Methods for determining control sequences for qubit interactions, and related quantum circuits, quantum devices, and methods for solving problems.

Abstract

A computer-implemented method for determining a control sequence for performing a sequence of qubit interactions on a plurality of qubits on a quantum device to simulate a quantum many-body Hamiltonian. The method includes determining a sequence of two-qubit interactions based on decomposing a multi-qubit interaction term (120) into a sequence of three interaction terms, where a first interaction term (111) includes a primal operator O, a second interaction term (112) includes an ancillary operator H, and a third interaction term (113) includes a negative of the primal operator, where H and O are each a tensor product of at least two Pauli matrices, and where the first interaction term, the second interaction term, and the third interaction term produced by the decomposition include fewer qubits than the original multi-qubit interaction term. The method further includes iterative decomposition of the interaction terms in terms of the primal and ancillary operators until the multi-qubit interaction term (110) is decomposed into a sequence of two-qubit interaction terms (120). The present invention further relates to computer programs, quantum circuits, quantum devices, and methods for solving problems involving many-body interactions.

Description

[Technical Field]

[0001] This invention relates in general to quantum computing. More specifically, it relates to a computer implementation method for determining a control sequence for simulating a quantum many-body Hamiltonian by performing a series of qubit interactions on multiple qubits on a quantum device. [Background technology]

[0002] A quantum computer, or quantum device, is a machine that uses the properties of quantum physics to store data and perform calculations. Compared to classical computers, which encode information in the form of bits, such as 0s or 1s, quantum computers use quantum bits (qubits), which can be in a coherent superposition of two states simultaneously. A qubit can refer to the fundamental unit of quantum information, or to a quantum device (such as a two-level quantum machine system) used to store a unit of quantum information. Thus, a quantum computer generally includes an array of qubits and hardware for manipulating these qubits.

[0003] There are three basic quantum computing methods: the analog quantum model, the universal quantum gate model (also known as the digital quantum computing model or quantum circuit model), and quantum annealing. In the quantum gate model, the manipulation of qubits or interactions between qubits are called gates, and a sequence of one or more gates configured to be applied to qubits constitutes a control sequence for a quantum device, which may be called a quantum circuit, corresponding to instructions for manipulating units of quantum information to perform a desired computation. Quantum gates are sometimes further called unitary operators, represented by unitary matrices. Quantum gates acting on multiple qubits are sometimes further called interactions between multiple qubits. Implementing gates acting on multiple qubits on a quantum device corresponds to performing qubit interactions on multiple qubits. Thus, unitary operators are sometimes also called interaction terms.

[0004] Many-body problems are a broad term for physical problems concerning the properties of systems containing many interacting objects, such as particles, where interactions between three or more objects are called many-body interactions. Because the complexity of classical simulations of quantum many-body systems typically increases exponentially with the system's dimension, quantum computers can offer advantages for simulating many-body problems. Many-body interactions arise naturally in the simulation of problems in fields such as quantum chemistry, finance, optimization, and high-energy physics. Thus, simulating many-body problems on a quantum computer requires the simulation of many-body interactions using qubits.

[0005] Quantum computers are well-suited for simulating quantum many-body systems, but current quantum computers are limited by the number of available qubits, as well as errors in the form of noise, faults, and loss of quantum coherence. The accuracy of quantum computation results can decrease rapidly as the number of gate operations, circuit depth, and / or measurements increases.

[0006] Furthermore, quantum error correction can be used in quantum computing to protect quantum information from errors caused by decoherence and other quantum noise; however, quantum error correction requires additional qubits, the number of which is limited by the total number of available qubits. A problem associated with some known methods for simulating many-body interactions is the use of auxiliary qubits, which reduces the number of remaining available qubits.

[0007] Some conventional methods also utilize multiple CNOT gates to solve the many-body problem. One problem with such approaches in digital quantum computing is that CNOT gates are not necessarily native to all qubit pairs in the quantum system currently available. Non-system-specific gates must be decomposed into a sequence of native gates, and therefore a larger number of gates are used. [Overview of the Initiative]

[0008] An object of the present invention is to mitigate at least some of the problems in the prior art. An object of the present invention is to provide an alternative and / or improved method or device for determining quantum gates applied to multiple qubits to simulate a quantum many-body Hamiltonian or interaction. Some embodiments of the present invention can be considered to provide a quantum compiler for providing quantum circuits. According to one aspect of the present invention, a series of qubit interactions are performed on multiple qubits on a quantum device to create a quantum many-body Hamiltonian H containing M qubits. d A computer implementation method is provided for determining a control sequence to simulate the many-body Hamiltonian H d It can be expressed as a tensor product of M Pauli matrices, and the method involves determining the sequence of 2-qubit interactions based on decomposing a multi-qubit interaction term into a sequence of three interaction terms, the first interaction term being a unitary e of the principal operator O. iuO Described by, where u is the coupling intensity coefficient of O between qubits on which the primary operator O acts, and the second interaction is the unitary e of the auxiliary operator H. iγH Described by, the third interaction is a unitary e of negative -O of the main operator -iuO Described by, where H and O are each the tensor product of at least two Pauli matrices, and the first, second, and third interaction terms generated by the decomposition contain fewer qubits than the original multi-qubit interaction term. The method involves iterative decomposition of interaction terms related to the main operator and auxiliary operator until the multi-bit interaction terms are decomposed into a sequence of two-bit interaction terms each related to an operator containing the tensor product of two Pauli matrices. The multi-bit interaction term in the first decomposition step is the multi-body Hamiltonian H d unitary [Number] described by, where γ is the coupling strength coefficient of H d and the multi-bit interaction terms in any subsequent decomposition step(s) are related to the main operator O or the auxiliary operator H, and the decomposition is based at least on the known or determined multi-body Hamiltonian H d identifying the qubits and types of corresponding Pauli matrices involved.

[0009] In other words, a computer-implemented method for determining a control sequence for simulating a multi-body quantum Hamiltonian H d comprising M qubits that can be represented as the tensor product of M Pauli matrices by performing a series of one or more interactions on a plurality of qubits on a quantum device may be provided, and the method is to determine a sequence of two-bit interactions based on decomposing the multi-bit interaction terms described by the multi-body Hamiltonian H d unitary [Number] where γ is the coupling strength coefficient of H d to a sequence of three interaction terms, the first interaction term is described by the unitary e iuO of the main operator O, where u is the coupling strength coefficient of O between the qubits on which the main operator O acts, the second interaction term is described by the unitary e iγH of the auxiliary operator H, and the third interaction term is the unitary e of the negative of the main operator -O-iuO Described by, where H and O are M H Quantum bits and M O The determination is the tensor product of at least two Pauli matrices, each containing a qubit, M O If >2, the first interaction term is repeatedly decomposed, and each becomes M O This involves creating a sequence of three subsequent interaction terms containing less than a certain number of qubits, and obtaining a third interaction term that has been repeatedly decomposed by either repeatedly decomposing the third interaction term or by changing the sign of the result of repeatedly decomposing the first interaction term. M H If >2, the second interaction term is M H This involves repeatedly decomposing the system into a sequence of three subsequent interaction terms, each containing fewer qubits than in the previous case, This includes iterating through the decomposition of the resulting terms until a multi-qubit interaction term is decomposed into a sequence of two-qubit interaction terms, each described by an operator containing the tensor product of two Pauli matrices, This decomposition involves known or determined many-body Hamiltonians H d Based at least on the many-body Hamiltonian H d This identifies the types of qubits included and their corresponding Pauli matrices.

[0010] In this specification, operator decomposition may refer to the decomposition of the corresponding interaction term.

[0011] The present invention provides a solution for providing many-body gates by using native two-qubit gates (TQGs) (or, in some cases, nearly native TQGs meaning TQGs and single-qubit gates (SQGs)). This may be advantageous compared to solutions that utilize CNOT gates.

[0012] In embodiments of the method, the final control sequence may include both SQG and TQG.

[0013] The number of gates, particularly preferably TQGs, required to simulate M-body interactions can be reduced by at least one compared to prior art approaches. Additionally or alternatively, the circuit depth of the quantum circuit provided through the control sequence can be reduced compared to prior art methods in which the same M-body problem is solved. For example, a method utilizing CNOT gates may result in the requirement of 2(M-1) TQGs and 2(M-1) circuit depths. However, the present invention can provide a solution that utilizes 2(M-1)-1 TQGs, resulting in a circuit depth of M-1 (for even M) or M (for odd M). In some embodiments, the circuit depth or the number of TQGs can be reduced to fewer than those considered above, and the avoidance of using CNOT gates can still offer advantages over the prior art.

[0014] The depth of a quantum circuit can refer to the number of time steps required for its completion. Therefore, since quantum computation can involve increasing errors as the time required for computation increases, reducing circuit depth can lead to a reduction in errors in addition to a reduction in computation time.

[0015] The present invention can also provide a solution that does not require auxiliary qubits, and therefore the number of qubits required can be reduced or optimized.

[0016] The quantum compiler or method according to the present invention can provide an automated and faster method for determining the optimal circuit for a particular available quantum device.

[0017] The method according to the present invention may be advantageous in use cases where the quantum device does not provide connectivity / coupling possibilities between selected qubits that may be included in the desired interaction term.

[0018] As an example, within a parity coding scheme, any optimization problem can be decomposed into a Hamiltonian having simplex and four-body bracket terms mapped to a square lattice. Implementing the associated Hamiltonian using CNOT gates can be done in a circuit with a depth of 6 (focusing only on TQG) and can be performed in 4 layers. However, according to the present invention, the circuit depth can be reduced to 4 layers and can be performed in 2 layers.

[0019] According to one embodiment of the present invention, a computer implementation method can be provided for determining a control sequence for simulating the Hamiltonian of an optimization problem by performing a series of qubit interactions on a plurality of qubits on a quantum device, wherein the Hamiltonian includes a plurality of four-body bracket terms mapped to a square lattice. The method may include first decomposing the four-body bracket terms according to the method of the present invention described above, and then optimizing the circuit by utilizing Pauli matrix exchange and identity relations, in order to reduce the circuit depth and / or the number of TQGs of the quantum circuit simulating the Hamiltonian of the optimization problem.

[0020] The method can be performed at least partially by a first computing device, typically a classical computer, and as a result, the method may further include providing a two-qubit interaction obtained through the method as a control sequence, which is a computer-readable output deliverable for implementation on a second computing device, which is a quantum device. The quantum device and computing device on which the method of the present invention is performed may be completely separate devices, or they may be coupled devices, and the control sequence may be deliverable directly to the quantum device, or may be implemented directly on it. The control sequence may include, or correspond to, a sequence of quantum gates.

[0021] Preferably, the square of the main operator O is equal to the identity matrix.

[0022] In one embodiment, decomposition can be based on known or determined qubit coupling paths, where the qubit coupling paths are many-body Hamiltonians H that are coupled when executing a determined control sequence on the quantum device. d This shows a qubit connection link that at least shows the qubits.

[0023] A qubit coupling path may be acquired as input, or a qubit coupling path may be determined, and when the control sequence is executed on the quantum device, the determined qubit coupling path may result in a selected circuit depth. The selected circuit depth may be the optimal or minimum circuit depth available, based on the many-body Hamiltonian and identified qubits, and taking into account the qubit connectivity of the quantum device.

[0024] The method may include obtaining information about the qubit connectivity of a quantum device. The obtained or determined qubit coupling paths may be based on qubit connectivity such that qubit connection links are available in qubit connectivity.

[0025] In some embodiments, it is known or can be determined that multiple qubit coupling paths may be available, and the qubit coupling path may be selected such that the selected qubit coupling path contains the minimum number of qubit connection links from the set of available qubit coupling paths. This qubit coupling path may be considered the optimal or shortest qubit coupling path. If multiple shortest qubit coupling paths are available, one may be selected, for example, based on user input, a predetermined preference of the method, or randomly.

[0026] Many-body Hamiltonians H for which the qubit coupling pathway is known or determined d If it contains more qubits than that, the method is - Many-body Hamiltonian H in decomposition da replacement that includes M'>M qubits within the qubit coupling path

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[0027] The qubit coupling pathway is a known or determined many-body Hamiltonian H d If it contains more than one qubit, the many-body Hamiltonian term H d This may include qubits for which connectivity is not available on the selected quantum device (disconnected qubits). The qubit coupling path may be determined based on the known connectivity of the device and provides qubit connectivity links between disconnected qubits via connected or connectable qubits. The present invention relates to a desired many-body Hamiltonian term H d To arrive at the corresponding solution, we can provide a method for determining a quantum circuit that exhibits qubit interactions that can occur with connected qubits on a selected quantum device.

[0028] Known or determined many-body Hamiltonian H d Qubit coupling paths involving more qubits than the many-body Hamiltonian H d To obtain the many-body Hamiltonian H, and then based on the qubit connectivity of the quantum device, dDetermining that a qubit coupling path is not possible, which involves determining only the qubits identified by, can also be obtained or determined by a method including. An alternative qubit coupling path with additional qubits in the quantum device may then be determined, such alternative qubit coupling path preferably involves a minimum number of qubits. The minimum number of qubits can be obtained based on the shortest available path to obtain the alternative qubit coupling path. Such a method may be advantageous because it allows the implementation of interactions between disconnected qubits without using a SWAP gate, thus reducing computation time and therefore computation error, since a SWAP gate implementation is about three times slower than a single native TQG implementation.

[0029] Alternatively, the many-body Hamiltonian H d If it is determined that a qubit coupling path containing only the qubits identified by is not possible, a SWAP gate can be applied to connect the required qubits, in which case the many-body Hamiltonian H d The control sequence for carrying out the procedure is a many-body Hamiltonian H according to the method of the present invention. d In addition to the control sequence determined by decomposing it, it may include a SWAP gate.

[0030] In embodiments of the present invention, the method is - Selecting at least one of the identified qubits as the central qubit, - The first auxiliary operator H is M H This involves selecting a tensor product of individual Pauli matrices, where each Pauli matrix in the tensor product acts on a different qubit, and that qubit is a many-body Hamiltonian H d Selected from those specified by, the selection includes at least one central qubit, and M H This involves selecting fewer than the number of Pauli matrices of the multi-qubit interaction being decomposed, - The first main operator O is M O This involves selecting the tensor product of individual Pauli matrices, MO The number of Pauli matrices of the multi-qubit interaction being decomposed is less than the number of Pauli matrices of the tensor product, each Pauli matrix of the tensor product acts on a different qubit, and the Pauli matrices of that qubit and the first principal operator O are such that at least one qubit is one of at least one central qubit, and H d The selection is made in proportion to the commutators of the main operator O and the auxiliary operator H. -To isolate a single M-field term, the coupling intensity coefficient of O is selected as u = π / 4 + a × π, where a is an integer, and this selection may include... - The main operator O and auxiliary operator H are chosen to be anticommutative, Iterative decomposition involves repeatedly selecting subsequent primary and auxiliary operators until the primary and auxiliary operators are the tensor product of two Pauli matrices, and thus correspond to a two-qubit interaction.

[0031] According to one embodiment of the present invention, selecting at least one of the identified qubits as the central qubit may be based on the qubit connectivity of the quantum device and / or known or determined qubit coupling paths.

[0032] A selected circuit depth may be provided by selecting at least one central qubit (one if M is odd, or two if M is even) and decomposing it with respect to the central qubit(s). The selected circuit depth may be optimized or minimal for a particular use case.

[0033] Choosing the coupling strength as u = π / 4 + a × π may result in the separation of the M-field term. For other coupling strengths, terms up to the M-field are generated.

[0034] In some embodiments, the coupling strength coefficient of the auxiliary operator may be selected as π / 4 + a × π.

[0035] The method may also involve obtaining information about the qubit connectivity of a quantum device. If the connectivity exhibits linear connectivity of qubits and one qubit can be coupled with at most two other qubits, then decomposing a previously determined interaction term related to a primary operator O may involve re-selecting a central qubit before selecting subsequent primary and auxiliary operators, where the central qubit(s) is selected from the qubits of operator O related to the interaction term being decomposed.

[0036] The method may further include obtaining information indicating the native interactions or gates of a quantum device. The method may also include applying a single-qubit gate in relation to a two-qubit interaction in a control sequence that does not correspond to the native interactions of the quantum device to obtain a two-qubit interaction corresponding to the native interactions of the quantum device.

[0037] One or more properties of a quantum device may be known or acquired in different embodiments of the present invention, and the properties include, for example, the device's native gate or the device's connectivity, and one or more properties may be taken into consideration in the method.

[0038] According to one aspect of the present invention, a computer program product is also provided which comprises program code means adapted to perform a method item of an embodiment of the present invention when executed on a computer.

[0039] Furthermore, according to one embodiment, a quantum circuit is provided which includes a sequence of qubit interactions determined by an embodiment of a method for determining a control sequence, which can be run on a quantum device containing at least M qubits for simulating a quantum many-body Hamiltonian.

[0040] Alternatively, a quantum device comprising at least M qubits may be provided, configured to perform a sequence of qubit interactions determined according to a method of an embodiment of the present invention for providing a control sequence.

[0041] A method for determining at least one characteristic of the system described in independent claim 15 is also provided.

[0042] Novel features that are considered characteristic of the present invention are described in particular in the appended claims. However, the present invention itself, with respect to both its structure and its method of operation, along with its additional purposes and advantages, will be best understood from the following description of specific exemplary embodiments when read in conjunction with the appended drawings.

[0043] Next, the present invention will be described in more detail with reference to exemplary embodiments shown in the accompanying drawings. [Brief explanation of the drawing]

[0044] [Figure 1] A flowchart of the method according to one embodiment of the present invention and related schematic decomposition is shown. [Figure 2] A flowchart of the method according to one embodiment of the present invention is shown. [Figure 3] A flowchart of the method according to one embodiment of the present invention is shown. [Figure 4] This demonstrates exemplary connectivity of a qubit. [Figure 5] This provides an example of decomposition related to quantum circuits. [Figure 6] An example of a control sequence as a quantum circuit is shown. [Figure 7] This provides an example of decomposition related to quantum circuits. [Figure 8] An example of a control sequence as a quantum circuit is shown. [Figure 9] The linear connectivity of qubits is described, and exemplary qubit coupling paths are shown. [Figure 10]An example of a control sequence as a quantum circuit is shown. [Modes for carrying out the invention]

[0045] Regarding the many-body problem, one of the challenges considered in quantum computation is the Hamiltonian H d The goal is to optimally implement this, which involves a string of Pauli terms acting on some different qubits, where the Hamiltonian describes the behavior of a many-body system, and quantum computation aims to implement the dynamics generated by the Hamiltonian. The unitary being implemented is unitary

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[0046] A non-commutative unitary transformation can be performed, where H d =e iO He -iOHere, O is the primary operator and H is the auxiliary operator, and the primary and auxiliary operators are not commutative and contain a string of Pauli interactions. The primary operator O and the auxiliary operator H may also be anticommutative with respect to each other.

[0047] The task at hand is to find a suitable method for obtaining the H and O terms and decomposing them into TQGs, preferably such that the number and / or depth of the TQGs is optimal. The following identities may be used for matrices P and R, where equation (1) is the Baker-Campbell-Hausdorff expansion and the second identity (equation (2)) is P -1 The case where = -P is derived from the fact that P is invertible, leading to a third identity (equation (3)).

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[0048] In the above,

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[0049] [O,H]≠0 and {O,H}=0 are used together with the O,H Hermitian operators, O 2 and H 2 is an identity e iαO = cos(α)I + isin(α)O (6) It shall be considered equal to. Combine the above characteristics.

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[0050] Using the above, a sequence of terms can be generated, and by appropriately setting the coupling strength of the main operator O, separated many-body terms can be generated. These terms can then be rotated individually to obtain the required combinations of Pauli matrix types X, Y, and Z operations for the many-body terms. Such a protocol can be implemented on a quantum computer without auxiliary qubits, using a digital algorithm in which unitary transformations are represented by gates.

[0051] In some embodiments, after decomposition, the final control sequence of the interaction is H d =e iO’ H'e -iO’It can be expressed as follows, where H' is the central final auxiliary operator H, and O' is the sum of all final main operators O and final auxiliary operators H except the central final auxiliary operator, where H' and O' do not commutate, and any adenand of O' commutates with any other adenand of O'. Examples of such final control sequences expressed as digital quantum circuits are shown later herein in Figures 6, 8, and 10.

[0052] The present invention relates to a computer implementation method for determining a control sequence for simulating a quantum many-body Hamiltonian by performing a series of qubit interactions on a plurality of qubits on a quantum device, and includes a method for decomposing a multi-qubit interaction term 110 into a sequence of three interaction terms 111, 112, and 113, as schematically shown in Figure 1B. The first interaction term 111 is a unitary e of the principal operator O. iuO Described by, where u is the coupling intensity coefficient of O between qubits on which the principal operator O acts. The second interaction term 112 is the unitary e of the auxiliary operator H. iγH Described by, the third interaction term 113 is the unitary e of the negative number -O of the main operator. -iuO Described by , where H and O are each the tensor product of at least two Pauli matrices. The first interaction term 111, the second interaction term 112, and the third interaction term 113 generated by decomposition 104 contain fewer qubits than the original multi-qubit interaction term 110.

[0053] A desired many-body Hamiltonian H containing M qubits d The many-body Hamiltonian H is either known or can be determined. d This can be expressed as the tensor product of M Pauli matrices, which identifies the qubits included and the types of the corresponding Pauli matrices. Referring to Figure 1, with respect to one embodiment of the many-body Hamiltonian H d This can be obtained, for example, as input to a first computing device having at least one processor used to perform a method for determining a control sequence (102). dThe system may be known or determined based on the problem to be solved, which involves a system containing M objects, and certain properties of the system are determined.

[0054] The inputs(s) considered may be acquired by a first computing device, which may be provided, for example, by a user of the first computing device, or by another computing program that determines the inputs(s) based on additional information provided by the user, which may be, for example, properties describing a quantum device on which a particular many-body interaction is performed and / or desired properties of the many-body interaction. At least part of the methods discussed herein may be performed without providing all the possible information discussed herein as possible inputs. The methods according to the present invention may be performed with any M, H d It can be understood by those skilled in the art that it may be possible to implement such a system that enables qubit connectivity or qubit coupling paths.

[0055] In one embodiment, the input obtained in the method may include at least a number M that identifies the size of the many-body term, i.e., the number of fields it contains; a qubit number vector [i,j,k,l…] that identifies a desired number of qubits contained in the many-body Hamiltonian; and a corresponding Pauli operator number given by a vector of the same size as the qubit number vector, such as [1,2,3,2…], where numbers 1, 2, and 3 represent Pauli matrix types X, Y, and Z, respectively.

[0056] Some of the information used in the method can be determined from the possible inputs obtained. For example, in one embodiment, the inputs obtained in the method may include at least a qubit number vector [i,j,k,l…] identifying a desired number of qubits contained in a many-body Hamiltonian, and a corresponding Pauli operator number given by a vector of the same size as the qubit number vector, such as [1,2,3,2…], where the numbers 1, 2, and 3 represent Pauli matrix types X, Y, and Z, respectively. Then, a number M identifying the size of the many-body term can be determined from the size of the qubit number vector or the Pauli operator type vector. Alternatively, the inputs obtained in the method may include at least a number M identifying the size of the many-body term, i.e., the number of bodies and qubit connectivity of the quantum device. Then, based on the number M and qubit connectivity of the quantum device, the qubit number vector can be determined. The Pauli operator type can be randomly assigned or determined based on further inputs including the qubit connectivity of the quantum device and the set of available native gates.

[0057] The term "quantum bit number" can refer to the number used to identify a particular qubit contained within a given quantum device. Of course, other identifiers can be used instead of numbers.

[0058] The method may involve an iterative decomposition of interaction terms, including primary and auxiliary operators, until the multi-qubit interaction term 110 is decomposed into a sequence of two-qubit interaction terms 120, each described by operators that include the tensor product of two Pauli matrices.

[0059] The multi-qubit interaction term 110 of the first decomposition step 104 is the many-body Hamiltonian H d Unitary

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[0060] Operators involving the tensor product of two Pauli matrices are:

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[0061] In step 104 of Figure 1A, the many-body Hamiltonian H d This is taken in the first decomposition step, and the corresponding multi-qubit interaction is decomposed using at least a first primal operator O and a first auxiliary operator H.

[0062] In some embodiments, the method may include a check step 106 for checking whether previously acquired principal operators O and auxiliary operators H correspond to a two-qubit interaction. If not, the method may include repeating at least steps 104 and 106 with the last acquired principal operator(s) O and / or auxiliary operator(s) H that do not correspond to a two-qubit interaction until the principal operator(s) O and auxiliary operators H include only the two-qubit interaction term 120.

[0063] Therefore, the method is the many-body Hamiltonian H dUnitary

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[0064] In the method, M O If >2, the decomposition is repeated for the first interaction term 111, and each of them is M O The interaction term can be decomposed into a sequence of three subsequent interaction terms containing less than 100 qubits. The method may then also include obtaining an iteratively decomposed third interaction term 113 by repeating the decomposition for the third interaction term 113, or by changing the sign of the result of repeating the decomposition for the first interaction term 111.

[0065] M H If >2, the decomposition is repeated for the second interaction term 112, and each of them is M H It can be decomposed into a sequence of three subsequent interaction terms, each containing less than 100 qubits.

[0066] The resulting decomposition of terms is multi-qubit interaction e iγH This may be iterated until it is decomposed into a series of at least three 2-qubit interaction terms 120.

[0067] Figure 1B schematically illustrates how the method can be used to decompose interaction terms 110, 111, 112, and 113 to finally arrive at the two-qubit interaction 120. The rectangular boxes in Figure 1B represent gates, unitary transforms, or interaction terms. The lines in Figure 1B indicate that each interaction term, which involves interactions between three or more qubits, can be decomposed into three further interaction terms, and then further decomposed into three subsequent interaction terms, and so on, until all interaction terms are decomposed into two-qubit interaction terms. Thus, the upper row in Figure 1B shows the many-body interaction term 110 being decomposed, and the lower row in Figure 1B shows the resulting control sequence, which includes multiple two-qubit interaction terms or TQG 120.

[0068] In some embodiments, referring to Figure 2, the method first involves many-body Hamiltonian H d This may include selecting at least one of the identified qubits within as the central qubit. If M is odd, one central qubit may be selected; if M is even, two central qubits may be selected. For odd M, in addition to the central qubit, a second qubit adjacent to the central qubit may be selected.

[0069] The central qubit is a many-body Hamiltonian H d It may also be defined as at least one of the identified qubits within the many-body Hamiltonian H d The central qubit is connected to the many-body Hamiltonian H via the qubits. d The longest chain, determined from the set of shortest chains linking to any other qubits of the many-body Hamiltonian H, has the maximum number of qubit-connected links. d Any non-central qubit can be accessed through the qubits of a many-body Hamiltonian H d It involves fewer or equal numbers of qubits compared to the longest chain, which is determined from the set of shortest chains that link to any other qubits of the same qubit.

[0070] The first auxiliary operator H is M HIt can be selected as a tensor product of individual Pauli matrices, where each Pauli matrix in the tensor product acts on a different qubit, and that qubit is a many-body Hamiltonian H d Selected from those specified by, the selection includes at least one central qubit, M H This is less than the number of Pauli matrices of the multi-qubit interaction that are decomposed.

[0071] The first main operator O is M O It can be chosen as the tensor product of individual Pauli matrices, where M O The number of Pauli matrices of the multi-qubit interaction being decomposed is less than the number of Pauli matrices of the tensor product, each Pauli matrix of the qubits and the first principal operator O is such that at least one qubit is one of at least one central qubit, and H d The commutators of the main operator O and the auxiliary operator H are selected in proportion to the commutators of the main operator O and the auxiliary operator H.

[0072] Furthermore, the coupling strength coefficient of the main operator O may be chosen as u = π / 4 + a × π (204), and the main operator O and the auxiliary operator H may be chosen to be anticommutative (206). Steps 204 and / or 206 may be performed in a different order.

[0073] Next, the iterative decomposition may involve repeatedly selecting subsequent principal and auxiliary operators until the final principal and auxiliary operators are the tensor product of two Pauli matrices and thus correspond to a two-qubit interaction.

[0074] Decomposition may take into account the qubit connectivity or topology of the quantum device, which may be known or acquired.

[0075] In some embodiments, qubit connectivity may be taken as input, for example, as a list of two-element vectors indicating which qubits are connected. For example, [2,3], [3,4], [4,5] may indicate that the qubits identified by numbers 2, 3, 4, and 5 are linearly connectable / coupled.

[0076] Figure 2 shows a general embodiment of the method, which may be applicable, for example, to branched topologies or qubit connectivity as described further below. However, Figure 3 shows a flowchart of the method applied to linear qubit connectivity, such as that found in square lattice qubit topologies. In such linear topologies, all qubits can be connected to form qubit coupling paths such that any one qubit in a path is coupled with at most two other qubits.

[0077] Here, the selection of the central qubit(s) may be performed each time a decomposition is carried out. Thus, if the known or acquired connectivity of the quantum device indicates linear connectivity of the qubits, the previously determined decomposition of the primal operator O may include re-selecting the central qubit(s) before selecting subsequent primal and auxiliary operators, where the central qubit(s) are selected from the qubits of the decomposed operator O.

[0078] In some embodiments, the qubit connectivity of a quantum device may exhibit qubit branch connectivity comprising at least a first qubit, a second qubit, and a plurality of additional qubits, wherein the first qubit is couplingable to the second qubit, and the additional qubits are couplingable to only one of the first or second qubits.

[0079] Thus, one embodiment of the present invention may also relate to a quantum device comprising at least M qubits, the device including branched qubit connectivity and configured to perform a sequence of qubit interactions determined according to the method herein.

[0080] FIG. 4 schematically shows in 4A a branched qubit connectivity 400a, where the circles represent qubits 402 (not all qubits are labeled), the solid lines refer to ZZ links, and the dashed lines refer to XX links. A link 401 (or qubit connectivity / coupling) may be any tensor product of two Pauli matrices selected from the X, Y, and Z Pauli matrices. The choice of a particular embodiment may depend on the native gates or interactions available in the quantum device. In this example, ZZ and XX are selected. In one example, the quantum device may be a superconducting quantum chip device. Any other type of quantum device may also be selected so that other native gates are possible. It is also possible that only one of the gates is native and the other is obtained from such native gates and native SQGs.

[0081] In one embodiment, the method may include information indicating native interactions or gates of the quantum device. This information may be used, if necessary, in a method for converting non-native TQGs to native gates. Information characterizing the native gates of the quantum device (or a second computing device) may be obtained, for example, as an input or may be known.

[0082] In the example of FIG. 4, it is also conceivable that the solid lines link two central qubits 402a (here numbered as qubits 0 and 1, e.g., q 0 and q 1 which may be referred to) of a possible M-body interaction. In branched connectivity, there may always be two central qubits, regardless of whether M is even or odd.

[0083] Figure 4B shows an example of linear connectivity for qubit 402. Here, if M is even, there can be two central qubits 402a (numbered qubits 0 and 1). Furthermore, if M is odd, there can be only one central qubit.

[0084] One exemplary method for determining the control sequence is as follows: Consider six objects.

number

[0085] In the first step, the central qubits, identified here as qubits numbered 2 and 3, may then be selected. Next, the second step is:

number

number

number

[0086] As you can see, the obtained primary operator O and auxiliary operator H are not a two-qubit interaction. The decomposition may then be performed iteratively. In the third step of the method, the decomposition is repeated, and the first auxiliary operator H may be used as a new multi-qubit interaction to be decomposed, here

number

number

number

[0087] In the fourth step of the method, the first principal operator O and the second principal operator O are three-field terms. n is the selected third primary operator

number

number

number

number

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[0088] Multi-qubit interaction H d This is then decomposed using the primary operator O and auxiliary operator H, which are two-qubit interactions.

[0089] The obtained term is the initial desired many-body Hamiltonian term H dcan be compared. In this example, these are equivalent, and it can be seen that no further SQG is required. However, if not, one layer can be applied before and after the interaction being applied to obtain the correct Pauli operator. Further, further SQG can be applied to convert any non-native TQG to a native TQG. Since XX and ZZ type gates are also considered native in this example, a third auxiliary operator

Number

Number

[0090] The above decomposition, i.e., the sequence of steps algorithmically executed using the operators and their unitaries, is shown for the quantum circuit of FIG. 5. FIG. 6 shows a parallelized quantum circuit that can be determined for the example considered above where the number of TQGs and / or the circuit depth is optimized.

[0091] Each horizontal line in FIG. 5 starting with the marking q i indicating qubit number i corresponds to qubit q i . Each rectangle in FIG. 5 refers to a gate,

Number

Number

number

[0092] The first quantum circuit in Figure 5 shows the control sequence represented by the quantum circuit obtained after the second step of the decomposition described above. The first gate is the interaction term.

number

number

number

number

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number

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[0093] As shown in Figure 6, the final quantum circuit in Figure 5 is further optimized to reduce the number of TQGs and / or circuit depth of the quantum circuit, and additional gates are applied to convert all required gates to gates specific to the available quantum device. In a particular example in Figure 6, all ZZ-type gates in the middle of the fourth quantum circuit in Figure 5 (at circuit depths 3-5) are interchangeable, and therefore gates with opposite signs but acting on the same qubit cancel each other out, thus reducing the number of TQGs and circuit depth of the quantum circuit. Furthermore, in a particular example in Figure 6, ZZ-type and XX-type gates are considered native, along with single-qubit gates (SQGs) marked as S in the quantum circuit. If YY-type gates are not native, Figure 6 shows the conversion of YY-type gates to XX-type gates by applying SQGs.

[0094] Further exemplary methods for determining the control sequence are given below when linear connectivity of qubits is used.

[0095] Also, in this example, as in the example above, we consider six objects.

number

[0096] In the first step, the central qubits, which are identified here as qubits numbered 2 and 3, may then be selected.

[0097] The second and third steps may be the same as those for the branch connectivity of qubits.

[0098] In the fourth step of the method, the first principal operator O and the second principal operator O are three-field terms. n is the selected third primary operator

number

number

number

number

number

[0099] Next, multi-qubit interaction H d It is decomposed into a primary operator O and an auxiliary operator H, which are two-qubit interactions.

[0100] The obtained term is the initial desired many-body Hamiltonian term H dThis can be compared to the following. In this example, these are equivalent and no further SQG is needed. However, if not, one layer can be applied before and after the interaction being applied to obtain the correct Pauli operator. Furthermore, additional SQG can be applied to convert any non-native TQG to a native TQG. Since the XX and ZZ gates are considered native in this example as well, no SQG is needed in this example.

[0101] The above decomposition, i.e., the sequence of steps performed algorithmically, is shown for the quantum circuit in Figure 7.

[0102] The quantum circuit in Figure 7 has a third and fourth principal operator and an auxiliary operator O p H p , O r H r However, each algorithm follows the same notation and procedure as the quantum circuit in Figure 5, except that the selection differs from that in Figure 5 as described above.

[0103] As shown in Figure 8, the final quantum circuit in Figure 7 is further optimized to reduce the number of TQGs and the circuit depth of the quantum circuit by canceling out any adjacent gates with opposite signs and acting on the same qubit. Commutation relations can be used when necessary when reducing the number of TQGs and / or circuit depth. In this example, since the XX and ZZ type gates are considered native, no further SQGs are needed. However, if any of the gates are not native, further SQGs can be applied to convert the non-native gate to a native gate.

[0104] In some embodiments, a qubit coupling path is used. The qubit coupling path is a many-body Hamiltonian H d It can be determined based on the following. The qubit coupling path can be obtained, for example, as an input. In some embodiments, the qubit coupling path can be determined as part of the method.

[0105] The qubit coupling path is obtained H d or many-body Hamiltonian H d It can be directly obtained or determined based on other inputs, such as information characterizing it. For example, in the above example, the many-body Hamiltonian H d but

number

[0106] In some embodiments, the many-body Hamiltonian H d The method can then obtain a list of available connectivitys between qubits. The method may then include providing a set of possible qubit coupling paths. A qubit coupling path containing the minimum number of qubit connectivity links can be selected to provide the shortest possible circuit depth.

[0107] Figure 9 shows some possible qubit coupling paths 901 on a quantum device using a square lattice topology (linear connectivity 900). The dots represent the device's qubits 402, the thin lines represent qubit connection links 401, and the thick lines represent qubit coupling paths 901. Qubit coupling paths A and A' can be used to generate 4-body terms, and B and B' can be used to generate 8-body terms. A and A', as well as B and B', are alternatives that provide similar circuit depth. Therefore, in both cases, either qubit coupling path can be selected. With respect to the 4-body term in Example C, there are no available alternative paths. However, in many cases, while multiple different qubit coupling paths are available in relation to a desired many-body term, the path that gives the minimum circuit depth may be favorably selected in the method.

[0108] In some embodiments, the central qubit may be defined based on the acquired or determined qubit coupling path. For example, the central qubit may be a many-body Hamiltonian H d The central qubit may be defined as at least one of the identified qubits within, and along the determined or acquired qubit coupling path, the many-body Hamiltonian H d The longest chain, determined from the set of shortest chains linking to any other qubits of the many-body Hamiltonian H, having the maximum number of qubit-connected links, is determined or acquired by linking any non-central qubits along the qubit coupling path. d It involves fewer or equal numbers of qubits compared to the longest chain, which is determined from the set of shortest chains that link to any other qubits of the same qubit.

[0109] Embodiments of the present invention are H d A quantum many-body Hamiltonian H that includes qubits that cannot be directly coupled (cut qubits) in a quantum device where this is performed. d It can be used to determine the control sequence for simulating the operation.

[0110] As an example, the desired many-body Hamiltonian H d teeth

number

[0111] The method may include determining the shortest path connecting the qubits of the many-body Hamiltonian H via additional qubits of a quantum device. If multiple paths of the same length are available, one may be selected by the user based on a preconfigured selection principle by a first computing device or randomly by the first computing device. If a qubit coupling path connecting the qubits of the many-body Hamiltonian H via additional qubits of the quantum device cannot be found, for example, an error message may be generated. d In the example considered above, the found possible qubit coupling path is shown by path D in FIG. 9. Qubit coupling path D includes six qubits labeled 0 to 5, and each qubit is connected only to its nearest neighboring qubit. d If a qubit coupling path connecting the qubits of the many-body Hamiltonian H via additional qubits of the quantum device cannot be found, for example, an error message may be generated.

[0112] As considered above

Number

[0113] Next, the substituted many-body Hamiltonian

Number

Number

Number

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[0114] Next, the above decomposition is

Number

Number

[0115] Therefore, the control sequence of the many-body Hamiltonian H d can be obtained by complementing the sequence obtained by decomposing the substitution many-body Hamiltonian

Number

Number

Number

[0116] In the example described here, [Number] and a new multi - qubit interaction having N = 5 Pauli matrices [Number] Using the fact that, the number of Pauli matrices in the new multi - qubit interaction can be reduced.

[0117] In the second step, the sequence of two - qubit interactions obtained by decomposing the substitutional many - body Hamiltonian [Number] is complemented by the two - qubit interactions described by the unitary [Number] preceding the sequence and the interactions described by the unitary [Number] following the sequence. [Number] In the third step, the substitutional many - body Hamiltonian [Number] is a new multi - qubit interaction

[0118] In the third step, the substitutional many - body Hamiltonian [Number] is the new multi - qubit interaction

number

[0119] In this example, the third step is:

number

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[0120] Figure 10 shows H dSome of the qubits are not directly connected (disconnected qubits), and therefore, the qubit coupling paths are known or determined for many-body Hamiltonians H d When there are more qubits than the many-body Hamiltonian H d An exemplary quantum circuit is shown demonstrating the method described above for determining the control sequence of the two-qubit interaction to simulate the following. Terms enclosed in dashed lines represent the substitution many-body Hamiltonian.

number

number

[0121] In general, the control sequences, and therefore quantum circuits (sequences of quantum gates), that may be provided in the present invention can be implemented in relation to various types of quantum devices. As is known to those skilled in the art, various methods for controlling the interaction between qubits to implement quantum gates are also available. In relation to a particular quantum device, and so that the determined control sequence can be implemented, instructions for applying a given gate, such as details regarding the sequence and / or duration of applied voltages, may be provided.

[0122] Embodiments of the present invention may also relate to a method for determining at least one property of a system. The system may relate to an M-body interaction problem, the system comprising at least M objects, and the property of the system to be determined is characterized by an M-body interaction problem. The system may be, for example, a molecule, the M-body interaction problem may be solving the electronic Schrödinger equation that gives the electronic structure of the molecule, and the electronic structure is the property to be determined.

[0123] Many-body Hamiltonian H d This can be determined based on the system and the M-body interaction problem to be solved. The control sequence can then be determined according to the method described herein. A user of the first computing device can initiate the determination of the control sequence, which can then be delivered to a second (quantum) computing device to include at least M qubits for implementation.

[0124] In the use of a second computing device, measurement gates may be applied to determine the characteristics of the system. The characteristics of the system may be the intrinsic energy of the system. The method can be repeated multiple times to measure the characteristics of the system multiple times. Multiple measurements can give different intrinsic energies of the system. The ground energy of the system can be estimated by finding the minimum intrinsic energy from those obtained by multiple measurements. Further characteristics of the system's ground state can be found by further repeating the method until an estimated ground state energy is found, and then applying further gates to obtain further characteristics.

[0125] The present invention has been described above with reference to the embodiments described above, and several advantages of the present invention have been demonstrated. It is clear that the present invention is not limited to these embodiments and includes the spirit of the invention and all possible embodiments within the scope of the following claims.

[0126] The features described in the dependent claims may be freely combined with each other unless explicitly stated otherwise.

Claims

1. A series of qubit interactions are performed on multiple qubits (402) on a quantum device to create a quantum many-body Hamiltonian H containing M qubits that can be expressed as the tensor product of M Pauli matrices. ｄ A computer implementation method for determining a control sequence for simulating the following, the method is This involves determining the sequence of two-qubit interactions (120) based on decomposing a multi-qubit interaction term (110) into a sequence of three interaction terms (104), wherein the first interaction term (111) is a unitary e of the principal operator O. ｉｕＯ Described by, where u is the coupling intensity coefficient of O between the qubits on which the principal operator O acts, and the second interaction term (112) is the unitary e of the auxiliary operator H. ｉγＨ Described by, the third interaction term (113) is the unitary e of the negative number -O of the main operator －ｉｕＯ Described by, where H and O are each the tensor product of at least two Pauli matrices, and the first interaction term (111), the second interaction term (112), and the third interaction term (113) generated by the decomposition (104) contain fewer qubits than the original multi-qubit interaction term (110), The method includes an iterative decomposition (104) of interaction terms (111, 112, 113) associated with a main operator and an auxiliary operator until the multi-qubit interaction term is decomposed into a sequence of two-qubit interaction terms (106, 120) associated with operators, each of which includes the tensor product of two Pauli matrices, wherein the multi-qubit interaction term (110) in the first decomposition step (104) is the many-body Hamiltonian H ｄ Unitary [Math 1] Described by, γ is H ｄ The coupling strength coefficient is such that the multi-qubit interaction term (110) in any subsequent decomposition step (104) is related to the principal operator O or the auxiliary operator H. The decomposition (104) is a known or determined many-body Hamiltonian H that identifies the type of the qubits and corresponding Pauli matrices included. ｄ A computer implementation method based at least on (102).

2. The method according to claim 1, comprising providing the two-qubit interaction (120) obtained through the method as a control sequence, as a computer-readable output deliverable for implementation on a quantum device.

3. The method according to any one of claims 1 to 2, wherein the square of the principal operator O is equal to the identity matrix.

4. The decomposition (104) is further based on a known or determined qubit coupling path (901), the qubit coupling path (901) is the many-body Hamiltonian H that is coupled when the determined control sequence is executed on the quantum device. ｄ The method according to any one of claims 1 to 3, wherein a qubit connection link (401) showing at least the qubits is provided.

5. The method according to claim 4, comprising taking the qubit coupling path (901) as input or determining the qubit coupling path (901), wherein when the control sequence is executed on the quantum device, the determined qubit coupling path (901) results in a selected circuit depth.

6. The method according to claim 4 or 5, comprising obtaining information relating to the qubit connectivity (400) of the quantum device, wherein the qubit coupling path (901) is based on the qubit connectivity (400a, 400b, 900) so that the qubit connection link (401) is available in the qubit connectivity (400a, 400b, 900).

7. The method according to claim 6, wherein when multiple qubit coupling paths (901) are available, the qubit coupling paths (901) are selected such that the selected qubit coupling path (901) includes the minimum number of qubit connection links (401) from the set of available qubit coupling paths (901).

8. where the qubit coupling path (901) includes more qubits (402) than the known or determined many-body Hamiltonian H ｄ then the method comprises - The many-body Hamiltonian H in the decomposition (104) ｄ The substitutions in the qubit coupling path (901) include M' > M qubits (402). [Math 2] The substitute for replacement [Math 3] The decision is to make, - The disassembly (104) is the replacement [Math 4] The decision to be made is based on the following: - The original many-body Hamiltonian H is reassembled by complementing the decomposed sequence of the 2-qubit interaction (120) with at least two additional qubit interactions. ｄ This involves determining the sequence of two-qubit interactions for carrying out the action, 〇 [Math 5] And A is anticommutative, and the square of A is equal to the identity matrix, and A and [Math 6] The commutator of operator A yields a new multi-qubit interaction involving the tensor product of N Pauli matrices, and the unitary e ｉＡ A step of selecting at least two additional qubit interaction terms described by, where N < M', and the selection of A is further based on the property that the square of the Pauli matrix is ​​equal to the identity matrix, The steps include: 1) Complementing the decomposed sequence of the two-qubit interaction with the selected two-qubit interaction term on one side of the sequence, and with the negative number of the selected two-qubit interaction term on the other side; 〇 [Number 7] of 【Number 8】 The method according to any one of claims 4 to 7, comprising determining, the step of replacing the aforementioned replacer with A obtained in the above step and repeating the above step until N = M.

9. The aforementioned method, - Selecting at least one of the identified qubits as the central qubit (402a) (202), - The first auxiliary operator H is M Ｈ (104) Selecting as a tensor product of individual Pauli matrices, where each Pauli matrix in the tensor product acts on a different qubit, and the qubit is the many-body Hamiltonian H ｄ Selected from those specified by, the selection includes the at least one central qubit (402a), M Ｈ This involves selecting (104) a number less than the number of Pauli matrices of the multi-qubit interaction that are decomposed, - The first main operator O is M Ｏ The selection of the tensor product of the individual Pauli matrices (104) is M Ｏ The number of Pauli matrices of the multi-qubit interaction to be decomposed is less than the number of Pauli matrices of the multi-qubit interaction to be decomposed, and each Pauli matrix of the tensor product acts on a different qubit, and the Pauli matrices of the qubit and the first principal operator O are such that at least one qubit is one of the at least one central qubit (402a), and H ｄ However, the main operator O and the auxiliary operator H are selected in proportion to the commutators (104), - Selecting the coupling strength coefficient of the principal operator O as u = π / 4 + a × π (204) in order to separate a single M-body term, where a is an integer, and (204) - The main operator O and auxiliary operator H are selected to be anticommutative (206), The method according to claim 8, wherein the iterative decomposition (104) is the tensor product of two Pauli matrices, and the method involves repeatedly selecting subsequent principal and auxiliary operators until the final principal and final auxiliary operators corresponding to a two-qubit interaction are obtained (106).

10. The method according to claim 9, wherein the method comprises obtaining information regarding the qubit connectivity (400a, 400b, 900) of the quantum device, and if the connectivity (400a, 400b, 900) indicates linear connectivity of the qubits (400b, 900) and one qubit can be coupled with at most two other qubits, the decomposition of a previously determined principal operator O comprises re-selecting the central qubit(s) before selecting subsequent principal and auxiliary operators, wherein the central qubit(s) (402a) is selected from the qubits of the decomposed operator O.

11. The method according to any one of claims 1 to 10, further comprising obtaining information indicating the native interaction of the quantum device, wherein the method comprises applying a single qubit gate in relation to a two-qubit interaction in the control sequence that does not correspond to the native gate of the quantum device to obtain a two-qubit interaction corresponding to the native gate of the quantum device.

12. A computer program for causing a computer to perform the method described in any one of claims 1 to 11.

13. A quantum circuit comprising a sequence of qubit interactions determined according to the method of any one of claims 1 to 11, executable on a quantum device comprising at least M qubits for simulating a quantum many-body Hamiltonian.

14. A quantum device comprising at least M qubits, wherein the quantum device is configured to perform the sequence of qubit interactions determined according to the method of any one of claims 1 to 11.

15. A method for determining at least one characteristic of a system, wherein the method is - To determine an M-body interaction problem relating to a system containing at least M objects, wherein at least one characteristic of the system is characterized by the M-body interaction problem. - Determining the control sequence described in any one of claims 1 to 11, - Implement the determined control sequence on a quantum device containing at least M qubits, A method comprising: applying a measuring gate to determine the characteristics of the system.

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