Computing excited state properties of molecular systems using a hybrid classical-quantum computing system

Abstract

A method for computing excited state properties of a molecular system using a hybrid classical-quantum computing system includes determining a ground state wave function of a combination of quantum logic gates using a quantum processor and a memory. In one embodiment, the method includes forming a set of excitation operators. In one embodiment, the method includes forming a set of commutators from the set of excitation operators and a Hamiltonian operator. In one embodiment, the method includes mapping the set of commutators to a set of qubit states corresponding to a set of qubits of the quantum processor. In one embodiment, the method includes evaluating the set of commutators using the quantum processor and the memory. In one embodiment, the method includes causing a quantum readout circuit to measure excited state energies from the set of computed commutators.

Description

Technical Field

[0001] The present invention generally relates to optimization using quantum computing. More specifically, the present invention relates to a system and method for computing excited state properties of a molecular system using a hybrid classical-quantum computing system. Background Art

[0002] In the following, unless explicitly distinguished when used, the "Q" prefix in a phrase word indicates a reference to the word or phrase in the context of quantum computing.

[0003] Molecules and subatomic particles obey the laws of quantum mechanics. Quantum mechanics is a branch of physics that explores how the physical world works at its most fundamental level. At this level, particles behave in strange ways, are in multiple states at once, and interact with other particles that are very far away. Quantum computing exploits these quantum phenomena to process information.

[0004] The computers we commonly use today are called classical computers (also referred to herein as "conventional" computers or conventional nodes or "CNs"). Conventional computers use conventional processors, semiconductor memories, and magnetic or solid-state storage devices made using semiconductor materials and technologies, which is the so-called von Neumann architecture. In particular, the processors in conventional computers are binary processors, that is, they operate on binary data represented by 1s and 0s.

[0005] Quantum processors (Q-processors) use the unique properties of entangled quantum bit devices (referred to in this article as "qubits" or "qubits") to perform computing tasks. In the specific field where quantum mechanics operates, particles of matter can exist in multiple states - such as an "on" state, an "off" state, and both "on" and "off" at the same time. Where binary computing using semiconductor processors is limited to using only the on and off states (equal to 1 and 0 in binary code), quantum processors use these quantum states of matter to output signals that can be used for data computing.

[0006] Conventional computers encode information in bits. Each bit can take on a value of 1 or 0. These 1s and 0s act as on / off switches that ultimately drive the computer's functions. Quantum computers, on the other hand, are based on qubits, which operate according to two key principles of quantum physics: superposition and entanglement. Superposition means that each qubit can represent a 1 and a 0 between possible outcomes of an event. Entanglement means that qubits in superposition can be related to each other in a nonclassical way; that is, the state of one (whether it is a 1 or a 0 or both) can depend on the state of the other, and two qubits contain more information when entangled than they would as two independent qubits.

[0007] Using these two principles, qubits act as processors of information, enabling quantum computers to work in a way that allows them to solve certain difficult problems that are difficult to solve using conventional computers.

[0008] There is a class of problems called optimization problems. An optimization problem is a computational problem in which the best or optimal solution is to be determined for a different problem, where the different problem has multiple possible solutions. For example, the different problem could be the famous traveling salesman problem, where a route between several cities must be determined so that the traveling salesman covers each city without revisiting any city. This problem has many possible solutions - routes between cities. The optimization problem related to the traveling salesman problem is to find the shortest route, i.e. the best or optimal route, out of many possible routes that each satisfy the requirements of the traveling salesman problem.

[0009] Configuring an optimization problem for execution on a computer so that the computer can compute an optimal solution in a finite time is a difficult problem in itself. Until recently, the only computing resources available for executing optimization problems were conventional computers as described herein. Many optimization problems are too difficult or too complex for conventional computers to compute in a reasonable time with reasonable resources. Typically, in such cases, an approximate solution that can be computed in a reasonable time and with reasonable resources is considered to be a solution that is close to optimal.

[0010] The advent of quantum computing presents the potential for advancement in many areas of computing, including the computation of optimization problems. Because quantum computing systems can evaluate many solutions in a solution space at once, the illustrative embodiments recognize that such systems are particularly well suited for solving optimization problems.

[0011] The exemplary embodiments recognize that several quantum computing methods have been proposed to solve the electronic structure of molecular systems. The electronic structure of a molecular system is governed by the interactions between the electron orbitals in the molecule. The state of a molecular system is the arrangement of molecular orbitals that describes the wave-like behavior of electrons in the molecule. The ground state corresponds to the lowest energy state of the molecular system. Excited energy states correspond to discrete energy differences greater than the ground state.

[0012] The illustrative embodiments recognize that quantum processors can perform variational algorithms that currently available conventional processors either cannot perform or can only perform with undesirable accuracy or computational resource consumption. Variational algorithms use varying trial wave functions to determine an upper bound on the energy of the ground state of a quantum system. A wave function is a mathematical description of a quantum state such as a quantum system. A quantum state is represented on a quantum processor as a series of quantum logic gates acting on qubits. Each quantum state of a quantum system contains a corresponding energy value.

[0013] The ground state energy of a quantum system corresponds to the minimum possible value of the ground state energy of the quantum system. The Hamiltonian is a matrix operator that describes the ground state energy of a quantum state. The Hamiltonian operator acting on the wave function determines the value corresponding to the ground state energy of the quantum state.

[0014] To calculate an upper bound on the ground state energy of a quantum system, a variational algorithm performs many evaluations starting from an initial wave function. Each evaluation calculates the ground state energy of the quantum state corresponding to the wave function being evaluated. The variational algorithm can then change the parameters of the wave function being evaluated to generate a new wave function, for example, changing at least one quantum logic gate in a set of quantum logic gates to perform a rotation on a qubit. The evaluation of the new wave function will calculate the ground state energy of the new quantum state corresponding to the new wave function. The variational algorithm compares the ground state energy of the previous wave function to the ground state energy of the new wave function.

[0015] The conventional processor executes an optimization algorithm that changes the parameters of the wave function. The quantum processor calculates the corresponding ground state energy of the wave function. The optimization algorithm determines how to change the parameters of the wave function to minimize the calculated ground state energy of the quantum system based on a comparison between the ground state energies of the new wave function and the previous wave function.

[0016] The variational algorithm can continue to perform evaluations until the calculated ground state energy is relatively stable, e.g., successive evaluations of the calculated ground state energy are within a threshold percentage. The stable calculated ground state energy resulting from the final evaluation corresponds to an upper bound on the minimum energy of the ground state of the quantum system. The corresponding wave function represents an approximation of the eigenfunctions of the quantum system.

[0017] The illustrative embodiments recognize that currently available methods are inaccurate for determining excited state energies of quantum systems. Quantum Subspace Extension (QSE) is a method for calculating excited state energies of quantum systems. The illustrative embodiments recognize that QSE is based on perturbation theory. Perturbation theory involves starting with a simple system with a known mathematical solution and then adding other elements to approximate a more complex system. The illustrative embodiments recognize that QSE exhibits certain disadvantages. The illustrative embodiments recognize that for certain quantum systems that do not have similar simple systems, QSE is not applicable. The illustrative embodiments further recognize that QSE is computationally very expensive in terms of the number of measurements required.

[0018] The Witness-Assisted Variational Eigenspectral Solver (WAVES) is another method for determining excited state energies of a quantum system. WAVES involves a variational search for the ground state of the quantum system, a second variational search for excited states, and an Iterative Phase Estimation Algorithm (IPEA) for approximating excited state energies. The illustrative embodiments recognize that WAVES has certain disadvantages. The illustrative embodiments recognize that WAVES requires a time-consuming variational optimization for each excited state. Summary of the invention

[0019] The illustrative embodiments provide a method, system, and computer program product for computing excited state properties of a molecular system using a hybrid classical-quantum computing system. Embodiments of the method include determining a ground state wave function of a combination of quantum logic gates using a quantum processor and a memory, the quantum processor including a set of quantum logic gates configured to perform a single qubit rotation.

[0020] In one embodiment, the method includes forming a set of excitation operators configured to determine an excited state wave function from a ground state wave function. In one embodiment, the method includes forming a set of commutators from the set of excitation operators and a Hamiltonian operator configured to determine a ground state energy of the wave function.

[0021] In one embodiment, the method includes mapping the set of commutators to a set of qubit states corresponding to a set of qubits of a quantum processor. In one embodiment, the method includes evaluating the set of commutators using a quantum processor and a memory. In one embodiment, the method includes causing a quantum readout circuit to measure excited state energies from the calculated set of commutators.

[0022] In one embodiment, the method includes extracting a set of Pauli strings, each Pauli string in the set of Pauli strings including a set of Pauli operators. In one embodiment, the method includes evaluating the set of Pauli strings to calculate the set of commutators.

[0023] In one embodiment, the method includes combining a subset of the set of Pauli strings, the subset corresponding to at least one of the set of commutators. In one embodiment, the method includes mapping the set of excitation operators using a Jordan-Wigner transformation.

[0024] In one embodiment, the set of commutators is mapped using a Jordan-Wigner transformation. In one embodiment, the method includes calculating a set of matrix elements of a secular matrix equation corresponding to a set of eigenvalue equations for excitation energies.

[0025] Embodiments include a computer usable program product. The computer usable program product includes a computer readable storage device and program instructions stored on the storage device.

[0026] Embodiments include a computer system. The computer system includes a processor, a computer-readable memory and a computer-readable storage device, and program instructions stored on the storage device for execution by the processor via the memory. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] The novel features which are believed to be characteristic of the invention are set forth in the appended claims. However, the invention itself and its preferred mode of use, other objects and advantages will be best understood from the following detailed description of illustrative embodiments read in conjunction with the accompanying drawings, in which:

[0028] Figure 1 depicts a block diagram of a network of data processing systems in which the illustrative embodiments may be implemented;

[0029] Figure 2 depicts a block diagram of a data processing system in which the illustrative embodiments may be implemented;

[0030] Figure 3 depicts a block diagram of an example hybrid quantum / classical optimization algorithm for solving a combinatorial optimization problem using a classical processor and a quantum processor in accordance with an illustrative embodiment;

[0031] Figure 4 depicts a block diagram of an example configuration for solving a mixed integer optimization problem on a hybrid classical-quantum computing system in accordance with an illustrative embodiment;

[0032] Figure 5 depicts a block diagram of an example universal quantum circuit for computing excited state energies using a hybrid classical-quantum computing system in accordance with an illustrative embodiment;

[0033] Figure 6 depicts a block diagram of an example configuration for computing excited state energies of a molecular system on a hybrid classical-quantum computing system in accordance with an illustrative embodiment;

[0034] Figure 7 depicts, by way of illustrative embodiment, a graph of excitation energies computed on a hybrid classical-quantum computing system; and

[0035] Figure 8 A flow chart of an example method for computing excited state energies of a molecular system on a hybrid classical-quantum computing system is shown in accordance with an illustrative embodiment. DETAILED DESCRIPTION

[0036] The illustrative embodiments used to describe the present invention generally relate to and solve the above-mentioned problem of using quantum computing to solve computational problems. The illustrative embodiments provide a method and system for computing excited state properties of a molecular system using a hybrid classical-quantum computing system.

[0037] Solution optimization is a recognized area of ​​technical effort. Quantum computing using processors formed of quantum qubits is another recognized area of ​​technical effort. The prior art has certain disadvantages and limitations in combining these two areas. The operation and / or configuration of the illustrative embodiments is to improve upon the prior art in these areas of technical effort, particularly in configuring optimization problems to be performed in a quantum computing environment, and to impart additional or new capabilities.

[0038] Exemplary embodiments recognize that solving optimization problems in quantum computing often requires converting the optimization problem and its inputs into a Hamiltonian, which is then passed to a quantum variational algorithm, such as the Variational Quantum Eigensolver (VQE) algorithm and the Quantum Approximate Optimization Algorithm (QAOA).

[0039] One embodiment provides a method for calculating excited state properties of a molecular system using a hybrid classical-quantum computing system. Another embodiment provides a conventional or quantum computer usable program product, which includes a computer readable storage device and program instructions stored on the storage device, the stored program instructions including a method for calculating excited state properties of a molecular system using a hybrid classical-quantum computing system. The instructions can be executed using a conventional or quantum processor. Another embodiment provides a computer system, which includes a conventional or quantum processor, a computer readable memory and a computer readable storage device, and program instructions stored on the storage device for execution by the processor via the memory, the stored program instructions including a method for calculating excited state properties of a molecular system using a hybrid classical-quantum computing system.

[0040] The illustrative embodiments recognize that a quantum processor can perform a variational algorithm to calculate an approximation of the ground state energy of a quantum system, such as an electron orbital configuration of a molecule with a given atomic spacing. A variational quantum eigensolver (VQE) is an example of a variational algorithm performed using a quantum processor. VQE varies parameters to prepare a quantum state and determines properties of the prepared quantum state. The quantum state is prepared on the quantum processor as a series of quantum logic gates acting on qubits.

[0041] The illustrative embodiments recognize that because the eigenfunctions of the quantum system are unknown or too complex, an initial quantum state (wave function) can be randomly selected. A variational algorithm evaluates the quantum state to determine properties of the initial quantum state (e.g., ground state energy), varies parameters to prepare a new quantum state (e.g., varies the rotation angle of a set of single qubit quantum logic rotation gates), determines properties of the new quantum state, and compares the properties of the initial quantum state and the new quantum state.

[0042] The variational algorithm iteratively generates new quantum states and minimizes properties corresponding to the quantum states. The variational algorithm includes an optimizer to minimize the properties corresponding to the quantum state. Each evaluation performed by the variational algorithm includes changing parameters (e.g., a single qubit rotation) to generate a new quantum state, calculating the properties of the new quantum state, comparing the properties of the new quantum state to the properties of the previous quantum state, and based on the comparison, determining how to change the parameters in successive evaluations to calculate the ground state energy of the quantum system. For example, the variational algorithm can perform an evaluation to determine an upper limit on the ground state energy of the quantum system.

[0043] The variational algorithm changes one or more parameters to produce one or more new quantum states and compares the ground state energy of the new quantum state with the ground state energy of one or more previous quantum states. The optimization program determines which (some) parameters to change and / or how to change the parameters to reduce the calculated ground state energy of the generated quantum state. The variational algorithm continues to perform evaluations until the calculated ground state energy reaches a minimum and becomes relatively stable. The calculated ground state energy of the final evaluation corresponds to an upper limit on the ground state energy of the quantum system.

[0044] This can be done by applying the excitation matrix operator on the ground state |0> of the quantum system to create excited states |n>. The excitation matrix operator is defined as Similarly, the deexcitation operator is defined as n =|0><n|. The commutator of the excitation operator and the Hamiltonian operator is defined as The commutator and excitation operator of the Hamiltonian are derived to obtain the excitation energy ΔE of the system 0n =E n -E 0 The expression is

[0045] By operating the previous equation from the left side using the excitation operator, the excitation energy ΔE is obtained 0n A useful form of: The excitation operator can be expressed as a linear expansion of the basic operator and the optimization of the expansion coefficient. The simplest basis is the fermionic creation and annihilation operator a composition, of which a i represents the excitation of a single electron from an occupied orbital i to a virtual orbital m. The Fermionic operator can be mapped to the Pauli operator by the Jordan-Wigner transformation defined by:

[0046]

[0047]

[0048] Where N is the total number of qubits.

[0049] The excitation operator can then be expressed as where α is the degree of excitation and Relative to The variation of parameters leads to the long-term matrix equation for determining the eigenvalues ​​(excitation energies) of the molecular system in:

[0050]

[0051]

[0052]

[0053]

[0054] And ω n It is excitation energy.

[0055] One or more embodiments provide a hybrid classical and quantum method for computing excited state properties of a molecular system. In one or more embodiments, a classical computer is used to optimize parameters and evaluate matrix equations, while a quantum computer is used to compute matrix elements.

[0056] In one embodiment, a commutator of Hamiltonian and excitation operators is mapped to a set of qubit states. In a specific embodiment, the commutator is mapped from a fermion operator to a Pauli operator using a Jordan-Wigner transformation. In this embodiment, a ground state wave function is determined. In this embodiment, the ground state wave function is parameterized as a set of angles.

[0057] For the sake of clarity of description, and without any limitation thereto, some example configurations are used to describe the illustrative embodiments. Based on this disclosure, a person of ordinary skill in the art will be able to think of many changes, adaptations and modifications to the configurations to achieve the described purpose, and these are all within the scope of the illustrative embodiments.

[0058] In addition, simplified diagrams of data processing environments are used in the drawings and illustrative embodiments. In a real computing environment, there may be additional structures or components not shown or described herein, or structures or components different from those shown but used for functions similar to those described herein, without departing from the scope of the exemplary embodiments.

[0059] Furthermore, the illustrative embodiments are described only with respect to specific actual or hypothetical components as examples. The steps described by the various illustrative embodiments may be adapted to calculate excited state energies of molecular systems using a variety of components that may be purposed or repurposed to provide the described functionality in a data processing environment, and such adaptations may be considered within the scope of the illustrative embodiments.

[0060] The illustrative embodiments are described only for certain types of steps, applications, classical processors, quantum processors, quantum states, classical feature spaces, quantum feature spaces, and data processing environments as examples. Any particular manifestation of these and other similar articles is not intended to limit the invention. Any suitable manifestation of these and other similar articles may be selected within the scope of the illustrative embodiments.

[0061] The examples in this disclosure are only for clarity of description and are not limited to the illustrative embodiments. Any advantages listed herein are only examples and are not intended to limit the illustrative embodiments. Additional or different advantages can be achieved through specific illustrative embodiments. In addition, specific illustrative embodiments may have some, all or none of the advantages listed above.

[0062] Reference is made to the accompanying drawings, especially to Figure 1 and 2 , these figures are exemplary diagrams of data processing environments in which illustrative embodiments may be implemented. Figure 1 and Figure 2 This is only an example and is not intended to assert or imply any limitation on the environments in which different embodiments may be implemented. Specific implementations may make many modifications to the illustrated environment based on the following description.

[0063] Figure 1 A block diagram of a network of data processing systems in which the illustrative embodiments may be implemented is depicted. Data processing environment 100 is a network of computers in which the illustrative embodiments may be implemented. Data processing environment 100 includes network 102. Network 102 is a medium for providing communication links between various devices and computers connected together within data processing environment 100. Network 102 may include connections such as wired, wireless communication links, or fiber optic cables.

[0064] Client or server are merely example roles for certain data processing systems connected to network 102 and are not intended to exclude other configurations or roles of these data processing systems. A classical processing system 104 is coupled to network 102. Classical processing system 104 is a type of classical processing system. Software applications can be executed on any quantum data processing system in data processing environment 100. Figure 1Any software application executed in the classic processing system 104 of can be configured to be executed in another data processing system in a similar manner. Figure 1 Any data or information stored or generated in the classical processing system 104 of the present invention may be configured to be stored or generated in another data processing system in a similar manner. A classical data processing system such as the classical processing system 104 may contain data and may have software applications or software tools that perform classical computing processes thereon.

[0065] Server 106 is coupled to network 102 along with storage device 108. Storage device 108 includes database 109 configured to store parameters of quantum states. Server 106 is a conventional data processing system. Quantum processing system 140 is coupled to network 102. Quantum processing system 140 is a quantum data processing system. The software application can be executed on any quantum data processing system in data processing environment 100. Figure 1 Any software application executed in the quantum processing system 140 of can be configured to execute in another quantum data processing system in a similar manner. Figure 1 Any data or information stored or generated in quantum processing system 140 can be configured to be stored or generated in another quantum data processing system in a similar manner. A quantum data processing system such as quantum processing system 140 can contain data and can have software applications or software tools that perform quantum computing processes on it.

[0066] Clients 110, 112, and 114 are also coupled to network 102. Conventional data processing systems, such as server 106, or clients 110, 112, or 114 may contain data and may have software applications or software tools that perform conventional computing processes thereon.

[0067] This is for example purposes only and does not imply any limitation on this architecture. Figure 1 Certain components that can be used in example implementations of the embodiments are depicted. For example, server 106 and clients 110, 112, 114 are depicted as servers and clients only as examples and do not imply limitations on client-server architectures. As another example, one embodiment can be distributed across several conventional data processing systems, quantum data processing systems, and data networks as shown, while another embodiment can be implemented on a single conventional data processing system or a single quantum data processing system, all within the scope of the illustrative embodiments. Conventional data processing systems 106, 110, 112, and 114 also represent example nodes in clusters, partitions, and other configurations suitable for the embodiments.

[0068] Device 132 is an example of a conventional computing device as described herein. For example, device 132 may take the form of a smartphone, a tablet computer, a laptop computer, a fixed or portable client 110, a wearable computing device, or any other suitable device. Figure 1 Any software application that is executed in another conventional data processing system in is configured to be executed in device 132 in a similar manner. Figure 1 Any data or information stored or generated in another conventional data processing system in may be configured to be stored or generated in device 132 in a similar manner.

[0069] Server 106, storage device 108, classical processing system 104, quantum processing system 140, and clients 110, 112, and 114, and device 132 may be coupled to network 102 using wired connections, wireless communication protocols, or other suitable data connectivity. Clients 110, 112, and 114 may be, for example, personal computers or network computers.

[0070] In the example shown, server 106 may provide data such as boot files, operating system images, and applications to clients 110, 112, and 114. Clients 110, 112, and 114 in this example may be clients to server 106. Clients 110, 112, 114, or some combination thereof, may include their own data, boot files, operating system images, and applications. Data processing environment 100 may include additional servers, clients, and other devices not shown.

[0071] In the example shown, memory 124 can provide data such as boot files, operating system images, and applications to classical processor 122. Classical processor 122 can include its own data, boot files, operating system images, and applications. Data processing environment 100 may include additional memory, quantum processors, and other devices not shown. Memory 124 includes application 105, which can be configured to implement one or more of the classical processor functions described herein to calculate excited state energies of molecular systems on a hybrid classical-quantum computing system according to one or more embodiments.

[0072] In the example shown, memory 144 can provide data such as boot files, operating system images, and applications to quantum processor 142. Quantum processor 142 may include its own data, boot files, operating system images, and applications. Data processing environment 100 may include additional memory, quantum processors, and other devices not shown. Memory 144 includes applications 146, which may be configured to implement one or more of the quantum processor functions described herein according to one or more embodiments.

[0073] In the example shown, data processing environment 100 can be the Internet. Network 102 can represent a collection of networks and gateways that communicate with each other using Transmission Control Protocol / Internet Protocol (TCP / IP) and other protocols. The core of the Internet is the backbone of data communication links between major nodes or hosts, which include thousands of commercial, government, educational and other computer systems that route data and messages. Of course, data processing environment 100 can also be implemented as many different types of networks, such as an intranet, a local area network (LAN), or a wide area network (WAN). Figure 1 This is intended as an example, not as an architectural limitation to the different illustrative embodiments.

[0074] Among other uses, data processing environment 100 may be used to implement a client-server environment in which illustrative embodiments may be implemented. A client-server environment enables software applications and data to be distributed across a network, thereby enabling applications to be run using interactivity between conventional client data processing systems and conventional server data processing systems. Data processing environment 100 may also employ a service-oriented architecture in which interoperable software components distributed over a network may be packaged together into a consistent business application. Data processing environment 100 may also take the form of a cloud and employ a cloud computing model of service delivery to enable convenient, on-demand network access to a shared pool of configurable computing resources (e.g., networks, network bandwidth, servers, memory, storage, applications, virtual machines, and services) that can be quickly configured and released with minimal management effort or interaction with a service provider.

[0075] refer to Figure 2 , which depicts a block diagram of a data processing system in which the illustrative embodiments may be implemented. Data processing system 200 is an example of a conventional computer, such as Figure 1 A conventional processing system 104, server 106 or clients 110, 112 and 114, or another type of device, in which computer usable program code or instructions implementing the processes of the illustrative embodiments may be located.

[0076] Data processing system 200 also represents a conventional data processing system or configuration thereof, such as a Figure 1 A conventional data processing system 132 in which computer usable program code or instructions implementing the processes of the illustrative embodiments may be located. Data processing system 200 is described as a computer only by way of example and is not limited thereto. Figure 1Implementations in other device forms, such as device 132 in FIG. 1 , may modify data processing system 200 by, for example, adding a touch interface, or even removing certain depicted components from data processing system 200 without departing from the general description of the operation and functionality of data processing system 200 described herein.

[0077] In the example shown, data processing system 200 employs a hub architecture that includes north bridge and memory controller hub (NB / MCH) 202 and south bridge and input / output (I / O) controller hub (SB / ICH) 204. Processing unit 206, main memory 208, and graphics processor 210 are coupled to north bridge and memory controller hub (NB / MCH) 202. Processing unit 206 may include one or more processors and may be implemented using one or more heterogeneous processor systems. Processing unit 206 may be a multi-core processor. In some embodiments, graphics processor 210 may be coupled to NB / MCH 202 via an accelerated graphics port (AGP).

[0078] In the example shown, a local area network (LAN) adapter 212 is coupled to the south bridge and I / O controller hub (SB / ICH) 204. An audio adapter 216, a keyboard and mouse adapter 220, a modem 222, a read-only memory (ROM) 224, a universal serial bus (USB) and other ports 232, and a PCI / PCIe device 234 are connected to the south bridge and I / O controller hub 204 via a bus 238. A hard disk drive (HDD) or solid-state drive (SSD) 226 and a CD-ROM 230 are connected to the south bridge and I / O controller hub 204 via a bus 240. The PCI / PCIe devices 234 may include, for example, Ethernet adapter add-in cards and PC cards for notebook computers. PCI uses a card bus controller, while PCIe does not. ROM 224 may be, for example, a flash binary input / output system (BIOS). Hard disk drive 226 and CD-ROM 230 may use, for example, an integrated drive electronics (IDE), a serial advanced technology attachment (SATA) interface, or variations such as external SATA (eSATA) and micro SATA (mSATA). Super I / O (SIO) device 236 may be coupled to south bridge and I / O controller hub (SB / ICH) 204 via bus 238.

[0079] Memories such as main memory 208, ROM 224, or flash memory (not shown) are some examples of computer usable storage devices. Hard disk drive or solid state drive 226, CD-ROM 230, and other similarly available devices are some examples of computer usable storage devices that include computer usable storage media.

[0080] An operating system runs on processing unit 206. The operating system coordinates and provides Figure 2 The operating system controls the various components in data processing system 200. The operating system may be a commercially available operating system for any type of computing platform, including but not limited to server systems, personal computers, and mobile devices. Object-oriented or other types of programming systems may operate in conjunction with the operating system and provide calls to the operating system from programs or applications executing on data processing system 200.

[0081] for operating systems, object-oriented programming systems, and such Figure 1 Instructions for an application or program, such as application 105 in , are located on a storage device, such as in the form of code 226A on hard drive 226, and may be loaded into at least one of one or more memories, such as main memory 208, for execution by processing unit 206. The processes of the illustrative embodiments may be performed by processing unit 206 using computer-implemented instructions, which may be located in a computer memory, such as main memory 208, read-only memory 224, or memory in one or more peripheral devices.

[0082] In addition, in one case, code 226A can be downloaded from remote system 201B via network 201A, where similar code 201C is stored on storage device 201D. In another case, code 226A can be downloaded to remote system 201B via network 201A, where the downloaded code 201C is stored in storage device 201D.

[0083] Figure 1-2 The hardware in the may vary depending on the implementation. In addition to or instead of Figure 1-2 The hardware shown may use other internal hardware or peripheral devices, such as flash memory, equivalent non-volatile memory, or optical disk drives, etc. In addition, the processes of the illustrative embodiments may be applied to a multi-processor data processing system.

[0084] In some illustrative examples, data processing system 200 may be a personal digital assistant (PDA), which is typically configured with flash memory to provide non-volatile memory for storing operating system files and / or user-generated data. The bus system may include one or more buses, such as a system bus; an I / O bus, and a PCI bus. Of course, the bus system may be implemented using any type of communication structure or architecture that provides for the transfer of data between different components or devices attached to the structure or architecture.

[0085] The communication unit may include one or more devices for sending and receiving data, such as a modem or a network adapter. A memory may be, for example, main memory 208 or a cache, such as that found in north bridge and memory controller hub 202. A processing unit may include one or more processors or CPUs.

[0086] Figure 1-2 The examples described in and above are not meant to imply architectural limitations. For example, data processing system 200 may be a tablet computer, laptop computer, or telephone device in addition to taking the form of a mobile or wearable device.

[0087] Where a computer or data processing system is described as a virtual machine, virtual device, or virtual component, the virtual machine, virtual device, or virtual component operates in the manner of data processing system 200 using virtualized representations of some or all of the components described in data processing system 200. For example, in the virtual machine, virtual device, or virtual component, processing unit 206 is represented as a virtualized instance of all or some number of hardware processing units 206 available in the host data. In the processing system, main memory 208 is represented as a virtualized instance of all or some portion of main memory 208 that may be available in the host data processing system, and disk 226 is represented as a virtualized instance of all or some portion of disk 226 that may be available in the host data processing system. In this case, the host data processing system is represented by data processing system 200.

[0088] refer to Figure 3 , which depicts a block diagram of an example hybrid quantum / classical optimization algorithm 300 for solving a combinatorial optimization problem using a classical processor 302 and a quantum processor 304. In this example, the classical processor 302 runs a classical optimization scheme to generate update parameters for the combinatorial optimization problem and sends the update parameters to the quantum processor 304. The quantum processor 304 prepares a quantum state according to the specific combinatorial problem to be solved and the given update parameters. The quantum processor 304 executes the prepared quantum state and measures the quantum state multiple times to sample from the solution space to generate samples. The classical processor 302 receives the samples from the quantum processor 304 and evaluates the received samples to determine whether the parameters of the classical optimization scheme are to be updated.

[0089] If the classical processor 302 determines that the parameters of the combinatorial optimization problem are to be updated, the classical processor 302 runs the classical optimization scheme using the updated parameters to generate further updated parameters. The classical processor 302 then sends the further updated parameters to the quantum processor 304. Typically, this process is repeated until convergence within an acceptable threshold is obtained.

[0090] Reference Figure 4, which depicts a block diagram of an example configuration 400 for solving a mixed integer optimization problem on a hybrid classical-quantum computing system according to an illustrative embodiment. The example embodiment includes a classical processing system 104 and a quantum processing system 140. The classical processing system 104 includes an application 402. In a particular embodiment, the application 402 is Figure 1 The application 402 includes a classical optimizer component 404, a decision variable determination component 406, a quantum circuit angle derivation component 408, a classical parameter derivation component 410, and a quantum state sample evaluation component 412.

[0091] Quantum processing system 140 includes quantum processor 142 and application 414. In certain embodiments, application 414 is Figure 1 The application 414 includes a quantum state preparation component 416 and a quantum state measurement component 418.

[0092] In an embodiment, the classical optimizer component 404 is configured to execute a classical optimization scheme and utilize the optimization parameter determination component 406 to generate decision variables of a classical objective function associated with the combinatorial optimization problem. In a specific embodiment, the decision variables include at least one discrete decision variable (e.g., a binary decision variable) and at least one continuous decision variable. The quantum circuit angle derivation component 408 is configured to derive a quantum angle of a quantum circuit within the quantum processor 142 of the quantum processing system 140 from one or more decision variables. In a specific embodiment, the quantum circuit angle derivation component 408 derives the quantum angle of the quantum circuit based on at least one discrete decision variable.

[0093] The classical parameter derivation component 410 is configured to derive classical parameters from one or more determined decision variables. In a specific embodiment, the classical parameter derivation component 410 is configured to derive classical parameters based on at least one continuous decision variable. The application 402 is also configured to provide the angular parameters to the quantum processing system 140.

[0094] The quantum state preparation component 416 is configured to prepare one or more quantum states for the quantum circuit of the quantum processor 142 based on the quantum angle of the solution space as the problem. The quantum processor 142 then executes using the prepared quantum state as the initial state of the qubit. The quantum state measurement component 418 is configured to measure the intermediate quantum state multiple times to generate samples representing the intermediate quantum state. In one or more embodiments, a fixed set of quantum state parameters are used to prepare the same quantum state, but the measurement of the quantum state results in probabilistic results generated by multiple samples. In a specific embodiment, the sample includes a discrete value (e.g., -1 or +1). The application 414 is further configured to send the quantum state measurement sample to the application 402 of the classical processing system 104.

[0095] In an embodiment, the quantum state sample evaluation component 412 evaluates the quantum state measurement sample using a classical aggregation function to determine an aggregate quantum state measurement value from the sample. The aggregation function determines multiple values ​​from a single value. In a specific embodiment, the classical aggregation function averages the quantum state measurement sample to determine a single aggregate value. In one or more embodiments, the quantum state sample evaluation component 412 evaluates multiple samples to obtain a measure of the quality of the quantum state and a measure of the solution to the mixed integer optimization problem that can be sampled from such a quantum state. In an embodiment, the quantum state sample evaluation function returns the aggregate value to the classical optimizer 404. In an embodiment, the application 402 determines updated decision variables based on the aggregate value, generates updated quantum state parameters from the updated decision variables, and provides the new quantum state parameters to the quantum processing system 140, and the process is repeated until a predetermined acceptable level is reached (e.g., after convergence).

[0096] refer to Figure 5 , which depicts a block diagram of an example universal quantum circuit 500 for computing excited state energies using a hybrid classical-quantum computing system, in accordance with an illustrative embodiment. The quantum circuit 500 includes a set of n qubits corresponding to the first n excitation energies. The ground state wave function is determined by VQE. The ground state wave function is parameterized in a set of angles. A set of Pauli strings is extracted and evaluated using a circuit parameterized in the set of angles. Measurements are made on the set of qubits and the measurements are then passed to a classical processor to determine eigenvalues ​​and excitation energies.

[0097] In the illustrated example, quantum circuit 500 is shown as a general example of a quantum circuit for implementing one or more methods described herein for calculating excited state energy. In a specific implementation, according to one or more embodiments, a quantum circuit may include any number of quantum gate arrangements to implement a corresponding method for calculating excited state energy of a molecular system.

[0098] refer to Figure 6 , which depicts a block diagram of an example configuration 600 for computing excited state energies of a molecular system on a hybrid classical-quantum computing system according to an illustrative embodiment. The example embodiment includes a classical processing system 104 and a quantum processing system 140. The classical processing system 104 includes an application 602. In a particular embodiment, the application 602 is Figure 1 The application 602 includes an operator preparation component 604, a commutator preparation component 606, an excitation energy determination component 608, and a quantum state sample evaluation component 610.

[0099] Quantum processing system 140 includes quantum processor 142 and application 612. In certain embodiments, application 612 is Figure 1The application 612 includes a commutator evaluation component 614 and a quantum state measurement component 616.

[0100] In this embodiment, operator preparation component 604 prepares excitation and de-excitation operators to determine excited states according to previously determined ground state wave functions. In one embodiment, commutator preparation component 606 prepares commutators from excitation operators and Hamiltonian operators. In a specific embodiment, component 606 maps the excitation operators to a set of qubit states using the Jordan-Wigner transformation. Application 602 sends the prepared commutator to application 612 of quantum processing system 140.

[0101] The commutator evaluation component 614 evaluates a set of commutators for the quantum circuit of the quantum processor 142 based on the quantum angle of the solution space as the problem. The quantum state measurement component 616 measures the intermediate quantum state multiple times to generate samples representing the intermediate quantum state. The application 612 is further configured to send the quantum state measurement samples to the application 602 of the classical processing system 104.

[0102] In an embodiment, quantum state sample evaluation component 610 evaluates the quantum state measurement sample using a classical aggregate function to determine an aggregate quantum state measurement value from the sample. The aggregate function determines multiple values ​​from a single value. In a specific embodiment, the classical aggregate function averages the quantum state measurement sample to determine a single aggregate value. In one or more embodiments, quantum state sample evaluation component 610 evaluates the multiple samples to obtain a solution to a matrix element. In this embodiment, excitation energy determination component 608 uses the solution to the matrix element to determine a set of excitation energies.

[0103] refer to Figure 7 , which depicts a graph of excitation energies calculated on a hybrid classical-quantum computing system in an illustrative embodiment. It can be seen that the difference (deviation) between the calculated excitation energies and the expected theoretical excitation energies in the baseline graph at a given internuclear distance is small.

[0104] refer to Figure 8 , which depicts a flow chart of an example method for computing excited state energies of a molecular system on a hybrid classical-quantum computing system according to an illustrative embodiment. In block 802, the quantum processor 142 determines a ground state wave function of the quantum system. In a particular embodiment, the ground state wave function is parameterized in a set of angles.

[0105] In block 804, the classical processor 122 prepares a set of excitation operators for determining excited states of the quantum system. In block 806, the classical processor 122 maps the excitation operators to a set of qubit states. In a particular embodiment, the excitation operators are mapped to a set of qubit states using a Jordan-Wigner transformation.

[0106] In block 808, the classical processor 122 computes a set of commutators from the mapped set of excitation operators and the mapped Hamiltonian operator. In a particular embodiment, the commutators form a set of Pauli strings, each of which includes a set of Pauli operators. In a particular embodiment, the Hamiltonian operator is mapped to a set of qubit states using a Jordan-Wigner transformation.

[0107] In block 810, the quantum processor 142 extracts the set of Pauli strings. The quantum processor 142 combines a subset of the set of Pauli strings. In a particular embodiment, the subset of the set of Pauli strings corresponds to at least one of the set of commutators. In block 812, the quantum processor 142 evaluates the set of Pauli strings using a quantum circuit parameterized in the set of angles corresponding to the ground state wave function.

[0108] In block 814, the classical processor 122 computes a set of matrix elements from the set of commutators. In block 816, the classical processor 122 determines a set of excitation energies from the set of matrix elements. In certain embodiments, the classical processor 122 computes a set of eigenvalues ​​to determine the set of excitation energies.

[0109] Thus, in illustrative embodiments, a computer-implemented method, system or apparatus, and computer program product are provided for computing excited state energies of molecular systems using a hybrid classical-quantum computing system and other related features, functions, or operations. Where embodiments are described for a device type, the computer-implemented method, system or apparatus, computer program product, or a portion thereof, is adapted or configured to be used with an appropriate and comparable representation of that device type.

[0110] Where embodiments are described as being implemented in an application, delivery of the application in a Software as a Service (SaaS) model is contemplated within the scope of the exemplary embodiments. In the SaaS model, the capabilities of the application implementing the embodiments are provided to users by executing the application in a cloud infrastructure. Users can access the application through a thin client interface, such as a web browser (e.g., web-based email) or other lightweight client application, using a variety of client devices. Users do not manage or control the underlying cloud infrastructure, including the network, servers, operating system, or storage of the cloud infrastructure. In some cases, users may not even manage or control the functionality of the SaaS application. In certain other cases, the SaaS implementation of the application may allow limited possible exceptions to user-specific application configuration settings.

[0111] The present invention may be a system, method and / or computer program product at any possible level of technical detail integration. The computer program product may include a computer-readable storage medium having computer-readable program instructions thereon for causing a processor to perform various aspects of the present invention.

[0112] A computer-readable storage medium may be a tangible device capable of retaining and storing instructions used by an instruction execution device. A computer-readable storage medium may be, for example, but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. A non-exhaustive list of more specific examples of computer-readable storage media includes the following: a portable computer floppy disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), a static random access memory (SRAM), a portable compact disk read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanical encoding device (such as a punch card or a raised structure in a groove on which instructions are recorded), and any suitable combination of the foregoing. Computer-readable storage media, including but not limited to computer-readable storage devices used herein, should not be interpreted as being transient signals themselves, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through waveguides or other transmission media (e.g., light pulses passing through optical fibers), or electrical signals transmitted through wires.

[0113] The computer-readable program instructions described herein may be downloaded from a computer-readable storage medium to a corresponding computing / processing device, or downloaded to an external computer or external storage device via a network (e.g., the Internet, a local area network, a wide area network, and / or a wireless network). The network may include copper transmission cables, optical transmission fibers, wireless transmissions, routers, firewalls, switches, gateway computers, and / or edge servers. The network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards the computer-readable program instructions to be stored in a computer-readable storage medium within the corresponding computing / processing device.

[0114] The computer-readable program instructions for performing the operation of the present invention may be assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state setting data, integrated circuit configuration data, or source code or object code or similar programming languages ​​written in any combination of one or more programming languages ​​(including object-oriented programming languages ​​such as Smalltalk, C++, etc. and procedural programming languages ​​such as "C" programming languages). The computer-readable program instructions may be executed entirely on a user's computer, partially on a user's computer, as an independent software package, partially on a user's computer and partially on a remote computer or entirely on a remote computer or server. In the latter case, the remote computer may be connected to the user's computer via any type of network (including a local area network (LAN) or a wide area network (WAN)), or may be connected to an external computer (e.g., via the Internet using an Internet service provider). In some embodiments, an electronic circuit including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA) may be personalized electronic circuits by executing computer-readable program instructions using the state information of the computer-readable program instructions to perform aspects of the present invention.

[0115] Various aspects of the present invention are described herein with reference to flowcharts and / or block diagrams of methods, devices (systems) and computer program products according to embodiments of the present invention. It will be understood that each frame of the flowchart illustration and / or block diagram and the combination of frames in the flowchart illustration and / or block diagram can be implemented by computer-readable program instructions.

[0116] These computer-readable program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing device to produce a machine, so that the instructions are executed via a computer or other processor to execute a programmable data processing device to create a device for implementing the functions / actions specified in the flowchart and / or block diagram box. These computer-readable program instructions can also be stored in a computer-readable storage medium, which can instruct a computer, a programmable data processing device, and / or other devices to function in a specific manner, so that the computer-readable storage medium in which the instructions are stored contains an article of manufacture including instructions for implementing various aspects of the functions / actions specified in the flowchart and / or block diagram or multiple boxes.

[0117] Computer-readable program instructions may also be loaded onto a computer, other programmable data processing device, or other device, so that a series of operating steps are performed on the computer, other programmable device, or other device to produce an implemented computer processing process, so that the instructions executed on the computer, other programmable device, or other device implement the functions / actions specified in the flowchart and / or block diagram blocks.

[0118] The flow chart and block diagram in the accompanying drawings show the possible implementation architecture, function and operation of the system, method and computer program product according to various embodiments of the present invention.In this regard, each frame in the flow chart or block diagram can represent a module, fragment or part of an instruction, which includes one or more executable instructions for realizing the specified logical function.In some alternative embodiments, the function pointed out in the frame may not occur in the order pointed out in the figure.For example, depending on the function involved, two frames shown in succession can actually be executed substantially at the same time, or these frames can be executed in reverse order sometimes.It should also be noted that each frame of the block diagram and / or flow chart and the combination of the frames of the block diagram and / or flow chart can be realized by a hardware-based special-purpose system that performs a specified function or action or performs a combination of special-purpose hardware and computer instructions.

Claims

1. A method, include: determining a ground state wave function of a combination of quantum logic gates using a quantum processor and a memory, the quantum processor comprising a set of quantum logic gates configured to perform a single qubit rotation; preparing, using a classical processor, a set of excitation operators configured to determine an excited state wave function from the ground state wave function; mapping, using the classical processor, an excitation operator in the set of excitation operators to a set of qubit states, the set of qubit states corresponding to a set of qubits of the quantum processor; computing, using the classical processor, a set of commutators from the mapped set of excitation operators and from a Hamiltonian operator mapped to a set of qubit states corresponding to the set of qubits of the quantum processor, the Hamiltonian operator being configured to determine a ground state energy of a wave function, wherein each commutator in the set of commutators is a commutator of a respective one of the set of excitation operators and the Hamiltonian operator; using the classical processor to map the computed set of commutators to a set of qubit states, the set of qubit states corresponding to the set of qubits of the quantum processor; the set of commutators evaluating computations using the quantum processor and memory; and A quantum readout circuit is enabled to measure excited state energies from the calculated set of commutators.

2. The method according to claim 1, further comprising: include: Extract the set of Pauli strings, each of which contains the set of Pauli operators.

3. The method according to claim 2, further comprising: include: The set of Pauli chords is evaluated to calculate the set of commutators.

4. The method according to claim 2, further comprising: include: A subset of the set of Pauli strings is combined, the subset corresponding to at least one of the set of commutators.

5. The method according to claim 1, further comprising: include: The set of excitation operators is mapped using the Jordan-Wigner transformation.

6. The method according to claim 1, in, The set of commutators is mapped using the Jordan-Wigner transformation.

7. The method according to claim 1, further comprising: include: calculating a set of matrix elements for a long-term matrix equation from the set of commutators; and The set of matrix elements is used to calculate a set of eigenvalues ​​corresponding to the set of excitation energies.

8. The method according to claim 1, in, The set of commutators is mapped to the set of qubit states using a Jordan-Wigner transformation and the classical processor before sending the mapped set of commutators from the classical processor to the quantum processor.

9. A computer usable program product comprising one or more computer readable storage devices and program instructions stored on at least one of the one or more computer readable storage devices, wherein the stored program instructions include: program instructions for determining a ground state wave function of a combination of quantum logic gates using a quantum processor and a memory, the quantum processor comprising a set of quantum logic gates configured to perform a single qubit rotation; program instructions for preparing, using a classical processor, a set of excitation operators, the set of excitation operators configured to determine an excited state wave function from the ground state wave function; program instructions for using the classical processor to map an excitation operator in the set of excitation operators to a set of qubit states corresponding to a set of qubits of the quantum processor; program instructions for computing, using the classical processor, a set of commutators from the mapped set of excitation operators and from a Hamiltonian operator mapped to a set of qubit states corresponding to a set of qubits of the quantum processor, the Hamiltonian operator configured to determine a ground state energy of a wave function, wherein each commutator in the set of commutators is a commutator for a respective one of the set of excitation operators and the Hamiltonian operator; program instructions for mapping the set of commutators computed using the classical processor to a set of qubit states corresponding to the set of qubits of the quantum processor; Program instructions for the set of commutators to evaluate computations using the quantum processor and memory; and Program instructions for causing a quantum readout circuit to measure excited state energies from the calculated set of commutators.

10. The computer usable program product according to claim 9, in, The program instructions are stored in a computer readable storage device in the data processing system and wherein the program instructions are transmitted from a remote data processing system over a network.

11. The computer usable program product according to claim 9, in, The program instructions are stored in a computer readable storage device in a server data processing system and wherein the program instructions are downloaded over a network to a remote data processing system for use in a computer readable storage device associated with the remote data processing system.

12. The computer usable program product of claim 9, further comprising: include: Program instructions for extracting a set of Pauli strings, each of which contains a set of Pauli operators.

13. The computer usable program product of claim 12, further comprising: include: Program instructions for evaluating the set of Pauli chords to calculate the set of commutators.

14. The computer usable program product of claim 12, further comprising: include: Instructions for combining a subset of the set of Pauli strings, the subset corresponding to at least one of the set of commutators.

15. The computer usable program product of claim 9, further comprising: include: Program instructions for mapping the set of excitation operators using a Jordan-Wigner transformation.

16. The computer usable program product of claim 9, further comprising: include: program instructions for calculating a set of matrix elements for a long term matrix equation from the set of commutators; and Program instructions for calculating a set of eigenvalues ​​corresponding to a set of excitation energies using the set of matrix elements.

17. A computer system comprising one or more processors, one or more computer-readable memories, and one or more computer-readable storage devices, and program instructions stored on at least one of the one or more computer-readable storage devices for execution by at least one of the one or more processors through at least one of the one or more computer-readable memories, the stored program instructions include: program instructions for determining a ground state wave function of a combination of quantum logic gates using a quantum processor and a memory, the quantum processor comprising a set of quantum logic gates configured to perform a single qubit rotation; program instructions for preparing, using a classical processor, a set of excitation operators, the set of excitation operators configured to determine an excited state wave function from the ground state wave function; program instructions for using the classical processor to map an excitation operator in the set of excitation operators to a set of qubit states corresponding to a set of qubits of the quantum processor; program instructions for computing, using the classical processor, a set of commutators from the mapped set of excitation operators and from a Hamiltonian operator mapped to a set of qubit states corresponding to a set of qubits of the quantum processor, the Hamiltonian operator configured to determine a ground state energy of a wave function, wherein each commutator in the set of commutators is a commutator for a respective one of the set of excitation operators and the Hamiltonian operator; program instructions for mapping the set of commutators computed using the classical processor to a set of qubit states corresponding to the set of qubits of the quantum processor; Program instructions for the set of commutators to evaluate computations using the quantum processor and memory; and Program instructions for causing a quantum readout circuit to measure excited state energies from the calculated set of commutators.

18. The computer system according to claim 17, further comprising: include: Program instructions for extracting a set of Pauli strings, each of which contains a set of Pauli operators.

19. The computer system according to claim 18, further comprising: include: Program instructions for evaluating the set of Pauli chords to calculate the set of commutators.

20. The computer system according to claim 18, further comprising: include: Program instructions for combining a subset of the set of Pauli strings, the subset corresponding to at least one of the set of commutators.

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