A hybrid quantum-classical computing system for the simulation of chemical systems using a chemical recognition state preparation strategy

Abstract

A chemical system is simulated using a hybrid quantum-classical computing system. The hybrid quantum-classical computing system comprises a classical computing component coupled to a quantum computing component. The hybrid quantum-classical computing system is configured to determine characterization parameters that describe the chemical system, where the characterization parameters are at least partially based on the chemical structure of the chemical system, generate an operator representation of the chemical system, including a single-electron excitation operator and a two-electron excitation operator, and reconfigure the order of appearance of the single-electron excitation operator and the two-electron excitation operator in the operator representation according to the characterization parameters. The simulation is executable using a smaller number of two-qubit quantum gates used in at least one quantum circuit executable on the quantum computing component used to perform the simulation.

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims priority to U.S. Application No. 63 / 344,629, filed May 22, 2022, the entire contents of which are incorporated herein by reference.

[0002] The present disclosure relates to a computational system for simulating chemical systems and a method for using such a computational system to simulate chemical systems. Exemplary embodiments relate to the use of a hybrid quantum-classical computational system to simulate chemical systems. [Background technology]

[0003] Without limitation, atoms, molecules, ions, periodic solids, periodic surface slabs, aggregates of one or more atoms, molecules, and / or ions, or combinations thereof, consist of one or more atomic nuclei and at least one electron, the quantum state of which is represented by an electronic wave function, which is conveniently constructed from molecular (or crystalline) orbitals and expressed as a linear combination of Slater determinants.

[0004] As the number of electrons in atoms and / or atoms in molecules, periodic solids, and / or periodic surface slabs increases, the computational complexity and processing costs of modeling such systems increases significantly, e.g., substantially exponentially. Through a great deal of effort, ingenuity, and innovation, many of the deficiencies in modeling and / or simulating such systems have been overcome by developing structured approaches in accordance with embodiments of the present invention, many examples of which are described in detail herein. Summary of the Invention [Means for solving the problem]

[0005] Various embodiments provide methods, systems, apparatus, computer program products, etc. for simulating chemical systems. Exemplary embodiments provide methods, systems, apparatus, computer program products, etc. for simulating chemical systems that are capable of simulating chemical systems, such as atoms, molecules, ions, periodic solids, periodic surface slabs, collections of one or more atoms, molecules, and / or ions, or combinations thereof, using a quantum computer through the execution of depth-reduced quantum circuits.

[0006] In various embodiments, hybrid quantum-classical computing techniques are used to accurately and efficiently model and / or simulate chemical systems. Specifically, at least one quantum circuit encoding the chemical system simulation is generated based on a transformation of a parameterized representation and / or operator representation of the chemical system from a fermion Hilbert space to a qubit Hilbert space for the qubits of a quantum computing component to be used to implement the at least one quantum circuit. The depth of the at least one quantum circuit (e.g., the number of quantum gates and / or the layers of quantum gates) is reduced based on the characterizing parameters of the chemical system and / or through a reordering of the occurrence of excitation operators in the operator representation of the chemical system. For example, in one exemplary embodiment, the at least one quantum circuit is configured to cause the implementation of excitation operators in the following order: a double excitation operator corresponding to a double excitation from a spatial orbital to a corresponding spatial orbital, any double excitation operator (e.g., corresponding to a double electron excitation other than one from a spatial orbital to a corresponding spatial orbital), and a single electron excitation operator.

[0007] In various embodiments, the arbitrary dual excitation operator is an arbitrary Jordan-Wigner encoded dual excitation operator configured to operate on four or more qubits (eg, four or more qubits).

[0008] In various embodiments, the number of excitation operators in the operator representation of the chemical system is reduced based on the symmetries of the chemical system. For example, excitations that are redundant due to the symmetries of the chemical system and / or that are disallowed based on the symmetries of the chemical system are removed from the operator representation to reduce the number of gates required to implement the resulting quantum circuit. In various embodiments, when performed by the quantum computing component of the hybrid quantum-classical computing component, a depth-reduced quantum circuit is determined and / or generated that determines the wave function of the chemical system, how the chemical system behaves in one or more interactions (e.g., with other chemical systems and / or electromagnetic radiation), structural properties of the chemical system, etc.

[0009] According to a first aspect, a hybrid quantum-classical computing system configured to perform a simulation of a chemical system is provided. The hybrid quantum-classical computing system includes a classical computing component coupled to a quantum computing component. The hybrid quantum-classical computing system is configured to determine characterization parameters describing the chemical system, the characterization parameters being based at least in part on a chemical structure of the chemical system; generate an operator representation of the chemical system including single-electron excitation operators and double-electron excitation operators; and reconfigure the order of appearance of the single-electron excitation operators and double-electron excitation operators in the operator representation in response to the characterization parameters. The simulation can be performed using a smaller number of two-qubit quantum gates used in at least one quantum circuit executable on the quantum computing component used to perform the simulation.

[0010] In an exemplary embodiment, the hybrid quantum-classical computing system is configured to rearrange the order of occurrence of single and double electronic excitation operators into (i) double excitation operators corresponding to double excitations from a spatial orbital to a corresponding spatial orbital, (ii) any double excitation operator, and (iii) any single excitation operator.

[0011] In an exemplary embodiment, the hybrid quantum-classical computing system is configured to: determine characterization parameters describing the chemical system, apply symmetry filtering to operator representations of the chemical system to evaluate its structural symmetry; generate a quantum state synthesis of the chemical system based on the symmetry; generate a spatial excitation synthesis of the chemical system from the quantum state synthesis; process the spatial excitation synthesis by converting from spatial orbitals to spin orbitals therein; introduce spin orbitals onto at least one quantum circuit; and generate a corresponding operator representation by using commuting sets of double excitation operators and single excitation operators to synthesize at least one quantum circuit, such that double excitation operators appear in the at least one quantum circuit before single excitation operators appear in the at least one quantum circuit, and each commuting set corresponds to an excitation of the chemical system.

[0012] In one exemplary embodiment, a hybrid quantum chemical computing system is configured to reduce the number of quantum gates required in quantum circuit synthesis depending on the structural symmetry of the chemical system.

[0013] In an exemplary embodiment, the hybrid quantum chemistry computational system is configured to apply symmetry filtering to the operator representation, which comprises at least one of defining a set of symmetries of the chemical system and identifying excitation operators of the operator representation of the chemical system that do not commute with one or more symmetries of the set of symmetries, or defining an Abelian point group of the chemical system and identifying redundant excitations of the chemical system based on the Abelian point group.

[0014] In an exemplary embodiment, generating a quantum state composition of a chemical system comprises generating a reference state (dual occupancy of spatial trajectories) based on dual occupancy spatial trajectories and virtual spatial trajectories of the chemical system.

[0015] In certain exemplary embodiments, spatial excitation synthesis comprises converting spatial inter-orbital operators into dual-excitation spatial inter-qubit operators.

[0016] In an exemplary embodiment, generating a quantum circuit composition comprises converting a dual excitation operator of a chemical system corresponding to a dual excitation other than an inter-spatial orbital dual excitation (e.g., any dual excitation operator) into any dual excitation qubit operator.

[0017] In an exemplary embodiment, the hybrid quantum-classical computing entity is configured to compile the quantum circuit composition into an executable quantum circuit, the quantum circuit composition comprising, in order, a double-excitation spatial inter-qubit operator, an optional double-excitation qubit operator, and a single-excitation qubit operator.

[0018] In an exemplary embodiment, the hybrid quantum-classical computing system is further configured to cause the quantum computing component to execute at least one quantum circuit to determine at least one of a wave function of the chemical system, a structural property of the chemical system, a chemical interaction property of the chemical system, or a reaction property of the chemical system.

[0019] In an exemplary embodiment, the hybrid quantum-classical computing system is further configured to cause the classical computing component to at least one of: (a) display a graphical representation of at least a portion of the simulation of the chemical system; or (b) generate and store in classical memory a file comprising one or more parameters of the simulation of the chemical system.

[0020] In one exemplary embodiment, the operator representation is generated using unitary coupled cluster singles and doubles (UCCSD) Ansatz.

[0021] According to another aspect, a method is provided for using a hybrid quantum-classical computing system to perform a simulation of a chemical system. The hybrid quantum-classical computing system includes a classical computing component coupled to a quantum computing component. In an exemplary embodiment, the method includes determining characterization parameters describing the chemical system, the characterization parameters being based at least in part on a chemical structure of the chemical system; generating an operator representation of the chemical system, the operator representation including single-electron excitation operators and double-electron excitation operators; and reconfiguring the order of appearance of the single-electron excitation operators and double-electron excitation operators in the operator representation in response to the characterization parameters. The simulation can be performed using a smaller number of two-qubit quantum gates used in at least one quantum circuit executable on the quantum computing component used to perform the simulation.

[0022] In an exemplary embodiment, the order of occurrence of the single and double electronic excitation operators is rearranged to: (i) a double excitation operator corresponding to a double excitation from a spatial orbital to a corresponding spatial orbital; (ii) an arbitrary double excitation operator; and (iii) an arbitrary single excitation operator.

[0023] In an exemplary embodiment, the system is configured to determine characterization parameters describing the chemical system, apply symmetry filtering to an operator representation of the chemical system to evaluate its structural symmetry, generate a quantum state synthesis of the chemical system based on the symmetry, generate a spatial excitation synthesis of the chemical system from the quantum state synthesis, process the spatial excitation synthesis by converting from spatial orbitals to spin orbitals therein, introduce spin orbitals onto at least one quantum circuit, and generate a corresponding operator representation by using exchange sets of double excitation operators and single excitation operators to synthesize at least one quantum circuit such that double excitation operators appear in the at least one quantum circuit before single excitation operators appear in the at least one quantum circuit, and each exchange set corresponds to an excitation of the chemical system.

[0024] In one exemplary embodiment, a hybrid quantum chemical computing system is configured to reduce the number of quantum gates required in quantum circuit synthesis depending on the structural symmetry of the chemical system.

[0025] In an exemplary embodiment, applying symmetry filtering to the operator representation comprises at least one of defining a set of symmetries of the chemical system and identifying excitation operators of the operator representation of the chemical system that do not commute with one or more symmetries of the set of symmetries, or defining an Abelian point group of the chemical system and identifying redundant excitations of the chemical system based on the Abelian point group.

[0026] In an exemplary embodiment, generating a quantum state composition of a chemical system comprises generating a reference state (dual occupancy of spatial trajectories) based on dual occupancy spatial trajectories and virtual spatial trajectories of the chemical system.

[0027] In certain exemplary embodiments, spatial excitation synthesis comprises converting spatial inter-orbital operators into dual-excitation spatial inter-qubit operators.

[0028] In an exemplary embodiment, generating a quantum circuit composition comprises converting a dual excitation operator of a chemical system corresponding to a dual excitation other than an inter-spatial orbital dual excitation (e.g., any dual excitation operator) into any dual excitation qubit operator.

[0029] In an exemplary embodiment, the method comprises compiling the quantum circuit composition into an executable quantum circuit, the quantum circuit composition further comprising, in order, a double-excitation spatial inter-qubit operator, an optional double-excitation qubit operator, and a single-excitation qubit operator.

[0030] In an exemplary embodiment, the method further comprises causing the quantum computing component to execute at least one quantum circuit to determine at least one of a wave function of the chemical system, a structural property of the chemical system, a chemical interaction property of the chemical system, or a reaction property of the chemical system.

[0031] In an exemplary embodiment, the method further comprises causing the classical computation component to at least one of: (a) display a graphical representation of at least a portion of the simulation of the chemical system; or (b) generate and store in the classical memory a file comprising one or more parameters of the simulation of the chemical system.

[0032] In one exemplary embodiment, the operator representation is generated using unitary coupled cluster singles and doubles (UCCSD) Ansatz.

[0033] According to another aspect, a computer program product is provided. In an exemplary embodiment, the computer program product comprises at least one non-transitory computer-readable medium storing executable instructions. The executable instructions, when executed by a hybrid quantum-classical computing system, are configured to cause the hybrid quantum-classical computing system to: determine characterizing parameters describing a chemical system, the characterizing parameters being based at least in part on a chemical structure of the chemical system; generate an operator representation of the chemical system, the operator representation including singly excited operators and doubly excited operators; and reconfigure an order of appearance of the singly excited operators and doubly excited operators in the operator representation as a function of the characterizing parameters; wherein the simulation is executable using a smaller number of two-qubit quantum gates used in at least one quantum circuit executable on a quantum computing component used to perform the simulation.

[0034] In an exemplary embodiment, the order of occurrence of the single and double excitation operators is rearranged to: (i) double excitation operators corresponding to excitation from one spatial orbital to a corresponding spatial orbital; (ii) any double excitation operator; and (iii) any single excitation operator.

[0035] In an exemplary embodiment, the executable instructions, when executed by the hybrid quantum-classical computing system, are further configured to cause the hybrid quantum-classical computing system to determine characterization parameters describing the chemical system, apply symmetry filtering to operator representations of the chemical system to assess its structural symmetry, generate a quantum state synthesis of the chemical system based on the symmetry, generate a spatial excitation synthesis of the chemical system from the quantum state synthesis, process the spatial excitation synthesis by converting from spatial orbitals to spin orbitals therein, generate a quantum circuit synthesis by exchanging a set of double-electron excitation operators from the processed spatial excitation synthesis, and generate a corresponding operator representation by modifying the quantum circuit synthesis by exchanging the set of single excitation operators such that double-excitation operators appear in the quantum circuit before single-excitation operators appear in the quantum circuit synthesis.

[0036] In an exemplary embodiment, the executable instructions, when executed by the hybrid quantum-classical computing system, are further configured to cause the hybrid quantum-classical computing system to reduce the number of quantum gates required in quantum circuit synthesis as a function of the structural symmetry of the chemical system.

[0037] In an exemplary embodiment, applying symmetry filtering to the operator representation comprises at least one of defining a set of symmetries of the chemical system and identifying excitation operators of the operator representation of the chemical system that do not commute with one or more symmetries of the set of symmetries, or defining an Abelian point group of the chemical system and identifying redundant excitations of the chemical system based on the Abelian point group.

[0038] In an exemplary embodiment, generating a quantum state composition of a chemical system comprises generating a reference state (dual occupancy of spatial trajectories) based on dual occupancy spatial trajectories and virtual spatial trajectories of the chemical system.

[0039] In certain exemplary embodiments, spatial excitation synthesis comprises converting spatial inter-orbital operators into dual-excitation spatial inter-qubit operators.

[0040] In an exemplary embodiment, generating a quantum circuit composition comprises converting a dual excitation operator of a chemical system corresponding to a dual excitation other than an inter-spatial orbital dual excitation (e.g., any dual excitation operator) into any dual excitation qubit operator.

[0041] In an exemplary embodiment, the executable instructions are further configured, when executed by the hybrid quantum-classical computing system, to cause the hybrid quantum-classical computing system to compile the quantum circuit composition into an executable quantum circuit, the quantum circuit composition comprising, in order, a double-excitation spatial inter-qubit operator, an optional double-excitation qubit operator, and a single-excitation qubit operator.

[0042] In an example embodiment, the executable instructions, when executed by the hybrid quantum-classical computing system, are further configured to cause the hybrid quantum-classical computing system to cause the quantum computing component to execute at least one quantum circuit to determine at least one of a wave function of the chemical system, a structural property of the chemical system, a chemical interaction property of the chemical system, or a reaction property of the chemical system.

[0043] In an example embodiment, the executable instructions, when executed by the hybrid quantum-classical computing system, are further configured to cause the hybrid quantum-classical computing system to cause the classical computing component to at least one of: (a) display a graphical representation of at least a portion of the simulation of the chemical system; or (b) generate and store in classical memory a file comprising one or more parameters of the simulation of the chemical system.

[0044] In one exemplary embodiment, the operator representation is generated using unitary coupled cluster singles and doubles (UCCSD) Ansatz.

[0045] Having described the invention in general terms, reference will now be made to the accompanying drawings, which are not necessarily drawn to scale. [Brief explanation of the drawings]

[0046] [Figure 1] FIG. 1 is a schematic diagram illustrating an exemplary hybrid quantum-classical computing system, according to an exemplary embodiment. [Figure 2] 1 is a flowchart illustrating processes, procedures, and / or operations performed by a hybrid quantum-classical computing system to provide output from a simulation of a chemical system, according to an example embodiment. [Figure 3] 1 is a flowchart illustrating processes, procedures, and / or actions performed by a hybrid quantum-classical computing system to generate quantum circuit composition for compilation to form a depth-reduced quantum circuit, in accordance with an example embodiment. [Figure 4] FIG. 1 is a schematic diagram of an example controller of a quantum component of a hybrid classical computing system configured to control the operation of one or more elements of the quantum component, according to various embodiments. [Figure 5] FIG. 1 is a schematic diagram of exemplary classical components of a hybrid quantum-classical computing system that may be used in accordance with certain exemplary embodiments. [Figure 6] 1 provides a plot showing a comparison of the number of gates in quantum circuits used to simulate various chemical systems using an exemplary embodiment of the present disclosure, the replacement set technique, and the naive technique. [Figure 7] 10 provides a set of plots illustrating a comparison of the number of gates for an exemplary embodiment of the present disclosure and a quantum circuit using the exchange set technique, and the improvement in the relative error of the ground state energy determined based on the quantum circuit. DETAILED DESCRIPTION OF THE INVENTION

[0047] The present invention will now be described more fully hereinafter with reference to the accompanying drawings, in which some, but not all, embodiments of the invention are shown. Indeed, the present invention may be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will satisfy applicable legal requirements. The term "or" (also written as " / ") is used herein in both the alternative and connective sense, unless otherwise indicated. The terms "exemplary" and "exemplary" are used as examples without denoting a level of quality. The terms "generally," "substantially," and "approximately" refer to within processing and / or manufacturing tolerances and / or within the user's measurement capabilities, unless otherwise indicated. Like numbers refer to like elements throughout.

[0048] Various embodiments provide methods, systems, apparatus, computer program products, etc. for simulating chemical systems. Exemplary embodiments provide methods, systems, apparatus, computer program products, etc. for simulating chemical systems using reduced-depth quantum circuits and / or quantum circuits that include fewer two-qubit quantum gates compared to conventionally produced quantum circuits for simulating chemical systems.

[0049] In general, quantum computers are expected to enable accurate modeling and / or simulation of chemical systems. For example, quantum computers are expected to enable accurate modeling and / or simulation of chemical systems that are too complex for traditional classical approaches. However, currently operational quantum computing components tend to contain a relatively small number of qubits (e.g., fewer than 100 qubits) and tend to be relatively noisy. As the depth (e.g., number of layers of quantum gates) and / or number of two-qubit quantum gates in a quantum circuit increases, the effect of noise on the resulting model or simulation also increases. In conventional techniques for generating quantum circuits to model and / or simulate chemical systems, the depth of the quantum circuit is O(n 4), where n is the number of spin orbitals included in the model and / or simulation. These conventional techniques lead to quantum circuits containing thousands of two-qubit quantum gates, which clearly exceed the capabilities of currently operational quantum computing components. Thus, a technical problem exists regarding how to generate quantum circuits for modeling and / or simulating chemical systems that can be executed by currently operational quantum computing components, i.e., noisy intermediate-scale quantum (NISQ) devices.

[0050] Various embodiments provide technical solutions to these technical problems. Specifically, various embodiments use the symmetry of a chemical system to reduce the number of two-qubit quantum gates required to accurately model and / or simulate the chemical system. In various embodiments, the number of two-qubit quantum gates in the resulting quantum circuit is at least ten times less than a quantum circuit for modeling and / or simulating the same chemical system generated through conventional techniques. Thus, various embodiments provide improvements in the fields of chemical system modeling and / or simulation, quantum chemistry, and / or quantum circuit generation.

[0051] In general, a quantum circuit is a sequence of ordered quantum operations (e.g., quantum operators applied to quantum data of one or more qubits through single- or two-qubit gates) performed on a set of qubits. A quantum computing component can execute the compiled quantum circuit to accomplish a quantum computation. In various scenarios, a quantum circuit includes layers (i.e., a temporal sequence) of quantum operations, with each layer including a collection of quantum operations performed at each time step of the quantum computation. Thus, the depth and / or number of layers or layer count in a quantum circuit provides an indication of the length of time required to perform a quantum computation and / or the number of quantum operations performed on individual qubits, both of which can affect the amount of noise in the result of each quantum computation.

[0052] In various embodiments, hybrid quantum-classical computing techniques are used to accurately and efficiently model and / or simulate chemical systems. In various embodiments, the wave function of the chemical system, structural properties of the chemical system, chemical interaction properties of the chemical system, and / or reaction properties of the chemical system are determined by executing a quantum circuit via a quantum computing component of a hybrid quantum-classical computing system. In various embodiments, executing the quantum circuit via the quantum computing component causes the quantum computing component to model and / or simulate the chemical system. In various embodiments, the model and / or simulation is a variational quantum eigensolver (VQE) model and / or simulation of the chemical system.

[0053] In various embodiments, generating and / or simulating a quantum circuit includes determining characterizing parameters that describe the chemical system. The characterizing parameters are determined based at least in part on the chemical structure of the chemical system. For example, in various embodiments, the characterizing parameters include nuclear information for each nucleus of the chemical system (e.g., the number of protons and neutrons present in each nucleus), the number of electrons in the chemical system, the Cartesian coordinates of the nuclei, various symmetries of the chemical system (e.g., but not limited to,

number

[0054] In various embodiments, generating and / or simulating the quantum circuit includes generating a parameterized representation of the chemical system. In various embodiments, the parameterized representation of the chemical system is a wave function of the chemical system. In various embodiments, the parameterized representation of the chemical system is determined based on an operator representation of the chemical system. For example, in certain exemplary embodiments, the parameterized representation is an eigenstate of the operator representation of the chemical system. In various embodiments, the operator representation of the chemical system is an effective Hamiltonian of at least a portion of the chemical system. In various embodiments, the operator representation is and / or includes a UCC operator (e.g., a UCCSD operator). In various embodiments, the operator representation comprises a plurality of operators configured to operate on various spatial orbitals and / or spin orbitals of the chemical system. As should be understood, an operator acting on a first spatial orbital and / or spin orbital can cause a change in the probability that the first spatial orbital and / or spin orbital and / or one or more second spatial orbitals and / or spin orbitals of the chemical system are occupied.

[0055] In various embodiments, the parameterization of the parameterized representation of the chemical system is determined based on the Ansatz to be used to represent the waveform of the chemical system and / or the operators to be combined to form an operator representation of the chemical system.

[0056] As should be understood, an operator is a function that acts on an element of one space to produce an element of the same or another space. For example, an excitation operator in the Hilbert space of electrons of a chemical system is configured to act on a representation of the electrons of the chemical system to produce one or more excitations thereof. An excitation operator that has been transformed and / or mapped into the Hilbert space of qubits of a quantum computing component acts on the quantum states of the qubits to cause the quantum states to represent, model, and / or simulate features of the chemical system (e.g., orbital occupancies).

[0057] In various embodiments, unitary coupled cluster (UCC) Ansatz are used to represent waveforms of chemical systems and / or operators of chemical systems (e.g., operators that are combined to provide operator representations). In one exemplary embodiment, the Ansatz is a UCCSD Ansatz, which is a UCC Ansatz that includes single and double excitation operators.

[0058] In various embodiments, the electron correlations of a chemical system are modeled and / or simulated using an operator representation of the chemical system (e.g., an effective Hamiltonian of at least a portion of the chemical system). In various embodiments, the basis and / or format of the operator representation is determined based on the Ansatz used. For example, in various embodiments using UCCSD Ansatz, the electron correlations are simulated by using USSCD operators. For example, in one exemplary embodiment, a parameterized representation of the chemical system is generated based on an operator representation of the chemical system (UCCSD operators or operators determined by another Ansatz).

[0059] In an exemplary embodiment, the operator expression is expressed as a product of Trotter-decomposed exponential functions. As is commonly understood in the art, Trotter decomposition of a product of exponential functions is a truncation of an infinite series used to evaluate a product of exponential functions. For example, in an exemplary embodiment where the operator expression includes the UCCSD operator, the operator expression is

number

number

number

[0060] In certain exemplary embodiments, a parameterized representation of a chemical system (e.g., a wave function of the chemical system, which in certain exemplary embodiments is determined and / or generated based on an operator representation of the chemical system) and / or an operator representation (e.g., an effective Hamiltonian of at least a portion of the chemical system) is transformed and / or mapped from the Hilbert space of the chemical system to the Hilbert space of the qubits of a quantum computing component. In various embodiments, the transformation and / or mapping of the parameterized representation and / or operator representation of the chemical system from fermionic operators to qubit operators is determined according to and / or depends on the type of qubits used by the quantum component of the hybrid quantum-classical computing system being used. For example, the quantum component may use photons, electrons, atomic nuclei, neutral atoms, ions, Josephson junctions, quantum dots, topological anions, and / or other quantum particles and / or systems as qubits. In certain exemplary embodiments, the transformation and / or mapping is performed using Jordan-Wigner coding, although various other codings are used in various other embodiments.

[0061] In embodiments using Jordan-Wigner encoding, for example, each qubit in a quantum circuit represents the electron number occupation of each spin orbital in a chemical system. For example, when a qubit is in the |1> state, the corresponding spin orbital is occupied, and when the qubit is in the |0> state, the corresponding spin orbital is unoccupied. Using Jordan-Wigner encoding, an operator expression including a UCCSD operator in one exemplary embodiment can be written as:

number

number

number

[0062] In various embodiments, generating and / or simulating a quantum circuit includes reconfiguring the order of occurrence of single-electron excitation operators and double-electron excitation operators in an operator representation of a chemical system (e.g., an effective Hamiltonian of at least a portion of the chemical system). In various embodiments, the order of occurrence of single-electron excitation operators and double-electron excitation operators in an operator representation of a chemical system (e.g., an effective Hamiltonian of at least a portion of the chemical system) is reconfigured. This reconfiguration reduces the number and / or lowers the count of two-qubit quantum gates (e.g., two-qubit gates such as controlled not gates (CX), ZZ-max gates, iSWAP gates, etc.) used in the quantum circuit to model and / or simulate the chemical system. In various embodiments, the reconfiguration of the occurrence of single-electron excitation operators and double-electron excitation operators is performed according to characterizing parameters of the chemical system. For example, knowledge of the symmetry of the chemical system, the spin orbitals and / or spatial orbitals of the chemical system, etc., is used to reconstruct the order of appearance of single-electron excitation operators and double-electron excitation operators in an operator representation of the chemical system (e.g., an effective Hamiltonian of at least a portion of the chemical system) in order to reduce the count of two-qubit quantum gates in the resulting quantum circuit.

[0063] After converting and / or mapping an operator representation of a chemical system (e.g., an effective Hamiltonian of at least a portion of the chemical system) to qubit operators and rearranging the order of occurrence of the single-electron excitation operators and double-electron excitation operators, a quantum circuit is compiled based on the operator representation. The quantum circuit is executed to model and / or simulate the chemical system using a quantum computing component. In an exemplary embodiment, a measurement operation is performed by the quantum computing component (e.g., as part of the quantum circuit) to extract a qubit representation of the chemical system.

[0064] The quantum bit representation of the chemical system is then processed to determine one or more of the wave function of the chemical system, properties indicative of how the chemical system behaves in one or more interactions (e.g., with other chemical systems and / or electromagnetic radiation), structural properties of the chemical system, etc.

[0065] As used herein, a classical computing component or computer is a computing entity that uses semiconductor-based computing techniques and hardware. A quantum computing component or computer uses the quantum states of quantum particles (called qubits) to perform computations.

[0066] Exemplary Hybrid Quantum-Classical Computing System 1 provides a block diagram of an exemplary hybrid quantum-classical computing system 100, according to various embodiments. In various embodiments, hybrid quantum-classical computing system 100 comprises a classical component, such as classical computing component 110, and a quantum component, such as quantum computing component 130.

[0067] Quantum computing component 130 comprises controller 132, qubits 134, qubit manipulation elements 136, and sensors 138. Controller 132 is configured to control the operation of qubit manipulation elements 136 to cause desired manipulation (e.g., controlled quantum state evolution) of qubits 134. Controller 132 is further configured to control the operation of sensors 138 configured to monitor, measure, and / or obtain measurements corresponding to the operation of the qubit manipulation components and to obtain measurements indicative of the respective quantum states of each qubit 134.

[0068] For example, in various embodiments, qubit manipulation elements 136 comprise voltage / current sources, laser sources, magnetic field sources (e.g., electromagnets and / or permanent magnets), and / or other hardware components configured for use in confining qubits and / or manipulating the quantum states of qubits. For example, in various embodiments, sensors 138 comprise photodetectors, voltage / current sensors, temperature sensors, pressure sensors, and / or other sensors that can be used to determine the quantum states of qubits and / or monitor the operation of one or more of qubit manipulation elements 136.

[0069] In various embodiments, classical computing component 110 is in communication with controller 132 of quantum computing component 130 via one or more wired or wireless networks 120 and / or via direct wired and / or wireless communication. For example, classical computing component 110 is configured to generate and provide quantum circuits configured to model and / or simulate chemical systems to quantum computing component 130 (e.g., its controller 132), and to receive qubit representations of chemical systems provided by quantum computing component 130 (e.g., its controller 132) via one or more wired or wireless networks 120 and / or via direct wired and / or wireless communication between classical computing component 110 and quantum computing component 130. For example, quantum computing component 130 is configured to receive the quantum circuit provided by classical computing component 110 and provide a qubit representation of the chemical system for receipt by classical computing component 110 via one or more wired or wireless networks 120 and / or via direct wired and / or wireless communication between classical computing component 110 and quantum computing component 130.

[0070] In various embodiments, hybrid quantum-classical computing system 100 and / or portions thereof (e.g., classical computing component 110 and / or quantum computing component 130) are configured to use the InQuanto quantum chemistry application library and / or the tket software development kit to perform the various processes, operations, etc. described herein. As should be understood, in various other embodiments, various other quantum chemistry application libraries and / or software development kits may be used.

[0071] Exemplary Operation of a Hybrid Quantum-Classical Computing System In various embodiments, hybrid quantum-classical computing system 100 is used to simulate a chemical system. For example, in various embodiments, hybrid quantum-classical computing system 100 is used to generate a model of a chemical system that represents a wave function of the chemical system, one or more structural properties of the chemical system, one or more chemical interaction properties of the chemical system, one or more reaction properties of the chemical system, etc. In various embodiments, the model, portions thereof, and / or graphical representations thereof are displayed via a display (e.g., of classical computing component 110), stored in a file that can be used as input to and / or directly provided as input to other simulations or models that use the interactions, structural properties, and / or reaction properties of the chemical system, such as to perform one or more functions thereof.

[0072] In various embodiments, classical computing component 110 obtains information corresponding to a chemical system. For example, user input (e.g., received via a user input interface of classical computing component 110) may provide, select, and / or cause access of information corresponding to a chemical system. In one exemplary embodiment, the information corresponding to a chemical system includes the chemical formula (H, HO, NH4 +, OH, CH4, a specified transition state of a chemical reaction, etc.), and / or other designation of a chemical system. In an exemplary embodiment, the information corresponding to the chemical system includes nuclear information for each nucleus of the chemical system (e.g., the number of protons and neutrons present in each nucleus), the number of electrons in the chemical system, the Cartesian coordinates of the nuclei, and / or any other information used to define the chemical system.

[0073] In various embodiments, obtaining information corresponding to the chemical system prompts and / or causes classical computation component 110 to determine and / or generate a model of the chemical system representing the (fermionic) wave function of the chemical system, one or more structural properties of the chemical system, one or more chemical interaction properties of the chemical system, one or more reaction properties of the chemical system, etc.

[0074] For example, classical computing component 110 may determine and / or identify characterizing parameters of a chemical system, perform quantum circuit synthesis using the characterizing parameters, and provide the quantum circuit to quantum computing component 130. Quantum computing component 130 may execute the quantum circuit (e.g., using multiple qubits thereof) to determine a qubit representation of the chemical system. For example, in various embodiments, the qubit representation may provide orbital occupancy information for multiple orbitals of the chemical system. In various embodiments, the qubit representation may provide one or more reduced density matrices (RDMs) for the chemical system (e.g., spinless (i.e., spin-tracked) RDMs such as one-particle RDM (1-RDM), two-particle RDM (2-RDM), etc.). In various embodiments, the qubit representation may provide a parameterized representation (e.g., a wave function) of the chemical system. In various embodiments, the qubit representation may be processed to determine one or more RDMs and / or other properties or characteristics of the chemical system.

[0075] The classical computation component 110 may then utilize and / or process the qubit representation of the chemical system to determine a wave function and / or structural, interaction, and / or other properties of the chemical system. For example, the classical computation entity utilizes and / or processes the qubit representation of the chemical system to determine approximations to properties of eigenstates of the system's overall electronic Hamiltonian, such as expectation values ​​of quantum operators acting on such states, to complete a simulation of the chemical system, to determine how the chemical system behaves in one or more interactions (e.g., with other chemical systems and / or electromagnetic radiation), to determine structural properties of the chemical system, etc.

[0076] FIG. 2 provides a flowchart of various processes, procedures, actions, etc., performed by hybrid quantum-classical computing system 100 in various embodiments. Beginning at step / action 202 of FIG. 2, classical computing component 110 determines characterization parameters describing a chemical system. For example, classical computing component 110 may receive information (e.g., through user input) that identifies and / or corresponds to a chemical system. For example, classical computing component 110 may determine the chemical formula (H, HO, NH4 + , OH, CH4, a specified transition state of a chemical reaction, etc.), and / or other designation of the chemical system. In an exemplary embodiment, the information corresponding to the chemical system includes nuclear information for each nucleus of the chemical system (e.g., the number of protons and neutrons present in each nucleus), the number of electrons in the chemical system, the Cartesian coordinates of the nuclei, and / or any other information used to define the chemical system.

[0077] Based on the information identifying and / or corresponding to the chemical system, the classical computation component 110 determines characterizing parameters of the chemical system. In various embodiments, the characterizing parameters are determined based at least in part on the chemical structure of the chemical system. In various embodiments, the characterizing parameters are determined using one or more of a lookup table relating to the structure of one or more chemical systems, processing information corresponding to the geometric arrangement of the chemical system (e.g., Cartesian coordinates of nuclei), determining one or more orbitals of the chemical system using Hartree-Fock and / or perturbation theory techniques, etc.

[0078] For example, in various embodiments, the characterizing parameters include nuclear information for each nucleus of the chemical system (e.g., the number of protons and neutrons present in each nucleus), the number of electrons in the chemical system, the Cartesian coordinates of the nuclei, various symmetries of the chemical system (e.g., but not limited to,

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[0079] In step / act 204, classical computation component 110 generates a parameterized and / or operator representation of the chemical system. For example, an Ansatz is identified and / or selected (e.g., UCCSD and / or another suitable Ansatz), and a corresponding operator representation describing the electron correlations of the chemical system is determined based on the identified and / or selected Ansatz. In various embodiments, the Ansatz is identified based on user input (e.g., received via a user input device), selected based on a default Ansatz, etc.

[0080] In various embodiments, the operator representation of the chemical system comprises operators that describe electron correlations of the chemical system. For example, in various embodiments, the operator representation of the chemical system includes operators that include sums over each excitation operator acting over all of the orbitals of the chemical system. For example, the operator representation of the chemical system is an integrated operator formed by summing over each excitation operator acting over all of the spin orbitals and / or spatial orbitals of the chemical system. In various embodiments, the excitation operators included in the operator representation of the chemical system are determined based at least in part on one or more of the spin multiplicity of the chemical system, the number of electrons in the chemical system, and the basis set size of the chemical system.

[0081] In various embodiments, the parameterized representation of the chemical system is determined based on an operator representation of the chemical system. For example, in various embodiments, the parameterized representation is a wave function of the chemical system and the operator representation is an effective Hamiltonian of at least a portion of the chemical system. In various embodiments, the parameterized representation of the chemical system and / or the parameterization of the operator representation is determined based on an Ansatz to be used to represent the waveform of the chemical system and / or the operator of the chemical system. In certain exemplary embodiments, the Ansatz used to represent the waveform of the chemical system and / or the operator of the chemical system is a Unitary Coupled Cluster (UCCSD) Ansatz containing single-electron excitation operators and double-electron excitation operators.

[0082] In various embodiments, the electronic correlations of a chemical system are modeled and / or simulated using an operator representation of the chemical system that is and / or includes an Ansatz-determined operator. For example, in various embodiments that use a UCCSD Ansatz, the electronic correlations are simulated by using a UCCSD operator. For example, in one exemplary embodiment, the operator representation of the chemical system is a UCCSD operator or another Ansatz-determined operator.

[0083] In one exemplary embodiment, the operator representation includes a UCCSD operator expressed as a product of Trotter-decomposed exponential functions. As commonly understood in the art, the Trotter decomposition of a product of exponential functions is a truncation of an infinite series used to evaluate a product of exponential functions. For example, in one exemplary embodiment, the UCCSD operator is

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[0084] In certain exemplary embodiments, operator representations of the chemical system (e.g., UCCSD operators in certain exemplary embodiments) are transformed and / or mapped from the Hilbert space of the chemical system to the Hilbert space of the qubits of the quantum computing component. In various embodiments, the transformation and / or mapping of the operator representation of the chemical system from fermion operators to qubit operators is determined by and / or depends on the type of qubits used by the quantum component of the hybrid quantum-classical computing system being used. For example, the quantum component may use protons, electrons, atomic nuclei, neutral atoms, ions, Josephson junctions, quantum dots, topological anions, and / or other quantum particles and / or systems as qubits. In certain exemplary embodiments, the transformation and / or mapping is performed using Jordan-Wigner encoding, although various other encodings are used in various other embodiments.

[0085] In embodiments using Jordan-Wigner encoding, for example, each qubit in a quantum circuit represents the electron number occupation of each spin orbital in a chemical system. For example, when a qubit is in the |1> state, the corresponding spin orbital is occupied, and when the qubit is in the |0> state, the corresponding spin orbital is unoccupied. Using Jordan-Wigner encoding, operator representations (e.g., including UCCSD operators or other Ansatz-determined operators) can be expressed as:

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[0086] In step / operation 206, classical computation component 110 reconfigures the order of occurrence of single-electron excitation operators and double-electron excitation operators in the operator representation of the chemical system. This reconfiguration reduces the number and / or count of two-qubit quantum gates (e.g., two-qubit gates) used in the quantum circuit to model and / or simulate the chemical system. In various embodiments, the reconfiguration of the occurrence of single-electron excitation operators and double-electron excitation operators is performed as a function of characterizing parameters of the chemical system. For example, knowledge of the symmetry of the chemical system, the spin orbitals and / or spatial orbitals of the chemical system, etc. is used to reconfigure the order of occurrence of single-electron excitation operators and double-electron excitation operators in the operator representation of the chemical system to reduce the count of two-qubit quantum gates in the resulting quantum circuit.

[0087] In various embodiments, the rearrangement of the order of occurrence of single-electron excitation operators and double-electron excitation operators in the operator representation of the chemical system is performed based at least in part on characterizing parameters of the chemical system. For example, the rearrangement of the occurrence of single-electron excitation operators and double-electron excitation operators in the operator representation of the chemical system is performed based on filtering of the excitation operators based on one or more symmetries of the chemical system, based on the type of excitation (inter-spatial orbital double excitation, any double excitation, any single excitation), through the definition of a reference state (double occupancy of spatial orbitals), through transforming at least a portion of the spatial orbitals into spin orbitals, through commuting sets of electronic excitation operators, etc.

[0088] In various embodiments, reconstructing the order of occurrence of the single-electron excitation operators and the double-electron excitation operators in the operator representation of the chemical system includes applying symmetry filtering to the waveform representation of the chemical system to assess its structural symmetry.

[0089] For example, in one exemplary embodiment, the classical computation component 110 applies symmetry filtering by defining a set of symmetries for the chemical system and identifying one or more excitation operators in the operator representation of the chemical system that do not commute with one or more symmetries in the symmetry set. One or more excitation operators that do not commute with one or more symmetries in the symmetry set are removed from the operator representation of the chemical system. In one exemplary embodiment, the symmetry set is

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[0090] In an exemplary embodiment, the classical computation component 110 applies symmetry filtering by defining an Abelian point group for the chemical system and identifying redundant excitations of the chemical system based on the Abelian point group. For example, in an exemplary embodiment, the largest and / or highest Abelian point group for the chemical system is defined and used to identify redundant excitations of the chemical system. Excitation operators corresponding to excitations of the chemical system identified as being redundant excitations are removed from the operator representation of the chemical system.

[0091] Conventionally, in a Trotter decomposed operator representation (e.g., in one exemplary embodiment, in the UCCSD operator thereof), a single-electron excitation operator is applied before the application of a double-electron excitation operator. According to various embodiments, the order of appearance of excitation operators in an operator representation of a chemical system is such that a double-electron excitation operator, corresponding to an electron pair being excited from one spatial orbital to another, is applied before the application of a double-electron excitation operator.

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[0092] For example, in various embodiments, the classical computation component 110 may be configured such that in an embodiment using UCCSD Ansatz the effective Hamiltonian is

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[0093] In various embodiments, rearranging the order of occurrence of the single-electron excitation operators and the double-electron excitation operators in the operator representation of the chemical system includes generating a quantum state synthesis of the chemical system based on the symmetry of the chemical system. In various embodiments, generating the quantum state synthesis comprises generating a double occupancy of the spatial orbital reference state by processing a reference HF state of the chemical system to extract spatial orbital occupancies. In an exemplary embodiment, only doubly occupied spatial orbitals and virtual spatial orbitals are included in the double occupancy of the spatial orbital reference state. For example, an HF state |110000> (where the first two 1s indicate a pair of occupied spin orbitals corresponding to the same spatial orbital and the four 0s correspond to two pairs of unoccupied virtual spin orbitals of the chemical system) that defines a spin-orbital occupancy is processed to provide a double occupancy of the spatial orbital reference state |100>, indicating that the first spatial orbital is doubly occupied and the second and third spatial orbitals are unoccupied. In an exemplary embodiment, these spatial orbital occupancies are mapped to even-indexed qubits of a quantum circuit by applying a Pauli-X gate to provide the qubit state |100000>. In one exemplary embodiment, quantum state synthesis causes qubits (e.g., qubits with even indices) to encode the (dual) occupation of the spatial orbitals of a chemical system, rather than the occupation of the spin orbitals of the chemical system.

[0094] In various embodiments, rearranging the order of appearance of the single-electron excitation operator and the double-electron excitation operator in the operator representation of the chemical system includes generating a spatial excitation composition of the chemical system from a quantum state composition. In various embodiments, the double-electron inter-spatial orbital operator performs an operation to excite a pair of electrons from a first spatial orbital to a second spatial orbital. Because the electron pair is excited together (e.g., as a pair), the electron pair may be approximated as a hard core boson. In other words, a double-electron excitation operator corresponding to an electron pair being excited from one spatial orbital to another spatial orbital:

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[0095] In various embodiments, rearranging the order of appearance of the singly and doubly excited operators in the operator representation of the chemical system involves modifying the representation so that the qubits of the quantum circuit convey spin-orbital occupancy information. For example, qubits corresponding to spin orbitals that are part of a doubly occupied spatial orbital are each made to indicate that the spin orbital is occupied. qubits corresponding to spin orbitals that are part of an unoccupied spatial orbital are made to indicate that the spin orbital is unoccupied. qubits corresponding to spin orbitals that are part of a singly occupied spatial orbital can be switched to indicate the appropriate single occupancy of the respective spatial orbital, as appropriate.

[0096] In various embodiments, rearranging the order of occurrence of the single-electron excitation operators and the double-electron excitation operators in the operator representation of the chemical system includes generating a quantum circuit synthesis by using exchange sets of operators. In various embodiments, each exchange set of operators represents an excitation. The operators in the exchange set of operators are configured to operate on and / or in the qubit Hilbert space of a quantum computing component to be used to implement the resulting at least one quantum circuit. In various embodiments, the exchange sets are generated and / or determined using tket. For example, in various embodiments, the exchange set technique may resemble and / or use the framework disclosed by U.S. Patent No. 11,144,689, issued October 12, 2021, the entire contents of which are incorporated herein by reference.

[0097] For example, the quantum circuit synthesis is generated in part by exchanging a set of double-electron excitation operators from the processed spatial excitation synthesis. For example, the spatial excitation synthesis may contain double-electron excitation operators corresponding to electron pairs being excited from one spatial orbital to another.

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[0098] For example, in one exemplary embodiment, any doubly excited operator

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[0099] In various embodiments, rearranging the order of appearance of the singly excited operators and the doubly excited operators in the operator representation of the chemical system includes modifying the quantum circuit synthesis by a swap set of the singly excited operators such that the doubly excited operators appear in the quantum circuit synthesis before the single excited operators appear in the quantum circuit synthesis. For example, in various embodiments, any singly excited operator

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[0100] In various embodiments, rearranging the order of appearance of the single-electron excitation operators and the double-electron excitation operators in the operator representation of the chemical system results in a quantum circuit that, when executed by quantum computing component 130, causes quantum computing component 130 to simulate the chemical system. Moreover, the resulting quantum circuit includes a lower count of two-qubit quantum gates, such as, for example, controlled-NOT (CX) gates, than traditional quantum circuit synthesis techniques.

[0101] In step / act 208, classical computing component 110 or controller 132 of quantum computing component 130 compiles the quantum circuit to generate a compiled and / or executable quantum circuit that is executable by quantum computing component 130. For example, the compiled and / or executable quantum circuit comprises a set of commands executable by controller 132 to cause controller 132 to control various components of quantum computing component 130 to cause quantum computing component 130 to execute the quantum circuit, etc. In one exemplary embodiment, the quantum circuit is compiled using tket, although various other quantum circuit compilers may be used in various other embodiments.

[0102] In step / operation 210, hybrid quantum-classical computing system 100 executes the compiled and / or executable quantum circuit. For example, controller 132 controls various components of quantum computing component 130 (e.g., qubit manipulation elements 136 and / or sensors 138) to evolve the quantum states of qubits 134 in a controlled manner such that measurements obtained as part of the execution of the compiled and / or executable quantum circuit provide a qubit representation of the chemical system.

[0103] In various embodiments, executing the compiled and / or executable quantum circuit comprises measuring the quantum states of one or more qubits to determine a qubit representation of the chemical system. In various embodiments, controller 132 and / or classical computation component 110 process the qubit representation of the chemical system to determine at least one of a wave function of the chemical system, a structural property of the chemical system, a chemical interaction property of the chemical system, or a reaction property of the chemical system. For example, controller 132 and / or classical computation component 110 perform post-processing of a model and / or simulation of the chemical system encoded by the qubit representation of the chemical system to determine at least one of a wave function of the chemical system, a structural property of the chemical system, a chemical interaction property of the chemical system, or a reaction property of the chemical system. In various embodiments, the format or structure of the wave function corresponds to an Ansatz used in generating the parameterized and / or operator representation of the chemical system.

[0104] In various embodiments, structural properties of a chemical system characterize and / or provide information about the structure of the chemical system. For example, structural properties can indicate the shape of the chemical system, the order and / or spatial distribution of constituent atoms or groups of atoms in the chemical system, the properties of various bonds in the chemical system, the relative positions of nuclei in the chemical system, etc.

[0105] In various embodiments, the chemical interaction properties of a chemical system characterize and / or provide information about how the chemical system interacts with one or more other chemical systems of the same or different type. For example, the chemical interaction properties of a chemical system may provide information about how the chemical system interacts with other chemical systems of the same type / formula as the chemical system or of a different type / formula than the chemical system.

[0106] For example, the structural and / or chemical interaction properties may be dissociation curves, bond energies, reaction energies, reaction barriers, binding energies, absorption energies, and the like.

[0107] In various embodiments, a reaction property of a chemical system characterizes and / or provides information about how the chemical system interacts with electromagnetic radiation. For example, the reaction property of a chemical system can be an excitation energy, a singlet-triplet gap, a photodissociation energy, a photoionization energy, an absorption cross section, a dielectric constant, a dielectric function, an oscillator strength, etc.

[0108] In step / act 212, classical computation component 110 provides at least a portion of a model and / or simulation of the chemical system, representing wave functions, structural properties, chemical interaction properties, and / or reaction properties of the chemical system. In various embodiments, providing at least a portion of the model and / or simulation of the chemical system comprises displaying, storing, transmitting (e.g., over one or more wired and / or wireless networks), providing, etc., a call response (e.g., an application program interface (API) call response).

[0109] For example, in one exemplary embodiment, classical computing component 110 causes a display (e.g., display 516 shown in FIG. 5 and / or another display) to display a representation of a model and / or simulation of a chemical system. For example, a graphics processing unit (GPU) of classical computing component 110 may generate a graphical representation of at least a portion of the model and / or simulation of the chemical system that provides, for example, a visualization of wave functions, structural properties, chemical interaction properties, and / or reaction properties of the chemical system. Classical computing component 110 may then cause the graphical representation of at least a portion of the model and / or simulation of the chemical system to be displayed via the display for review and / or viewing by a human user.

[0110] In another example, the classical computing component 110 may generate and store (e.g., in memory 522, 524) a file comprising at least a portion of a model and / or simulation of a chemical system. For example, the file may comprise the chemical system's wave function, structural properties, chemical interaction properties, reaction properties, chemical formula, etc. The file may then be provided to one or more programs, applications, modules, etc. running on the classical computing component 110 or another computing entity as input for one or more functions and / or calculations performed thereby. For example, a file storing and / or encoding at least a portion of a model and / or simulation of a chemical system may be used by various programs, applications, modules, etc. to generate a graphical representation and / or visualization of the chemical system and / or portions thereof, perform a simulation including the interaction of the chemical system with one or more other chemical systems (of the same or different chemical formula), perform a simulation of a bulk material including the chemical system, perform a simulation including the interaction of the chemical system with one or more biological systems, perform a simulation to determine the optical spectrum of the chemical system, etc.

[0111] 3 provides a flowchart illustrating various processes, procedures, actions, etc., performed by hybrid quantum-classical computing system 100 to reconstruct the order of appearance of single-electron excitation operators and double-electron excitation operators in an operator representation of a chemical system. For example, in an exemplary embodiment, the steps / actions of the flowchart illustrated in FIG. 3 are performed as part of step / action 208. As described elsewhere herein, the operator representation of a chemical system includes an effective Hamiltonian for the chemical system (or for at least a portion of the electrons of the chemical system). The order in which the operators comprising the effective Hamiltonian are ordered and / or expressed (e.g., within a quantum circuit) affects the number of two-qubit quantum gates required to model and / or simulate the chemical system via a quantum circuit to be executed by quantum computing component 130.

[0112] In various embodiments, symmetry filtering of excitation operators and / or treating double electron inter-spatial orbital excitations as bosonic excitations and corresponding qubit reference frame transformations enable modeling and / or simulation of chemical systems via quantum circuits that require significantly (e.g., more than a factor of 10) fewer and / or fewer two-qubit quantum gates (such as CX gates) compared to conventional quantum circuits for modeling and / or simulating chemical systems.

[0113] Beginning at step / action 302, classical computation component 110 applies symmetry filtering. Symmetry filtering removes and / or excludes excitation operators from the operator representation of the chemical system that are impermissible and / or redundant based on the symmetries of the chemical system. This reduces the overall number of excitation operators present in the operator representation of the chemical system.

[0114] For example, in various embodiments, applying symmetry filtering comprises filtering a set of excitations present in an operator representation of the chemical system using symmetries (e.g., molecular symmetries, etc.) of the chemical system to identify forbidden terms in the operator representation of the chemical system. In various embodiments, a first symmetry filtering is performed. The first symmetry filtering is performed using a set of symmetries (e.g., molecular symmetries, etc.) of the chemical system to identify forbidden terms in the operator representation of the chemical system.

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[0115] In various embodiments, symmetry filtering is performed and / or applied by identifying and / or defining the symmetries of the chemical system. In various embodiments, the symmetries of the chemical system are identified and / or defined based at least in part on characterizing parameters of the chemical system. For example, the geometry of the chemical system, as indicated by the characterizing parameters of the chemical system, may be processed to identify and / or determine the symmetries of the chemical system.

[0116] For example, a symmetry set and / or point group having the same symmetry as the chemical system is identified. The identified symmetry set and / or point group corresponds to the degree of symmetry of the chemical system. Based on the identified symmetry set and / or point group, excitation operators that perform and / or correspond to excitations that are disallowed and / or redundant are removed from the excitation operators present in the operator representation of the chemical system. In other words, excitation operators that perform and / or correspond to excitations that are disallowed and / or redundant are removed from the operator representation of the chemical system, thereby reducing the number of operations included therein.

[0117] For example, in one exemplary embodiment, the classical computation component 110 applies a first symmetry filtering by defining a set of symmetries for the chemical system and identifying one or more excitation operators of the operator representation of the chemical system that do not commute with one or more symmetries in the set of symmetries. The one or more excitation operators that do not commute with one or more symmetries in the set of symmetries perform and / or correspond to impermissible excitations. Thus, the one or more excitation operators that do not commute with one or more symmetries in the set of symmetries are removed from the operator representation of the chemical system. In one exemplary embodiment, the set of symmetries for the chemical system

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[0118] In an exemplary embodiment, the classical computation component 110 applies the second symmetry filtering by defining an Abelian point group of the chemical system and identifying redundant excitations of the chemical system based on the Abelian point group. For example, in an exemplary embodiment, the largest and / or highest Abelian point group of the chemical system is defined and used to identify redundant excitations of the chemical system. Excitation operators corresponding to excitations of the chemical system identified as being redundant excitations are removed from the operator representation of the chemical system. For example, in various embodiments, the second symmetry filtering may be performed similarly to the scheme described in "Use of molecular symmetry in coupled-cluster theory" by Petr Carsky et al.

[0119] In various embodiments, the symmetry filtering includes only one of (a) defining a set of symmetries for the chemical system and identifying disallowed excitations based thereon, or (b) defining a maximal Abelian group for the chemical system and identifying redundant excitations based thereon. In various embodiments, the symmetry filtering includes both (a) defining a set of symmetries for the chemical system and identifying disallowed excitations based thereon, and (b) defining a maximal Abelian group for the chemical system and identifying redundant excitations based thereon.

[0120] After performing and / or applying symmetry filtering, the remaining excitation operators of the operator representation of the chemical system are processed to generate and / or synchronize a quantum circuit for modeling and / or simulating the chemical system, for example, the quantum circuit includes preparing a set of qubits to encode the spin-orbital occupation of the chemical system through application of an effective Hamiltonian for the chemical system (or an effective Hamiltonian for at least a portion of the electrons in the chemical system) through transformation of the effective Hamiltonian into an effective qubit operator via a quantum gate performed on the corresponding qubit.

[0121] In various embodiments, a quantum circuit is synthesized via steps / acts 304-312 such that qubit operators that are transformed and / or mapped representations and / or versions of double-electron inter-orbital excitation operators appear in the quantum circuit before qubit operators that are transformed and / or mapped representations and / or versions of any double-electron excitation operators, and such that qubit operators that are transformed and / or mapped representations and / or versions of any double-electron excitation operators appear in the quantum circuit before qubit operators that are transformed and / or mapped representations and / or versions of any single-electron excitation operators. In one exemplary embodiment, at least a portion of steps / acts 304-312 are performed by classical computation component 110 using tket or another quantum circuit generation and / or compilation program.

[0122] In step / operation 304, the classical computation component 110 generates the quantum state synthesis portion of the quantum circuit. In various embodiments, generating the quantum state synthesis comprises generating a (dual occupancy of spatial orbitals) reference state by processing a reference HF state of the chemical system to extract and / or determine the occupancy of the reference spatial orbitals. For example, in one exemplary embodiment, the reference HF state of the chemical system is provided by characterization parameters of the chemical system. The reference HF state is processed and / or analyzed to identify doubly occupied spatial orbitals (e.g., both spin orbitals of the spatial orbital are occupied) and / or unoccupied virtual spatial orbitals. In one exemplary embodiment, only doubly occupied spatial orbitals and virtual spatial orbitals are included in the (dual occupancy of spatial orbitals) reference state.

[0123] An initial portion of a quantum circuit is generated that initializes a set of qubits. Each qubit is indexed by an index p. In various embodiments, the qubits are indexed by p=2q and p=2q+1, where q is an integer corresponding to the two spin orbitals of each spatial orbital q. The initial portion of the quantum circuit is generated to include operations configured to, for a qubit corresponding to a doubly occupied spatial orbital q, initialize the corresponding qubit indexed by p=2q to state |1>, indicating doubly occupied the corresponding spatial orbital, and to initialize all other qubits to state |0>. For example, the initial portion of the quantum circuit may include initializing operations that cause the states of the qubits to be initialized based on the doubly occupied spatial orbitals as indicated by the reference HF state.

[0124] For example, an HF state |110000> (where the first two 1s indicate a pair of occupied spin orbitals corresponding to the same spatial orbital and the four 0s correspond to two pairs of unoccupied virtual spin orbitals of the chemical system) that defines a spin-orbital occupation is processed to provide a dual occupation of the spatial orbital reference state |100>, indicating that the first spatial orbital is dual occupied and the second and third spatial orbitals are unoccupied. In an exemplary embodiment, these spatial orbital occupations are mapped to even-indexed qubits (indexed by p=2q) of a quantum circuit by applying a Pauli-X gate to provide the qubit state |100000>. In an exemplary embodiment, quantum state synthesis causes qubits (e.g., even-indexed qubits) to encode the (dual) occupation of the spatial orbitals of the chemical system, rather than the occupation of the spin orbitals of the chemical system.

[0125] In step / operation 306, the classical computation component 110 generates the spatial excitation synthesis portion of the quantum circuit. In various embodiments, the operator that excites a pair of electrons from a first spatial orbital to a second spatial orbital is transformed from a fermionic operator acting on individual spin orbitals to a bosonic operator acting on spatial orbitals (e.g., corresponding to a pair of spin orbitals). The electron pair is excited together (e.g., as a pair), and the electron pair may be approximated as a hard-core boson (e.g., as an entity with integer spin). In other words, the double-electron excitation operator corresponding to the electron pair being excited from one spatial orbital to another spatial orbital is

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[0126] Therefore, the double electron space inter-orbital excitation operators in the operator representation (e.g., effective Hamiltonian) of a chemical system

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[0127] In step / operation 308, classical computation component 110 processes spatial excitation synthesis. An initial portion of the quantum circuit initialized qubits with even indices to represent dual occupancies of spatial orbitals. A second portion of the quantum circuit performs excitation operations corresponding to exciting electrons from dual-occupied spatial orbitals to previously unoccupied spatial orbitals. Now, classical computation component 110 generates a third portion of the quantum circuit that alters the representation of the qubits. For example, the third portion of the quantum circuit alters the semantic meaning of the qubit states so that each qubit corresponds to the occupancy of a respective spin orbital.

[0128] For example, if the state of the qubit indexed by p=2q is |1> at the end of the second part of the quantum circuit, then the corresponding spatial orbital q is doubly occupied, and the third part of the quantum circuit causes the qubit indexed by p=2q+1 to be set to state |1> but keeps the qubit indexed by p=2q in state |1> to indicate that both spin orbitals corresponding to spatial orbital q are occupied.

[0129] In reality, at the end of the second part of the quantum circuit, the state of the qubit indexed by p = 2q is a superposition of the |1> and |0> states given by a|0> + b|1>, and a 2 +b 2= 1. The third part of the quantum circuit ensures that for qubits corresponding to spatial trajectories q that were doubly occupied or unoccupied in the HF reference state, the qubit indexed by p = 2q + 1 is in the same state as the qubit indexed by p = 2q.

[0130] In step / action 310, the classical computation component 110 introduces spin orbitals on the quantum circuit. For example, the classical computation component 110 generates a fourth portion of the quantum circuit that introduces spin orbitals on the quantum circuit. The fourth portion of the quantum circuit initializes qubits corresponding to spatial orbitals that were singly occupied in the HF reference state. The initial, second, and third portions of the quantum circuit have not acted on and / or performed quantum gates on qubits corresponding to singly occupied spatial orbitals (according to the HF reference state) other than initializing those qubits to the |0> state. The fourth portion of the quantum circuit includes an operation that causes the states of qubits corresponding to singly occupied spatial orbitals in the HF reference state to be switched to the |1>0 state to indicate the occupation of those spin orbitals.

[0131] In step / act 312, classical computation component 110 uses an exchange set of double excitation operators and single excitation operators (e.g., as provided by tket in one exemplary embodiment) to synthesize a quantum circuit. For example, classical computation component 110 generates a quantum circuit synthesis that includes a fifth and sixth portion of the quantum circuit. In various embodiments, each chemical excitation corresponds to an exchange set of operators. In various embodiments, the exchange sets of operators are sets of Pauli operators that commute with each other and are transformed versions of the excitation operators from the operator representation of the chemical system.

[0132] For example, using a commutative set of operators, the classical computation component 110 can compute any doubly excited operator

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[0046] A sixth portion of the quantum circuit is generated, comprising a qubit operator corresponding to

[0133] In various embodiments, any doubly excited operator

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[0134] In various embodiments, any single electron excitation operator

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[0135] In various embodiments, classical computation component 110 generates a seventh portion of the quantum circuit corresponding to performing a measurement operation to measure the state of the qubit after performing the sixth portion of the quantum circuit. For example, the results of the measurement may form and / or be processed to form a qubit representation of the chemical system.

[0136] Thus, classical computation component 110 generates a quantum circuit comprising an initial quantum circuit portion, a second quantum circuit portion, a third quantum circuit portion, a fourth quantum circuit portion, a fifth quantum circuit portion, a sixth quantum circuit portion, and optionally a seventh quantum circuit portion, where these portions of the quantum circuit are ordered as listed herein such that qubit operations corresponding to double-electron inter-orbital excitation operators are performed before any double-electron excitation operators, and any double-electron excitation operators are performed before any single-electron excitation operators.

[0137] Technical Advantages In general, quantum computers are expected to enable accurate modeling and / or simulation of chemical systems. For example, quantum computers are expected to enable accurate modeling and / or simulation of chemical systems that are too complex for traditional classical techniques. However, currently operating quantum computing components, referred to as NISQ devices as discussed above, tend to contain a relatively small number of qubits (e.g., fewer than 100 qubits) and tend to be relatively noisy. As the depth (e.g., number of layers of quantum gates) and / or number of two-qubit quantum gates in a quantum circuit increases, the effect of noise on the resulting model or simulation also increases. In conventional techniques for generating quantum circuits for modeling and / or simulating chemical systems, the depth of the quantum circuit is O(n 4 ), where n is the number of spin orbitals included in the model and / or simulation. These lead to quantum circuits containing thousands of two-qubit quantum gates, which clearly exceeds the capabilities of currently operational quantum computing components. Thus, a technical problem exists as to how to generate quantum circuits to model and / or simulate chemical systems that can be performed by currently operational quantum computing components.

[0138] Various embodiments provide technical solutions to these technical problems. Specifically, various embodiments use the symmetry of a chemical system to reduce the number of two-qubit quantum gates required to accurately model and / or simulate the chemical system. In various embodiments, the number of two-qubit quantum gates in the resulting quantum circuit is at least ten times less than a quantum circuit for modeling and / or simulating the same chemical system generated through conventional techniques. Thus, various embodiments provide improvements in the fields of modeling and / or simulating chemical systems, quantum chemistry, and / or quantum circuit generation.

[0139] FIG. 6 shows the results of experiments on various chemical systems (e.g., OH, CH, H) for chemically-aware quantum circuit synthesis of an exemplary embodiment, conventional exchange set quantum circuit synthesis, and conventional naive quantum circuit synthesis. 2O 6 provides an illustration of the controlled-NOT (CX) gate count of quantum circuits used to model and / or simulate chemical systems (CH, CH, and the transition state (TS) of CH and OH). As can be seen in FIG. 6, the number of two-qubit quantum gates (CX gates in the illustrated example) for each of the chemical systems is significantly reduced using chemically aware quantum circuit synthesis of the illustrative embodiments as compared to traditional quantum circuit synthesis techniques. Thus, as shown in FIG. 6, various embodiments provide improvements in the fields of modeling and / or simulating chemical systems, quantum chemistry, and / or quantum circuit generation.

[0140] 7 provides a set of plots showing a comparison of the number of gates in a quantum circuit and the improvement in the relative error of the ground-state energy determined based on the quantum circuit using an exemplary embodiment of the present disclosure and the exchange set technique. Panel a of FIG. 7 shows the scaling of the two-qubit gate count for the chemical system CH4 against various activity spaces of the chemical system (e.g., the number of qubits used to represent the activity space) for one exemplary embodiment (labeled "chemically aware"), for the exchange set technique, and for individual techniques in which an operational representation of the chemical system is directly or naively converted to a quantum circuit.

[0141] Panel b of Figure 7 shows a comparison of two-qubit gate counts between an exemplary embodiment of the present disclosure (labeled "chemical recognition"), the exchange set technique, and a separately synthesized quantum circuit for a 10-qubit model. Panel c of Figure 7 shows a comparison of two-qubit gate counts between an exemplary embodiment of the present disclosure (labeled "chemical recognition"), the exchange set technique, and a separately synthesized quantum circuit for a 6-qubit model. Panel d of Figure 7 shows the improvement in relative error of ground-state energies calculated on a noisy intermediate scale quantum (NISQ)-era quantum computer using quantum circuits generated through an exemplary embodiment of the present disclosure (labeled "chemical recognition") and quantum circuits generated through a separately synthesized technique.

[0142] Exemplary Controller In various embodiments, hybrid quantum-classical computing system 100 includes quantum computing component 130. Quantum computing component 130 is configured to perform various quantum computations and / or operations through the execution of one or more quantum circuits and / or algorithms. In various embodiments, quantum computing component 130 is configured to control the operation of one or more components of quantum computing component 130 (e.g., qubit manipulation element 136, sensor 138), receive sensor signals indicative of measurements captured by sensor 138, and / or communicate with classical computing component 110. Quantum computing component 130 optionally includes, but is not limited to, trapped-ion qubits, cryogenically cooled Josephson junction qubits, or optical qubits.

[0143] 4, in various embodiments, the controller 132 may comprise various controller elements, including a processing element 405, a memory 410, a driver controller element 415, a communication interface 420, an analog-to-digital converter element 425, etc. For example, the processing element 405 may comprise one or more processing devices, such as a programmable logic device (CPLD), a microprocessor, a coprocessing entity, an application-specific instruction set processor (ASIP), an integrated circuit, an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), a programmable logic array (PLA), a hardware accelerator, other processing devices and / or circuits, etc. The term circuit may refer to an entirely hardware embodiment or a combination of hardware and a computer program product. In one exemplary embodiment, the processing element 405 of the controller 132 is a clock and / or is in communication with a clock.

[0144] For example, memory 410 may comprise non-transitory memory such as volatile and / or non-volatile memory storage, such as one or more of a hard disk, ROM, PROM, EPROM, EEPROM, flash memory, MMC, SD memory card, memory stick, CBRAM, PRAM, FeRAM, RRAM, SONOS, racetrack memory, RAM, DRAM, SRAM, FPM DRAM, EDO DRAM, SDRAM, DDR SDRAM, DDR2 SDRAM, DDR3 SDRAM, RDRAM, RIMM, DIMM, SIMM, VRAM, cache memory, register memory, etc. In various embodiments, memory 410 may store a queue of commands to be executed (e.g., an executable queue) to cause a quantum algorithm and / or circuit to run, qubit records corresponding to qubits of a quantum computing component (e.g., in a qubit record data store, qubit record database, qubit record table, etc.), calibration tables, computer program code (e.g., in one or more computer languages, dedicated controller languages, etc.), etc. In an example embodiment, execution of at least a portion of the computer program code stored in memory 410 (e.g., by processing element 405) causes controller 132 to perform one or more steps, operations, processes, procedures, etc. described herein, such as for controlling the operation of one or more qubit manipulation elements 136, processing sensor signals indicative of measurements captured by sensor 138, and / or communicating with classical computing component 110 of hybrid quantum-classical computing system 100.

[0145] In various embodiments, driver controller element 410 may include one or more driver and / or controller elements each configured to control one or more drivers. In various embodiments, driver controller element 410 may comprise a driver and / or driver controller. For example, a driver controller may be configured to cause one or more corresponding drivers to be operated according to executable instructions, commands, etc. scheduled and executed by controller 132 (e.g., by processing element 405). In various embodiments, driver controller element 415 may enable controller 132 to operate various of qubit manipulation elements 136 and / or sensors 138. In various embodiments, the drivers may comprise laser drivers configured to operate one or more lasers, drivers for controlling the operation of one or more voltage / current sources to cause the generation and provision of one or more voltage and / or current signals, and / or various other drivers configured to control the operation of each qubit manipulation element 136 of quantum computing component 130.

[0146] In various embodiments, the controller 132 comprises means for communicating and / or receiving signals from one or more sensors (e.g., photodetectors, voltage / current sensors, temperature sensors, pressure sensors, and / or other sensors). For example, the controller 132 may comprise one or more analog-to-digital converter elements 425 configured to receive signals from the one or more sensors.

[0147] In various embodiments, controller 132 comprises a communications interface 420 for interfacing with and / or communicating with classical computing component 110 of hybrid quantum-classical computing system 100. For example, controller 132 may comprise a communications interface 420 for receiving one or more quantum circuits that model and / or simulate a chemical system, executable instructions, instruction sets, etc. from classical computing component 110, and for providing outputs received from quantum computing component 130 (e.g., via sensors 138) and / or results of processing the outputs to determine qubit representations of the chemical system to classical computing component 110. In various embodiments, classical computing component 110 and controller 132 may communicate via a direct wired and / or wireless connection and / or via one or more wired and / or wireless networks 120.

[0148] Exemplary classical computation components 5 provides an exemplary schematic diagram depicting an exemplary computational entity 10 that may be used with embodiments of the present invention. In various embodiments, classical computing component 110 is configured to interface with quantum computing component 130. For example, classical computing component 110 is configured to interface with quantum computing component 130 to enable efficient and accurate modeling of chemical systems through the execution of quantum circuits having a reduced depth and / or a reduced number of two-qubit quantum gates (compared to corresponding quantum circuits generated via traditional quantum circuit synthesis techniques) using quantum computing component 130. For example, classical computing component 110 may be configured to communicate with quantum computing component 130 to enable a user (e.g., a human user or a program running on classical computing component 110) to provide input to quantum computing component 130 and receive, display, analyze, etc., output from quantum computing component 130.

[0149] 5, classical computing component 110 may include antenna 512, transmitter 504 (e.g., wireless), receiver 506 (e.g., wireless), and processing element 508 that provide signals to and receive signals from transmitter 504 and receiver 506, respectively. The signals provided to and received from transmitter 504 and receiver 506, respectively, may include signaling information / data in accordance with an applicable wireless system interface standard for communicating with various entities, such as controller 132, other classical computing components 110, etc. In this regard, classical computing component 110 may be capable of operating with one or more air interface standards, communication protocols, modulation types, and access types.

[0150] For example, classical computing component 110 may be configured to receive and / or provide communications using a wired data transmission protocol, such as fiber distributed data interface (FDDI), digital subscriber line (DSL), Ethernet, asynchronous transfer mode (ATM), frame relay, data over cable service interface specification (DOCSIS), or any other wired transmission protocol.Similarly, the classical computing component 110 may support general packet radio service (GPRS), Universal Mobile Telecommunications System (UMTS), Code Division Multiple Access 2000 (CDMA2000), CDMA2000 1X (1xRTT), Wideband Code Division Multiple Access (WCDMA), Global System for Mobile Communications (GSM), Enhanced Data rates for GSM Evolution (EDGE), Time Division-Synchronous Code Division Multiple Access (TD-SCDMA), Long Term Evolution (LTE), Evolved Universal Terrestrial Radio Access Network (E-UTRAN), Evolution-Data Optimized (EVDO), High Speed ​​Packet Access (HSPA), High-Speed ​​Downlink Packet Access (HSDPA), IEEE 802.11 (Wi-Fi), Wi-Fi Direct, 802.16 (WiMAX), ultra wideband It may be configured to communicate over a wireless external communications network using any of a variety of protocols, such as Ultra Wideband (UWB), infrared (IR) protocol, near field communication (NFC) protocol, Wibree, Bluetooth protocol, wireless universal serial bus (USB) protocol, and / or any other wireless protocol.The classical computing component 110 may use such protocols and standards to communicate using Border Gateway Protocol (BGP), Dynamic Host Configuration Protocol (DHCP), Domain Name System (DNS), File Transfer Protocol (FTP), Hypertext Transfer Protocol (HTTP), HTTP over TLS / SSL / Secure, Internet Message Access Protocol (IMAP), Network Time Protocol (NTP), Simple Mail Transfer Protocol (SMTP), Telnet, Transport Layer Security (TLS), Secure Sockets Layer (SSL), Internet Protocol (IP), Transmission Control Protocol (TCP), User Datagram Protocol (UDP), Datagram Congestion Control Protocol (DCCP), Stream Control Transmission Protocol (SCTP), HyperText Markup Language (HTML), and the like.

[0151] Through these communication standards and protocols, classical computing component 110 can communicate with various other entities using concepts such as Unstructured Supplementary Service information / data (USSD), Short Message Service (SMS), Multimedia Messaging Service (MMS), Dual-Tone Multi-Frequency Signaling (DTMF), and / or Subscriber Identity Module Dialer (SIM Dialer). Classical computing component 110 can also download modifications, add-ons, and updates to its firmware, software (including, e.g., executable instructions, applications, program modules), and operating system, for example.

[0152] In various embodiments, classical computing component 110 may comprise a network interface 520 for interfacing and / or communicating with, for example, controller 132. For example, classical computing component 110 may comprise a network interface 520 for providing qubit constraint information, executable instructions, instruction sets, etc. for receipt by controller 132 and / or for receiving output and / or results of processing output (e.g., measurements corresponding to active orbitals) provided by quantum computing component 130. In various embodiments, classical computing component 110 and controller 132 may communicate via a direct wired and / or wireless connection and / or via one or more wired and / or wireless networks 120.

[0153] In various embodiments, the processing element 508 may comprise one or more processing devices, such as a programmable logic device (CPLD), a microprocessor, a co-processing entity, an application specific instruction set processor (ASIP), an integrated circuit, an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a programmable logic array (PLA), a hardware accelerator, a graphics processing unit (GPU), a central processing unit (CPU), other processing devices and / or circuits, etc. The term circuitry may refer to an entirely hardware embodiment or a combination of hardware and a computer program product.

[0154] Classical computing component 110 may also comprise user interface devices comprising one or more user input / output interfaces (e.g., a display 516 and / or speaker / speaker driver coupled to processing element 508, as well as a touchscreen, keyboard, mouse, and / or microphone coupled to processing element 508). For example, the user output interface may be configured to provide an application, browser, user interface, interface, dashboard, screen, webpage, page, and / or similar terms used herein, that runs interchangeably on and / or is accessible via computing entity 10 to cause a display or audible presentation of information / data and to interact with it via one or more user input interfaces. The user input interface may comprise any of a number of devices that enable computing entity 10 to receive data, such as a keypad 518 (hard or soft), a touch display, a mouse, a voice / speech or motion interface, a scanner, a reader, or other input device. In embodiments that include a keypad 518, the keypad 518 may include (or cause the display of) conventional numeric (0-9) and related keys (#, *), and other keys used to operate the classical computing component 110, or may include a set of keys that can be activated to provide a full set of alphabetic keys or a full set of alphanumeric keys. In addition to providing input, the user input interface may be used to enable or disable certain features, such as, for example, a screen saver and / or sleep mode. Through such input, the classical computing component 110 can gather information / data, user interaction / input, etc.

[0155] Classical computing component 110 may also include volatile storage or memory 522 and / or nonvolatile storage or memory 524, which may be embedded and / or removable. For example, nonvolatile memory may be ROM, PROM, EPROM, EEPROM, flash memory, MMC, SD memory card, memory stick, CBRAM, PRAM, FeRAM, RRAM, SONOS, racetrack memory, etc. Volatile memory may be RAM, DRAM, SRAM, FPM DRAM, EDO DRAM, SDRAM, DDR SDRAM, DDR2 SDRAM, DDR3 SDRAM, RDRAM, RIMM, DIMM, SIMM, VRAM, cache memory, registered memory, etc. Volatile and nonvolatile storage or memory may store databases, database instances, database management system entities, data, applications, programs, program modules, scripts, source code, object code, byte code, compiled code, interpreted code, machine code, executable instructions, etc. to implement the functionality of classical computing component 110.

[0156] In any of the above aspects, the various features may be implemented in hardware, or as software modules running on one or more processors / computers.

[0157] The present invention also provides a computer program or computer program product comprising instructions that, when executed by a computer (or a hybrid quantum-classical computer), cause the computer to perform any of the methods / method steps described herein, and a non-transitory computer-readable medium comprising instructions that, when executed by a computer (or a hybrid quantum-classical computer), cause the computer to perform any of the methods / method steps described herein. A computer program embodying the present invention may be stored on a non-transitory computer-readable medium, or may be in the form of a signal, such as, for example, a downloadable data signal provided from an Internet website, or in any other form.

[0158] For example, the disclosure extends to a computer program comprising executable instructions that, when executed by a hybrid quantum-classical computing system, are configured to cause the hybrid quantum-classical computing system to (i) determine characterizing parameters describing a chemical system, the characterizing parameters being based at least in part on a chemical structure of the chemical system; (ii) generate an operator representation of the chemical system, the operator representation including singly excited operators and doubly excited operators; and (iii) reconfigure an order of appearance of the singly excited operators and doubly excited operators in the operator representation in response to the characterizing parameters, wherein the simulation is executable using a smaller number of two-qubit quantum gates used in at least one quantum circuit executable on a quantum computing component used to perform the simulation.

[0159] conclusion Many modifications and other embodiments of the inventions described herein will come to mind to one skilled in the art to which these inventions pertain having the benefit of the teachings presented in the foregoing descriptions and the associated drawings. It is to be understood, therefore, that the inventions are not to be limited to the particular embodiments disclosed, but that modifications and other embodiments are intended to be included within the scope of the appended claims. Although specific terms are employed herein, they are used in a general and descriptive sense only and not for purposes of limitation. [Explanation of symbols]

[0160] 10 Computational Entities 100 Hybrid quantum-classical computing system 110 Classical Computation Components 120 Wireless Network 130 Quantum Computing Components 132 Controller 134 qubits 136 qubit manipulation elements 138 Sensors 405 Processing Elements 410 memory 415 Driver Controller Elements 420 Communication Interface 425 Analog to Digital Converter 504 Transmitter 506 Receiver 508 Processing Elements 512 Antenna 516 Display 518 keypad 520 network interface 522 Volatile Memory 524 Non-volatile memory

Claims

1. 1. A hybrid quantum-classical computing system configured to perform a simulation of a chemical system, the hybrid quantum-classical computing system comprising a classical computing component coupled to a quantum computing component, the hybrid quantum-classical computing system comprising: (i) determining characterization parameters describing the chemical system, the characterization parameters being based at least in part on a chemical structure of the chemical system; and (ii) generating an operator representation of said chemical system, including a single-electron excitation operator and a double-electron excitation operator; (iii) rearranging the order of appearance of the singly excited operators and the doubly excited operators in the operator representation according to the characterizing parameters; wherein the simulation is executable using a smaller number of two-qubit quantum gates used in at least one quantum circuit executable on the quantum computing component used to perform the simulation.

2. The hybrid quantum-classical computing system may further define the order of occurrence of the single-electron excitation operator and the double-electron excitation operator as: (i) a double excitation operator corresponding to a double excitation from one spatial orbital to the corresponding spatial orbital, (ii) any double excitation operator, and (iii) any single excitation operator or configured to reconfigure into the hybrid quantum-classical computing system determines the characterizing parameters describing the chemical system; (i) applying symmetry filtering to the operator representation of the chemical system to assess its structural symmetry; (ii) generating a quantum state composition of the chemical system based on the symmetry; (iii) generating a spatial excitation combination of the chemical system from the quantum state combination using hard core boson operators; (iv) processing said spatial excitation combination by converting from spatial orbitals to spin orbitals therein; (v) introducing a spin orbital onto the at least one quantum circuit; and (vi) using exchange sets of the double excitation operator and the single excitation operator to synthesize the at least one quantum circuit such that a double excitation operator appears in the at least one quantum circuit before a single excitation operator appears in the at least one quantum circuit, and each exchange set corresponds to an excitation of the chemical system. and generating a corresponding operator expression by 2. The hybrid quantum-classical computing system of claim 1, wherein the hybrid quantum-classical computing system is at least one of:

3. the hybrid quantum-classical computing system is configured to reduce the number of quantum gates required in the quantum circuit synthesis according to the structural symmetry of the chemical system; or applying the symmetry filtering to the operator representation, defining a set of symmetries of the chemical system and identifying excitation operators of the operator representation of the chemical system that do not commute with one or more symmetries of the set of symmetries; or defining an Abelian point group of the chemical system and identifying redundant excitations of the chemical system based on the Abelian point group; or generating the quantum state composition of the chemical system comprises generating a reference state based on a dual occupied spatial orbital and a virtual spatial orbital of the chemical system; or generating the spatial excitation combination comprises converting a spatial inter-orbital operator to a dual-excitation spatial inter-qubit operator; The hybrid quantum-classical computing system of claim 2 , wherein the hybrid quantum-classical computing system is at least one of:

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