Methods for determining control sequences for quantum bit interactions and related quantum circuits, quantum devices, and methods for solving problems - Patents.com

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] The present invention relates generally to quantum computing, and more particularly to a computer-implemented method for determining control sequences for performing a sequence of qubit interactions on a plurality of qubits on a quantum device to simulate a quantum many-body Hamiltonian. [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, e.g., 0 or 1, quantum computers use quantum bits (qubits) that can be in a coherent superposition of two states at the same time. Qubits can refer to the basic unit of quantum information or quantum devices (such as two-level quantum mechanical systems) used to store units of quantum information. Thus, a quantum computer generally includes an array of qubits and hardware for manipulating these qubits.

[0003] There are three fundamental quantum computing methods: the analog quantum model, the universal quantum gate model (also known as the digital quantum computing model or the quantum circuit model), and quantum annealing. In the quantum gate model, the manipulation of qubits or interactions between qubits are referred to as 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 referred to as a quantum circuit, that corresponds to instructions for manipulating units of quantum information to perform a desired computation. Quantum gates may also be referred to as unitary operators represented by unitary matrices. Quantum gates that operate on multiple qubits may also be referred to as interactions between multiple qubits. Implementing gates that operate on multiple qubits on a quantum device corresponds to performing qubit interactions on the multiple qubits. Thus, unitary operators may also be referred to as interaction terms.

[0004] The many-body problem is a general name for a wide range of physical problems that concern the properties of systems that include many interacting objects, such as particles, where interactions between three or more objects are referred to as many-body interactions. Because the complexity of classical simulations of quantum many-body systems typically grows exponentially with the dimensionality of the system, quantum computers can offer advantages for simulating many-body problems. Many-body interactions arise naturally in the simulation of problems related to fields such as quantum chemistry, finance, optimization, and high-energy physics. Thus, in order to simulate many-body problems on quantum computers, the simulation of many-body interactions using qubits is necessary.

[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, glitches, and loss of quantum coherence. The accuracy of quantum computation results can degrade rapidly as the number of gate operations, circuit depth, and / or number of measurements increase.

[0006] 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 available number of qubits. A problem associated with some of the known methods for simulating many-body interactions is the use of auxiliary qubits, which reduces the number of remaining available qubits.

[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 necessarily native to all of the qubit pairs in currently available quantum systems. Gates that are not system specific need to be decomposed into a sequence of unique gates, thus resulting in a larger number of gates being used. Summary of the Invention

[0008] It is an object of the present invention to alleviate at least some of the problems in the prior art. It is an object of the present invention to provide an alternative and / or improved method or device for determining quantum gates to be applied to a plurality of qubits to simulate a quantum many-body Hamiltonian or interactions. Some embodiments of the present invention may be viewed as providing a quantum compiler for providing a quantum circuit. According to one aspect of the present invention, a series of qubit interactions are performed on a plurality of qubits on a quantum device to generate a quantum many-body Hamiltonian H comprising M qubits. d A computer-implemented method for determining a control sequence for simulating a multi-body Hamiltonian H d can be expressed as a tensor product of M Pauli matrices, and the method includes determining a sequence of two-qubit interactions based on decomposing the multi-qubit interaction term into a sequence of three interaction terms, the first interaction term being a unitary e of the primal operator O. iuO where u is the coupling strength coefficient of O between the qubits acted upon by the primary operator O, and the second interaction is the unitary e iγH The third interaction is described by the unitary e of the negative of the principal operator -O. -iuO where H and O are each a tensor product of at least two Pauli matrices, and 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 includes an iterative decomposition of interaction terms associated with the primary and ancillary operators until the multi-qubit interaction terms are decomposed into a sequence of two-qubit interaction terms each associated with an operator comprising a tensor product of two Pauli matrices, the multi-qubit interaction terms in the first decomposition step being ... d Unitary

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[0009] In other words, a series of one or more interactions can be performed on multiple qubits on a quantum device to obtain a quantum many-body Hamiltonian H containing M qubits that can be expressed as a tensor product of M Pauli matrices. d A computer-implemented method for determining a control sequence for simulating Many-body Hamiltonian H d Unitary

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[0010] As used herein, decomposition of an operator may refer to the decomposition of the corresponding interaction terms.

[0011] In the present invention, a solution is provided for providing many-body gates by using native two-qubit gates (TQGs) (or in some cases near-native TQGs, meaning TQGs and single-qubit gates (SQGs)), which can be advantageous compared to solutions where CNOT gates are utilized.

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

[0013] The number of gates, particularly preferably TQGs, required to simulate M-body interactions may be reduced by at least one when compared to prior art approaches. Additionally or alternatively, the circuit depth of the quantum circuit provided through control sequences may 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 a requirement of 2(M-1) TQGs and a circuit depth of 2(M-1). However, the present invention may provide a solution utilizing 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 number of TQGs may be reduced less than discussed above, but the avoidance of using CNOT gates may still provide an advantage over the prior art.

[0014] The depth of a quantum circuit can refer to the number of time steps required for its completion. Thus, because quantum computation can involve increasing error as the time required for computation increases, reducing the circuit depth can result in reduced error in addition to reduced computation time.

[0015] The present invention can also provide a solution in which auxiliary qubits are not required, thus reducing or optimizing the number of qubits required.

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

[0017] Methods according to the invention may be advantageous in use cases where quantum devices do not provide connectivity / coupling possibilities between selected qubits that can be included in the desired interaction terms.

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

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

[0020] The method may be at least partially performed by a first computing device, typically a classical computer, such that the method may further include providing the two-qubit interactions obtained through the method as a control sequence 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 coupled 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.

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

[0022] In one embodiment, the decomposition may be further based on known or determined qubit coupling paths, which may be a multi-body Hamiltonian H that is coupled when executing the determined control sequence on the quantum device. d qubit connecting links representing at least one qubit of

[0023] A qubit coupling path may be taken as an 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 the identified qubits and taking into account the qubit connectivity of the quantum device.

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

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

[0026] A many-body Hamiltonian H with known or determined qubit coupling paths d For a system that includes more qubits than -decomposing the many-body Hamiltonian H da replacement that includes M'>M qubits within the qubit coupling path

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[0027] The qubit coupling paths are determined by a known or determined many-body Hamiltonian H d If it contains more qubits than d may include qubits for which connectivity is not available on the selected quantum device (disconnected qubits). Qubit coupling paths may be determined based on the known connectivity of the devices, providing qubit connection links between disconnected qubits via connected or connectable qubits. The present invention provides a method for determining the desired many-body Hamiltonian terms H d In one embodiment, a method may be provided for determining a quantum circuit that indicates qubit interactions that may take place involving qubits having connectivity on a selected quantum device to arrive at a solution corresponding to

[0028] A known or determined many-body Hamiltonian H d The qubit coupling pathway involving more qubits than d Then, based on the qubit connectivity of the quantum device, we obtain the many-body Hamiltonian H dand determining that a qubit coupling path involving only the qubit identified by is not possible. Alternative qubit coupling paths involving additional qubits of the quantum device may then be determined, such alternative qubit coupling paths preferably involving a minimum number of qubits. The minimum number of qubits may be based on the shortest path available to obtain the alternative qubit coupling path. Such an approach may be advantageous as it allows for the implementation of interactions between disconnected qubits without using SWAP gates, thus reducing computation time and therefore computation errors, as SWAP gate implementation is about three times slower than one native TQG implementation.

[0029] Alternatively, the many-body Hamiltonian H d If it is determined that a qubit coupling path involving only qubits identified by is not possible, a SWAP gate can be applied to connect the required qubits, in which case the multi-body Hamiltonian H d The control sequence for implementing the multibody Hamiltonian H d may include a SWAP gate in addition to the control sequence determined by decomposing

[0030] In an embodiment of the invention, the method comprises the steps of: - selecting at least one of the identified qubits as a central qubit; and -The first auxiliary operator H is M H a tensor product of Pauli matrices, each of which acts on a different qubit, the qubit being determined by the many-body Hamiltonian H d , where the selection includes at least one central qubit, and M H is smaller than the number of Pauli matrices of the multi-qubit interaction to be decomposed, and -The first primary operator O is M O tensor product of Pauli matrices MO is less than the number of Pauli matrices of the multi-qubit interaction being decomposed, each Pauli matrix of the tensor product acts on a different qubit, 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 is chosen to be proportional to the commutator of the primary operator O and the auxiliary operator H; - selecting a coupling strength coefficient for O as u=π / 4+a×π to separate a single M-body term, where a is an integer; - the primary operator O and the auxiliary operator H are chosen to be anticommutative, The iterative decomposition involves repeatedly selecting successive primal and auxiliary operators until the primal and auxiliary operators are tensor products of two Pauli matrices and therefore correspond to a two-qubit interaction.

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

[0032] The selection of at least one central qubit (one if M is odd, or two if M is even) and the decomposition taking into account the central qubit(s) may provide a selected circuit depth, which may be optimized or minimal for a particular use case.

[0033] Choosing the coupling strength as u=π / 4+a×π can result in the M-body terms being able to be separated. For other coupling strengths, up to M-body terms 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 include obtaining information regarding qubit connectivity of the quantum device. If the connectivity indicates linear connectivity of the qubits, such that one qubit can be coupled with at most two other qubits, decomposing the previously determined interaction term associated with a primal operator O may include reselecting a central qubit before selecting subsequent primal and ancillary operators, where the central qubit(s) is selected from the qubits of the operator O associated with the interaction term being decomposed.

[0036] The method may further include obtaining information indicative of a native interaction or gate of the quantum device. The method may also include applying a single-qubit gate in conjunction with a two-qubit interaction in the control sequence that does not correspond to a native interaction of the quantum device to obtain a two-qubit interaction that corresponds to the native interaction of the quantum device.

[0037] One or more characteristics of the quantum device may be known or obtained in different embodiments of the invention, including, for example, the native gates of the device or the connectivity of the device, and the one or more characteristics may be taken into account in the method.

[0038] According to an aspect of the present invention there is also provided a computer program product comprising program code means adapted to perform the method items of the present invention when executed on a computer.

[0039] Further, according to one aspect, there is provided a quantum circuit executable on a quantum device including at least M qubits for simulating a quantum many-body Hamiltonian, the quantum circuit including a sequence of qubit interactions determined according to an embodiment of the method for determining a control sequence.

[0040] There may also be provided a quantum device including at least M quantum bits, the quantum device configured to implement a sequence of quantum bit interactions determined in accordance with a method of an embodiment of the present invention for providing a control sequence.

[0041] There is also provided a method for determining at least one characteristic of a system according to independent claim 15.

[0042] The novel features which are believed to be characteristic of the invention are set forth with particularity in the appended claims. The invention itself, however, both as to its organization 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.

[0043] The invention will now be explained in more detail with reference to exemplary embodiments according to the accompanying drawings. [Brief description of the drawings]

[0044] [Figure 1] 1 shows a flow chart of a method and associated schematic decomposition according to one embodiment of the present invention. [Diagram 2] 2 shows a flow chart of a method according to one embodiment of the present invention. [Diagram 3] 2 shows a flow chart of a method according to one embodiment of the present invention. [Figure 4] 1 illustrates an exemplary connectivity of qubits. [Diagram 5] We provide an example of decomposition for quantum circuits. [Figure 6] 1 shows an example of a control sequence as a quantum circuit. [Figure 7] We provide an example of decomposition for quantum circuits. [Figure 8] 1 shows an example of a control sequence as a quantum circuit. [Figure 9] 1 depicts the linear connectivity of qubits and shows exemplary qubit coupling paths. [Figure 10]1 shows an example of a control sequence as a quantum circuit. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0045] For many-body problems, one of the issues to be considered in quantum computing is the Hamiltonian H d The objective of quantum computing is to optimally implement the Hamiltonian, which contains a string of Pauli terms acting on some number of different qubits, where the Hamiltonian describes the behavior of the many-body system, and quantum computing aims to implement the dynamics generated by the Hamiltonian.

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

[0047] The task at hand is to find a suitable scheme for taking the H and O terms and decomposing them into TQGs, preferably such that the number of TQGs and / or the circuit depth is optimal. The identities given below for the matrices P and R may be utilized, where equation (1) is the Baker-Campbell-Hausdorff expansion, and the second identity (equation (2)) is the P -1 = -P, this is derived by the fact that P is invertible, resulting in the third identity (equation (3)).

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[0048] In the above,

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[0049] [O,H] ≠ 0 and {O,H} = 0 with the O,H Hermitian operators, 2 and H 2 is the identity e iαO =cos(α)I+isin(α)O (6) is equal to Combine the above qualities.

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[0050] Using the above, a sequence of terms can be generated, and the coupling strength of the main operator O can be appropriately set to generate separated many-body terms. The terms can then be rotated individually to obtain the required combination of Pauli matrix type X, Y, and Z operations of the many-body terms. Such a protocol can be implemented on a quantum computer without auxiliary qubits, using digital algorithms where unitary transformations are represented by gates.

[0051] In some embodiments, after decomposition, the final control sequence of interactions is H d =e iO’ H'e -iO’where H' is a central final ancillary operator H, O' is the sum of all final primal operators O except for the central final ancillary operator and the final ancillary operators H, where H' and O' do not commute, and any augend in O' commutes with any other augend in O'. Examples of such final control sequences represented as digital quantum circuits are shown later in this specification in Figures 6, 8, and 10.

[0052] The present invention includes a computer-implemented method for determining a control sequence for performing a series of qubit interactions on a plurality of qubits on a quantum device to simulate a quantum many-body Hamiltonian, the method comprising decomposing a multi-qubit interaction term 110 into a sequence of three interaction terms 111, 112, 113, as shown generally in FIG. 1B. 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 the unitary e of the negative of the main operator -O. -iuO where H and O are each a tensor product of at least two Pauli matrices. The first interaction term 111, the second interaction term 112, and the third interaction term 113 produced by decomposition 104 include fewer qubits than the original multi-qubit interaction term 110.

[0053] A desired many-body Hamiltonian H containing M qubits d is known or can be determined. The multi-body Hamiltonian H d can be expressed as a tensor product of M Pauli matrices, identifying the types of qubits and corresponding Pauli matrices involved. With reference to FIG. 1 and for one embodiment of the present invention, the multi-body Hamiltonian H d may be obtained as an input by a first computing device that includes at least one processor that is utilized to execute the method for determining a control sequence (102). dmay be known or determined based on the problem to be solved involving a system including M objects, and certain characteristics of the system are determined.

[0054] The input(s) considered may be obtained by the first computing device, they may be provided, for example, 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, which may be, for example, properties describing the quantum device in which the particular many-body interaction is implemented and / or desired attributes of the many-body interaction. At least some of the methods discussed herein may be performed without being provided with all possible information discussed herein as possible inputs. The methods according to the invention may be implemented using any M, H d It will be appreciated by those skilled in the art that any number of quantum bit connections, qubit connectivity, or qubit coupling paths may be possible.

[0055] In one embodiment, 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 fields involved, a qubit number vector [i,j,k,l...] ​​identifying the desired number of qubits to be included in the many-body Hamiltonian, and corresponding Pauli operator numbers 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 indicate the Pauli matrix types X, Y, and Z, respectively.

[0056] Part of the information utilized in the method may 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...] ​​that identifies the desired number of qubits to be included in the many-body Hamiltonian, and the corresponding Pauli operator numbers 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 indicate the Pauli matrix types X, Y, and Z, respectively. A number M that identifies the size of the many-body terms can then 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 that identifies the size of the many-body terms, i.e., the number of bodies and the qubit connectivity of the quantum device. The qubit number vector can then be determined based on the number M of quantum devices and the qubit connectivity. The Pauli operator types may be randomly assigned or determined based on further inputs including the qubit connectivity of the quantum devices and the set of available native gates.

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

[0058] The method may include an iterative decomposition of the 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 an operator including a tensor product of two Pauli matrices.

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

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[0060] The operators involving the tensor product of two Pauli matrices are

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

[0062] 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 two-qubit interaction terms 120.

[0063] Thus, the method consists in considering the many-body Hamiltonian H dUnitary

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[0064] In the method, M O >2, the decomposition is repeated for the first interaction term 111, which is O The method may then 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 a result of repeating the decomposition for the first interaction term 111.

[0065] M H >2, the decomposition is repeated for the second interaction term 112, each of which is M H can be decomposed into a sequence of three subsequent interaction terms involving fewer than qubits.

[0066] The decomposition of the obtained terms gives the multi-qubit interaction e iγH may be iterated until {overscore (x)} is decomposed into a series of at least three two-qubit interaction terms 120.

[0067] FIG. 1B shows diagrammatically how the method can be used to decompose the interaction terms 110, 111, 112, 113 and finally arrive at a two-qubit interaction 120. The rectangular boxes in FIG. 1B indicate gates, unitary transformations or interaction terms. The lines in FIG. 1B indicate that each interaction term involving 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 top row of FIG. 1B shows the many-body interaction term 110 being decomposed, and the bottom row of FIG. 1B shows the resulting control sequence involving multiple two-qubit interaction terms or TQGs 120.

[0068] In some embodiments, with reference to FIG. 2, a method includes first computing a multi-body Hamiltonian H d The method may include selecting 202 at least one of the identified qubits in as a central qubit. If M is odd, one central qubit may be selected, and if M is even, two central qubits may be selected. For an odd M, in addition to the central qubit, a second qubit adjacent to the central qubit may be selected.

[0069] The central qubit is expressed by the many-body Hamiltonian H d may be defined as at least one of the identified qubits in the many-body Hamiltonian H d The central qubit is transformed into the many-body Hamiltonian H d The longest chain, determined from the set of shortest chains that link to any other qubit in, has the maximum number of qubit-connecting links and is given by the many-body Hamiltonian H d We can apply the qubits of d The longest chain involves a smaller or equal number of qubits compared to the longest chain determined from the set of shortest chains that link to any other qubit in the set.

[0070] The first auxiliary operator H is M Hcan be chosen as a tensor product of Pauli matrices, each of which acts on a different qubit, and the qubits are expressed by the many-body Hamiltonian H d , where the selection includes at least one central qubit, and M H is smaller than the number of Pauli matrices of the multi-qubit interaction being decomposed.

[0071] The first main operator O is M O can 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 of the tensor product acts on a different qubit, the Pauli matrices of the qubits and 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.

[0072] Additionally, a coupling strength coefficient for the primary operator O may be selected as u=π / 4+a×π (204), and the primary operator O and the auxiliary operator H may be selected to be anti-commutative (206). Steps 204 and / or 206 may be performed in a different order.

[0073] 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.

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

[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 / coupleable.

[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 flow chart of the method as applied to linear qubit connectivity, such as found in a square lattice qubit topology. In such a linear topology, all qubits may be connected to form qubit coupling paths, such that any one qubit in the path is coupled to at most two other qubits.

[0077] Here, the selection 202 of the central qubit(s) may be performed each time a decomposition is performed. Thus, if the known or obtained connectivity of the quantum device indicates linear connectivity of the qubits, the decomposition of a previously determined primal operator O may include reselecting the central qubit(s) before selecting subsequent primal and ancillary operators, where the central qubit is selected from the qubits of the operator O being decomposed.

[0078] In some embodiments, the qubit connectivity of a quantum device may exhibit qubit branching connectivity, comprising at least a first qubit and a second qubit and a number of additional qubits, where the first qubit can be coupled to the second qubit and the additional qubit can be coupled to only one of the first or second qubits.

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

[0080] FIG. 4, at 4A, shows a schematic of a branch 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. The 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 selection of a particular implementation may depend on the available native gates or interactions of 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 be selected, as other native gates may be possible. It may also be possible that only one gate is native and the other is obtained from such native gates and native SQG.

[0081] 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 the non-native TQG to native gates, if desired. Information characterizing the native gates of the quantum device (or second computing device) may be obtained, for example, as input, or may be known.

[0082] In the example of FIG. 4, the solid lines represent the two central qubits 402a (here numbered qubits 0 and 1, e.g., q 0 and q 1 , which may be referred to as M. In branched connectivity, there can always be two central qubits, regardless of whether M is even or odd.

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

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

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

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

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[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 third main operator selected

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

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

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[0090] The above decomposition, i.e., the sequence of steps that are algorithmically performed using operators and their unitaries, is shown with respect to the quantum circuit of Figure 5. Figure 6 shows a parallelized quantum circuit that can be determined for the example considered above, in which the number of TQGs and / or the circuit depth are optimized.

[0091] Marking q indicating quantum bit number i i Each horizontal line in Figure 5, starting with i Each rectangle in Figure 5 corresponds to a gate.

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[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

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

[0094] A further exemplary method for determining the control sequence is given below when linear connectivity of the qubits is used.

[0095] Also, in this example, as in the previous example, six objects are considered.

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

[0097] The second and third steps may be the same as for the branching 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 third main operator selected

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

[0100] The obtained terms are the initial desired multibody Hamiltonian terms H dIn this example, these are equivalent and no further SQG is needed. However, if this is not the case, one layer can be applied before and after the applied interaction to obtain the correct Pauli operator. Furthermore, a further SQG can be applied to convert any non-native TQG to a native TQG. The XX and ZZ gates are also considered to be native in this example, so no SQG is needed in this example.

[0101] The above decomposition, i.e. the sequence of steps that are algorithmically performed, is illustrated with respect to a quantum circuit in FIG.

[0102] The quantum circuit of FIG. 7 includes the third and fourth main operators and the auxiliary operator O p , H p , O r , H r We follow the same notation and procedure as the quantum circuit of FIG. 5, except that is selected differently from FIG. 5 by each algorithm as described above.

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

[0104] In some embodiments, a qubit coupling pathway is utilized. The qubit coupling pathway is based on the multi-body Hamiltonian H d The qubit coupling paths may be obtained, for example, as an input. In some embodiments, the qubit coupling paths may be determined as part of the method.

[0105] The qubit coupling pathway is obtained by H d or the many-body Hamiltonian H d For example, in the above example, the multi-body Hamiltonian H d but

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[0106] In some embodiments, the multi-body Hamiltonian H d A list of available connectivity between qubits may be obtained, and a list of available connectivity between qubits may also be obtained. The method may then include providing a set of possible qubit coupling paths. The qubit coupling path that includes the smallest number of qubit connection links may be selected to provide the shortest possible circuit depth.

[0107] FIG. 9 shows some possible qubit coupling paths 901 on a quantum device using a square lattice topology (linear connectivity 900). The dots represent the qubits 402 of the device, the thin lines represent the qubit connecting links 401, and the thick lines represent the 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. Thus, in both cases, either qubit coupling path can be selected. For the 4-body term of example C, there are no alternative paths available. However, in many cases, while multiple different qubit coupling paths are available in connection with a desired many-body term, the path that provides the minimum circuit depth can be advantageously selected in the method.

[0108] In some embodiments, the central qubit may be defined based on an obtained or determined qubit coupling path. For example, the central qubit may be defined based on a multi-body Hamiltonian H d and coupling the central qubit along the determined or obtained qubit coupling path with the multi-body Hamiltonian H d qubits along the determined or obtained qubit coupling path, the longest chain is determined from the set of shortest chains linking any non-central qubit to any other qubit in the set of shortest chains, the longest chain having the greatest number of qubit connecting links being determined by the multi-body Hamiltonian H d The longest chain involves a smaller or equal number of qubits compared to the longest chain determined from the set of shortest chains that link to any other qubit in the set.

[0109] An embodiment of the present invention is d A quantum many-body Hamiltonian H that includes qubits that are not directly coupleable in the quantum device in which d The control sequence for simulating the above-mentioned

[0110] As an example, the desired multibody Hamiltonian H d teeth

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[0111] The method further comprises the step of: d The method may include determining the shortest path connecting the qubits of the quantum device. If multiple paths of the same length are available, one may be selected by a user, based on a preconfigured selection principle by the first computing device, or randomly by the first computing device. d If a qubit coupling path cannot be found that connects the qubits of, for example, an error message may be generated.

[0112] The above considered

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[0113] Next, the permutation many-body Hamiltonian

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[0114] Next, the above decomposition

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[0115] Therefore, the many-body Hamiltonian H d The control sequence of is the permutation multibody Hamiltonian

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[0116] In the example described here,

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[0117] In the second step, the permutation many-body Hamiltonian

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[0118] In the third step, we define the permutation many-body Hamiltonian

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

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[0120] Figure 10 shows the H d Some of the qubits in are not directly connected (disconnected qubits), and therefore the qubit coupling paths are known or determined by the many-body Hamiltonian H d When the number of qubits is larger than 1, the multi-body Hamiltonian H d 1 shows an example quantum circuit illustrating the above-described method for determining a control sequence for two qubit interactions to simulate the permutation many-body Hamiltonian

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[0121] In general, the control sequences, and thus quantum circuits (sequences of quantum gates), that may be provided in 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. Details regarding the instructions for applying the specified gates, e.g., the order and / or duration of the applied voltages, may be provided in conjunction with a particular quantum device and such that the determined control sequence is executable.

[0122] An embodiment of the invention may also relate to a method for determining at least one property of a system. The system may be associated with an M-body interaction problem, the system including at least M bodies, and the property of the system to be determined is characterized by the M-body interaction problem. The system may be, for example, a molecule, and the M-body interaction problem may be solving an electronic Schrodinger 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 can be determined based on the system and the M-body interaction problem to be solved. A control sequence can then be determined according to the methods described herein. A user of a first computing device can initiate the determination of the control sequence, and the control sequence can be distributed to a second (quantum) computing device for inclusion with at least M qubits for implementation.

[0124] Using the second computing device, a measurement gate may be applied to determine a characteristic of the system. The characteristic of the system may be an intrinsic energy of the system. The method may be repeated multiple times to measure the characteristic of the system multiple times. The multiple measurements may give 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 characteristics of the ground state of the system may 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 aforementioned embodiments, and some advantages of the present invention have been demonstrated. It is clear that the present invention is not limited to these embodiments only, but includes all possible embodiments within the spirit of the present invention and the scope of the following claims.

[0126] Features recited in dependent claims may be freely combined with one another, unless otherwise expressly stated.

Claims

1. A series of qubit interactions are performed on a plurality of qubits (402) on a quantum device to generate a quantum many-body Hamiltonian H including M qubits, which can be expressed as a tensor product of M Pauli matrices. ｄ 1. A computer-implemented method for determining a control sequence for simulating a determining a sequence of two-qubit interactions (120) based on decomposing (104) the multi-qubit interaction term (110) into a sequence of three interaction terms, the first interaction term (111) being a unitary e ｉｕＯ where u is the coupling strength coefficient of O between the qubits on which the primary operator O acts, and the second interaction term (112) is the unitary e ｉγＨ The third interaction term (113) is the unitary e －ｉｕＯ where H and O are each a 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) include 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 primary and ancillary operators until the multi-qubit interaction term is decomposed into a sequence of two-qubit interaction terms (106, 120) each associated with an operator comprising a tensor product of two Pauli matrices, the multi-qubit interaction term (110) in a first decomposition step (104) being decomposed into the multi-body Hamiltonian H ｄ Unitary [0010] and γ is expressed by H ｄ where the multi-qubit interaction term (110) in any subsequent decomposition step(s) (104) is related to a primary operator O or an ancillary operator H, The decomposing (104) may involve generating a known or determined many-body Hamiltonian H ｄ (102). A computer-implemented method based at least on (102).

2. 2. The method of claim 1 , further comprising providing the two-qubit interaction (120) obtained through the method as the 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 main operator O is equal to an identity matrix.

4. The decomposing (104) is further based on known or determined qubit coupling paths (901), the qubit coupling paths (901) being determined by the multi-body Hamiltonian H ｄ The method according to any one of claims 1 to 3, further comprising: a qubit connecting link (401) indicative of at least said qubits of said first qubit.

5. 5. The method of claim 4, wherein the method comprises taking as an input the qubit coupling path (901) 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. 6. The method of claim 4 or 5, wherein the method comprises obtaining information about a qubit connectivity (400) of the quantum device, and wherein the qubit coupling paths (901) are based on the qubit connectivity (400a, 400b, 900) such that the qubit connection links (401) are available in the qubit connectivity (400a, 400b, 900).

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

8. The qubit coupling path (901) is determined by the known or determined multi-body Hamiltonian H ｄ In the case where the number of qubits (402) is more than The multi-body Hamiltonian H in the decomposition (104) ｄ a permutation including M′>M qubits (402) in the qubit coupling path (901) [0025] The substitute for replacing [0030] determining - the decomposition (104) is [0045] and determining, by complementing the decomposed sequence of −2 qubit interactions (120) with at least two additional qubit interactions to obtain the original multi-body Hamiltonian H ｄ determining a sequence of two-qubit interactions to implement 〇 [0050] and A is anticommutative, the square of A is equal to the identity matrix, and A and [006] The unitary e of the operator A is denoted as ｉＡ where N<M′, and said selection of A is further based on the property that the square of the Pauli matrix is ​​equal to the identity matrix; Complementing the decomposed sequence of two-qubit interactions with the selected two-qubit interaction term on one side of the sequence and with the negative of the selected two-qubit interaction term on the other side of the sequence; 〇 [0070] of [0080] and replacing the commutator of N with A obtained in the above step and repeating the above steps until N=M.

9. The method comprises: - selecting (202) at least one of the identified qubits as a central qubit (402a); - The first auxiliary operator H is Ｈ a tensor product of Pauli matrices, each Pauli matrix of the tensor product acting on a different qubit, the qubits being determined by the multi-body Hamiltonian H ｄ wherein the selection includes the at least one central qubit (402a), and M Ｈ is less than the number of Pauli matrices of the multi-qubit interaction to be decomposed (104); - The first main operator O is M Ｏ tensor product of Pauli matrices M Ｏ is less than the number of Pauli matrices of the multi-qubit interaction being decomposed, each Pauli matrix of the tensor product acts on a different qubit, the Pauli matrices of the qubits and the first principal operator O are such that at least one qubit is one of the at least one central qubits (402a), and H ｄ is selected to be proportional to the commutator of the primary operator O and the auxiliary operator H; - selecting (204) the coupling strength coefficient of the main operator O as u=π / 4+a×π, where a is an integer, in order to separate a single M-body term; - the primary operator O and the auxiliary operator H are chosen (206) to be anti-commutative; 9. The method of any one of claims 1 to 8, wherein the iterative decomposition (104) comprises iteratively selecting subsequent main and auxiliary operators until a final main and auxiliary operator is obtained (106), which is a tensor product of two Pauli matrices and thus corresponds to a two-qubit interaction.

10. 10. The method of claim 1, wherein the method comprises obtaining information about qubit connectivity (400a, 400b, 900) of the quantum device, where the connectivity (400a, 400b, 900) indicates a linear connectivity of qubits (400b, 900), where one qubit can be coupled with at most two other qubits, and where the decomposing a previously determined primal operator O comprises re-selecting the central qubit(s) before selecting subsequent primal and ancillary operators, the central qubit (402a) being selected from the qubits of the operator O being decomposed.

11. 11. The method of claim 1, further comprising obtaining information indicative of a native interaction of the quantum device, the method comprising applying a single-qubit gate in conjunction with a two-qubit interaction in the control sequence that does not correspond to a native gate of the quantum device to obtain a two-qubit interaction that corresponds to a native gate of the quantum device.

12. A computer program product comprising program code means adapted to carry out the method items according to any one of the preceding claims when said computer program product is executed on a computer.

13. 12. 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. 12. A quantum device comprising at least M qubits, the quantum device being configured to perform the sequence of qubit interactions determined according to a method according to any one of claims 1 to 11.

15. 1. A method for determining at least one characteristic of a system, the method comprising: - determining an M-body interaction problem associated with a system including at least M bodies, wherein at least one property of the system characterizes the M-body interaction problem; - determining a control sequence according to any one of claims 1 to 11, - implementing said determined control sequence on a quantum device comprising at least M qubits; - applying a measurement gate to determine said characteristic of said system.

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