080] The multi-qubit interaction term 110 of the first decomposition step 104 is expressed as the many-body operator H d, Unitary

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[0081] The operators involving the tensor product of two Pauli matrices are

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[0082] In step 104 of FIG. 3, a many-body operator H dis obtained in a first decomposition step, and the corresponding multi-qubit interactions are decomposed using at least a first primary operator O and a first ancillary operator H.

[0083] In some embodiments, the method may include a checking step 106 to check whether the previously obtained primary operator O and ancillary operator H correspond to a two-qubit interaction. If not, the method may include repeating at least steps 104 and 106 with one or more last obtained primary operator(s) O and / or ancillary operator(s) H that do not correspond to a two-qubit interaction, until the primary operator O and ancillary operator H include only one-qubit or two-qubit interaction terms 120.

[0084] Therefore, the decomposition can be expressed as a multi-qubit or many-body operator H d Unitary

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[0085] M O If >2, the decomposition may be repeated on the first interaction terms 111, each of which may be decomposed into M OThe decomposition may then also include repeating the decomposition on the third interaction term 113 or changing the sign of the result of repeating the decomposition on the first interaction term 111 to obtain an iteratively decomposed third interaction term 113.

[0086] M H If >2, the decomposition may be repeated for the second interaction terms 112, each of which may be decomposed into M H qubits.

[0087] The resulting decomposition of terms gives the multi-qubit interactions e iγH may be iterated until x is decomposed into a series of at least three one-qubit or two-qubit interaction terms 120.

[0088] FIG. 2 shows generally how the method can be used to decompose interaction terms 110, 111, 112, 113, ultimately arriving at a two-qubit interaction 120. The rectangular boxes in FIG. 2 illustrate gates, unitary transformations, or interaction terms. The lines in FIG. 2 indicate that each interaction term involving interactions between three or more qubits can be decomposed into three further interaction terms, which can then be further decomposed into three subsequent interaction terms, and so on, until all interaction terms are decomposed into one-qubit or two-qubit interaction terms. Thus, the top row of FIG. 2 shows the multi-qubit interaction term 110 being decomposed, and the bottom row of FIG. 2 shows the resulting control sequence, which includes multiple two-qubit interaction terms or TQGs 120.

[0089] In some embodiments, with reference to FIG. 4 , a method includes first determining a multi-qubit operator H dselecting 202 at least one of the identified qubits in as a central qubit. The central qubit may be considered as a qubit where a multi-qubit operator being decomposed is split into a primary operator and an ancillary operator, and the central qubit participates in both the primary operator and the ancillary operator.

[0090] Therefore, the multi-qubit operator H d When involves three qubits, the central qubit can be chosen in several different ways. The choice of the central qubit at each decomposition step in the iterative decomposition can affect the "shape" of the resulting quantum circuit. Thus, depending on the choice of the central qubit, the same multi-qubit operator H d There may be given several different decompositions that can be performed with

[0091] The first auxiliary operator H is M H can be chosen as a tensor product of Pauli matrices, where each Pauli matrix in the tensor product acts on a different qubit, which is then treated as a multi-qubit operator H d and a selection range including at least one central qubit, M H is less than the number of Pauli matrices of the multi-qubit interaction being decomposed.

[0092] The first main operator O is M O may be chosen as the tensor product of Pauli matrices, where M O is less than the number of Pauli matrices of the multi-qubit interaction being decomposed, each Pauli matrix in the tensor product acts on a different qubit, the qubits and Pauli matrices of the first principal operator O are such that at least one qubit is one of at least one central qubits, and H d is chosen to be proportional to the commutator of the primary operator O and the auxiliary operator H.

[0093] Additionally, a coupling strength coefficient of the primal operator O may be selected 204 as u=π / 4+a*π, and the primal operator O and the auxiliary operator H may be selected 206 to be anti-commutative. Steps 204 and / or 206 may also be performed in a different order.

[0094] The iterative decomposition may then involve iteratively selecting subsequent primal and auxiliary operators until the final primal and auxiliary operators are tensor products of two Pauli matrices and thus correspond to a two-qubit interaction.

[0095] Here, selection 202 of the central qubit(s) may occur each time a decomposition is performed: decomposition of a previously determined primal operator O or ancillary operator H may involve reselecting the central qubit(s) before selecting subsequent primal and ancillary operators, where the central qubit(s) are this time selected from the qubits of the operator being decomposed.

[0096] The primary operator in each decomposition step includes a tensor product of Pauli matrices acting on a qubit on a first side, such as the left or right side, of the central qubit in the considered multi-qubit interaction term, and includes a Pauli matrix acting on the central qubit, and the auxiliary operator in each decomposition step includes a tensor product of Pauli matrices acting on a qubit on a second side of the central qubit in the considered multi-qubit interaction term, and includes a Pauli matrix acting on the central qubit.

[0097] The Pauli matrices acting on the central qubit in the considered multi-qubit interaction may be considered to be of a first type, whereby the Pauli matrices acting on the central qubit in the primary operators are selected to be of a second type, and the Pauli matrices acting on the central qubit in the auxiliary operators are selected to be of a third type.

[0098] Thus, at each decomposition step, there are two different possibilities for the choice of the Pauli matrix type acting on the central qubit in the primary and auxiliary operators. Thus, in addition to the choice of the central qubit, there may also be multiple alternative ways in which the decomposition can be performed in this respect.

[0099] Alternative decompositions (or decomposition paths) can result in several different "shapes" of the circuit, such as V-shaped or X-shaped.

[0100] The decomposition may be performed separately for each qubit entity that is a multi-qubit operator. The result of the decomposition may be a decomposed sequence of steps that operate on qubits, which may be a sequence of one-qubit and / or two-qubit operators. The iterative decomposition results in a sequence of two-qubit operators, but two-qubit operators that correspond to non-native gates of a given quantum device can be converted to native operations using one-qubit rotations.

[0101] Here is an example of a decomposition with a first qubit entity that is an empty circuit, and a second qubit entity that is a multi-qubit operator:

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[0102] In a first step, a central qubit, now identified as the second qubit, may then be selected.

[0103] The second step may then involve decomposition, with the first prime operator

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[0104] In this way, the obtained primary operator O and ancillary operator H are not two-qubit interactions. The decomposition can then be performed iteratively. In a third step of the method, the decomposition is repeated and the first ancillary operator H can be used as the new multi-qubit interaction to be decomposed, where

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[0105] In a fourth step of the method, a first primal operator O and a second primal operator O n is a three-field term, and the selected third principal operator

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[0106] Next, the multi-qubit interaction H d is decomposed into a main operator O and an auxiliary operator H, which are two-qubit interactions.

[0107] The obtained term is the initial desired multi-qubit operator term H d can be compared to . In this example, they are equivalent and no further SQG is required. However, if this is not the case, one layer can be applied before and after the applied interaction to obtain the correct Pauli operator. On top of that, a further SQG can be applied to convert any non-native TQG to a native TQG. The XX and ZZ gates are considered to be native in this example and no SQG is required.

[0108] After the above decomposition, the obtained sequence or series of steps or steps operating on the decomposed qubit can be combined with the first qubit entity to obtain a combined sequence of steps operating on the qubit. The combined sequence of steps is compressed to obtain a compressed sequence of steps.

[0109] Figure 5 shows a parallelized quantum circuit that can be determined for the example considered above, where the number of TQGs and / or the circuit depth are optimized. Adjacent gates of opposite sign and acting on the same qubit are cancelled, and commutative relations can be used as necessary when reducing the number of TQGs and / or the circuit depth. In the current example, the XX and ZZ type gates are considered to be native, and no further SQGs are required. However, if any gates are not native, further SQGs could be applied to convert the non-native gates to native gates.

[0110] Figure 6, 6A-6C, shows a diagrammatic notation that can be used to illustrate gates in a quantum circuit, now considering a first qubit q0 and a second qubit q1. The diagram shows three different types of two-qubit gates (or operators) that arise during the decomposition, depending on the exponential prefactor, where σ and ρ are the Pauli operators acting on the qubits q0 and q1, respectively. Figure 6A shows

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[0111] Figure 7 shows a multi-qubit operator with three qubits (starting with qubit number 1).

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[0112] Starting with a multi-qubit operator with four qubits, choosing how to select the Pauli matrices for the primary and auxiliary operators at each decomposition step leads to more alternative decompositions. The alternative decompositions lead to X-shaped and V-shaped circuits. Figure 8 shows the multi-qubit operator with four qubits.

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[0113] If the obtained quantum bit entity includes two or more single multi-qubit operators, the previous description of the second quantum bit entity being decomposed into a decomposed sequence of steps acting on the quantum bits combined with the first quantum bit entity may first be performed to determine the combined sequence of steps acting on the quantum bits, which is finally compressed (if applicable, the compression may not include any steps if the compression step is not feasible) to determine the compressed sequence of steps acting on the quantum bits.

[0114] Further multi-qubit operators, e.g. a second multi-qubit operator, may then also be decomposed into a sequence of steps operating on qubits, including one-qubit and / or two-qubit operators. The compressed sequence of steps operating on qubits resulting from the decomposition, combination and compression involving the first qubit entity and the second qubit entity may then be used such that the compressed sequence of steps operating on qubits is combined with a decomposed sequence of steps operating on qubits resulting from a further decomposition of the (e.g. second) multi-qubit operator to obtain a further combined sequence of steps operating on qubits. The further combined sequence of steps operating on qubits may be compressed once more to obtain a further compressed sequence of steps operating on qubits.

[0115] For any further multi-qubit operators, the condensed sequence of steps operating on qubits obtained in connection with the previously decomposed multi-qubit operator may be used as the first qubit entity, or as the one with which the now decomposed multi-qubit operator is combined and condensed.

[0116] 10A-10C illustrate a schematic sequence of steps for operating on qubits, in which three qubits are shown: q0, q1 and q2.

[0117] FIG. 10A shows a sequence of steps including step 1 of reference number 402, which includes only one step acting on a qubit. Step 1 does not include any interaction between qubits. Such a sequence of steps without any interaction between qubits corresponds to an empty circuit and may be a first qubit entity. FIG. 10B shows a sequence of steps acting on a qubit including two steps 2 and 3 of reference number 402 and two two-qubit interactions 404 between qubits. FIG. 10C shows the result of a combination of the sequences of steps acting on qubits (or circuits) of FIG. 10A and FIG. 10B. The combination then includes a first step (in time) acting on a qubit without any qubit interaction, a second step acting on a qubit including one (first) two-qubit interaction, and a third step acting on a qubit including one (second) two-qubit interaction.

[0118] 11A-11I illustrate in a schematic manner different techniques of compression that can be used, such as shifting one two-qubit operator / interaction, exchanging through one two-qubit operator / interaction, canceling one two-qubit operator / interaction, reconstructing one two-qubit operator / interaction, and merging into one two-qubit operator / interaction. These techniques can be performed on a combined sequence of steps acting on qubits obtained from previous decomposition and combination steps. The compression can include at least one of the techniques cited above, i.e., any one of shifting one two-qubit operator / interaction, exchanging through one two-qubit operator / interaction, reconstructing one two-qubit operator / interaction, canceling one two-qubit operator, and merging into one two-qubit operator / interaction, or any combination thereof. One or more of these actions / techniques can result in a compressed sequence of steps acting on qubits.

[0119] 11A-11B show one two-qubit interaction 404 of type A that is shifted 502 in the sequence of steps 402 acting on qubits from the third step to the second step. For a further exemplary scenario of a shift with, for example, four qubits and two steps, in step 1 the TQG may act on qubits 1 and 2, and in step 2 the TQG may act on qubits 3 and 4. The second TQG may then be shifted to step 1, so that in step 2 the qubits are unaffected by the action and that step can be removed. In this way, one step in depth may be removed. Note that any step where there is no action on a qubit may be deleted.

[0120] Figure 11C shows a sequence of steps operating on qubits, including a two-qubit interaction 404 of type A in step 2 and a two-qubit interaction of type B in step 3, where the interactions operate on the same qubit. After a swap operation 504, which is only possible if the two gates are commutative, the two-qubit interaction of type B is performed in step 2 and the two-qubit interaction of type A is performed in step 3, as shown in Figure 11D.

[0121] Figure 11E shows a sequence of steps operating on qubits, including a two-qubit interaction of type A 404 in step 2 and a two-qubit interaction of type B in step 3, where the interactions operate on the same qubit. After the merge operation 506, the sequence of steps operating on the qubits in Figure 11F includes a two-qubit interaction of type AB performed in step 2.

[0122] Figure 11G shows a sequence of steps operating on qubits, including a two-qubit interaction 404 of type A in step 2 and a two-qubit interaction of type B in step 3, where the interactions operate on the same qubit. After the reduction / cancellation operation 508, the sequence of steps operating on the qubits in Figure 11H includes a two-qubit interaction of type C performed in step 1. Alternatively, after the cancellation operation 510, the sequence of steps operating on the qubits in Figure 11I does not include an interaction between the qubits.

[0123] For compression, it is sufficient to focus on operations between successive two-qubit gates, since all one-qubit gates that appear in between can be reconstructed to the edge. Note also that non-adjacent gates can be made adjacent, for example by applying one or more shift operations.

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[0124] The operations that can be performed in compression are explained in more detail below. Given a commutative pair of gates operating on the same qubit, if they differ only in the prefactor of the sign of the exponential function, they can be swapped through them, or shifted, reconstructed or cancelled, or merged into one gate if the gates are considered native. Thus, for example, a first qubit entity and a second qubit entity may include steps or interactions that operate on at least a subset of the same qubits. The operations can be illustrated as in FIG. 13, where 13A illustrates two qubit gates operating on the same qubit through which they are swapped and cancelled, and FIG. 13B shows two qubit gates operating on the same qubit that are merged into one two qubit gate. In FIG. 13B, the last two gates are

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[0125] Any combination of two gates shown in Figures 6A-6C with the same prefactor in the exponential function will result in a set of three native gates, i.e.

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[0126] For anticommutative two-qubit gates operating on the same qubit, if they are exponential and have prefactors of ±π / 4, it is possible to combine them into one two-qubit gate, which is often the case. This is shown in Figure 14, where

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[0127] It is also possible to swap through a two-qubit gate that overlaps on one qubit if they are acting with the same Pauli operator.

[0128] Below we provide some examples of how the decomposition, combination and compression techniques, including swapping, cancelling, shifting and / or merging, may be used in practice.

[0129] FIG. 15 shows at 15A and 15B a first multi-qubit operator

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[0130] The first multi-qubit operator H d,1 and the second multi-qubit operator H d,2 For the decomposition being performed on H, one alternative decomposition may be selected. Once a suitable sequence of steps acting on the qubits has been determined after the decomposition, a combined sequence of steps may be determined; one example of such a combined sequence of steps is d,1 and the second multi-qubit operator H d,2 Utilizing an alternative decomposition chosen for x, y, and z, is shown in Figure 15C. Here, two multi-qubit operators (both being 4-qubit operators) are decomposed in such a way that the resulting 2-qubit operators are swapped as long as all operators can be paired and merged into an iSWAP gate, thereby reducing the depth of the circuit. Figure 15D shows the result of pairing the operators of Figure 15C into native 2-qubit operators. This reduces the depth of the quantum circuit by half.

[0131] In addition to the above examples, FIG. 16 also illustrates, at 16A-16D, a first multi-qubit operator H d,1 (16A) and the second multi-qubit operator H d,2(16B), and the third multi-qubit operator

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[0132] In general, the control sequences, and therefore quantum circuits (sequences of quantum gates), that may be provided with the present invention may be implemented in conjunction with various types of quantum devices. As known to those skilled in the art, various methods are also available for controlling the interactions between qubits to implement quantum gates. Instructions for applying specified gates, e.g., details of the order and / or duration of voltages to be applied, may be provided in conjunction with a particular quantum device and to enable the determined control sequence to be implemented.

[0133] An embodiment of the present invention may also relate to a method for determining at least one property of a system. The system may relate to a multi-body interaction problem, the system includes at least M bodies, where M is equal to the number of qubits involved in at least one of the associated multi-qubit operators, and the property of the system to be determined may be characterized, for example, by a Hamiltonian, where the multi-qubit operator may be a Hamiltonian or the Hamiltonian may include multiple terms, where at least one term is a first qubit entity (first multi-qubit operator). The system may be, for example, a molecule, and the multi-body problem may be a solution of an electronic Schrodinger equation that gives the electronic structure of the molecule, where the property of the electronic structure is determined.

[0134] Multi-qubit operator H d (or any additional multi-qubit operators or qubit entities) may be determined based on the system and the many-body interaction problem to be solved. A control sequence may then be determined according to the methods described herein. A user of a first computing device may initiate the determination of the control sequence, and the control sequence may be distributed to a second (quantum) computing device containing at least M qubits for implementation.

[0135] Using the second computing device, a measurement gate may be applied to determine a property of the system. The property of the system may be an intrinsic energy of the system. The method may be repeated multiple times to measure the property of the system multiple times. The multiple measurements may result in different intrinsic energies of the system. The ground energy of the system may be estimated by finding the minimum intrinsic energy from those obtained in the multiple measurements. Further properties of the ground state of the system may be found by further repeating the method until the ground state energy can be estimated, and then applying further gates to obtain further properties.

[0136] The present invention can be conveniently used in connection with qubit entities that may be obtained when considering fermionic Hamiltonians and their simulation on quantum devices, where qubit entities that are multi-qubit operators may correspond to different terms in the fermionic Hamiltonian.

[0137] The electronic structure Hamiltonian H es is expressed as follows in the second quantization format:

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[0138] When characterizing a physical system for which a Hamiltonian equation is to be solved or a physical system to be simulated, a Hamiltonian can be considered, where selected terms, connectivity, and / or spin are taken into account, leading to a sparser Hamiltonian, i.e., the Hamiltonian H for the electronic structure above: es is less dense than 0(M 4 ) terms, where M is the total number of fermionic modes considered. An example of a sparse Hamiltonian comes from the Fermi-Hubbard model that may be used in condensed matter physics, where the number of terms is 0(M 2 ) Hamiltonian is

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[0139] Fermions are particles of half-integer spin that obey the Pauli exclusion principle, so that only one fermion can occupy a particular quantum state at a given time. Hence, quantum states should be antisymmetric under exchange, and the annihilation and creation operators should be anticommutative, as follows:

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[0140] The fermion Hamiltonian can then be rewritten or expressed in the following form:

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[0141] Further properties of the Hamiltonians that characterize the systems to which they relate include, for example, the fermion lattice dimension and the number of lattice points (L = L x L y L z , where L x is the number of lattice points in the first dimension, and L y is the number of lattice points in the second dimension, and L zis the number of lattice points in the third dimension), the total number of fermion modes (M) and the number of modes per lattice point (M / L).

[0142] Hamiltonians can range widely in terms of the degree of connectivity, i.e., which interactions between fermions are considered. Connectivity can include nearest neighbor interactions only, nearest neighbor and next nearest neighbor interactions, or nearest neighbor, next nearest neighbor and higher neighbor interactions).

[0143] The considered lattice points (one qubit) on the qubit layout can be assigned to represent the modes of the considered Hamiltonian. The fermion-qubit mapping is the set of edges (E pq =-E qp ) operator and vertex (V p ) operators first. After defining such edge and vertex operators, any fermion operator can be transformed into a product of edge and vertex operators and then expressed in terms of operators acting on qubits. The edge and vertex operators can be defined as follows:

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[0144] These relationships may be summarized by stating that any two different vertex or edge operators are anticommutative with each other if they act on a common fermionic mode and are otherwise commutative. Finding a suitable fermion-qubit mapping may then involve determining a set of Pauli operators that correspond to the vertex and edge operators such that the above relationships are satisfied. The Pauli weight associated with the mapping may be determined as the maximum Pauli weight associated with either the edge or vertex operator. The Pauli weight may refer to multiple different qubits associated with the operator being utilized.

[0145] A further condition to be considered is the edge operator on the closed path {p1, p2, …}, i.e., E 12 , E 23 , and E 31 The remark is that the product of edge operators, which connect qubits to form a closed path such as x, y ...

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[0146] Suitable edge and vertex operators may be determined, for example, by determining a number of different combinations (such as all possible combinations) of tensor products of Pauli operators and selecting a set of operators that satisfy selected criteria, such as at least anticommutative and commutative criteria.

[0147] After determining a set of suitable edge and vertex operators, these can be utilized in mapping each interaction between fermionic modes to qubit operators and finding an expression corresponding to the fermionic operators of the Hamiltonian of the problem being considered.

[0148] To determine a set of qubit interactions corresponding to the interactions between fermionic modes, we consider the relationship between the edge and vertex operators, the fermionic Majorana operators, JPEG2025509459000070.jpg1217, and fermion annihilation and creation operators can be used. Such a relation is given by:

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[0149] There are various considerations or criteria that may be taken into account in the selection of edge and vertex operators and thus in the determination of the fermion-qubit mapping. Favoring some criteria may lead to weaknesses related to other aspects. The criteria may relate to performance aspects in terms of computational resources required to perform the simulation / computation on the quantum device utilizing the determined mapping and the control sequence determined therefrom (e.g., circuit depth, number of two-qubit gates, qubit-to-mode ratio, i.e., number of qubits required to encode one fermion mode, and / or Pauli weights), limitations imposed by hardware aspects (such as the number of qubits and / or the connectivity of the qubit layout utilized, and / or the types of one-qubit and / or two-qubit gates allowed depending on which gates are native to the hardware in question), aspects related to known or estimated errors, or requirements for error correction related to the computation performed via the quantum device, and / or versatility aspects related to the type of Hamiltonian to which the mapping may be applied. It may therefore be impossible to determine a fermion-qubit mapping whose performance is optimal in relation to any type of Hamiltonian, any type of fermion system, and any type of hardware.

[0150] In the following, a fermion-qubit mapping is introduced that is well suited for use on hardware including a square lattice qubit layout in which one qubit is connectable with up to four other qubits, and the mapping further has low weight composite edge operators for the far-neighbor (beyond NN) edges, enabling the simulation of a broad class of two-dimensional fermion lattices, for example, from condensed matter physics. Furthermore, the selected mapping is conveniently utilized in the methods of the present disclosure, including decomposition, combination, and / or compression, to provide less circuitry in depth for simulating fermionic Hamiltonian systems.

[0151] The fermion-qubit mapping presented here provides an effective strategy for mapping square fermion lattice geometries into square lattice qubit layouts and also embedding the fermion lattice geometries shown in Figures 17A-17I. The figures show alternative fermion lattice geometries (2-dimensional) and projections, associations of fermion lattice points with physical qubits, and associated connectivity between lattice points that can be efficiently embedded into a square lattice qubit layout using the mappings of the present invention. Figure 17A shows a square lattice with NNN connectivity, Figure 17B shows a square lattice with NN connectivity, Figure 17C shows a Shastry-Sutherland lattice, Figure 17D shows a checkerboard lattice, Figure 17F shows a triangular lattice, Figure 17G shows a honeycomb lattice, Figure 17H ​​shows a Kagome lattice, and Figure 17I shows a tetrakis lattice. The methods of the present invention can make maximum use of available qubits, or qubits available to associate with fermion lattice points present in a square lattice qubit layout. Within at least a selected subset or selected square lattice portion of the square lattice layout of the quantum device, all (physical) qubits may be utilized in the projected lattice geometry.

[0152] The invention may allow for determining edge operators that connect two physical qubits via a vertical stack of ancilla qubits, where multiple vertical edges may be configured to reach further neighbors (not connected by the directly determined edge operator), and the cost may increase by one ancilla qubit per unit of distance on the fermionic graph.

[0153] A method for providing fermion-to-qubit mapping can include receiving parameters of a fermion Hamiltonian to be simulated, the parameters including at least a number of fermion lattice points L, a number of fermion modes m in the fermion lattice, and a fermion operator corresponding to an interaction between the fermion modes m.

[0154] In the mapping, the fermionic lattice may be projected onto the qubit layout of the quantum device such that all fermionic modes are assigned to qubits of the quantum device, which are referred to as physical qubits P, and the projection between the fermionic modes and the physical qubits is one-to-one. Additionally, to solve the anticommutative relation, a number of further qubits of the quantum device are referred to as ancilla qubits A, which are not assigned to any fermionic modes.

[0155] Each physical qubit P may then be associated with at least one edge operator E and one vertex operator V. Each fermion operator may then be mapped to a qubit operator based on the edge operator E and the vertex operator V. These qubit operators may correspond to qubit entities, and at least one of the qubit operators is a second qubit entity.

[0156] The physical qubits P and ancilla qubits A may be arranged on single horizontal lines of a two-dimensional square lattice of the qubit layout, such that each single horizontal line includes at least one column P' and at least one ancilla qubit A, each column P' includes one or more physical qubits P, and the arrangement of the physical qubits P and ancilla qubits A in each single horizontal line of the qubit layout is the same.

[0157] Preferably, in a row of qubits there are at most two consecutive ancilla qubits A. More may be utilized, but at most two consecutive ancilla qubits may be more efficient.

[0158] The qubits may further be arranged in a pattern, the pattern being repeated consecutively within a single horizontal line of qubits, the pattern being selected from the group of P'A, P'P'A, P'P'AA, and P'AA.

[0159] A row of qubits can be obtained by selecting a pattern and repeating the pattern in the horizontal dimension to obtain a desired / determined total number of columns P' and ancilla qubits A in the row. The pattern may be repeated a selected number of times, or the pattern may be repeated such that the pattern is only partially provided at the beginning and / or end of a row or horizontal single line of qubits. For example, a row of qubits having pattern P'P'A may include columns P' and ancilla qubits A arranged as follows: P'AP'P'AP'P'AP', i.e., the pattern is repeated twice in its entirety and the sequence is "cut" as P'A at the beginning of the row and P' at the end of the row.

[0160] The plurality of consecutive physical qubits P that the column P' corresponds to or includes can be obtained based on the fermionic system and / or Hamiltonian considered. For example, if the fermionic system includes two fermionic modes per fermionic lattice point, then the pattern of P'AA corresponds to PPAA when implemented in rows of qubits in a qubit layout.

[0161] The horizontal rows of qubits may correspond to a sequence of qubits that may be obtained through further consideration of the fermionic system. The obtained or known information may indicate a plurality of fermionic lattice points in two or three dimensions, including a plurality of fermionic lattice points in a first dimension L1, a plurality of fermionic lattice points in a second dimension L2, and optionally a plurality of fermionic lattice points in a third dimension L3, where the total number of fermionic lattice points is L=L1L2 when two dimensions are considered, and L=L1L2L3 when three dimensions are considered.

[0162] Each fermionic mode in the fermionic Hamiltonian can be identified by indices i, j, and k that indicate the location of the fermionic lattice points along the respective dimensions, where i = [1,L1], j = [1,L2], and k = [1,L3]. The number of fermionic modes at each of the fermionic lattice points identified by indices i, j, and k is M ijk and the total number of fermionic modes in the fermionic lattice is M =

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[0163] A physical qubit P is assigned to fermionic modes, each of which is identified by indices i, j, k, and l. ijkl may be placed in the qubit layout as follows: -Each row of the qubit layout contains L1 columns P′, and each column is P′ ij where P is a number of physical qubits P that are assigned to fermionic modes associated with fermionic lattice points having position indices i and j. ijkl where P′ is the sequence ij The number of physical qubits in

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[0164] The physical qubits in any column P' in the horizontal dimension may be arranged such that the order of physical qubits with indices k and l is reversed in successive columns P'.

[0165] When a fermion Hamiltonian describes a system with two or more spin types in the fermion lattice points, the mode M ijk may further be assigned to a number of orbitals o, each of which includes a number of spins s, and M ijk=o*s, where physical qubits assigned fermionic modes with spin of one orbit are each separated by one physical qubit, and where if the l index of the physical qubit assigned to the orbit is odd, then the fermionic modes associated with the physical qubit with index l may be arranged such that as the index l increases, the spin type s also increases, whereas if the index l of the physical qubit representing the orbit is even, then the fermionic modes associated with the physical qubit with index l may be arranged such that as the index l increases, the spin type s decreases.

[0166] The association of physical qubits with edge and vertex operators may take into account the native gates of the quantum device on which the obtained control sequence is implemented. In one embodiment, the physical qubits in a row of qubits in the horizontal dimension may be grouped into pairs of physical qubits alternately labeled even or odd, where ancilla qubits are not taken into account, and adjacent even and odd pairs share a physical qubit, and then associating each physical qubit with at least one edge operator includes: For each even pair of physical qubits p and q and one or more ancillary qubits between them (if present), an even horizontal edge operator associated with this qubit

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[0167] FIG. 18 shows an advantageous selection of edge and vertex operator types, where the selection of the Pauli operator type for the vertex operator is V p =Z p and Z p is the Pauli operator σ acting on the quantum bit p. ZFIG. 18A illustrates vertex operators, FIG. 18B illustrates possible horizontal edge operators, and FIG. 18C illustrates possible vertical edge operators that may be used in connection with most qubits and qubit layouts. FIG. 18D illustrates vertical edge operators that may be used in the rare case where two associated physical qubits P may be connected without an ancilla qubit A. This situation may arise at the boundaries of finite systems, e.g., in connection with honeycomb-type lattice geometries, where it may not be necessary to use an ancilla qubit between at least some possible pairs of physical qubits at the edges of the qubit lattice.

[0168] The different possibilities of the edge operators depend on the placement of ancilla qubits near the considered physical qubit. In Fig. 18, the physical qubits (or the string of physical qubits P') are illustrated as boxes and the ancilla qubits as circles, but the notation X / Y and X / Y relating to physical qubits within the same operator means that both qubits are inversely related to σ X Operator or σ Y (In the following figures, the notations X / Y and Y / X relating to physical qubits within the same operator indicate that one qubit is acted on by an operator corresponding to σ X or σ Y , and the other qubit is acted on by a different operator σ X or σ Y The choice of these operators indicates that the native gate is

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[0169] FIG. 18B shows that the Pauli sequence corresponding to the horizontal edge operator is σ X Operator or σ Y One of the operators, σ Z We show that an operator can be formed by combining it with at most two intermediate ancilla qubits acting on it. σ X or σ Y By varying between adjacent horizontal edges, one can ensure that the horizontal edges are mutually anticommutative. Thus, depending on the choice of pattern used, one can choose the Pauli weights between horizontal edges to be between 2 and 4.

[0170] As can be seen from Figure 18C, all vertical edge operators depicted in relation to qubit layouts typically have a square-like shape that includes two physical qubits and two ancilla qubits. The physical qubits can be on either the left or right side of the ancilla qubit, and both Pauli operators acting on the physical qubits in the vertical edge operators as well as the ancilla qubits and horizontal edge operators acting on the physical qubits are σ X or σ Y , the choice should be the same as the horizontal edge operator used in the same mapping, and they should be anticommutative with the horizontal edge operator.

[0171] 19-21 illustrate an example of applying the method of the present invention for the simulation of a spinless Fermi-Hubbard model (one mode per fermion lattice point) with NN and NNN hopping / interactions. The example considered is a 3x3 fermion lattice (L1=3, L2=3, L3=1) projected onto a qubit layout with three horizontal rows, one row having the following arrangement of physical and ancilla qubits: PAPAPA (pattern of PA used subsequently, P' is equal to P since only one mode per lattice point is considered).

[0172] Figure 19 shows the fermion lattice in 19A and the connectivity considered. Figures 19B-19I specify (highlighted with bold lines) the different hopping terms on the fermion lattice to be considered: the first horizontal interaction in 19B, the second horizontal interaction in 19C, the first vertical interaction in 19D, the second vertical interaction in 19E, the first diagonal interaction in 19F, the second diagonal interaction in 19G, the third diagonal interaction in 19H, and the fourth diagonal interaction in 19I. In Figure 19, the circles represent different fermion lattice points. In separate subfigures, operators that can be applied in parallel are illustrated on the same lattice.

[0173] Figure 20 illustrates a qubit layout that may be used to simulate the 3×3 fermion lattice and interacting fermion Hamiltonian shown in Figure 19. In Figure 20, squares represent physical qubits and circles represent ancilla qubits, and Figure 20 also indicates the qubit numbers that may be used.

[0174] 21A-21H illustrate hopping operators (i.e., operators including one edge operator multiplied by one vertex operator) that may be determined to be utilized in determining fermion-qubit mappings and associated qubit operators that include the qubit layout of FIG. 20 and correspond to the fermion lattice and fermion interactions of FIG. 19.

[0175] The hopping operator illustrated in FIG. 21A corresponds to the first horizontal interaction in FIG. 19B, FIG. 21B corresponds to the second horizontal interaction in FIG. 19C, FIG. 21C corresponds to the first vertical interaction in FIG. 19D, FIG. 21D corresponds to the second vertical interaction in FIG. 19E, FIG. 21E corresponds to the first diagonal interaction in FIG. 19F, FIG. 21F corresponds to the second diagonal interaction in FIG. 19G, FIG. 21G corresponds to the third diagonal interaction in FIG. 19H, and FIG. 21H corresponds to the fourth diagonal interaction in FIG. 19I.

[0176] The horizontal or vertical hopping operator is V i E ij where E ijare the corresponding edge operators in the horizontal or vertical direction, and the diagonal hopping operators can be obtained as composite operators, and V i E jk E kj where one of the edge operators is a corresponding horizontal edge operator and the other edge operator is a corresponding vertical edge operator.

[0177] The following procedure may be followed to arrive at the circuit of FIG. 22, which may be determined as a final sequence of steps acting on qubits, which may serve as a control sequence for performing qubit interactions on a quantum device.

[0178] First, as many parallel blocks of gates as possible can be placed. For example, all horizontal hoppings of the type shown in FIG. 21A can be grouped, then horizontal hoppings of the type shown in FIG. 21B can be grouped, and then groups can be added until all groups are included. Each of these hoppings consists of a sum of Hermitian conjugates. The operator pairs obtained from mapping the sum of Hermitian conjugate hopping terms can be combined. As can be seen in FIG. 22, those hopping terms that lead to maximum cancellation and compression of the gate can be put together, and all the operators are also rotated (with one-qubit gates) to a form that can be considered as a native gate. Then, parallel operators acting on different sets of qubits can be collected in layers and applied simultaneously.

[0179] In addition to the compression considered above, some further cancellations can be made between Trotter layers. In this way, additional reductions can be obtained when simulating many Trotter steps. This reduction depends on the depth of the two longest hopping terms and whether they can place a boundary without losing compression.

[0180] To show how the mapping should be applied when more than one mode per fermionic lattice point is considered, FIG. 23 illustrates an example of edge and vertex operators that may be utilized when the pattern P′AA is selected and the fermionic system contains two modes per lattice point. Thus, in this case, the column P′ corresponds to two physical qubits, namely PP. The squares represent the physical qubits P and the circles represent the ancilla qubits A. In this example, each lattice point is considered to contain one orbital, each with two spin types, here “up” and “down”. Filled squares (black) correspond to one spin type mode (e.g., up), while empty squares (no fill) correspond to the other spin type mode (e.g., down).

[0181] 23A and 23C show horizontal edge operators, 23B and 23D show vertical edge operators, and 23E shows vertex operators. The groups of operators in each separate figure act on different qubits and can therefore be implemented in parallel. The operators are illustrated by marking the associated qubit with a letter corresponding to the associated Pauli operator type and showing the associated qubits and connections in bold. Selected regions 220 and 226 focus on horizontal edge operators, 224 and 228 focus on vertical edge operators, and 222 focus on vertex operators, each shown for both up and down spin modes, respectively. Thus, each edge operator acts on or involves physical qubits that are assigned to corresponding fermionic modes on different fermionic lattice points.

[0182] It may also be noted that if the fermionic lattice is three-dimensional, each lattice point in the third dimension (L3) may be treated in the projection as an additional mode or orbit of the fermionic lattice point in the first dimension. Thus, a column P' may include a number of physical qubits P corresponding to the modes contained in the associated lattice point in the first dimension and a number of physical qubits corresponding to the modes contained in each fermionic lattice point in the third dimension. However, it should be noted that in this case, the obtained circuit depth may not be constant or scale linearly with the size of the fermionic system, but the circuit depth scales linearly with the third dimension and the number of fermionic modes per lattice point.

[0183] When more than one fermionic mode per fermionic lattice point is considered, the different modes may be associated with different orbitals and / or different spin types of the same orbital. In these cases, however, the column P' may then include multiple consecutive physical qubits P forming a chain of modes. A mode associated with a physical qubit at the end of one chain or column P' may readily engage in interaction with one or more modes of further fermionic lattice points associated with physical qubits of further columns P', such interactions being made possible through edge and vertex operators that may connect the qubits associated with the modes.

[0184] In some cases, therefore, there may be a chain of fermionic modes associated with qubits in a column, with one or more modes initially located at intermediate lattice points in a row, and internal modes associated with internal physical qubits in column P'. Such internal modes may be advantageously relocated such that they may be located at end positions of a chain of fermionic modes in order to participate in one or more interactions.

[0185] The relocation of fermionic modes may be accomplished by utilizing fSWAP operators such that one or more internal modes are shifted in a first direction or a second direction (left or right) along a row of qubits. A network of fSWAP operators including M / L-2 parallel fSWAP layers may swap fermionic modes in a column P' of M / L modes by alternating two layers of fSWAP between every adjacent qubit pair, which may be alternately labeled as either even (e.g., physical qubit numbers 2n, 2n+1) or odd (e.g., physical qubit numbers 2n+1, 2n+2).

[0186] The assignment of modes to the arrangement of rows of qubits may include an ordering of the modes such that an fSWAP network may be efficiently used to bring related modes next to each other on a connection graph corresponding to a qubit layout projection. The assignment of modes as disclosed herein may result in efficient simulation of fermionic Hamiltonians since multiple operations may be performed in parallel. Advantageously, there may be no physical qubits assigned to fermionic modes that are not involved in qubit interactions during a layer of a control sequence or quantum circuit where a qubit interaction corresponding to a fermionic interaction is applied.

[0187] Table 1 below shows how the mappings presented herein can utilize the decomposition and compaction techniques described to provide less circuit depth than the prior art Derby-Klassen (DK) mapping. The table considers the circuit depth per Trotter step and shows the results of several different spin-full fermion lattices with the selected pattern P'AA for the mappings of the present disclosure. The decomposition and compaction techniques presented herein are also performed herein for DK mapping (and can also be readily used with further prior art mappings). However, Table 1 shows how the mappings disclosed above can result in even less circuit depth than, for example, DK mapping, even when decomposition and compaction are applied to both. [Table 1] Table 1: Summary of depth for the implementation of Trotter step circuits for different spin-full fermion lattices using fSIM(θ,φ) native two-qubit gates.

[0188] In summary, it would therefore be advantageous to determine a fermion-to-qubit mapping that provides qubit operators that can be conveniently used to obtain efficiently reduced circuit depths using the presently disclosed decompositions and compressions.

[0189] Thus, the methods of the present invention may be used to determine a control sequence for simulating a fermion Hamiltonian, the method may further include determining a plurality of fermion-qubit mappings, determining a plurality of qubit entities associated with each fermion-qubit mapping for which a final control sequence is determined, and selecting the control sequence that provides the least circuit depth.

Claims

1. 1. A computer-implemented method for determining a control sequence for executing a series of steps operating on a plurality of qubits on a quantum device, the method comprising: - obtaining a plurality of qubit entities including at least a first qubit entity and a second qubit entity, the qubit entities comprising a sequence of steps acting on qubits, and at least the second qubit entity comprising one multi-qubit interaction term; - decomposing the multi-qubit interaction term of the second qubit entity such that the multi-qubit interaction term is decomposed into a sequence of one-qubit and / or two-qubit interaction terms and determining a decomposed sequence of steps acting on qubits; - combining a first sequence of steps acting on qubits of the first qubit entity with the decomposed sequence of steps acting on qubits of the second qubit entity to obtain a combined sequence of steps acting on qubits; compressing the combined sequence of steps operating on qubits to provide a compressed sequence of steps operating on qubits including only -1 qubit and / or 2 qubit interactions, wherein a number of steps operating on qubits in the compressed sequence of steps operating on qubits is less than or equal to the number of steps operating on qubits in the combined sequence of steps operating on qubits; providing a control sequence for executing a sequence of steps operating on quantum bits, the control sequence including the compressed sequence of steps operating on at least quantum bits.

2. 2. The method of claim 1 , wherein the first qubit entity comprises a first sequence of steps operating on qubits, the first qubit entity comprising steps including one-qubit and / or two-qubit interaction terms and / or steps not including interactions between qubits.

3. 3. The method of claim 1 or claim 2, wherein the decomposition comprises iterative decomposition steps of multi-qubit interaction terms, each decomposition step including at least one interaction term that involves fewer qubits than the interaction term of a previous decomposition step.

4. The method of any one of claims 1 to 3, wherein the final decomposition step of the decomposition of the multi-qubit interaction terms comprises a final decomposition sequence of steps including at least a sequence of three two-qubit interaction terms, the final decomposition step being the last decomposition step performed before combining.

5. 5. The method of claim 1, wherein the method additionally comprises obtaining information indicative of a native interaction of the quantum device, the method comprising applying a one-qubit interaction in conjunction with a two-qubit interaction that does not correspond to a native gate of the quantum device to obtain a one-qubit and / or two-qubit interaction term corresponding to a native interaction of the quantum device.

6. 6. The method of claim 1, wherein the compression comprises compressing at least two two-qubit gates operating on the same qubit into one two-qubit gate.

7. 7. The method of claim 1, wherein the compressing comprises compressing at least two two-qubit gates operating on at least one common qubit into one one-qubit gate and one two-qubit gate.

8. The method according to any one of claims 1 to 7, wherein the compressing step is performed recursively.

9. 9. The method of any one of claims 1 to 8, wherein the compacting comprises performing at least one action selected from the group of swapping through a two-qubit operator, cancelling the one two-qubit operator, shifting the one two-qubit operator, reconstructing the one two-qubit operator, or merging into the one two-qubit operator, wherein the action is optionally performed on at least one pair of adjacent one-qubit or two-qubit operators acting on at least one same qubit and / or on adjacent steps acting on qubits.

10. 9. The method of any one of claims 1 to 8, wherein the compression comprises at least one of the following actions: merging and / or cancelling into a two-qubit operator, optionally combined with at least one further action of cancelling a two-qubit operator, shifting the one two-qubit operator, reconstructing the one two-qubit operator, or swapping through the one two-qubit operator, or merging into the one two-qubit operator.

11. 11. The method according to claim 9 or 10, wherein the act of shifting comprises performing a shift of an operator to at least an adjacent step.

12. 11. The method of claim 9 or 10, wherein the act of exchanging comprises exchanging at least two gates.

13. 13. The method of any one of claims 6 to 12, wherein the compressing comprises reconstructing at least one two-qubit gate and at least one one-qubit gate.

14. the plurality of qubit entities includes at least one subsequent qubit entity, the subsequent qubit entity including one multi-qubit interaction term, and the method further comprising: - decomposing the subsequent qubit entity to determine a subsequent decomposed sequence of steps acting on the qubits of the subsequent qubit entity; - combining said subsequent decomposed sequence with said compressed sequence of a step operating on quantum bits determined in a previous compression; - compressing said combined sequence of steps; - determining a final sequence of steps acting on quantum bits as the compressed sequence of steps acting on quantum bits determined in a last performed compression.

15. 15. The method of claim 14, wherein the subsequent qubit entities are decomposed using the same decomposition as used for the compressed sequence of steps acting on qubits determined in a previous compression.

16. 15. The method of claim 14, wherein the subsequent qubit entities are decomposed using a different decomposition than the decomposition used for the compressed sequence of steps operating on qubits determined in a previous compression.

17. The multi-qubit interaction term involves M qubits and can be expressed as a tensor product of Pauli matrices, and the type of qubits and corresponding Pauli matrices to be involved is specified, and the decomposition is a multi-qubit interaction term expressed as a unitary e of the primal operator O. ｉｕＯ where u is a coupling strength coefficient of O between the quantum bits on which the primary operator O acts; and a unitary e ｉγＨ The second interaction term described by the negative −O unitary e －ｉｕＯ and a third interaction term written as: The method includes iteratively decomposing interaction terms associated with a primary operator and an ancillary operator until the multi-qubit interaction term is decomposed into a sequence of two-qubit interaction terms, the multi-qubit interaction terms of the first decomposition step being unitary of the first multi-qubit operator and / or the second multi-qubit operator. [0010] where H ｄ denotes the multi-qubit operator being decomposed, and γ denotes H ｄ and the multi-qubit interaction terms in any subsequent decomposition step(s) are associated with a primary operator O or an ancillary operator H.

18. The method further comprising: - selecting at least one of said qubits identified as a central qubit; - The first auxiliary operator H is Ｈ a tensor product of Pauli matrices, each of which acts on a different qubit, and the qubit is selected according to the operator H ｄ wherein the selection includes at least one of the central qubits, and M Ｈ is less than the number M of Pauli matrices of the multi-qubit operator being decomposed; -M Ｏ Selecting a first principal operator O as a tensor product of Pauli matrices, Ｏ is less than the number of Pauli matrices of the multi-qubit operator being decomposed, each Pauli matrix in the tensor product acts on a different qubit, the Pauli matrices of the qubits and the first principal operator O are selected such that at least one qubit involved in the first principal operator O is one of at least one of the central qubits, and H ｄ is proportional to a commutator of the primary operator O and the auxiliary operator H; - selecting the coupling strength coefficient of the main operator O as u=π / 4+a*π, where a is an integer, to separate one M-body term; the primal operator O and the auxiliary operator H are chosen to be anti-commutative, the square of the primal operator O being equal to the identity matrix, 18. The method of claim 17, wherein the iterative decomposition comprises iteratively selecting subsequent principal and ancillary operators until a final principal and ancillary operator is obtained that is a tensor product of two Pauli matrices and thus corresponds to a two-qubit interaction, and wherein the decomposition of a previously determined principal operator O comprises re-selecting the central qubit before selecting subsequent principal and ancillary operators, the central qubit being selected from the qubits of the operator being decomposed.

19. 20. The method of claim 18, wherein the primary operator in each decomposition step acts on a qubit on a first side of the central qubit of the multi-qubit operator being considered and comprises a tensor product of Pauli matrices with a Pauli matrix acting on the central qubit, the ancillary operator in each decomposition step acts on a qubit on a second side of the central qubit of the multi-qubit operator being considered and comprises a tensor product of Pauli matrices with a Pauli matrix acting on the central qubit, the Pauli matrices acting on the central qubit of the multi-qubit operator being considered are selected to be of a first type, the Pauli matrices acting on the central qubit of the primary operator are selected to be of a second type, and the Pauli matrices acting on the central qubit of the ancillary operator are selected to be of a third type.

20. The decomposition of at least one of the multi-qubit interaction terms is performed multiple times with multiple alternative decompositions to obtain multiple alternative decomposed sequences of steps operating on qubits, the method additionally comprising: - determining a plurality of alternative combined sequences of steps operating on the qubits; and determining a plurality of alternative condensed sequences of steps operating on the qubits; - selecting a compressed sequence of steps operating on quantum bits that provides the least circuit depth of a plurality of alternative compressed sequences of steps operating on quantum bits as the compressed sequence of steps operating on quantum bits to be used in a subsequent combination and compression step or as the compressed sequence of steps operating on quantum bits to be used as a final sequence of steps operating on quantum bits.

21. 21. The method of claim 20, wherein the alternative decompositions are performed by selecting a different qubit as a central qubit and / or selecting different types of Pauli matrices for the primary and auxiliary operators in at least one of the iterations in the alternative decompositions.

22. 22. The method of claim 1, wherein the method is used to determine a control sequence for simulating a Hamiltonian operator including a plurality of interaction terms corresponding to a multi-qubit operator, the plurality of qubit entities including at least a multi-qubit operator corresponding to the interaction terms of the Hamiltonian operator.

23. The method is used to determine a control sequence for simulating a fermion Hamiltonian, the method comprising: - obtaining the parameters of a simulated fermion Hamiltonian, said parameters comprising at least: ○Multiple fermion lattice points L, a plurality of fermionic modes m in the fermionic lattice, and a fermionic operator corresponding to an interaction between the fermionic modes m; - projecting the fermionic lattice onto a qubit layout of the quantum device such that all fermionic modes are assigned to qubits of the quantum device, the qubits being referred to as physical qubits P, the projection between the fermionic modes and the physical qubits being one-to-one, and a number of further qubits of the quantum device being referred to as ancilla qubits A, the ancilla qubits A not being assigned to any fermionic modes, The physical qubits P and the ancilla qubits A are arranged on single horizontal lines of a two-dimensional square lattice, each single horizontal line including at least one column P′ and at least one ancilla qubit A, each column P′ including one or more physical qubits P; a projection in which the arrangement of the physical qubits P and the ancilla qubits A on each single horizontal line of the qubit layout is the same; and - associating each physical qubit P with at least one edge operator E and one vertex operator V, Each physical quantum bit P is treated as a vertex operator V ｐ and V ｐ is a first type of Pauli operator selected from Pauli operator types X, Y, and Z acting on the physical qubit p; In the horizontal dimension, for any pair p and q of physical qubits that are directly adjacent with no physical qubits or ancilla qubits between them or that are separated by one or two ancilla qubits, a horizontal edge operator associated with said qubits [0025] Define [0030] but, At least two Pauli operators, each of a second or third type selected from Pauli operator types X, Y, and Z, acting on quantum bits p and q, respectively; ■ If any ancilla qubits exist between the physical qubits p and q along the horizontal dimension, additional Pauli operators, each of a first type acting on each of the ancilla qubits if the ancilla qubits exist; and and when two horizontal edge operators act on the same qubit q, the first of the two horizontal edge operators [0045] acts on the quantum bit q with a second type of Pauli operator, then the second of the two horizontal edge operators [0050] acts on the qubit q with a Pauli operator of a third type, and vice versa, ○ for any pair of physical qubits p and q, the physical qubits p and q are immediately adjacent in the vertical dimension, the pair of physical qubits are immediately adjacent to a pair of ancilla qubits a and b, the ancilla qubits a and b are immediately adjacent in the vertical dimension, the ancilla qubits a and b are located adjacent to the qubits p and q, respectively, and a vertical edge operator [006] Define where: [0070] is a product of four Pauli operators, each of the second or third type, and each of the four Pauli operators acting on one of the qubits p, q, a, b, such that each of the four Pauli operators acts on a different qubit; ■ A Pauli operator acting on an ancilla qubit a that has no different type, ■ the Pauli operator acting on the physical quantum bit p is of the same type as the Pauli operator acting on the physical quantum bit q and forms part of the horizontal edge operator acting on at least the physical quantum bit p and the ancilla quantum bit a; similarly, the Pauli operator acting on the physical quantum bit q is of the same type as the Pauli operator acting on the physical quantum bit q and forms part of the horizontal edge operator acting on at least the physical quantum bit q and the ancilla quantum bit b; A first vertical edge operator is formed when the ancilla qubits a and b are disposed on a first side of the physical qubits p and q, respectively, along the horizontal dimension. [0080] or, when the ancilla qubits a and b are disposed on a second side of the physical qubits p and q along the horizontal dimension, a second vertical edge operator [0090] It is called, when two vertical edge operators act on the same ancilla qubit, if one of the two vertical edge operators acts on the ancilla qubit with a Pauli operator of a second type, then the other of the two vertical edge operators acts on the ancilla qubit with a Pauli operator of a third type, and vice versa; The method according to any one of claims 1 to 22, further comprising: mapping each fermion operator to a qubit operator based on the edge operator E and the vertex operator V, wherein at least one of the determined qubit operators is utilized as a second qubit entity.

24. A computer program product comprising program code means adapted to carry out the method according to any one of claims 1 to 23, when the program product is run on a computer.

25. 24. A quantum circuit comprising a sequence of qubit interactions determined according to the method of any one of claims 1 to 23.

26. 1. A method for determining at least one property of a system, the method comprising: - determining a multi-body interaction problem associated with a system, wherein at least one property of said system is characterized by said multi-body interaction problem; - determining a control sequence according to any one of claims 1 to 23, - implementing the determined control sequence on a quantum device comprising at least M qubits and applying measurement gates to determine the property of the system.

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