Method and apparatus for locally simulating resource-optimized fermions on a quantum computer for quantum chemistry - Patents.com

Abstract

Aspects of the present disclosure describe a method for performing a simulation of a system, the method including predicting a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; minimizing the energy of the system based on the first set of Ansatz terms and the first initial amplitudes; calculating perturbation corrections using one or more Ansatz wave functions; determining whether the energy of the system has converged; and predicting a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms in response to determining that the energy of the system has not converged.

Description

[Technical field]

[0001] (CROSS REFERENCE TO RELATED APPLICATIONS) This application is a joint venture between U.S. Non-provisional Application No. 17 / 177,813, filed February 17, 2021 and entitled "METHODS AND APPARATUSES FOR RESOURCE-OPTIMIZED FERMIONIC LOCAL SIMULATION ON QUANTUM COMPUTER FOR QUANTUM CHEMISTRY", U.S. Provisional Application No. 62 / 979,974, filed February 21, 2020 and entitled "METHODS AND APPARATUSES FOR RESOURCE-OPTIMIZED FERMIONIC LOCAL SIMULATION ON QUANTUM COMPUTER FOR QUANTUM CHEMISTRY", and U.S. Provisional Application No. 62 / 979,974, filed December 23, 2020 and entitled "METHODS AND APPARATUSES FOR RESOURCE-OPTIMIZED FERMIONIC LOCAL SIMULATION ON QUANTUM COMPUTER FOR QUANTUM CHEMISTRY". This application claims priority to and benefit of U.S. Provisional Application No. 63 / 130,088, entitled "METHOD FOR USE IN HYDROXY-BASED METHODS AND CHEMISTRY," the contents of which are incorporated herein by reference in their entireties.

[0002] (Government License Rights) This invention was made with Government support under a DOE Basic Energy Science award (DOE BES award de-sc0019449). The Government has certain rights in this invention.

[0003] Aspects of the present disclosure relate generally to improving resource utilization in quantum computers. [Background technology]

[0004] Atomic fluxes may be generated and used as a source of neutral atoms or ions for certain systems. Some of these systems may include, for example, quantum information processing (QIP) systems. Trapped ions are one of the representative implementations of QIP systems. Atom-based qubits may be used as quantum memories, as quantum gates in quantum computers and simulators, and may also function as nodes in quantum communication networks. Qubits based on trapped atomic ions enjoy a rare combination of attributes. For example, qubits using atomic ions have exceptional coherence properties, can be prepared and measured with nearly 100% efficiency, and can be easily mutually entangled using appropriate external control fields such as light or microwaves. These properties make atom-based qubits suitable for extensive quantum operations such as quantum computing and quantum simulation.

[0005] One application of quantum simulation is to use QIP systems to simulate fermionic matter, such as fermions that undergo local interactions. Simulation of fermionic systems using quantum computers may include two approaches: a variational quantum-classical hybrid strategy suitable for imperfect and pre-fault-tolerant (pre-FT) quantum computers, and Hamiltonian dynamics simulation based on quantum simulation algorithms suitable for fault-tolerant quantum computers. The former exploits the fact that quantum computers enable both efficient preparation of good ansatz states for the ground state required in estimating the ground state energy of a fermionic system, and a means to evaluate the expectation value of the system's Hamiltonian for the prepared ansatz states. In contrast, the latter exploits the ability of quantum computers to efficiently simulate quantum systems with local Hamiltonians, and can evaluate the ground state energy of the system in combination with quantum phase estimation. In the pre-FT regime, the cost of quantum computation may be dominated by multi-qubit gates. In contrast, in the FT regime, where circuits are typically described by sets of Clifford+T gates, the quantum computation cost is

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[0006] The following presents a simplified summary of one or more aspects in order to provide a basic understanding of such aspects. This summary is not an extensive overview of all contemplated aspects, nor is it intended to identify key or critical elements of all aspects or to delineate the scope of any or all aspects. Its sole purpose is to present some concepts of one or more aspects in a simplified form as a prelude to the more detailed description that is presented later.

[0007] One aspect of the disclosure includes a method including predicting a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; minimizing an energy of the system based on the first set of Ansatz terms and the first initial amplitudes; calculating perturbation corrections using one or more Ansatz wave functions; determining whether the energy of the system converges; and in response to determining that the energy of the system does not converge, predicting a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms.

[0008] Another aspect of the disclosure includes a quantum computing device configured to perform the steps of predicting a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; minimizing an energy of the system based on the first set of Ansatz terms and the first initial amplitudes; calculating perturbation corrections using one or more Ansatz wave functions; determining whether the energy of the system converges; and in response to determining that the energy of the system does not converge, predicting a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms.

[0009] Aspects of the present disclosure include predicting a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; minimizing an energy of the system based on the first set of Ansatz terms and the first initial amplitudes; calculating at least one of the energy or perturbation corrections in the wave functions based on one or more Ansatz wave functions; determining whether the energy of the system converges; and in response to determining that the energy of the system does not converge, predicting a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms.

[0010] Aspects of the disclosure include a non-transitory computer-readable medium that, when executed by a processor, causes the processor to store therein instructions for predicting a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; minimizing an energy of the system based on the first set of Ansatz terms and the first initial amplitudes; calculating at least one of the energy or wave function perturbation corrections based on one or more Ansatz wave functions; determining whether the energy of the system has converged; and, in response to determining that the energy of the system has not converged, predicting a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms.

[0011] Aspects of the disclosure include a system having a memory and a processor configured to: predict a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; minimize an energy of the system based on the first set of Ansatz terms and the first initial amplitudes; calculate at least one of the energy or wave function perturbation corrections based on one or more Ansatz wave functions; determine whether the energy of the system converges; and in response to determining that the energy of the system does not converge, predict a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms.

[0012] To the accomplishment of the foregoing and related ends, the one or more aspects comprise the features hereinafter fully described and particularly pointed out in the claims. The following description and the annexed drawings set forth in detail certain illustrative features of the one or more aspects. These features are indicative, however, of only a few of the various ways in which the principles of the various aspects may be employed and the description is intended to include all such aspects and their equivalents. [Brief description of the drawings]

[0013] The disclosed aspects are described below in conjunction with the accompanying drawings, which are provided to illustrate, but not to limit, the disclosed aspects, and in which like designations refer to like elements.

[0014] [Figure 1] FIG. 1 illustrates an example of a standard two-body interaction circuit according to some aspects of the present disclosure. [Diagram 2] FIG. 1 illustrates an example of an FT regime optimization circuit for two-body terms in accordance with some aspects of the present disclosure. [Diagram 3] FIG. 13 illustrates an example of a triple-controlled RX(θ) gate implementation with a relative phase triple-controlled NOT gate. [Figure 4]1 illustrates an example quantum circuit and optimization structure in accordance with certain aspects of the present disclosure. [Diagram 5] FIG. 13 illustrates another example of an FT regime optimization circuit for two-body terms in accordance with some aspects of the present disclosure. [Figure 6] FIG. 1 is a flow diagram of a method for performing a simulation according to some aspects of the present disclosure. [Figure 7] 1 is an example of a chart comparing the ground state energy of a water molecule in its equilibrium geometry using the STO-3G basis set, calculated by various methods, according to some embodiments of the present disclosure. [Figure 8] 1 illustrates an example circuit of an adjacent Pauli ring circuit according to some aspects of the present disclosure. [Figure 9] 1 shows example diagrams of partial improvements according to some aspects of the present disclosure. [Figure 10] 1 illustrates an example histogram of the excess number of CNOT gates in accordance with some aspects of the present disclosure. [Figure 11] FIG. 1 is a block diagram illustrating an example of a quantum information processing (QIP) system according to some aspects of the present disclosure. [Figure 12] FIG. 2 illustrates an example of a computing device according to some aspects of the present disclosure. [Figure 13] 13 shows an example chart illustrating the reduction in measurement times according to some aspects of the present disclosure.

[0015] Attached hereto are appendices, the contents of which are incorporated by reference in their entirety. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0016] The detailed description set forth below in conjunction with the accompanying drawings is intended as a description of various configurations and is not intended to represent the only configurations in which the concepts described herein may be practiced. The detailed description includes specific details for the purpose of providing a thorough understanding of the various concepts. However, it will be apparent to one skilled in the art that these concepts may be practiced without these specific details. In some instances, well-known components are shown in block diagram form in order to avoid obscuring such concepts.

[0017] As mentioned above, trapped ions can be used to implement quantum bits for use in quantum information processing (QIP) systems. A typical ion trap shape or structure used for quantum information and metrology purposes is a linear radio frequency (RF) Paul trap (also called RF trap or simply Paul trap), where nearby electrodes hold static and dynamic potentials that result in the confinement of the ions. Qubits based on trapped ions can be used in different kinds of devices, including, but not limited to, quantum memories, quantum gates in quantum computers and simulators, and nodes in quantum communication networks.

[0018] As used in this disclosure, the terms "atoms" and "neutral atoms" are interchangeable, and the terms "ions" and "atomic ions" may be interchangeable. Additionally, the terms "atomic ions," "ions," "atoms," and "neutral atoms" may describe particles that are generated from a source material and that are to be or are actually confined or trapped in a trap to form a crystal, lattice, or similar arrangement or configuration. The term "plasma" may refer to a flux of neutral atoms, ions, or both.

[0019] One aspect of the present disclosure optimizes quantum circuits implementing Fermionic simulations in both the pre-FT and FT regimes. The cost functions considered are the number of multi-qubit gates in the pre-FT regime and the R that needs to be applied in the simulation circuit in the FT regime. z R of the number of gates or time steps z The number of gates. Other aspects of the present disclosure include a product formula (PF)-based approach for the FT regime and a unitary coupled cluster (UCC) Ansatz-based variational approach for the pre-FT regime. In other aspects, the fermion simulation may include using a second-order perturbation-based approach and qubit operator transformations from generalized fermions.

[0020] In some implementations, simulation results in the FT regime may be utilized in chemical, medical, physical, and other applications. The simulation may be in real time.

[0021] In certain implementations, the results of the simulation in the pre-FT regime may be utilized to calculate the ground states of the particles, which may be used to determine chemical reactions, bonding properties, energy states, etc.

[0022] In one aspect of the present disclosure, in the FT regime, R z The (θ) and T gate counts may be optimized along with the ancilla qubit count, assuming the implementation uses a product formula algorithm. Savings ratios of 2 in gate count and 11 in the number of ancilla qubits may be achieved in some cases. In the pre-FT regime, the number of two-qubit gate counts may be optimized.

[0023] One aspect of the present disclosure presents a framework that allows for bootstrapping the VQE progression towards convergence of the ground state energy of a fermionic system. This framework, based on perturbation theory, may be capable of improving the energy estimates at each cycle of the VQE progression. The improved energy approach may result in a saving of quantum resources (e.g., the number of qubits and / or quantum gates) required to reach a pre-specified tolerance from a known ground state energy.

[0024] For the PF regime, one aspect of the present disclosure is to z (θ) gates may utilize more than one T gate, which is resource intensive, so R z (θ) Number of gates and R z (θ) The purpose is to reduce the gate depth.

[0025] In one implementation, the fermionic system is characterized by a time-independent local second quantized Hamiltonian H,

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[0026] Evolution operator U evolution =e -iHt To implement on a quantum computer using a quantum mechanical simulation approach (where H is the Hamiltonian of the system and t is the duration for evolving the system forward in time), U evolution The algorithm to implement

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[0046] This includes asymptotically optimal quantum signal processing where ∈ is the algorithm approximation error.

[0027] One aspect of the present disclosure includes a hybrid quantum-classical approach based on the technique of variational quantum eigenvalue solving. This technique can be implemented on a quantum computer.

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[0028] In an alternative implementation, the unitarian that evolution operator U ansatz may be implemented on a quantum computer (e.g., unitary coupled cluster Ansatz). ansatz One or more variational parameters of

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[0029] In certain embodiments, a complementary approach to the hybrid method described above, based on perturbation theory, may be used to estimate the ground state energy.

[0030] Figure 1 shows

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[0031] In some implementations, the 2k-th order PF algorithm may be utilized to simulate fermionic systems. After the JW transformation is performed on the Hamiltonian H in equation (1), the time evolution operator is

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[0032] In a particular implementation, the two-body Trotter term

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[0033] Figure 2 Triple Control R X The gates may be implemented using single qubit gates and / or two qubit gates. For example, triple controlled R X The gate is Hadamard Gate, R z In one implementation, a triple-controlled R gate and / or a triple-controlled NOT gate may be used, as shown in FIG. X The gate may be a relative phase triple controlled-NOT gate. The circuit 300 shown in FIG. 3(a) has R z Reduces the count from the standard 8 to 2.

[0034] In some aspects of the present disclosure, the Trotter terms above may be implemented using an initialized ancilla qubit in the |0> state, as shown in FIG. 3(b). The circuit 350 shown in FIG. 3(b) provides an R of 1 for each two-body Trotter term. z This configuration is optimal for implementing the two-body Trotter term for FT regime fermion simulations. z It can be depth.

[0035] In some implementations, triple control R X R of a gate (such as circuit 300) z The gate and T-gate optimized implementation includes a relative-phase Toffoli gate instead of a traditional Toffoli gate to reduce the T-gate count. In some embodiments, a relative-phase Toffoli gate with three controls may not require an ancilla qubit.

[0036] Circuit 300 includes two R z Includes: R z To reduce the depth, an ancilla qubit prepared at |0> may be introduced. X Gate R z An example of a depth 1 configuration is shown in circuit 350. In some cases, when multiple qubits rotate by the same angle simultaneously (as would be the case if the molecule being simulated has symmetry), it is possible to achieve a depth of 1 with a reasonable tradeoff in the number of qubits and T gates. z To exponentially reduce the number of gates, a weighted-sum method is sometimes employed. Two R z The gate may not benefit from the weighted sum method. Figure 4 shows an example of a triple-controlled relative-phase Toffoli gate 400. The triple-controlled relative-phase Toffoli gate 400 is a logical NOT gate. Figure 4 also shows a simplex operator

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[0037] In a particular implementation, the techniques described above may be implemented as a generalization of single-field terms, which may then be applied to any many-body terms.

[0038] In certain cases, for example, by the JW transformation, σ acting on several other qubits x,y,z In the presence of operators, we may modify the two-body circuit as follows, e.g., σ acting on an additional qubit. z There exists an operator, in this case

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[0039] In a particular implementation, the simplex operator

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[0040] In the pre-FT regime, the energy of the ground state of a system whose Hamiltonian is given in the form (1) may be calculated, for example, by the variational quantum eigensolver (VQE) approach, which involves creating an Ansatz state on a quantum computer and evaluating its energy. In the VQE approach, the quantum computer is repeatedly called to calculate the energies of parametric Ansatz states, thereby minimizing the energy (locally or globally) to obtain the ground state of the target system.

[0041] In another aspect of the present disclosure, the embodiments of the present disclosure may be used in the pre-FT regime to calculate the energy of the ground state, a kind of Ansatz state |Ψ ansatz > is |Ψ ansatz >=U ansatz A parametrizable unitarian evolution operator U such that |0> ansatz The initial state |Ψ0> may be transformed using the operator U ansatz is in exponential form U ansatz =e (Z-Z†) By changing the parameter Z, the Ansatz state energy <Ψ ansatz │H|Ψ ansatz >=<Ψ0│U † HU|Ψ0〉 can be minimized variationally. With an appropriate transformation from the fermion basis to the qubit basis, this energy expectation value can be evaluated efficiently on a quantum computer (the quantum resource cost is calculated by the operator U ansatz The following procedure is the implementation cost of the unitary evolution operator U ansatz As an example, the Unitary Coupled Cluster Ansatz with Single and / or Double Excitations (UCCSD) may be used. Other examples include the k-product of exponentials of distinct pair-coupled cluster double excitation operators (k-UpCCGSD) or a simple unitary extension of the Generalized Unitary Coupled Cluster Ansatz (UCCGSD).

[0042] In some implementations, the approach may start with a ground-state Hartree-Fock (HF) wavefunction |Ψ0>, which can be calculated on a classical computer and implemented on a quantum computer. The wavefunction |Ψ0> is then transformed into a unitary operator

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[0043] In some embodiments of the present disclosure, two procedures may be used to reduce the circuit complexity in the approach detailed above, expressed in terms of the total number of multi-qubit gates. A first-order PF algorithm may be used to implement the Ansatz, although extension to any higher-order PF algorithm is straightforward. Alternatively or additionally, a hybrid framework based on perturbation theory may be used to reduce the circuit complexity. In the hybrid framework, based on a small instance of the VQE circuit, significant Ansatz terms may be included in a large instance of the VQE circuit, effectively bootstrapping the VQE progression towards an energy estimate of the ground state of the simulated system.

[0044] In a specific implementation, the general framework can optimize VQE-based quantum simulations by utilizing the power of perturbation theory to predict which Ansatz terms to include in the UCCSD Ansatz and then modifying the VQE results by post-processing. The framework can optimize both the total number of VQE runs and the size of the Ansatz state preparation circuit used until the ground state energy estimates converge. A derivation of a simple perturbation scheme that can be easily implemented in the framework is sometimes called the Hybrid Second-Order Möller-Plesset Perturbation (HMP2) method.

[0045] In some aspects of the present disclosure, any general (i.e., not specific to any system) unitarian thats operator may be expressed as:

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[0046] Next, the Ansatz evolution operator

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[0047] In one aspect of the present disclosure, convergence can be achieved by starting with an operator with a small set of variational parameters that can achieve an error threshold and iteratively growing that set, thereby reducing the complexity of the circuit. Going from the mth iteration to the (m+1)th iteration, the set of variational parameters of the operator grows, T m ⊂T m+1 One aspect for achieving rapid convergence may be the selection of additional variational parameters to include between iterations. For example, one aspect of the present disclosure may include iteratively selecting the next variational parameters to include in the Ansatz state preparation based on the size of the perturbatively predicted wavefunction correction amplitude for a given Ansatz state for which the variational parameters have already been optimized (e.g., via conventional means of VQE without perturbative energy correction).

[0048] FIG. 6 shows a flow diagram of the proposed method 600. The method 600 may include a systematic method of using perturbation methods to improve the quantum resources required for the VQE approach. Specifically, the method 600 may rapidly converge the system to the ground state energy of the quantum circuit. The rapid convergence may depend on the selection of the individual excitation terms in equation (8) used to prepare a subset of the variational parameters in the UCCSD example.

[0049] In one exemplary implementation, similar to conventional VQE approaches for fermion simulations, the method begins with the ground state Hartree-Fock wave function of the single-particle Hamiltonian as |Ψ>. The method 600 may be implemented by a classical computer, a quantum computer, or a hybrid classical-quantum computer. In one aspect, the method 600 may be implemented by a computing device 1200 as described below (FIG. 12).

[0050] At block 605, the method 600 may predict an initial guess for a starting set of Ansatz terms and associated amplitudes. For example, the first step may be to predict an initial evolution operator (EO) to use for the first iteration of the VQE.

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[0051] In one aspect of the disclosure, the method 600 may include selecting a method of perturbation, such as second-order Möller-Plesset (MP2) perturbation theory. The method 600 may include selecting a set of Ansatz. In one implementation, the set of Ansatz may be compatible with unitary classes for later use with a quantum computer. An example includes UCCSD. The set may include a finite number of Ansatz terms, each of which is distinct. The method 600 may execute the selected method of perturbation with the selected set of Ansatz on a classical computer. The execution may result in an initial strength of the perturbation associated with each Ansatz term, which may be used by the quantum computer.

[0052] At block 610, using the initial set of Ansatz terms and their initial variational parameter values, the method 600 calculates the energy

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[0053] At block 615, once the energy has been minimized and the Ansatz state has converged, method 600 may calculate perturbation corrections in the energy and / or wavefunction using the Ansatz wavefunction. For example, method 600 may calculate correlation energies and corrections to the wavefunctions. As a result, the total energy of the cycle may be the sum of the energy corrections and the VQE energy. Details of the hybrid perturbation method are described in more detail below.

[0054] In certain aspects of the present disclosure, the method 600 may configure a VQE quantum circuit that can create an Ansatz state for a sought ground state, with the Ansatz terms and initial perturbation strengths determined by running the selected perturbation method with the selected Ansatz set on a classical computer. Based on the Hamiltonian of the simulated system, the method 600 may append the required measurement basis conversion circuitry for each term of the Hamiltonian for which an expectation value is to be determined. In the case of a hybrid approach such as HMP2, the method 600 may additionally consider an energy correction operator, which tends to require the consideration of additional measurement basis conversion circuitry. In some implementations, the method 600 may compile and / or optimize the circuit and run the resulting circuit on a quantum computer to evaluate the expectation value of the Hamiltonian. In the hybrid case, the method 600 may additionally or alternatively evaluate the expectation value of the energy correction term on a quantum computer. The method 600 may use a classical optimizer to feed forward new variational parameters to try, which may replace the initial perturbation strengths used above. The above process may be repeated until the energy estimate converges to a minimum.

[0055] At block 620, the method 600 may determine whether the energy calculation has converged.

[0056] At block 625, method 600 may predict the Ansatz operator to be used next and the initial guesses for the newly added variational parameters based on the ability of the new operator to recover the wave function correction. For example, before starting the next cycle, method 600 may optionally identify a pool of operators that can be direct successors of the current operator. Method 600 may then evaluate how much each operator in the pool can explain towards the perturbative wave function correction. The operator that can recover the most wave function correction (if any) may be used for the next cycle, and the method of recovery of the correction may indicate an initial guess for the newly added variational parameters in the next cycle. Initial variational parameter values ​​for the Ansatz terms that will remain part of the next VQE simulation may be imported from the previous cycle.

[0057] In some implementations, if the energy estimate based on the selected Ansatz is still far from the actual ground state energy, or if there is no known ground state energy, the method 600 can make an educated guess by systematically increasing the Ansatz size (described below). The method 600 may include determining whether further increasing the Ansatz would significantly change the energy estimate. The method 600 may include determining which Ansatz terms to add to the existing Ansatz based on a hybrid perturbation framework. An initial guess of the perturbation strength associated with the new term is determined based on the hybrid framework as well.

[0058] At block 630, the method 600 may output the simulation results.

[0059] In one embodiment, the techniques described in method 600 may be performed on a classical computer to generate simulation parameters (e.g., Ansatz terms) for executing a quantum circuit on a quantum computer. In another embodiment, the techniques described in method 600 may be performed on a hybrid classical-quantum computer to generate simulation parameters for executing a quantum circuit on the hybrid classical-quantum computer.

[0060] The cycle repeats the outlined procedure of performing VQE simulations and hybrid perturbation calculations until total energy convergence is achieved (at block 630).

[0061] The Möller-Plesset perturbation method may be hybridized with VQE simulation, hereafter referred to as HMP2. The HMP2 method may improve each VQE cycle in terms of energy via perturbative energy corrections, and optimize the Ansatz operator (if any) to be used in the next cycle via perturbative wave function corrections. In the mth VQE cycle, the energy of the Ansatz state is

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[0062] VQE results (e.g., Ansatz wave functions

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[0063] next,

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[0064] The energy correction of VQE can be obtained using equations (18) and (21).

[0065] Terms in a space appropriately transformed from the fermion basis to the qubit basis

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[0066] In some implementations, equation (22) can be expressed as |Ψ ansatz >

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[0067] In some embodiments, the following aspects of the present disclosure may be used to determine the next Ansatz evolution operator to use in the (m+1)th round. m (S m After variably determining a set of values ​​for the parameters of

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[0068] identity

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[0069] For UCCSD, R m Each

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[0070] In some embodiments, the procedure introduced above for the mth round of a VQE cycle can also be used prior to the first cycle by setting m = 0. In such cases, the calculations can be classical and the energy corrections can be MP2 energy corrections. The wave function corrections can inform which operators to try first and suggest initial values ​​for the corresponding variational parameters.

[0071] The number of individual Pauli string operators (e.g., equation (22)) whose expectation values ​​must be evaluated using a quantum computer scales to O(n 8 ) (n is the number of qubits). Here, for a fixed Ansatz, Z is fixed and equation (21) takes the time required for computation to be O(n 8 ) Paulist ring measurement,

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[0072] One aspect of the disclosure involves the selection of qubits. Some qubits may be used to create an Ansatz state while others may operate classically, since only qubits that are coupled via the selected Ansatz are entangled. Another aspect may utilize an extension to a small, discrete set of qubits with a set size that allows for inexpensive classical post-processing. In some implementations, if there are two qubits that represent the same spatial orbital with two opposite spins, but whose energies cannot be distinguished by a particular choice of Ansatz, the two qubits may encode redundant information. This may allow for encoding information using only one qubit.

[0073] An aspect of the present disclosure includes measuring multiple Pauli rings simultaneously and / or in parallel to reduce the total number of measurements, provided that the Pauli rings commute with each other. Methods of measurement may include general commuting (GC) partitioning and qubit-wise commuting (QWC) partitioning. GC partitioning reduces the number of measurements and may incur the cost of additional two-qubit gates. QWC partitioning can reduce the number of measurements (to a lesser extent than GC partitioning) without increasing the number of two-qubit gates. In some aspects, QSR may be implemented before GC to reduce the additional two-qubit gates.

[0074] Figure 7 shows the D αAn example of a graph 700 showing the convergence of ground state energy using different approaches as the number of terms N increases is shown. The insertion order of the Ansatz terms for the conventional curve shown here is derived from prior knowledge of the order of determinant contributions in classical FCI calculations and closely mimics the ideal case, which is almost impossible to obtain in practice. Comparing the two convergence curves, the HMP2-assisted simulation captures the major Ansatz terms. Figure 7 shows that the FCI and HMP2 orders are in good agreement. The HMP2 correction to the ground state energy can help to accelerate the energy convergence to the FCI energy. Given that the implementation of each additional Ansatz term results in a significant accumulation of noise in noisy intermediate-scale quantum (NISQ) hardware, the faster energy convergence that the HMP2 method enables can be useful for quantum computers.

[0075] In some cases, various fermion-to-qubit transformations can be used to reduce the quantum resource requirements of pre-FT fermion simulations. Well-known fermion-to-qubit transformations, such as the Jordan-Wigner (JW) and Bravyi-Kitaev (BK) transformations, map fermion creation (annihilation) operators onto the Pauli ring. However, there are many other transformations that can in principle be used, such as the Generalized Transformation (GT) method, which includes the JW and BK transformations. Using the PF algorithm as a concrete example of an implementation of the UCCSD Ansatz, significant quantum resource savings can be achieved by appropriately choosing the mapping of a given cluster operator to the input and by carefully choosing the heuristic optimization method.

[0076] The transformation in the GT method may adhere to the relationship specified in equation (2). This may be achieved by considering the following invertible upper triangular basis transformation matrix β:

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[0077] The creation and annihilation operators based on GT are

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[0078] In our particular implementation, the UCCSD Ansatz implementation relies on the JW transformation and considers a series of carefully selected heuristics to optimize the resulting quantum circuit. Details of the heuristics considered are provided below. A summary of the steps in the order of their application is given below.

[0079] The outermost loop of the above approach considers different transformation matrices β. For a given mapping matrix, the following routine is executed to construct and optimize a circuit that implements the UCCSD Ansatz. This routine is designed to iteratively call a set of dedicated automatic circuit optimization tools. The efficiency of these tools allows for rapid evaluation of the cost function for different cases in each subroutine.

[0080] In one implementation, a first routine can be associated with labeling the fermion levels: U(j) is the empty set and P(j)=R(j). Unlike the JW transformation, in the GT approach, an infinite number of combinations of the sets U(j), P(j), and R(j) are possible. To exploit this, it is desirable to carefully select which fermion levels map to which qubit indices. However, exploring all possible mappings can be computationally prohibitively expensive. Instead, a greedy approach can be utilized, whereby the routine explores one permutation at a time from a given fermion level to qubit index mapping. Specifically, from a given mapping, the permutation that reduces the quantum resource requirements the most is applied. This process may be repeated until no single permutation reduces the quantum resource requirements. The following subroutine can illustrate a cost function evaluation of each permutation:

[0081] In some embodiments, the first routine can reduce the number of two-qubit gates by utilizing a different mapping from fermion level labels to qubit indices. The Pauli ring of the excitation operator is

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[0082] In some embodiments, for an n-qubit system representing n fermion levels, a greedy approach may be implemented. First, a set of unique permutation matrices for all possible k-swap operations {P i}, where the k-swap operation is done by j ,2l j +1) with the same subscript index j∈{0,…,k-1}, we define the k unique ordered pairs of (2m j ,2m j +1) with k other unique ordered pairs, where

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[0083] In some cases, once the labels are determined, they are applied to the relevant fermionic operators, including the molecular Hamiltonian, to maintain consistency during the simulation.

[0084] In another implementation, the second routine may involve inter-Trotter term ordering. Properly ordering the Trotter terms can result in significant savings in the two-qubit gate count due to gate cancellation between adjacent Trotter term circuits. While a similar approach may be possible, the GT approach allows for non-trivial modifications. Specifically, each Trotter term may be processed to determine whether it qualifies to be classified into the same equivalence class, whose elements have an opportunity for resource savings when placed adjacent to each other on a quantum circuit via the aforementioned gate cancellation. Once the equivalence classes according to the eligibility of each Trotter term have been determined, a simple greedy approach can be used to order the Trotter term elements of each equivalence class to reduce the two-qubit gate count.

[0085] In some aspects of the disclosure, a subroutine may be associated with Trotter inter-term ordering. This subroutine (also referred to as the second routine) uses heuristics to reduce the number of two-qubit gates by exploiting the freedom to arbitrarily order the Trotter terms. The arbitrary ordering respects the error bounds provided by the PF algorithm. In one aspect of the disclosure, a preprocessing step is performed based on information passed from the Trotter intra-term optimization step described below. Specifically, the operator

number

[0086] In one example, after determining ineligibility, the subroutine proceeds according to a greedy approach. The subroutine identifies the most frequently qualified index (e.g., p) across all terms. Terms with index p are then grouped as eligible target qubits and classify them into equivalence class [p]. Next, all group elements are removed from the list of one-body and two-body operators. This procedure is repeated until there are no more operators in the list.

[0087] In some aspects of the present disclosure, quantum resource cost reductions can occur between circuit representations of elements of the same equivalence class because the target qubits of two-qubit gates from each element of the same class are the same. Once some or all equivalence classes are specified, the subroutine rearranges the order in which the elements are implemented in the quantum circuit. Because considering all permutations would be cost prohibitive, the subroutine can implement a greedy approach by starting with the two elements that can reduce resource cost the most. The subroutine then concatenates the next element identified from the set of elements not implemented in the circuit based on the amount of resource cost reduction. This concatenation process is repeated until no elements are not implemented in the circuit. In some cases, each attempt to test which element may be optimal for a given iteration can include four cases. Circuit concatenation may be performed as a prefix or suffix, and the elements to be concatenated may be considered in their original intra-term order or inverse.

[0088] In some cases, the subroutines may be associated with intra-Trotter ordering. It is an efficient way to implement single-fermion or dual-fermion Trotter terms, where a JW transformation can be used for a given mapping from fermion labels to qubit indices. The method may include optimal circuit configurations that exploit full inter-qubit connectivity. The method relies on careful ordering of the implementation of intra-Trotter operators, e.g., the exemplary dual-fermion Trotter term

number

[0089] In one aspect, the Trotter subroutine can reduce the number of two-qubit gates by optimizing the ordering of the transformed Pauli rings that form the two-body operator. Since each unitary term contains two Pauli rings, up to inversion, one ordering can be derived. Thus, the unitary terms do not require a particular ordering. For two-body operators, applying the appropriate transformations and the PF algorithm, the eight subterms

number

number

[0090] In a particular implementation, for a selected target qubit t, the reordering of the Trotter intra terms may maximize the CNOT reduction (e.g., the CNOT reduction between adjacent Pauli string circuits). For each of the adjacent Trotter intra terms, i.e.,

number

[0091] In some cases, for circuit 900, if M0=M2, there is a reduction of two CNOTs. If M0≠M2, there is a reduction of one CNOT. For a selected target qubit t,

number

number

number

number

[0092] The resulting optimized circuit implements the UCCSD Ansatz in the chosen transformation basis defined by β. However, except for the JW transformation where β=1, the GT β matrix requires an initial mapping of the basis at the beginning of the circuit, which adds overhead in the gate count of two qubits.

number

[0093] Effectively, in the JW transformation, whenever a pair of adjacent qubits is excited to another pair of adjacent qubits exhibiting a different fermion level, regardless of the corresponding appropriate fermion level, the circuit implementing such an excitation term sometimes simplifies to require only two two-qubit gates, half the number of qubits that would otherwise be required. To take advantage of this, one can utilize bosonic circuits written by the JW transformation and non-bosonic circuits written by the GT approach in parallel. When going from the half-qubit space of the bosonic circuits back to the full-qubit space of the non-bosonic circuits, n / 2 CNOT gates can be expended.

[0094] In some implementations, a correspondence between the generalized bosonic and fermionic terms may be established. Fermionic biexcitation terms

number

number

number

number

[0095] Table II shows the circuit metrics, measured by the number of two-qubit gates used to implement UCCSD Ansatz circuits for different molecules, for the best GT transformations found by JW, BK, and the above heuristic toolchain. To find the best GT transformation, particle swarm optimization may be used. The advantage offered by the GT transformation varies across molecule combinations and ranges from 1.44% to 21.43%. This demonstrates the heuristic's ability to indeed further optimize the quantum circuits obtained via the JW transformation by considering custom-selected GT transformations for different input cases. The low reduction rate may be due to the classical optimization being unable to sample enough matrices with limited classical computational resources.

[0096] [Table 1]

[0097] In some examples, parallel implementation of multiple Trotter terms may be considered in reducing the Rz gate depth of a quantum simulation circuit. Based on the circuit configurations described above, a methodology for optimization for parallel implementation may be contemplated.

[0098] As mentioned above, the circuit implementing the Trotter term has three main parts: an initial CNOT gate network that computes linear functions of Boolean input variables; a triple-controlled R X Note that the CNOT gate network is the reverse of the initial CNOT gate network. If we denote the input Boolean variables as a, b, c, and d from top to bottom, the CNOT gate network is

number

number

[0099] For VQE simulations in the pre-FT regime, one embodiment of the present disclosure includes a general framework that exploits the predictive and corrective powers of perturbation theory. In one embodiment, in addition to the HMP2 method described herein, many more complex forms of perturbation theory can also be used to improve the efficiency and accuracy of the simulation. For example, it is possible to obtain coefficients of Ansatz terms that perturbatively include triple excitation terms or higher order excitation terms, perhaps similar to how classical CCSD(T) or CCSDT-1a / b methods include triple excitation terms. The above method is general enough that other perturbation methods can replace the HMP2 method in a perturbation-assisted quantum simulation cycle.

[0100] In a particular implementation, if the framework used to consider bosonic terms in the JW transformation and non-bosonic terms in the GT transformation is extended and generalized, the use of multiple fermion-to-qubit transformations β for a given set of excitation terms may be valuable in reducing the overall resource requirements. Drawing from the fact that different β transformations lead to different resource requirements in implementing the excitation terms of the target UCCSD Ansatz circuit, it is possible to anticipate that a certain subset of excitation terms may be implemented more efficiently by one β transformation and that certain other excitation terms may be implemented more efficiently by yet another β transformation. Thus, splitting the set of excitation terms required to prepare a UCCSD Ansatz state into subsets of excitation terms may prove more favorable in quantum resource requirements, which may be implemented more efficiently by a respective appropriate selection of β transformations for each subset. Optimizing against the expected trade-off between overhead costs incurred due to the switching of transformations and savings obtained through a tailor-made selection of transformations may be possible without departing from the spirit of the current application.

[0101] In another aspect of the present disclosure, a procedure using binary particle swarm optimization (PSO) may be implemented to optimize the binary matrix β of the GT. To encode the problem, the upper triangular entries of the n×n binary matrix β are of size

number

[0102] One aspect of the present disclosure begins by creating a group of particles, each of which is mapped to an initial position X i (t=0), except for k. A non-negative integer t represents a time step, and t=0 may be the starting point of the optimization. The total number of particles in the swarm is

number

number

[0103] At any or every time step t, the procedure computes the locally optimal position vector L for each particle i. i (t) and the corresponding locally optimal cost s i (t) and the global optimal position vector G(t). i (t) is the cost function value s among all the positions that particle i has moved to by time step t. i (t) can be defined as the smallest position vector. The global position vector is the vector that is the smallest among all particles. i (t) is the smallest L i (t). The velocity vector is from the tth step to the (t+1)th step.

number

number

number

[0104] Optimization is done when t is t max This may be continued until ∂t osc = 10, and if n ≤ 8, t max = 10000, otherwise t max = 100. In another example, t max The choice of t is t max may be large enough that the optimization begins to oscillate before reaching

[0105] Examples of GT results using binary PSO are shown below.

number

number

[0106] In one aspect of the present disclosure, the circuit optimization technique may include implementing a first routine related to fermion level labeling.

[0107] In another aspect, the circuit optimization technique may include implementing subroutines associated with Trotter inter-term ordering.

[0108] In some aspects, the circuit optimization techniques may include implementing subroutines associated with Trotter inner term ordering.

[0109] In certain aspects of the disclosure, one or more optimization techniques may be implemented. For example, the circuit optimization technique may include implementing a fermion level labeling routine and a Trotter inter-term ordering subroutine. In another example, the circuit optimization technique may include implementing a fermion level labeling routine and a Trotter intra-term ordering subroutine. In yet another example, the circuit optimization technique may include implementing a Trotter inter-term ordering subroutine and a Trotter intra-term ordering subroutine. In some examples, the circuit optimization technique may include implementing a fermion level labeling routine, a Trotter inter-term ordering subroutine, and a Trotter intra-term ordering subroutine.

[0110] FIG. 9 shows simulation diagrams 900, 910, 920, 930 of fractional improvement ρ as a function of fractional classical computational resources R used for binary PSO. The first simulation diagram 900 is associated with hydrogen fluoride (HF). The second simulation diagram 910 is associated with beryllium hydride (BeH2). The third simulation diagram 920 is associated with water (H2O). The fourth simulation diagram 930 is associated with ammonia (NH3). Different numbers of Ansatz terms may be used for the simulation diagrams (e.g., 4 to 6 Ansatz terms). Simulations are performed with the STO-3G basis set using the HMP2 method with UCCSD Ansatz. The number of excitation terms included in the final UCCSD operator is 3, 9, and 52 for HF, BeH2, and NH3, respectively. This may result in an estimate of the respective ground state energies within chemical accuracy from the known ground state energies. If the term n is small enough to explore a relatively large portion of the possible variations of the β matrix (R>10 -5 ), improvements range from less than 14% to more than 20%. The larger n, the less improvement is likely.

[0111] In another aspect of the disclosure, a qubit space reduction (QSR) technique procedure may be used to optimize the number of Pauli strings to be measured. In a first variant, the procedure may classically treat qubits in classically accessible states. In a second variant, the procedure may exploit bosonic states by unfolding only a single qubit for two spin orbitals in the same spatial orbit, with degenerate energies according to a specific choice of Ansatz.

[0112] For the first variant, in some examples, the measured Pauli ring S=A0A1...A (n-1) Consider, where A i ∈{σ x ,σ y ,σ z ,I}, where I is the 2 × 2 identity operator. If we conjugate the above to the quantum state |Ψ>=|ψ>, where |ψ> denotes the quantum state of a qubit that is entangled by the Ansatz method and |φ> denotes the quantum state of a qubit that is a classically accessible state (e.g., a computational basis state), then we obtain

number

[0113] An example of the implementation of the second variant is described below. The quantum state |ψ> is

number

[0114] Since the separated two-qubit states |00> and |11> do not encode more than one qubit of information, (31) becomes

number

[0115] Paulist Ring S=A0A1…A (n-1) can exist in the entire space spanned by |Ψ> in (31). Here, A0A1 can be the Pauli product that exists in the state space of the separated two qubits in (31). Conjugating S and |Ψ>, we obtain the following equation:

number

[0116] Compressed Paulistring S comp =A comp A2…A (n-1) is |Ψ in Eq.(32). comp > can exist in the entire space spanned by |Ψ>. By conjugating S and |Ψ>, we obtain the following equation:

number

[0117] Based on equations (33) and (34), the following condition

number

[0118] For all possible A0 and A1, Table II below shows the corresponding A comp Since many null matrices appear in Table II, they can be combined with a reduced set of operators to be measured, including the three single-qubit Pauli matrices and the identity matrix, to reduce the measurement overhead.

[0119] Another aspect of the present disclosure includes the distribution of extra CNOT when using GC in QSR. In GC, groups of Pauli strings are created, which can be drawn from the original set of strings required by the VQE simulation and measured simultaneously. Each group may contain a different number of extra CNOTs. Each group may require a different number of measurements, which may be less than the number of Pauli strings within the group. FIG. 10 shows diagrams 1000, 1010, 1020, 1030 of the distribution of the number of extra CNOTs in the number of measurements of the GC method for water molecules. The first diagram 1000 illustrates a HF with 1 extra CNOT. The second diagram 1010 illustrates a HF with 9 extra CNOTs. The third diagram 1020 illustrates a HF with 19 extra CNOTs. The fourth diagram 1030 illustrates a HF with 28 extra CNOTs. The number of extra CNOTs and the number of measurements to be performed can increase as a function of the size of the larger Ansatz.

[0120] 11 is a block diagram illustrating an example of a QIP system 1100 according to an embodiment of the present disclosure. The QIP system 1100 may also be referred to as a quantum computing system, a computing device, a trapped ion system, or the like.

[0121] The QIP system 1100 may include a source 1160 that provides atomic species (e.g., a plume or flux of neutral atoms) to a chamber 1150 having an ion trap 1170 that ionizes (e.g., photoionizes) and traps the atomic species. In one example, a power controller 1140 may supply the electrical energy used by the source 1160 to generate the atomic flux.

[0122] The imaging system 1130 can include a high resolution imager (e.g., a CCD camera) for monitoring the atomic ions while they are provided to the ion trap or after they are provided to the ion trap 1170. In one embodiment, the imaging system 1130 can be implemented separately from the optical controller 1120, although the use of fluorescence to detect, identify, and label the atomic ions using image processing algorithms may need to be coordinated with the optical controller 1120.

[0123] QIP system 1100 may include an algorithm component 1110 that may operate with other parts of QIP system 1100 (not shown) to perform quantum algorithms or operations, including combinatorial stacks or sequences of single-qubit and / or multi-qubit (e.g., two-qubit) operations, as well as extended quantum computation. In this manner, algorithm component 1110 may provide instructions to various components of QIP system 1100 (e.g., to light controller 1120) to enable implementation of quantum algorithms or operations.

[0124] Referring now to FIG. 12, an exemplary computing device 1200 according to aspects of the disclosure is illustrated. The computing device 1200 may represent, for example, a single computing device, multiple computing devices, or a distributed computing system. The computing device 1200 may be configured as a quantum computer (e.g., a quantum information processing (QIP) system), a classical computer, or a combination of quantum and classical computing capabilities. For example, the computing device 1200 may be used to process information using quantum algorithms based on trapped ion technology, and thus may implement methods for executing, optimizing, compiling, and / or executing quantum circuits as described above. A general example of a computing device 1200 as a QIP system that may implement various techniques described herein is illustrated in the example shown in FIG.

[0125] In one example, the computing device 1200 may include a processor 1210 for performing processing functions related to one or more features described herein. The processor 1210 may include a single or a set of multiple processors or a multi-core processor. Furthermore, the processor 1210 may be implemented as an integrated and / or distributed processing system. The processor 1210 may include a central processing unit (CPU), a quantum processing unit (QPU), a graphics processing unit (GPU), or a combination of these types of processors. In one aspect, the processor 1210 may refer to a general-purpose processor of the computing device 1200, which may also include additional processors 1210 for performing more specific functions, such as functions for individual beam control.

[0126] In one example, the computing device 1200 may include a memory 1220 that stores instructions executable by the processor 1210 to perform functions described herein. In one implementation, for example, the memory 1220 may correspond to a computer-readable storage medium that stores code or instructions for performing one or more of the functions or operations described herein. In one aspect of the present disclosure, the memory 1220 may be a non-transitory computer-readable medium that stores code or instructions for performing one or more of the functions, techniques, and / or operations described herein. In one example, the memory 1220 may include instructions for performing aspects of a method according to the present disclosure. As with the processor 1210, the memory 1220 may refer to a general memory of the computing device 1200, which may also include additional memory 1220 that stores instructions and / or data for more specific functions, such as instructions and / or data for individual beam control.

[0127] Additionally, computing device 1200 may include a communications component 1230 that provides for establishing and maintaining communications with one or more parties utilizing the hardware, software, and services described herein. Communications component 1230 may carry communications between components on computing device 1200 or between computing device 1200 and external devices, such as devices located over a communications network and / or devices connected serially or locally to computing device 1200. For example, communications component 1230 may include one or more buses and may further include transmit chain and receive chain components associated with transmitters and receivers, respectively, operable to interface with external devices.

[0128] Additionally, computing device 1200 may include data store 1240, which may be any suitable combination of hardware and / or software that provides mass storage of information, databases, and programs employed in connection with the implementations described herein. For example, data store 1240 may be a data repository for operating system 1260 (e.g., a classical OS or a quantum OS). In one implementation, data store 1240 may include memory 1220.

[0129] Computing device 1200 may include a user interface component 1250 operable to receive input from a user of computing device 1200 and further operable to generate output for presentation to the user or for provision to another system (directly or indirectly). User interface component 1250 may include one or more input devices, including but not limited to a keyboard, a number pad, a mouse, a touch-sensitive display, a digitizer, navigation keys, function keys, a microphone, a voice recognition component, any other mechanism capable of receiving input from a user, or any combination thereof. Additionally, user interface component 1250 may include one or more output devices, including but not limited to a display, a speaker, a haptic feedback mechanism, a printer, any other mechanism capable of presenting output to a user, or any combination thereof.

[0130] In one implementation, the user interface component 1250 may send and / or receive messages corresponding to the operation of the operating system 1260. Additionally, the processor 1210 may execute, and the memory 1220 or the data store 1240 may store, the operating system 1260 and / or applications or programs.

[0131] If the computing device 1200 is implemented as part of a cloud-based infrastructure solution, the user interface component 1250 may be used to enable a user of the cloud-based infrastructure solution to interact with the computing device 1200 remotely.

[0132] In some implementations, computing device 1200 may be implemented to perform classical computing, quantum computing, or both. Computing device 1200 may be implemented as a classical computer, a quantum computer, or a hybrid classical-quantum computer.

[0133] In one aspect of the disclosure, the techniques described above may be performed by a quantum computer, a classical computer, or a hybrid classical-quantum computer. For example, techniques for simulating a quantum circuit, optimizing a quantum circuit, compiling a quantum circuit, and / or executing a quantum circuit may be performed by computing device 1200.

[0134] In some aspects, computing device 1200 and / or one or more of the subcomponents of computing device 1200 may be configured to implement and / or define means for implementing the techniques described above.

[0135] FIG. 13 shows a chart 1300 showing the number of measurements of a water molecule using the various optimization schemes described above. Compared to the number of measurements optimized using only QSR, the number optimized using GC partitioning and QSR is more than two orders of magnitude smaller, while the number optimized using QWC partitioning with QSR is about one order of magnitude smaller. Chart 1350 shows the additional cost in terms of the number of two-qubit gates associated with using GC partitioning with QSR compared to the total number of two-qubit gates using JW transformation. The ratio of the two can decrease as the number of Ansatz terms increases.

[0136] In some implementations, the computation may be performed using the HMP2 framework with UCCSD Ansatz. D of the UCCSD Ansatz operator of the VQE cycle α The total number of measurements as a function of the number of terms N is shown in chart 1300 for three different optimization schemes.

[0137] In some implementations, the average number of additional two-qubit gates, nGC+QSR, resulting from using GC partitioning with QSR is shown in chart 1350. Chart 1350 also shows the total number of two-qubit gates, nGC+QSR, to induce a UCCSD Ansatz state implemented using the JW transformation. JW Also shown is n GC+QSR The distribution of is as described above. Also, chart 1350 shows the ratio of the above numbers.

number

[0138] In one implementation, a choice can be made between QWC+QSR and GC+QSR depending on a resource cost function. For example, if the two-qubit gate quality is tighter than the number of measurements divided by the total execution time, then QWC+QSR may be selected. Otherwise, GC+QSR may be selected.

[0139] Aspects of the disclosure include a method that includes predicting a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; minimizing an energy of the system based on the first set of Ansatz terms and the first initial amplitudes; calculating at least one of the energy or perturbation corrections in the wave functions based on one or more Ansatz wave functions; determining whether the energy of the system has converged; and in response to determining that the energy of the system has not converged, predicting a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms.

[0140] One aspect of the disclosure includes the method described above, further comprising minimizing the energy of the system based on the second set of Ansatz terms.

[0141] Aspects of the present disclosure include any of the methods described above, further including generating an output of the simulation in response to determining that the energy of the system converges.

[0142] Aspects of the disclosure include any of the methods described above, wherein predicting the first set of Ansatz terms and the first initial amplitude includes predicting using second-order Möller-Plesset (MP2) perturbation theory.

[0143] Aspects of the present disclosure include any of the methods above, wherein the first set of Ansatz includes unitary coupled cluster Ansatz having single or double excitations.

[0144] Aspects of the disclosure include any of the methods above, wherein minimizing the energy of the system includes calculating the energy using a variational quantum eigenvalue solver (VQE) approach.

[0145] Aspects of the present disclosure include any of the methods above, further including calculating an energy correction operator via a hybrid second-order Moller-Plesset perturbation (HMP2) method.

[0146] Aspects of the present disclosure include any of the methods described above, further including compiling a circuit associated with the simulation, optimizing the circuit associated with the simulation, and running the circuit via a quantum computer.

[0147] Aspects of the present disclosure include any of the methods above, wherein predicting the second set of Ansatz terms and the second initial amplitude includes increasing an Ansatz size of the second set of Ansatz terms.

[0148] Aspects of the disclosure include any of the methods above, wherein predicting the second set of Ansatz terms and the second initial amplitudes includes determining one or more additional Ansatz terms to add to the first set of Ansatz terms to generate the second set of Ansatz terms, and adding the one or more additional Ansatz terms to the first set of Ansatz terms to generate the second set of Ansatz terms, and further including minimizing the energy of the system based on the second set of Ansatz terms and the second initial amplitudes.

[0149] The previous description of the disclosure is provided to enable any person skilled in the art to make or use the disclosure. Various modifications of the disclosure will be readily apparent to those skilled in the art, and the common principles defined herein may be applied to other variations without departing from the spirit or scope of the disclosure. Furthermore, although elements of the described aspects may be described or claimed in the singular, the plural is contemplated unless limitation to the singular is expressly stated. Furthermore, all or a portion of any aspect may be utilized with all or a portion of any other aspect, unless otherwise stated. Thus, the disclosure is not intended to be limited to the examples and designs described herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. 1. A method for performing a simulation of a system, comprising: predicting a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; minimizing the energy of the system based on the first set of Ansatz terms and the first initial amplitudes; calculating at least one of the energy or wavefunction perturbation corrections based on one or more Ansatz wavefunctions; determining whether the energy of the system converges; in response to determining that the energy of the system has not converged, predicting a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms; A method comprising:

2. The method of claim 1 , further comprising minimizing the energy of the system based on the second set of Ansatz terms.

3. The method of claim 1 , further comprising the step of generating an output of the simulation in response to determining that the energy of the system has converged.

4. 2. The method of claim 1 , wherein predicting the first set of Ansatz terms and the first initial amplitude comprises predicting using second-order Moller-Plesset (MP2) perturbation theory.

5. The method of claim 1 , wherein the first set of Ansatz comprises unitary coupled cluster Ansatz having single or double excitations.

6. The method of claim 1 , wherein minimizing the energy of the system comprises calculating the energy using a Variational Quantum Eigenvalue Calculator (VQE) approach.

7. The method of claim 6 , further comprising the step of calculating an energy correction operator by a hybrid second-order Moller-Plesset perturbation (HMP2) method.

8. compiling a circuit associated with said simulation; optimizing a circuit associated with said simulation; running the circuit via a quantum computer; The method of claim 1 further comprising:

9. 2. The method of claim 1, wherein predicting the second set of Ansatz terms and the second initial amplitudes comprises increasing an Ansatz size of the second set of Ansatz terms.

10. The step of predicting the second set of Ansatz terms and the second initial amplitudes comprises: determining one or more additional Ansatz terms to add to the first set of Ansatz terms to generate the second set of Ansatz terms; adding the one or more additional Ansatz terms to the first set of Ansatz terms to generate the second set of Ansatz terms; minimizing the energy of the system based on the second set of Ansatz terms and the second initial amplitudes. The method of claim 1 further comprising:

11. When executed by a processor, the processor: instructions for predicting a first set of Ansatz terms and first initial amplitudes associated with the first set of Ansatz terms; instructions for minimizing the energy of a system based on the first set of Ansatz terms and the first initial amplitudes; instructions for calculating at least one of the energies or wavefunction perturbation corrections based on one or more Ansatz wavefunctions; instructions for determining whether the energy of the system has converged; instructions for predicting a second set of Ansatz terms and second initial amplitudes associated with the second set of Ansatz terms in response to determining that the energy of the system has not converged; A non-transitory computer readable medium having stored therein.

12. 12. The non-transitory computer-readable medium of claim 11, further comprising instructions for minimizing the energy of the system based on the second set of Ansatz terms.

13. 12. The non-transitory computer readable medium of claim 11, further comprising instructions for generating an output of a simulation in response to determining that the energy of the system has converged.

14. 12. The non-transitory computer-readable medium of claim 11, wherein the instructions for predicting the first set of Ansatz terms and the first initial amplitude comprise instructions for predicting using second-order Moller-Plesset (MP2) perturbation theory.

15. 12. The non-transitory computer readable medium of claim 11, wherein the first set of Ansatz comprises unitary coupled cluster Ansatz having single or double excitations.

16. 12. The non-transitory computer-readable medium of claim 11, wherein the instructions for minimizing the energy of the system include instructions for calculating the energy using a variational quantum eigensolver (VQE) approach.

17. 20. The non-transitory computer-readable medium of claim 16, further comprising instructions for computing an energy correction operator via a Hybrid Second Order Moller-Plesset Perturbation (HMP2) method.

18. instructions for compiling a circuit associated with the simulation; instructions for optimizing a circuit associated with said simulation; 12. The non-transitory computer-readable medium of claim 11, further comprising: instructions for executing the circuit via a quantum computer.

19. 12. The non-transitory computer-readable medium of claim 11, wherein the instructions for predicting the second set of Ansatz terms and the second initial amplitudes comprise instructions for increasing an Ansatz size of the second set of Ansatz terms.

20. The instructions for predicting the second set of Ansatz terms and the second initial amplitude include: instructions for determining one or more additional Ansatz terms to add to the first set of Ansatz terms to generate the second set of Ansatz terms; instructions for adding the one or more additional Ansatz terms to the first set of Ansatz terms to generate the second set of Ansatz terms; Including, instructions for minimizing the energy of the system based on the second set of Ansatz terms and the second initial amplitudes.

12. The non-transitory computer-readable medium of claim 11, further comprising:

Images