Quantum computing device, method of operation and non-transitory computer-readable storage medium

Abstract

A hybrid computer mechanism configured to perform a simulation of a chemical system includes a combination of a classical computer coupled to a quantum computer, and the hybrid computer mechanism, when in use, is configured to receive input data and generate corresponding processed output data from the input data, and is configured to (a) receive information describing the chemical system in the input data, (b) process the information describing the chemical system using a preentangler algorithm and a fixed circuit algorithm to generate a quantum Ansatz that defines initial values ​​for a quantum circuit calculation and a Hamiltonian from which a variational circuit algorithm is generated, (c) calculate a corresponding quantum circuit using the quantum Ansatz and variational circuit algorithm to generate a quantum calculation result, and (d) process the quantum calculation result to generate output data including information describing an electron trajectory simulation of the chemical system.

Description

[Technical Field]

[0001] The present disclosure relates to quantum computing devices, e.g., hybrid computing devices including a combination of a classical binary computer coupled to a quantum computer. The present disclosure also relates to methods for operating the quantum computing devices. Furthermore, the present disclosure relates to software products recorded on machine-readable media, the software products being executable on the quantum computing devices to perform the methods. [Background technology]

[0002] Quantum computers have recently become available as noisy intermediate-scale quantum (NISQ) devices [1], and quantum computers can be classically simulated (i.e., "emulated"). One technical problem with NISQ devices is that their currently available qubits are too noisy to perform the deep quantum circuits required for most useful applications of quantum computers. For example, qubit phase estimation is difficult to implement on NISQ devices. To bridge the gap between modern NISQ devices and future fault-tolerant quantum computing devices, many researchers have turned to variational quantum algorithms, such as variational quantum eigensolvers (VQEs), quantum optimization algorithms, variational imaginary time evolution algorithms, variational quantum adiabatic algorithms, and quantum neural network algorithms [2–8]. Variational quantum circuits that can be implemented using NISQ devices are typically shallower than many other types of quantum circuits implemented using such devices, and because of this shallowness, they beneficially exhibit a certain algorithmic resilience to noise. Furthermore, such variational quantum circuits function iteratively by evaluating a given objective function of a given optimization problem on a given NISQ device and updating the variational parameters of the variational quantum circuit using a classical optimization algorithm.

[0003] Unfortunately, devising sufficiently expressive quantum circuit ansatz suffers from exponentially vanishing gradients as a function of the number of qubits used, resulting in a barren plateau that takes exponential time to escape [9]. In the presence of noise, even less expressive problem-inspired ansatz suffers from the barren plateau as long as one attempts to train a superlinear number of parameterized gates

[10] . Therefore, reducing the number of variational parameters is essential for the success of variational quantum algorithms in the near future. Vanishing gradients also posed an early challenge to deep neural networks in classical machine learning.

[0004] In general, classical canonical transformation theory typically separates the electronic correlations of molecules into two components. Static correlations have been used to quantify the portion of the electronic correlations associated with multiple relevant determinants close in energy to the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO). Dynamic correlations describe corrections to the ground state resulting from excitations involving low-lying core orbitals or high-lying virtual orbitals. When static correlations are negligible, this is called single-reference chemistry. In such situations, classical coupled cluster (CC) methods, if performed with a sufficiently large basis set, are generally sufficient to describe the dynamic correlations.

[19] Conversely, multireference situations arise when a single determinant is insufficient to describe the chemical bond even qualitatively. In practice, these situations are found in chemical reactions, i.e., crossing the HOMO-LUMO gap where the valence configurations of the products and reactants are nearly vanishing, as well as in excited states and transition metal chemistry

[20] .

[0005] To deal with multi-reference situations, when conducting quantum chemical simulations, it is common practice to divide the molecular orbitals into core (c), active (a), and virtual (v) orbitals, as shown in Figure 1b. (i) The core orbit is defined as being completely filled; (ii) the orbitals in the active space are partially filled; (iii) The orbit in virtual space is empty. The optimal method for assigning active space is unknown and is initially based on chemical intuition. Modern computer software programs such as BLOCK

[21] and AutoCAS [22-25] use Fiedler vectors as a proxy for mutual information between orbitals and use machine learning algorithms to automate the assignment of active space. Then, the Hilbert space is

[0006]

number

[0007] Many methods have been developed to address the multi-reference situation, such as multi-reference coupled clusters

[26] , all of which are computationally expensive when implemented using NISQ devices. Below, we further discuss canonical transformation (CT) theory to aid in understanding the subject matter of this disclosure. This disclosure uses the CAS-DMRG Ansatz to define the initial values ​​of qubits used in implementing quantum circuits on NISQ devices. In CT theory, for a given qubit, the initial values ​​are given by equations (1) and (2).

[0008]

number

[0009]

number

[0010] where M is the number of spin orbitals in the active space and the physical index i j is in the range of 0 to 1, and α j is 1...D j It takes values ​​in the range of D j is the bond dimension of bond j. The bond dimension D of an MPS is the sum of all D j Objective A [j] is a matrix for j∈{1,M}, a 3-leg tensor for the given bulk. The optimization cost is O(MD 3 ), operations up to D = 1000 can be routinely performed, and the largest bond dimension reported for supercomputer chemical simulations is D = 65536

[27] . In practice, much lower bond dimensions can yield accurate results, especially in pseudo-1D geometries, where convergence has been achieved for active spaces of up to 100 orbitals [28, 29].

[0011] The main challenge of CAS-DMRG is to recover the dynamic correlations. The single-reference CC method is applicable because a given excitation operator assumes fixed orbital occupations of a given reference state. One proposal to overcome this problem is the canonical transformation (CT) theory mentioned above [14, 30-33]. In CT, the CC excitation operator is replaced by a unitary operator given in equation (3): U=e T (3) where:

[0012]

number

[0013]

number

[0014]

number

[0015]

number

[0016] On classical binary computers, an exact and efficient implementation of equation (5) is not possible. This is because expressions like [[H,T],T] contain N-point correlation functions. While one- and two-body reduced density matrices of the reference wave function can be computed, computing the general N-body terms necessarily requires exponential time. Therefore, in CT theory, higher-order terms are approximated by sums of products of one- and two-body terms, using so-called cumulant expansions

[34] . The convergence of cumulant expansions is generally fast in single-reference scenarios but slow for multi-reference systems. Ironically, these are precisely the systems for which CAS-DMRG is particularly well suited

[35] . Therefore, to overcome the key limitations of CT theory, an alternative method for computing (5) must be found. Summary of the Invention [Problem to be solved by the invention]

[0017] It is an object of the present disclosure to provide improved methods for configuring quantum computing devices to perform quantum chemical simulations, such simulations being usable in the manufacturing of chemical materials, pharmaceuticals, and in the design and control of chemical manufacturing machinery processes. [Means for solving the problem]

[0018] According to a first aspect, there is provided a method for configuring a hybrid computing arrangement to perform a simulation of a chemical system, the hybrid computing arrangement comprising a combination of a classical computer coupled to a quantum computer, the hybrid computing arrangement being configured, in use, to receive input data and generate corresponding processed output data from the input data, the method comprising: (a) configuring a classical computer to receive information describing a chemical system in input data; (b) configuring a classical computer to process information describing the chemical system using a pre-entangler algorithm and a fixed-circuit algorithm to generate quantum Ansatz defining initial values ​​for the quantum circuit computation and a Hamiltonian from which a variational circuit algorithm is generated; (c) configuring a quantum computer to use quantum Ansatz and variational circuit algorithms to compute the corresponding quantum circuit to generate a quantum computation result; (d) configuring a classical computer to process the quantum computation results to generate output data that includes information describing the electron trajectory simulation of the chemical system.

[0019] The present invention is advantageous in that the use of preentanglers, e.g., parameter-free preentanglers, allows for the offloading of computational load from quantum computers to classical computers. Furthermore, the present invention is advantageous in that it allows for the reduction of the number of parameters in variational quantum circuit algorithms, especially as the number of qubits used in quantum circuits increases.

[0020] Optionally, the method includes configuring a preentangler algorithm to function as a parameter-free preentangler. Further optionally, the method includes configuring the preentangler algorithm using a matrix product state (MPS) algorithm. Further optionally, the method includes generating a matrix product state (MPS) based on a linear combination of unitaries describing the chemical system. Optionally, the method includes configuring a hybrid computer system to generate the matrix product state (MPS) using a density matrix renormalization group (DMRG) algorithm to capture the complete active space (CAS) for one or more nonlinear transition metal complexes included in the chemical system. Optionally, the method includes configuring a hybrid computer system to generate the matrix product state (MPS) by using an MPS algorithm based on iterative generation using ancilla.

[0021] Optionally, the method includes finding a ground state of the Hamiltonian using a quantum circuit.

[0022] Optionally, the method includes configuring a variational quantum circuit algorithm as a variational quantum eigensolver based on one or more canonical transformations, and the method includes configuring a hybrid computer mechanism to generate a density matrix renormalization group (DMRG) algorithm using a classical computer to construct static correlations in wave functions describing the chemical system, the density matrix renormalization group (DMRG) algorithm being used to generate a corresponding DMRG-QCT portion of the quantum circuit.

[0023] Optionally, the method comprises: (i) assigning molecular orbitals that can occupy core space, virtual space, and active space to an active space based on a molecular system; (ii) algorithmically approximating the ground state of the Hamiltonian by generating a matrix product state (MPS) description |ψ0〉 of the ground state for a given coupling dimension D using the MSRG algorithm; (iii) Finding a quantum circuit that creates the MPS description |ψ0〉 on a quantum computer, and (iv) Constructing a variational quantum circuit that describes dynamic correlations by coupling active orbitals in the core space and the active space; and (v) generating an output result by minimizing the ground state energy from the result of executing the quantum circuit; and configuring a hybrid computer mechanism to:

[0024] According to a second aspect, there is provided a hybrid computing arrangement configured to perform a simulation of a chemical system, the hybrid computing arrangement comprising a combination of a classical computer coupled to a quantum computer, the hybrid computing arrangement being configured, in use, to receive input data and generate corresponding processed output data from the input data, the computing arrangement comprising: (a) configured to receive information describing a chemical system in input data; (b) configured to process information describing the chemical system using a preentangler algorithm and a fixed circuit algorithm to generate a quantum Ansatz that defines initial values ​​for the quantum circuit computation and a Hamiltonian from which a variational circuit algorithm is generated; (c) configured to compute a corresponding quantum circuit using quantum Ansatz and variational circuit algorithms to generate a quantum computation result; (d) configured to process the quantum computation results to generate output data including information describing the electron trajectory simulation of the chemical system.

[0025] Optionally, in a hybrid computer setup, a preentangler algorithm is configured to function as a parameter-free preentangler.

[0026] Optionally, in the hybrid computing system, the preentangler algorithm is configured to use a matrix product state (MPS) algorithm. Further optionally, the hybrid computing system is configured to generate matrix product states (MPS) using a density matrix renormalization group (DMRG) algorithm to capture a complete active space (CAS) for one or more nonlinear transition metal complexes included in the chemical system. Further optionally, the hybrid computing system is configured to generate matrix product states (MPS) by using an MPS algorithm based on iterative generation using an ancillary.

[0027] Optionally, a hybrid computer mechanism is configured to find the ground state of the Hamiltonian using quantum circuits.

[0028] Optionally, the hybrid computer mechanism is configured to include a variational quantum circuit algorithm as a variational quantum eigenvalue solver based on one or more canonical transformations, and the hybrid computer mechanism is configured to generate a density matrix renormalization group (DMRG) algorithm using a classical computer to construct static correlations in wave functions describing the chemical system, and the density matrix renormalization group (DMRG) algorithm is used to generate a corresponding DMRG-QCT portion of the quantum circuit.

[0029] Optionally, a hybrid computer mechanism (i) assigning molecular orbitals that can occupy core space, virtual space, and active space to an active space based on a molecular system; (ii) algorithmically approximating the ground state of the Hamiltonian by generating a matrix product state (MPS) description |ψ0〉 of the ground state for a given coupling dimension D using the MSRG algorithm; (iii) Finding a quantum circuit that creates the MPS description |ψ0〉 on a quantum computer, and (iv) constructing a variational quantum circuit that describes the dynamic correlation by coupling active orbitals in the core space and the active space; and (v) generating an output result by minimizing the ground state energy from the result of executing the quantum circuit; The device is configured to:

[0030] According to a third aspect, there is provided a non-transitory computer readable storage medium comprising unique computer readable instructions executable on data processing hardware, the unique computer readable instructions, when executed using the data processing hardware, performing the method of the first aspect.

[0031] Further aspects, advantages, features, and objects of the present disclosure will become apparent from the following detailed description of the drawings and illustrative embodiments, taken in conjunction with the appended claims.

[0032] It will be appreciated that features of the present disclosure can be combined in various combinations without departing from the scope of the present disclosure as defined in the appended claims.

[0033] Embodiments of the present disclosure will be described with reference to the following drawings. [Brief explanation of the drawings]

[0034] [Figure 1A] FIG. 1 is a diagram of a quantum computing device that includes a classical computing mechanism coupled with a quantum computing mechanism, the classical and quantum computing mechanisms working in conjunction to perform computational tasks that involve the creation and execution of quantum circuits. [Figure 1B] (a) is a diagram of the general structure of the pre-entangled variational quantum algorithm, and (b) is a diagram of the classification of spin-orbitals in the complete active space method. [Figure 2] 1 is a diagram of a quantum circuit for computing DMRG-QCT. [Figure 3](a) is a diagram of a water molecule with balanced bond angles, (b) is a diagram of a nitrogen dimer, (c) is a diagram of a BeH2 molecule, which is the sum of a Be atom and an H2 molecule, and (d) is a diagram of a P4 system containing two H2 molecules. [Figure 4] This figure shows the absolute energy error of different methods for approximating the ground-state energy using quantum circuits. The energy error is strictly positive for all methods except for CCSD, which is negative for some points. Figure 4 shows (a) a water molecule in the STO-6G basis set with 14 qubits and an active space used for DMRG calculations containing 5 × 2 = 10 spin orbitals; (b) a nitrogen dimer in STO-6G with an active space of 12 spin orbitals; (c) BeH2 in the STO-6G basis set with an active space of 10 spin orbitals; and (d) a P4 model system with a 6-31G basis set and an active space containing 12 spin orbitals. All plots shown in Figure 4 share the same common legend, shown separately in (a) and (b). [Figure 5] FIG. 4 shows the energy error as a function of the bond dimension D used in DMRG for a water molecule with R(OH)=108 Å, where D=16 is the exact active (i.e., no approximation) bond dimension used in FIG. [Figure 6] (a) Diagram of the increasing number of parameters for GUCC and DMRG-QCT Ansatz, with an inset showing the same given data but normalized by the number of parameters for the GUCC Ansatz; (b) and (c) are diagrams of the scaling of the required quantum circuit depth and number of quantum gates as a function of the number of variational parameters. [Figure 7] (a) Illustrates the fidelity of the approximate states produced by the SEQ algorithm for the ground state of the N2 molecule in the 20-qubit STO-6G basis set, with an interatomic distance N N = 1.8 Å, and (b) illustrates the energy error for the same molecule, with the gray region indicating errors below chemical precision. [Figure 8](i) (Top) Diagram of a nondeterministic circuit for constructing a linear combination of unitaries, and (ii) (Bottom) Diagram of a multi-qubit controlled unitary explained in terms of one-body and two-body gates. [Figure 9A] 1 is a table of circuit requirements for molecules considered for the molecules described in this disclosure. In the table, (●) represents the QCT-S and QCT-SD requirements normalized to GUCC-SD. The third column contains the coupling dimension D of the MPS by DMRG and the number of layers DSEQ used by the successive unitary algorithm to approximate the MPS to a given desired accuracy. [Figure 9B] This figure shows the probability of measuring zero with the LCU circuit as shown in Figure 8. Data is shown inset for the convergence of k as a function of 1 / DLCU. HAF represents the ground state data for the 16-point Heisenberg antiferromagnet (HAF) chain H = Σi(XiXi+1 + YiYi+1 + JZZiZi+1). HO represents the data for the water molecule shown in Figure 3(a) in the 14-qubit STO-6G basis set. Additionally, P4 system represents the results for the ground state of the P4 model system shown in Figure 3(c) in the 16-qubit STO-6G basis set. [Figure 10A] Figure 10A(I) shows the results for the ground state of the 20-spin HAF. Furthermore, (a, b) show the Schmidt values ​​corresponding to the two algorithms without any truncation of Dmax. Furthermore, (c) shows the bond dimension as a function of the number of layers. Furthermore, (d) shows the convergence of the fidelity corresponding to the two algorithms with truncation to Dmax = 3D. [Figure 10B] Figure 10B(II) shows data for the N2 molecule. Furthermore, (a, b) show the Schmidt values ​​corresponding to the two algorithms without any truncation of Dmax. Furthermore, (c) shows the bond dimension as a function of the number of layers. And (d) shows the convergence of fidelity corresponding to the two algorithms with truncation to Dmax=3D. [Figure 11]For a given basis state, the first row shows the fidelity and the second row shows the energy error. Panel (I) shows results for the P4 system shown in Figure 3a in the 6-31G basis set containing 16 qubits. Panel (II) shows data for the N2 molecule shown in Figure 3b with 20 qubits. Panel (III) shows results for the HAF with 32 spins. [Figure 12] 1 is a diagram of method steps for implementing an embodiment according to the present disclosure. [Figure 13] 1A-1C are diagrams of quantum gates and their equivalents to illustrate implementations of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0035] In the accompanying drawings, underlined numbers are used to identify the element in which the underlined number is located or to which the underlined number is adjacent. When a number is not underlined and has an associated arrow, the ununderlined number is used to identify the entire element to which the arrow is pointing.

[0036] As an overview, modern computing hardware—for example, those configured using silicon-based integrated circuits and associated data memory devices for data processing—becomes more computationally powerful and faster as the feature sizes of the integrated circuits decrease. As circuit feature sizes approach the nanometer scale, Heisenberg's uncertainty principle and quantum effects become more prominent in the design and operation of integrated circuits. At the limit, representing data using single particles, such as ions or photons, in the form of qubits, represents the limit of miniaturization and offers the fastest computing performance, especially when phenomena such as superposition and entanglement are exploited. Such techniques are utilized in modern quantum computers. An overview of quantum computing and quantum information is provided in the academic publication "Quantum Computations and Quantum Information" by Nagy et al. (The International Journal of Parallel, Emergent and Distributed Systems, Vol. 21, No. 1, February 2006, pp. 1–59).

[0037] In some cases, quantum computers and quantum computing systems may exploit the quantum properties of fields and particles to improve computational speed compared to classical computers, enabling the solution of computationally complex problems within timeframes suitable for practical applications. In classical data processing, where data represent, for example, real-world physical signals, many mathematical operations are used to process the data, such as calculating averages, correlations, Fourier transforms, and Hadamard transforms. The data to be processed can be quite large—terabytes, or even petabytes, of information, acquired serially over a period of time, for example, from a sensor array. The processing time and associated latency for processing the data are important factors, for example, in real-time control situations. Quantum computers have the potential to be configured to process large amounts of data very efficiently. However, a challenging contemporary technical problem is how to optimally configure quantum computers to implement such data processing. For example, given the level of noise introduced by various elements of quantum circuits, a technical problem is how to use a quantum computer to perform a computational task (e.g., a quantum computing task) such that the noise introduced during the calculation does not impose limits on the minimum computational error achievable.

[0038] While it is known to use a combination of classical and quantum computers when attempting to solve extremely complex problems, in this disclosure, such a combination is referred to as a quantum computing system. Classical computers use silicon integrated circuit devices configured to operate at substantially room temperature (approximately +20°C), whereas quantum computers are configured to be cooled to extremely low temperatures (approximately -273°C) to function most effectively. When addressing a given computational task, the task can be divided into execution on a classical computer and execution on a quantum computer. Furthermore, for some tasks, classical computers are more efficient than quantum computers at handling simple computational tasks, while quantum computers may be able to solve certain types of computational tasks that are inefficient to run on classical computers. However, transferring tasks between classical and quantum computers incurs time overhead, which is preferably reduced as much as possible to achieve optimal computational performance in a quantum computing system.

[0039] When using such a tandem configuration of classical and quantum computers, it may be advantageous to perform certain arithmetic calculations on the quantum computer rather than the classical computer, e.g., to increase data processing throughput. A technical problem that arises is how to most effectively configure a quantum computer to perform certain types of arithmetic calculations such that noise generated during the quantum calculations does not impose constraints on the total number of quantum operations that can be performed for the quantum computing task. Accordingly, some embodiments of the present disclosure are directed to addressing the technical problem of configuring a classical computing system tandemly coupled with a quantum computer to enable more efficient computation of input data to generate corresponding processed output data, and to reduce computation errors in the processed output data while keeping quantum noise generated within the quantum computer below a threshold.

[0040] FIG. 1A illustrates a schematic diagram of a quantum computing system 100. In some examples, the quantum computing system 100 may receive input data 101 from a data source and generate output data 102. In some cases, the input data 101 may include a real-valued function, and the output data 102 may include a predicted value of the real-valued function over a probability distribution. In some embodiments, the output data 102 may include an estimated quantum amplitude of error below a threshold error. In some implementations, the quantum computing system 100 may include at least a classical computing system 110 (also referred to as a classical computer or a binary data computer) and a quantum computer 130 in communication with the binary data computer. In some cases, the classical computing system 110 may be in communication with the quantum computer 130. In some examples, the classical computing system 110 may be combined and coupled with the quantum computer 130. The classical computing system 110 may exchange data with the quantum computer 130 via one or more data links, which may be provided, for example, using a data highway. In some examples, the classical computing system 110 may include non-transitory memory and at least one electronic processor configured to execute computer-executable instructions (e.g., software instructions or program instructions) stored in the non-transitory memory. In some examples, the electronic processor may be implemented using silicon integrated circuits that, when in use, perform binary digital calculations. The one or more data processors may be configured to execute software instructions to process input data 101 and generate output data 102, with assistance from a quantum computer 130 coupled to the classical computing system 110.

[0041] The memory may be non-volatile memory such as flash memory, a hard disk, magnetic disk memory, optical disk memory, or any other type of non-volatile memory. Additionally, memory types may include, but are not limited to, random access memory ("RAM") and read-only memory ("ROM"). In some examples, classical computing system 110 may be programmed to execute different procedures, each implemented based on a different set of instructions.

[0042] In some cases, the electronic processor of classical computing system 110 may execute computer-executable instructions to receive input data 101 from a data source (e.g., from a sensor or sensor network), process the input data 101 to generate quantum computer input data 103, and transmit the quantum computer input data 103 to quantum computer 130. In addition, the electronic processor of classical computing system 110 may transmit configuration data 105 that can be used to configure quantum computer 130 (e.g., to configure one or more quantum gates of quantum computer 130). In some cases, the configuration data 105 may be stored in memory of classical computing system 110. In some other cases, electronic processing of classical computing system 110 may generate the configuration data 105 based at least in part on data stored in memory of classical computing system 110. In some cases, the configuration data 105 may be provided by a user via a user interface (e.g., a user interface of classical computing system 110).

[0043] In some examples, quantum computer input data 103 may include any data related to instructions that can be executed or used by a controller of quantum computer 130 to control and manage certain operational aspects of quantum computer 130. In some examples, any data included in quantum computer input data 103 may include instructions usable by a compiler (e.g., a quantum compiler) executed by a controller of quantum computer 130 to, for example, mitigate errors and manage the placement of qubits.

[0044] In some cases, the data source may include, for example, one or more of a data memory in which data is stored, a sensor mechanism configured to stream sensor data, and a user interface. In some embodiments, the data memory source may include an electronic memory configured to store computer-executable data. The data may include data received from a user, another computing system (e.g., a classical or quantum computing system), or a sensor. In some examples, the sensor mechanism may include a sensor that generates sensor data in real time, data streamed from a satellite, a camera surveillance system, a genomic data PCR reader, an MRI 3-D imager, an encryption device, etc. Alternatively or additionally, the input data 101 may be provided from other sources, for example, financial transaction data, parameters of a physical system being modeled, etc.

[0045] Quantum computer 130 is used by classical computing system 110 to perform particularly computationally complex tasks that would take an unacceptably long time for classical computing system 110 to process. Quantum computer 130 may comprise one or more quantum circuits operating on qubits configured to perform any computationally complex task using quantum effects. In various implementations, quantum computer 130 may include one or more qubits (e.g., an array of qubits) and one or more quantum gates. In some cases, the quantum gates may comprise one or more rotation gates. In some cases, a quantum circuit may comprise at least a portion of one or more qubits and one or more quantum gates.

[0046] In some implementations, quantum computer 130 includes qubits in the range of 30-1000, more optionally 50-500, and various gates that allow for modification of quantum parameters such as qubit phase (i.e., rotation operation R) and for entanglement and superposition operations between qubits. In some examples, quantum computer 130 is configured for quantum noise reduction to reduce quantum computation errors arising from quantum noise. Furthermore, in certain configurations of quantum computer 130, its qubits and quantum gates are cooled to extremely low temperatures, e.g., within 1 Kelvin of absolute zero, during operation. Optionally, quantum computer 130 is implemented using photonic devices, cryogenic superconducting gates, or ion traps, or a combination thereof. Optionally, classical computing system 110 is spatially remote from quantum computer 130, and data is exchanged between them via one or more data communication links, e.g., Internet data links.

[0047] In some cases, the program instructions may include a quantum amplitude estimation and / or amplification algorithm (e.g., a maximum likelihood amplitude estimation algorithm), a variational approximation algorithm. In some cases, at least a portion of the quantum amplitude estimation and / or amplification algorithm and the variational approximation algorithm may be executed by quantum computer 130 using one or more quantum circuits. In some examples, one or more quantum circuits of the quantum computing system may be configured based at least in part on input data 101. Quantum computer 130 may further process outputs 104 received from one or more quantum circuits to produce results usable to generate output data 102.

[0048] In some implementations, quantum computer 130 may operate by processing a series of "shots." In some cases, an initial state may be defined (ansatz), and each shot may involve creating a qubit having the defined initial state and performing a temporal sequence of quantum operations on the qubit to generate a processed qubit having a final state. In some cases, the initial state may consist of a zero state. In some cases, quantum computer 130 may generate a processed qubit having a final state by modifying the initial state. In some cases, quantum computer 130 may read out the final state of the processed qubit using a measurement operation (e.g., a quantum measurement operation).

[0049] In some implementations, quantum computer 130 may include one or more quantum circuits configured to process qubits. In some cases, the number of quantum operations performed by a quantum circuit and / or the longest path through a quantum circuit may be referred to as the "quantum circuit depth." In some cases, a path through a quantum circuit may include a series of quantum operations performed to transform an initial quantum state into a final quantum state.

[0050] Each shot may have a temporal duration that may be limited by quantum noise generated in the qubit, which may manifest as qubit decoherence. In some examples, quantum noise generated in a qubit increases as more quantum operations are performed on the qubit. As such, quantum noise may increase with the corresponding quantum circuit depth. Despite such technical challenges, quantum computers 130 are highly effective for certain types of computational tasks. Some of the methods disclosed herein can reduce the quantum circuit depth of a quantum circuit configured to generate outputs usable to compute values ​​of arithmetic functions based on values ​​of one or more random variables associated with a probability distribution (e.g., a marginal distribution). Advantageously, the methods can reduce the quantum circuit depth without significantly increasing computation time (e.g., convergence time) and / or errors associated with the computed values.

[0051] In some cases, configuration data 105 may be generated during execution of the aforementioned software instructions prior to run-time of quantum computer 130. In some examples, configuration data 105 may include data usable to configure one or more quantum circuits of quantum computer 130 based at least in part on an mathematical function. In some such cases, at least a portion of the software instructions may include a specialized compiler that, when executed, generates configuration data 105 by compiling other portions of instructions (e.g., configuration instructions) that represent the configuration of quantum computer 130. In some examples, configuration data 105 may comprise data and / or instructions executable by quantum computer 130 to configure the quantum circuits therein in accordance with the configuration instructions.

[0052] For example, the TKET compiler provided by Cambridge Quantum Computing Ltd. can convert some of the instructions into Hamiltonian functions, which are then processed during compilation to generate Pauli sequences, from which corresponding Pauli gadgets are derived, which are then used to define configurational connections of quantum gates in quantum computer 130, e.g., to create a given quantum circuit. Such a process of ultimately converting Hamiltonian functions into configurational connections of quantum gates is described, for example, in the IBM publication "Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters" (Ewout van den Berg and Kristan Temme, IBM TJ Watson, Yorktown Height, NY, USA, 31 March 2020). The entire contents of this IBM publication are incorporated herein by reference and made a part of this specification. Additionally, the publication "A compact ion-trap quantum computing demonstrator" by Pogorelov et al. describes a practical implementation of a quantum computer 130, the entire contents of which are incorporated herein by reference and made a part of this specification.

[0053] In some cases, a quantum compiler can convert a first quantum circuit or a symbolic form of the first quantum circuit into a second quantum circuit or a symbolic form of the second quantum circuit, where the second quantum circuit or the symbolic form of the second quantum circuit constitutes an "intrinsic gate" of the target hardware. In implementation, the TKET compiler may reduce the corresponding quantum circuit depth (e.g., the quantum circuit depth of the second quantum circuit or its symbolic form). In some examples, the TKET compiler may reduce the quantum circuit depth by removing at least some apparent redundancies in the quantum circuit.

[0054] It will be appreciated that the aforementioned software instructions may relate to many types of computations that process data to produce a technical effect, such as, for example, implementing data encryption, data decryption, filtering measurement data representing measured physical parameters to reduce stochastic noise in the data, correlating measurement data to detect the occurrence of signal features that are masked by stochastic noise, etc., thereby enabling quantum computing system 100 to provide a technical effect in processing data.

[0055] In some applications, the computation may require the calculation of an arithmetic function. In some cases, it may be desirable for the arithmetic computation to be performed on the quantum computer 130 (e.g., to reduce computation time). When reading out such qubits, various methods can be used to reduce the qubit readout noise, such as quantum amplitude estimation (QAE), which requires repeated shots to calculate an average of the outputs from which a best estimate of the qubit value can be calculated. To achieve optimal data processing throughput of the quantum computing system 100, it is often useful to reduce the amount of task switching between the quantum computer 130 and the classical computing system 110.

[0056] Thus, from the foregoing, it will be appreciated that to obtain maximum performance from quantum computing system 100, it may be desirable for some of these shots to allow arithmetic calculations to be performed on quantum computer 130 in preference to using classical computing system 110. In the present disclosure, particularly effective and efficient methods are provided for performing such arithmetic calculations using quantum computer 130.

[0057] Generally, in embodiments of the present disclosure, quantum circuits are used to perform quantum chemistry simulations on a quantum computing device that includes a classical computing mechanism operatively coupled to a quantum computing mechanism, where the quantum circuit is configured to use parameter-free preentanglers as initial states for the quantum algorithm. Such use of quantum circuits is used to address electronic structure problems, where: (i) a quantized version of the canonical transformation proposed by Yanai and Chan [J. Chem. Phys. 124, 194106 (2006)]; and (ii) Full active space density matrix renormalization group and A new Ansatz is utilized, which is generated by a combination of the following: This new Ansatz allows for the shifting of computational load between quantum and classical computing mechanisms. When implementing quantum circuits near multi-reference points in the potential energy surfaces of H2O, N2, BeH2, and related P4 systems, such a strategy has been found to require 30% to 3000% fewer parameters than the corresponding generalized unitary coupled cluster quantum circuits. Thus, embodiments of the present disclosure utilize a new algorithm for creating matrix product states (MPSs) based on linear combinations of unitaries, which is compared to known sequential unitary algorithms, such as those proposed by Ran in Phys. Rev. A 101, 032310 (2020).

[0058] This disclosure describes an orthogonal approach to finding the ground state of a chemical Hamiltonian, among other things. Instead of pre-training the parameters of a quantum circuit, embodiments of the present disclosure use classical resources (e.g., provided by a classical quantum computer) to find a parameter-free quantum circuit to which a variational circuit is added. It will be appreciated that for problem sizes of practical use, this initialization brings the quantum circuit close enough to the "narrow gorge" of the optimization domain to ensure successful optimization.

[0059] The resulting quantum circuit used in the embodiments of the present disclosure has two stages: (a) a pre-entangler stage, i.e., a parameter-free circuit that can be optimized classically, efficiently, and scalably; and (b) The quantum Ansatz stage, i.e., the subsequent circuit of the parameterized gate shown in Figure 1B(a). A good preentangler meets two criteria: (I) It exists in a manifold of quantum states that are easy to classically optimize. (II) A shallow quantum circuit is found for the manifold of quantum states. In this work, we focus on matrix product states (MPSs) as preentanglers, and the canonical transformation (CT)

[14] is adapted from quantum chemistry as a quantum ansatz. We describe the application of such a procedure to four problems in quantum chemistry with strong electron correlations.

[0060] The present disclosure implements a method that beneficially utilizes a combination of pre-entanglement and quantum ansatz, involving the following features: (i) MPSs can be obtained efficiently by the density matrix renormalization group (DMRG) algorithm. DMRG is most efficient in the presence of local one-dimensional interactions, but this method has been shown to faithfully capture the complete active space (CAS) of nonlinear transition metal complexes

[15] and has been implemented for up to 100 orbitals. (ii) Considerable research has been done on the translation of MPSs into the quantum circuits that create them. For example, Ran developed an iterative MPS creation algorithm

[16] based on sequential generation using ancillaries

[17] . In this work, we extend the core of the MPS creation algorithm with a new strategy

[18] that uses linear combinations of unitaries introduced to perform Hamiltonian simulations on quantum computers. (iii) CT theory is a classical method designed to refine the wave functions of CAS-DMRG

[14] . It is formulated as a unitary transformation

[14] . The difficulty of implementing them classically can be overcome by approximations or by using quantum computer mechanisms to perform quantum computations as proposed by this disclosure.

[0061] Next, a variational quantum eigensolver based on canonical transformation theory is described in detail in the context of embodiments of the present disclosure. To overcome the intractability of Equation (5), we propose to optimize Equation (5) variationally on a quantum computer mechanism. After describing the DMRG-QCT method, we present a reconstructed version of the method that can address multi-reference problems and the associated resource requirements.

[0062] Next, we will explain DMRG-QCT in detail. DMRG-QCT uses DMRG to construct static correlations in a given wave function in a classical computational setup, and then uses a CT-inspired variational ansatz called quantum canonical transformation (QCT) to add dynamic correlations. As shown in Figure 2, the DMRG-QCT method consists of two steps (i.e., STEP 1 and STEP 2), as described below.

[0063] In the first step, STEP 1, given the appropriate assignment of molecular orbitals to a given active space, the algorithm used approximates the ground state of the Hamiltonian using the DMRG method. The output from the DMRG method is an MPS description of the ground state |ψ0〉 for a given bond dimension D. While exact diagonalization can be used to construct the preentangler, the embodiments of the present disclosure utilize DMRG because exact diagonalization is impractical in situations where scalable algorithms are usefully used to address real-world problems with many qubits. A quantum circuit is found to create the MPS on a quantum computer. It is important that the unitary description is reliable. That is, the energy error of the unitary approximation should be small enough so as not to ignore too many static correlations in the MPS description. The technique for creating an MPS on a quantum computer is described in detail in the next section.

[0064] The second step, STEP 2, is to construct a variational circuit that describes the dynamic correlation by connecting the active orbitals with the orbitals in the core space and the virtual space. The construction of the variational circuit is performed by using e as defined in Eq. (3). T This is achieved by implementing the quantum circuit e T The exact circuit description of is intractable, and therefore an approximation, namely the first-order Trotter-Suzuki decomposition

[36] , is used.

[0065]

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[0066]

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[37] . In

[37] , it has been pointed out that the variance in performance for different orderings is high when strong static correlations are present. However, in the DMRG-QCT method, the description of static correlations is already addressed by the DMRQ calculation of the active space, so the variance in performance corresponding to different orderings in the QCT case is expected to be small. Together with the approximation of U in (6), the goal of the DMRG-QCT method is to approximate the ground state energy given in (7), i.e., E(θ)=〈ψ0|U † (θ)HU(θ)|ψ0〉 (7) The goal is to minimize q by measuring it in a quantum computing setup and updating q until the minimum ground state energy is found.

[0067] To understand embodiments of the present disclosure, it is useful to view the DMRG-QCT method as an interpolation between purely classical and purely quantum calculations. By choosing the active space to include all orbitals, a purely classical DMRG calculation is obtained. Conversely, in the limit of an empty active space, a fully quantum Generalized Unitary Couple Cluster (GUCC) Ansatz is recovered, a commonly used variant of the Unitary Couple Cluster Ansatz that is suitable for multireference calculations.

[0068] Next, we explore the expressive power of DMRG-QCT in detail. To exploit the effectiveness of the DMRG-QCT Ansatz method, we describe four different molecular systems. These systems contain points on their potential energy surfaces (PESs) that exhibit strong multireference behavior [20, 38, 39]. The PESs of H2O and N2 describe typical bond-breaking situations. In the first example, we consider the one-parameter dissociation of a water molecule at an equilibrium angle (see Figure 3a), examining the ground-state energy when two hydrogen atoms simultaneously symmetrically separate from an oxygen atom. In the second example, we consider the ground-state PES of a nitrogen molecule, parameterized by extending the N-N bond distance (see Figure 3b). In the third example, we consider the transition state of the reaction between Be and H2

[40] . The parameterization of the PEW is shown in Figure 3c, where x = 0 indicates the equilibrium state of BeH2 in a linear configuration, and as values ​​of x increase, the system transitions across a transition point to two non-interacting systems (Be and H2). Finally, in the fourth example, a P4 model system

[41] with two hydrogen molecules is considered (see Figure 3d). The HH bond distance in the two molecules is kept fixed, while this distance in the two molecules, denoted a in the PES, is varied. As values ​​of a increase, with a = 2 Bohr, the system becomes D 4h Such high symmetry leads to pseudo-degeneracy of the configuration and therefore to the existence of strong static correlations in the associated wave functions

[20] .

[0069] Embodiments of the present invention can be implemented by calculating one- and two-electron integrals using the well-known, contemporary PySCF software package

[42] . For all quantum chemistry tasks, the software product ChemMPS2 is used to perform CAS-DMRG calculations [43-46]. The software product ChemMPS2 exploits SU(2), U(1), and point group symmetries to provide a symmetric MPS description of a given ground state. To explore the ability of the QCT variational circuit to represent dynamic correlations with high precision, the exact DMRG construction of a given ground state in the active space is considered with respect to the results provided below (i.e., the coupling dimension D used by DMRG is large enough to represent the exact state). Furthermore, once the MPS description of the ground state is obtained, a sequential unitary algorithm is used to obtain the corresponding quantum circuit that creates the MPS.

[0070] Assuming a unitary construction of the ground state in the active space, the known modern software product InQuanto

[47] is used. InQuanto is a software platform designed to perform calculations for solving chemical problems on quantum computing mechanisms, generate Jordan-Winger transformations of fermion Hamiltonians, and implement the corresponding QCT Ansatz. It will be recognized that QCT involves the application of the Trotter-Suzuki decomposition. As pointed out in

[37] , T To decompose , ordering of double excitations is performed before single excitations, and such ordering reduces the depth and complexity of the corresponding quantum circuits, thereby producing more accurate final calculation results using quantum computer mechanisms.

[0071]

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[0072]

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[48] , where the modern software product Ppytket is suitable for use in decomposing DMRG-QCT circuits into elementary CNOT gates and one-qubit unitary gates

[49] . Such Pytket compilations also perform certain significant optimizations, which lead to more efficient quantum circuits implemented by quantum circuit mechanisms. To optimize energy, in embodiments of the present disclosure, variational parameters are iteratively updated in QCT using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [50-53].

[0074] Examples of results using the DMRG-QCT method on the aforementioned molecules are shown in Figure 4, where the notation ( ● e, ● o) is used to specify the number of electrons and orbitals in the active space used in the DMRG calculation. Data is presented for Hartree-Fock (HF), where configuration interactions with single and double excitations (CISD) and coupled clusters with single and double excitations (CCSD) are used, e.g., as described above. The energy errors of the different methods are then calculated with respect to the full configuration interaction method in a given basis. As expected, the classical single-reference method performs poorly. The CISD error is significantly larger for elongated bonds, as shown in Figure 4(a, b), and for high-symmetry points, as shown in Figure 4(c, d). CCSD becomes non-variational near multi-reference points.

[0075] The data labeled DMRG in Figure 4 illustrate the energy error when no dynamic correlations are present, where the variational QCT circuit plays a minor role and the static correlations generated by the pre-entangled MPS circuit completely determine the energy approximation. Figure 4 also shows the results of second-order N-electron valence state perturbation theory (NEVPT2), applied in addition to the multireference wavefunction from DMRG. The energy error of DMRG-QCT for one electron, labeled QCT-S in Figure 4, shows the energy reduction when the QCT circuit includes only variational parameters for one-electron excitations. It should be noted that unitary coupled cluster ansatz involving only one-electron excitations are typically not considered in the literature. This is due to Thouless's theorem

[54] , which states that clusters of one-electron excitations cannot reduce the energy of a single-reference wavefunction. Lowering the energy by building dynamic correlations in addition to the static MPS quantum circuit is a unique feature of QCT-S.

[0076] DMRG-QCT with single and double excitations, called "QCT-SD," can find states with an error less than the chemical accuracy for H2O and BeH2. Although the energy error of NEVPT2 is small for Figure 4(a, b), this error cannot achieve the chemical accuracy for the BeH2 and P4 systems. Furthermore, even in Figure 4(a, b), the energy error of NEVPT2 is not satisfactory in the single-reference region. The nonparallelism error (NPE)

[20] , defined as the absolute difference between the maximum and minimum energy errors, is found to be 3.59 mH, 9.93 mH, 16.21 mH, and 1.75 mH for the NEVPT2 method, corresponding to the four plots in Figure 4. Conversely, for the QCT-SD method, the NPE is 0.30 mH and 1.15 mH for Figure 4(a, c). Due to computational and circuit requirements, this disclosure does not describe DMRG-QCT Ansatz (see Figure 4(b, d)) for single- and double-electron excitations of N2 and P4. In contrast, it will be recognized that the GUCC-SD algorithm is computationally intractable for all simulation examples illustrated in this disclosure.

[0077] This disclosure also describes the performance of QCT Ansatz for imperfect static constructions using the DMRG method. Figure 5 illustrates the effect on the performance of DMRG-QCT Ansatz as the bond dimension of the MPS describing the reference state |ψ0〉 in (1) is varied. It will be appreciated that both QCT-S and QCT-SD exhibit a degree of resilience to imperfect MPS constructions by DMRG, and that QCT-SD Ansatz maintains energy errors below chemical accuracy. With significant decreases in bond dimension D (which corresponds to Hartree-Fock single-reference in the D → 1 limit), QCT does not provide energies within chemical accuracy. At low bond dimensions, CT single-reference and CT-SD perform similarly, while at higher bond dimensions, CT-SD comes very close to being accurate. The data in Figure 5 also suggest that adding CT single-electron excitations is preferable to increasing the bond dimension utilized by DMRG; for example, the CT-S energy for D = 6 is already below the maximum active space energy.

[0078] Next, we discuss the efficiency of DMRG-QCT in detail. We demonstrate the circuit complexity of implementing the DMRG-QCT Ansatz and compare it with the GUCC Ansatz. Furthermore, we verify the scaling of the number of variational parameters as a function of the number of spin-orbitals or qubits, n. We expect that the number of variational parameters is the most important factor in determining the cost (both circuit requirements and time complexity) of a variational quantum eigensolver. A reduction in the number of parameters directly translates into a reduction in the depth and number of basic gates. Furthermore, we also reduce the number of required iterations, i.e., energy evaluation by a quantum computer and variational parameter updates by a classical optimizer. Energy estimation by a quantum computer is very expensive due to the large number of Pauli sequences in the chemical Hamiltonian; therefore, a reduction in the number of variational iterations leads to a substantial reduction in the algorithm's runtime.

[0079] Each variation parameter is e in (6). Tcorresponds to one excitation (one- or two-electron) in the first-order Trotter-Suzuki approximation of , so the number of parameters is equal to the number of excitations. Furthermore, for QCT Ansatz, the number of electrons in the active space or the total number of electrons does not change the number of excitations, and this is also true for GUCC Ansatz. Figure 6a shows a plot of the scaling of the variational parameters by counting the number of excitations for GUCC and DMRG-QCT Ansatz against the number of spin orbitals. If m is the number of orbitals (i.e., m = n / 2), in GUCC-D the number of excitations is given by

[0080]

number

[0081] The first (second) term corresponds to two - electron excitation, and in two - electron excitation, the complete action of excitation is in one (two) spin channels. The coefficient of 2 / 3 in the first term takes into account the reduction obtained by enforcing the symmetry of the azimuthal spin. In the case of the DMRG - QCT ansatz, different partitions (c, a, v) are considered by dividing n spin - orbitals into core (c) orbitals, active (a) orbitals, and virtual (v) orbitals. Note that for each partition (c, a, v), c = v (i.e., the number of core orbitals is equal to the number of virtual orbitals). As the size of the active space is increased, the number of variational parameters decreases (see Fig. 6a (insert)). This decrease is more pronounced for large n, for example, when a = 0.8n, where a four - fold decrease is achieved. Importantly, this phenomenon is more pronounced for the partition c < a << v, which represents typical MR calculations. For example, for the partition (0.05n, 0.15n, 0.8n), a ten - fold decrease in the number of variational parameters is achieved. Fig. 6a shows a plot of the scaling of the GUCC - S ansatz, which is the upper limit of the scaling of QCT - S. It is clearly seen that the increase in the number of parameters of GUCC - S (or CT - S) is almost negligible compared to the two - electron excitation ansatz. Thus, the QCT - S ansatz can be useful in certain multi - reference situations where it can lower the energy without any overhead. It is important to recognize that the minimal basis is used for three out of four multi - reference problems due to computational requirements. Using the minimal basis makes it difficult to strictly utilize the performance of multi - reference techniques for small (usually small) dynamic correlation energies and relatively easy to achieve chemical accuracy. However, no inconsistency in the performance of DMRG - QCT was observed for all multi - reference problems studied during the development of the embodiments of the present disclosure, and similar performance is expected to be observed when considering beyond the minimal basis.

[0082] In FIGS. 6(b, c), the circuit depth corresponding to the system size with a maximum n = 36 and the cost of implementing excitation / variational parameters as basic gates are shown. For each ansatz, almost completely linear scaling is obtained. The pre-factor of the linear scaling is very high, especially with respect to the number of gates, where it is 46. From the scaling of variational parameters, gates, and circuit depth, the DMRG-QCT ansatz is expected to obtain a significant cost reduction for GUC, especially for typical active spaces where c < a << v.

[0083] In FIG. 9A, the resource requirements of the problems considered in FIG. 4 are shown. The number of gates and circuit depth of QCT also include the cost of the MPS state creation circuit. Despite these additional costs, for the molecules considered, a reduction in the total cost metric of 20 - 40% for QCT-SD and 80 - 97% for QCT-S is observed. Note that the relative reduction increases with the size of the system.

[0084] Next, the unitary circuit required to create an MPS of approximately physical dimension d and virtual bond dimension D is described. To handle qubits, it is assumed here that d = 2. First, the sequential unitary (SEQ) algorithm required to construct the MPS is outlined. Then, the effect of unitary freedom in the SEQ description of the MPS is explained.

[0085] Next, the Linear Combination of Unitaries (LCU) algorithm as an alternative to the SEQ approach is described. Various aspects of the SEQ algorithm and the LCU algorithm are described and their relative performances are compared.

[0086] In FIG. 13, the second row defines the unitary corresponding to the intermediate tensor of the MPS. Each index has dimension 2, and the colors of the matching rows on both sides of each identifier are each G [j]Each input of NULL(...) is a vector in d×D=4-dimensional space, and the output represents an orthogonal base of the null space of the input vectors. The upper right index G [j] and the bottom left index G [j+1] The cascade action of unitaries obtained by contracting |ψ〉 = U|00...0〉 gives a global unitary U, and the action of U on the product state gives the MPS, i.e., |ψ〉 = U|00...0〉.

[0087] For a generic MPS |ψ〉 with D>d, Ran is U † This insight led to an iterative algorithm that starts by initializing |ψ0〉 = |ψ〉, and each iteration consists of compressing the MPS to joint dimension d and applying a disentangling unitary as follows:

[0088]

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[0089]

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[0090]

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[0091]

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[0092] An important aspect of the SEQ algorithm is the freedom of the definition of local tensors. [j] The choice of orthonormal basis for the null space used in the definition of is not unique. In what follows, the performance of the SEQ algorithm is analyzed for different choices of orthonormal basis in each unitary sheaf for the ground state |ψ〉 of the nitrogen dimer (N2). Figure 7(A,b) shows the unitary approximation as a function of the unitary sheaf.

[0093]

number

[0094] Next, the use of the Unitary Linear Combination (LCU) algorithm is described, which introduces a new way to create the MPS, which is approximated as:

[0095]

number

[0096] [Table 1]

[0097] The LCU algorithm works iteratively by estimating the residual between the target MPS and the current approximation (see Algorithm 1 above for the implementation). The algorithm starts by compressing |ψ〉 to D=2 and using it to initialize |ψ0〉. It is then possible to use the method defined in (8) to find an exact unitary representation of |ψ0〉 such that |ψ0〉 = U0|00...0〉. At the beginning of each iteration, the current approximation |ψ i-1 The projection of the goal state |ψ〉 onto 〉 is calculated, i.e.,

[0098]

number

[0099] Next, from |ψ〉 to |ψ i-1 〉 proj By subtracting and compressing the obtained MPS to D=d=2, the residual |r i 〉 is calculated. At this time, |ψ〉 and |ψ i-1 It is important to note that it is also possible to define the residual by taking the difference between |ψ i-1 〉 proj It is numerically found that the residuals obtained from |ψ〉 perform better when including features involved in achieving high fidelity in |ψ〉. Next, the variational parameters k i For |ψ〉 and |ψ i-1 〉+k i |ri The optimization is to maximize the normalized overlap between k and k. One-parameter optimization is very fast, since fidelity can be calculated very quickly for MPS. Furthermore, it is possible to optimize k without any loss of optimality. i >0. i Once you have obtained the new |ψ i 〉, and its bond dimension is D max If it exceeds this, it is beneficial to compress it.

[0100] Next, an important issue regarding the circuit implementation of linear combinations of unitaries is considered. Since linear combinations of unitaries are not unitary, realizing such behavior in quantum devices requires the use of nondeterministic implementations. Usefully, the implementation given in

[18] is adapted using Gray code constructions [59-61] (see Fig. 8). The dashed box in Fig. 8 shows the circuit implementations for unitaries ordered by their Gray code.

[0101]

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[0102]

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[62] . Using such a construction, the number of controls is linear and hence the number of layers D LCUMulti-controlled unitary gates can be implemented with a number of two-qubit gates that is logarithmic (increasing linearly with increasing coupling dimension). Although a significant number of ancillary qubits is required, it is possible to compute quantum chemical problems in NISQ devices with a moderate number of qubits. In Figure 8, the multi-qubit B in the first column is scaled by a factor k i The rest of B is set in an orthogonal subspace to make B unitary. The circuit in Figure 8 is

[0103]

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[0104] The cost of the LCU circuitry with respect to the elementary gates is determined by the cost of realizing the control of two-qubit unitaries. To implement the control of two-qubit gates, their corresponding isometric structures (i.e., involving only the first row of these unitaries, as mentioned above) are also beneficially utilized. Furthermore, the first row of multi-qubit gates B and B † Since only O(lgD LCU loglgD LCU ), CNOT gates and 1-qubit gates are required to implement B. The total cost of implementing the LCU circuit is D LCU is linear, but the pre-scaling factor is quite large: approximately 24nD, where n is the number of qubits. LCU CNOT gate and 31nD LUC In contrast, to implement the SEQ circuit, the number of CNOTS gates and 1-qubit gates is 2nD SEQ and 6nD SEQ The improved scaling of the SEQ circuit makes the SEQ algorithm more favorable in situations where it is necessary to achieve a desired energy error when performing quantum chemistry simulations.

[0105] Instructively, the probability of measuring zero with an ancilla qubit for various systems is analyzed. For a sufficiently large D LCU It can be seen that the probability of converging to a reasonable value for is provided in Figure 9. While the probability convergence can be attributed to the decay of the LCU coefficients (see Figure 9 (inset)), the fact that the limiting probability is quite large for electronic structure problems makes the LCU algorithm particularly useful when applied to creating the ground state of chemical Hamiltonians. Furthermore, such an approach also alleviates the need to use amplitude amplification or trivial amplitude amplification, which can incur significant overhead [64-66].

[0106] Next, we present the numerical analysis of the SEQ and LCU algorithms in more detail. An important property of the SEQ algorithm is the presence of a plateau, as also noted in

[16] . A similar behavior of the plateau has been observed in a different context involving finding refinements of mixed states

[67] . At each iteration (9), we apply a disentanglement unitary to increase the Schmidt rank and thus the joint dimension of the MPS. The latter is achieved by increasing the number of layers D SEQ grows exponentially with , and to keep it computationally feasible, we need to max D maxIt is instructive to analyze the behavior of the Schmidt value (with respect to the cut-access maximum bond dimension) for the SEQ algorithm without discarding any Schmidt values ​​(see graph (a) in Figures 10A and 10B(I, II)). The application of the disentanglement unitary to the preceding MPS introduces low-weight Schmidt values, and this behavior holds for all the different models studied in developing the embodiments of the present disclosure. It is important to note that the long-term behavior of the Schmidt value is rather unique for conventional many-body systems as opposed to chemical systems, as can be seen in Figures 10A and 10B(I, II)a. Figures 10A and 10B(I, II)c show the multiplicative increase in the bond dimension of the MPS before saturating to a maximum value. One would naively expect to discard the low-weight Schmidt values ​​(introduced by the disentanglement unitary) without any results. However, this is not the case, and as can be seen in Figures 10A and 10B(I, II)d, |ψ i 〉 to D max The losses incurred by compressing it to the sigma lead to the appearance of a plateau. The slowdown in convergence is more pronounced for the energy error (see Fig. 7b), where it flattens out well before achieving chemical accuracy, which contributes significantly to the usefulness of the SEQ algorithm for CT Ansatz.

[0107] The LCU algorithm described above is an efficient classical algorithm even without any truncation. During each iteration of the LCU algorithm, the joint dimension of the MPS, |ψ, is approximately i 〉 increases additively (i.e., |ψ i 〉 has a join dimension of only 2i). This is in contrast to the SEQ algorithm, where the join dimension is O(2 i ) is expected. Figures 10A and 10B(I, II)c illustrate the qualitative difference in the increase in joint dimension between the SEQ algorithm and the LCU algorithm. Figures 10A and 10B(I, II)b show the difference in the increase in joint dimension between the LCU algorithm and the SEQ algorithm. i〉, where the Schmidt value contains very specific features. Furthermore, Figures 10A and 10B(I, II)d show the long-term behavior of the LCU algorithm, which continues to increase its fidelity to the target state even for large D, and the existence of a plateau is not so obvious.

[0108] In developing embodiments of the present disclosure, benchmark performance of the SEQ and LCU algorithms for different chemical and spin systems was implemented. For example, for DMRG-QCT Ansatz, it was found that for a given MPS creation method to be used in the context of a chemical system, it is important that the method achieves an energy density error below the chemical precision. The energy error caused by the unitary approximation of the method provides a lower bound on the total energy error of the DMRG-QCT Ansatz. Figure 1 shows the behavior of the SEQ and LCU algorithms in terms of fidelity to the true ground state and energy error. For chemical systems (Figure 11(I, II)), data is shown in the multi-reference region, and for spin problems (Figure 11(III)), the behavior near the critical point is shown. The linear combination D LCU Although the resource overhead for implementing these unitary layers is very high, the LCU algorithm can always find energy errors below chemical accuracy. Furthermore, the LCU algorithm achieves low errors even in the HAF case.

[0109] As a summary and overview of the above disclosure, a general method is proposed to overcome some of the challenges of variational quantum algorithms. This general method involves two steps: Step 1: A classically found trial wave function is created on a quantum computer. Step 2: This trial wave function is then refined using a short variational quantum circuit.

[0110] This general method can be applied in the context of quantum chemistry, where one natural choice is to assign the static and variational parts of the circuit to the static and dynamic correlations of the wave function, respectively. In embodiments of the present disclosure, this general method has been found to lead to three types of advantages. (i) Existing Ansatz, e.g., the GUCC Ansatz, can be extended, which, in the case of embodiments of the present disclosure, leads to a reduction in variational parameters of 30-80%, independent of the size of the system, depending on the relative size of the active space. (ii) The method provides a process for shifting computational load between classical and quantum processors, e.g., the computation can be switched from purely classical (where the entire system is in the active space) to purely quantum (where no precomputation of the active space is performed). (iii) Importantly, this method opens up new, shallower Ansatz possibilities that have not been explored before. The availability of entanglement-active spatial trial wavefunctions allows the use of QCT-S Ansatz, which, for 100 spin orbitals, contains roughly 1000 times fewer variational parameters than the GUCC Ansatz. This can be a major advantage in that it makes simulations of much larger chemical systems feasible on modern NISQ computing devices.

[0111] One key step in the proposed general method is the approximate implementation of MPS on a given quantum computer, e.g., a NISQ-era quantum computer. A new algorithm for this purpose, the LCU algorithm, is therefore presented. The LCU algorithm is compared with the known SEQ algorithm from the literature, and it is found to be more expressive, although it requires a larger constant prefactor in its circuit decomposition. Furthermore, it is found that the unitary freedom present in the canonical form of MPS can be used for both disentangler optimization and circuit synthesis, concluding that the latter application is generally more powerful.

[0112] While the above describes an embodiment of the present disclosure, an optional implementation of the preentanglement technique using DMRG-QCT is provided, which is quite flexible and can be easily utilized to complement the capabilities of other techniques such as Adapt-VQE and symmetry-preserving Ansatz [68, 69]. Furthermore, this method, including the preentanglement technique, can be advantageously used in the fields of quantum optimization and machine learning.

[0113] Optionally, embodiments of the present disclosure may also create an MPS by using a hybrid combination of the SEQ and LCU procedures. A first approach to this combination is to modify Equation (10) to obtain a sequence of Ansatz operators, where each operator itself is a linear combination of unitaries, and thus is not necessarily unitary. A second approach to this combination is to use a linear combination of successive unitaries in Equation (11) (i.e., each unitary in the sum is a product of unitaries). Of these two approaches, the second approach can be implemented with minor modifications from Algorithm 1. In the era of NISQ device resources, these two generalized approaches may offer a better tradeoff between accuracy and circuit requirements for creating tensor states.

[0114] It will be recognized that there are other methods for creating MPSs. For example, in Equation (70), it has been proposed to optimize the unitary sheaf of the unitary-ansatz algorithm by automatic discrimination, leading to a more optimal approximation. Using Algorithm 1 instead in conjunction with a similar unitary optimization procedure is expected to result in significant improvements to the LCU algorithm. However, this incurs increased classical computational overhead, resulting in higher-quality states within the variational manifold defined by Equation (11). Optionally, embodiments of the present disclosure can use a preentangler, which implements a more general tensor network, such as that given in

[71] , via a quantum circuit tensor network. Optionally, a preentangler created using an adiabatic algorithm

[72] is used in embodiments of the present disclosure. Furthermore, the LCU algorithm described in this disclosure can be used in other contexts, for example, to create low-energy states for quantum algorithms based on time series

[73] .

[0115] Although CT theory has been described above in describing embodiments of the present disclosure, other methods exist for recovering dynamic correlations from the activity space. Additionally, n-electron valence second-order perturbation theory

[74] or adiabatic connections

[75] are optionally used in embodiments of the present disclosure.

[0116] 12, steps of a method according to the present disclosure for performing a simulation of a chemical system are shown. The method is implemented using a hybrid computing arrangement, the hybrid computing arrangement including a combination of a classical computer coupled to a quantum computer, the hybrid computing arrangement configured, in use, to receive input data and generate corresponding processed output data from the input data. The method includes steps 200-230. Step 200 involves configuring a classical computer to receive information describing a chemical system in input data. Step 210 involves configuring a classical computer to process information describing the chemical system using a preentangler algorithm and a fixed circuit algorithm to generate quantum Ansatz that define initial values ​​for the quantum circuit computation and a Hamiltonian from which the variational circuit algorithm is generated. Step 220 includes configuring a quantum computer to use quantum Ansatz and variational circuit algorithms to compute the corresponding quantum circuit to generate a quantum computation result. Step 230 includes configuring a classical computer to process the quantum computation results to generate output data that includes information describing the electron trajectory simulation of the chemical system.

[0117] Optionally, the method comprises: (i) assigning molecular orbitals that can occupy core space, virtual space, and active space to an active space based on a molecular system; (ii) algorithmically approximating the ground state of the Hamiltonian by generating a matrix product state (MPS) description |ψ0〉 of the ground state for a given coupling dimension D using the MSRG algorithm; (iii) Finding a quantum circuit that creates the MPS description |ψ0〉 on a quantum computer, and (iv) Constructing a variational quantum circuit that describes dynamic correlations by coupling active orbitals in the core space and the active space; and (v) generating an output result by minimizing the ground state energy from the result of executing the quantum circuit; and configuring a hybrid computer mechanism to:

[0118] Changes can be made to the embodiments of the present disclosure described above without departing from the scope of the disclosure, which is defined by the appended claims. As used to describe and claim the present invention, terms such as "including," "comprising," "incorporating," "consisting," "having," "is," and the like are intended to be construed non-exclusively, i.e., allowing for items, components, or elements not expressly stated to be present. References to the singular should also be construed to relate to the plural. As an example, "at least one of" means "one of" in one instance and "multiple" in another instance. Furthermore, "one or more" should be construed similarly.

[0119] The phrases "in one embodiment," "according to one embodiment," and the like generally mean that the particular feature, structure, or characteristic that follows the phrase is included in at least one embodiment of the present disclosure, and may be included in more than one embodiment of the present disclosure. Importantly, such phrases do not necessarily refer to the same embodiment.

[0120] The terms "computer" or "computation-based device" are used herein to refer to any device with processing capability to execute instructions. Those skilled in the art will recognize that such processing capability is incorporated in many different devices, and thus the terms "computer" and "computation-based device" each include personal computers (PCs), servers, mobile phones (including smartphones), tablet computers, set-top boxes, media players, game consoles, personal digital assistants, wearable computers, and many other devices.

[0121] The methods described herein may, in some examples, be performed by software in machine-readable form on a tangible, non-transitory storage medium, e.g., in the form of a computer program including computer program code adapted to perform one or more operations of the methods described herein when the program is run on a computer, the computer program may be embodied in a non-transitory computer-readable medium, and the software may be suitable for execution on a parallel or serial processor such that the operations of the methods are performed in any suitable order or simultaneously.

[0122] This recognizes that software is a valuable, separately tradable commodity. It is intended to encompass software that runs on or controls "dumb" or standard hardware to perform a desired function. It is also intended to encompass software that "describes" or defines the configuration of hardware, such as HDL (Hardware Description Language) software used to design silicon chips or configure general-purpose programmable chips to perform a desired function.

[0123] Those skilled in the art will recognize that storage devices used to store program instructions are optionally distributed over a network. For example, a remote computer can store an example of a process written as software. A local or terminal computer can access the remote computer, download some or all of the software, and execute the program. Alternatively, a local computer may download some software as needed, or execute some software instructions locally and some on the remote computer (or computer network). Those skilled in the art will also recognize that all or some of the software instructions may be executed by dedicated circuitry, such as a digital signal processor (DSP), programmable logic array, or the like, using conventional techniques known to those skilled in the art.

[0124] As will be apparent to those skilled in the art, the ranges or device values ​​given herein may be extended or modified without losing the effect sought.

[0125] Although the subject matter has been described in language specific to structural features and / or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the above-described features or acts. Rather, the above-described specific features or acts are disclosed as example forms of implementing the claims.

[0126] It will be understood that the benefits and advantages described above may relate to one embodiment or several embodiments. The embodiments are not limited to those that solve any or all of the stated problems or to those that have any or all of the stated benefits and advantages. No single feature or group of features is required or essential to every embodiment.

[0127] Conditional terms used herein, such as "can," "could," "might," "may," "for example," and the like, among others, are generally intended to convey that certain embodiments include certain features, elements, and / or steps, while other embodiments do not, unless otherwise indicated or understood differently within the context in which they are used. Thus, such conditional terms are generally not intended to imply that features, elements, and / or steps are in any way required for one or more embodiments, or that one or more embodiments necessarily include logic for determining, with or without author input or direction, whether those features, elements, and / or steps are included in or should be performed in a particular embodiment. The terms "comprise," "include," "have," and the like are synonymous and used open-endedly inclusively and do not exclude additional elements, features, acts, operations, blocks, etc. Additionally, the word "or" is used in its inclusive sense (and not its exclusive sense), so that, for example, when used to connect a list of elements, the word "or" may refer to one, some, or all of the elements in the list. Additionally, the articles "a," "an," and "the," as used in this application and the appended claims, unless otherwise specified, should be construed to mean "one or more" or "at least one."

[0128] As used herein, phrases referring to "at least one" of a list of items refer to any combination of those items containing only one element. As an example, "at least one of A, B, or C" is intended to cover A; B; C; A and B; A and C; B and C; and A, B, and C. Transitional language such as "at least one of X, Y, and Z," unless otherwise specified, is generally understood in accordance with the context in which it is used to convey that an item, term, etc. may be at least one of X, Y, or Z. Thus, such transitional language is generally not intended to suggest that a particular embodiment requires that at least one of X, at least one of Y, and at least one of Z, respectively, be present.

[0129] The actions of the methods described herein may be performed in any suitable order, or simultaneously where appropriate. Additionally, individual blocks may be deleted, combined with other blocks, or rearranged in any of the methods without departing from the scope of the subject matter described herein. Aspects of the examples described above may be combined with aspects of other examples described to form further examples without losing the desired effect.

[0130] It will be understood that the above description is given by way of example only, and that various modifications may be made by those skilled in the art. The above detailed description, examples, and data provide a complete description of the structure and use of the exemplary embodiments. Although various embodiments have been described above with a certain level of detail or with reference to one or more specific embodiments, those skilled in the art can make numerous modifications to the disclosed embodiments without departing from the scope of the present specification. Addendum: List of citations related to quantum computing [1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019). [2] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'brien, A variational eigenvalue solver on a photonic quantum processor, Nature communications 5, 1 (2014). [3] B. F. Schier, J. Tura, and J. I. Cirac, Adiabatic spectroscopy and a variational quantum adiabatic algorithm, arXiv preprint arXiv:2103.01226 (2021). [4] B. Bauer, D. Wecker, A. J. Millis, M. B. Hastings, and M. Troyer, Hybrid quantum-classical approach to correlated materials, Physical Review X 6, 031045 (2016). [5] K. M. Nakanishi, K. Fujii, and S. Todo, Sequential minimal optimization for quantum-classical hybrid algorithms, Physical Review Research 2, 043158 (2020). [6] S. McArdle, X. Yuan, and S. Benjamin, Error-mitigated digital quantum simulation, Physical review letters 122, 180501 (2019). [7] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al., Variational quantum algorithms, Nature Reviews Physics 3, 625 (2021). [8] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantumclassical algorithms, New Journal of Physics 18, 023023 (2016). [9] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature communications 9, 1 (2018).

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Claims

1. 1. A method for configuring a hybrid computing arrangement to perform a simulation of a chemical system, the hybrid computing arrangement including a combination of a classical computer coupled to a quantum computer, the hybrid computing arrangement being configured, in use, to receive input data and to generate corresponding processed output data from the input data, the method comprising: (a) configuring the classical computer to receive information describing a chemical system in the input data; (b) configuring the classical computer to process the information describing the chemical system using a preentangler algorithm and a fixed-circuit algorithm to generate quantum Ansatzes that define initial values ​​for quantum circuit computations and Hamiltonians from which variational circuit algorithms are generated; (c) configuring the quantum computer to use the quantum Ansatz and the variational circuit algorithm to compute a corresponding quantum circuit to generate a quantum computation result; (d) configuring the classical computer to process the quantum computation results to generate the output data, the output data including information describing an electron trajectory simulation of the chemical system; A method comprising:

2. The method of claim 1 , comprising configuring the preentangler algorithm to function as a parameter-free preentangler.

3. 3. The method of claim 2, comprising implementing the preentangler algorithm using a matrix product states (MPS) algorithm.

4. The method of claim 3 , comprising generating a matrix product state (MPS) based on a linear combination of unitaries describing the chemical system.

5. 4. The method of claim 3, comprising configuring the hybrid computer mechanism to generate matrix product states (MPS) using a Density Matrix Renormalization Group (DMRG) algorithm to capture a complete active space (CAS) for one or more nonlinear transition metal complexes included in the chemical system.

6. 4. The method of claim 3, further comprising configuring the hybrid computer system to generate matrix product states (MPS) by using the MPS algorithm based on sequential generation using ancillaries.

7. The method of claim 1 , comprising using the quantum circuit to find a ground state of the Hamiltonian.

8. 2. The method of claim 1, further comprising configuring the variational circuit algorithm as a variational quantum eigenvalue solver based on one or more canonical transformations, the method comprising configuring the hybrid computer arrangement to generate a density matrix renormalization group (DMRG) algorithm using the classical computer to construct static correlations in wave functions describing the chemical system, the density matrix renormalization group (DMRG) algorithm being used to generate a corresponding DMRG-QCT portion of the quantum circuit.

9. (i) assigning molecular orbitals that can occupy a core space, a virtual space, and an active space to an active space based on a molecular system; (ii) A matrix product state (MPS) description of the ground state |ψ for a given coupling dimension D using an algorithm that approximates the ground state of said Hamiltonian by using the MSRG algorithm. ０ 〉; and (iii) The MPS description |ψ on the quantum computer ０ 〉, and (iv) constructing a variational quantum circuit that describes the dynamic correlation by coupling active orbitals in the core space and the active space; (v) generating an output result by minimizing the ground state energy from the result of executing the quantum circuit; 2. The method of claim 1, further comprising configuring the hybrid computer facility to:

10. 1. A hybrid computing arrangement configured to perform a simulation of a chemical system, the hybrid computing arrangement comprising a combination of a classical computer coupled to a quantum computer, the hybrid computing arrangement configured, in use, to receive input data and to generate corresponding processed output data from the input data, the computing arrangement comprising: (a) configured to receive information describing a chemical system in the input data; (b) processing the information describing the chemical system using a preentangler algorithm and a fixed-circuit algorithm to generate quantum Ansatzes that define initial values ​​for quantum circuit computations and Hamiltonians from which variational circuit algorithms are generated; (c) configured to use the quantum Ansatz and the variational circuit algorithm to compute a corresponding quantum circuit to generate a quantum computation result; (d) a hybrid computing mechanism configured to process the quantum computation results to generate the output data, the output data including information describing an electron trajectory simulation of the chemical system.

11. The hybrid computer arrangement of claim 10 , wherein the preentangler algorithm is configured to function as a parameter-free preentangler.

12. 11. The hybrid computer arrangement of claim 10, wherein the preentangler algorithm is configured to use a matrix product state (MPS) algorithm.

13. 13. The hybrid computing mechanism of claim 12, wherein the hybrid computing mechanism is configured to generate matrix product states (MPS) using a density matrix renormalization group (DMRG) algorithm to capture a complete activity space (CAS) for one or more nonlinear transition metal complexes included in the chemical system.

14. 13. The hybrid computer system method of claim 12, wherein the hybrid computer system is configured to generate a matrix product state (MPS) by using the MPS algorithm based on sequential generation using an ancillary.

15. The hybrid computing mechanism of claim 10 , wherein the hybrid computing mechanism is configured to use the quantum circuit to find a ground state of the Hamiltonian.

16. 11. The hybrid computer mechanism of claim 10, wherein the hybrid computer mechanism is configured to include the variational circuit algorithm as a variational quantum eigenvalue solver based on one or more canonical transformations, and wherein the hybrid computer mechanism is configured to generate a density matrix renormalization group (DMRG) algorithm using the classical computer to construct static correlations in wave functions describing the chemical system, the density matrix renormalization group (DMRG) algorithm being used to generate a corresponding DMRG-QCT portion of the quantum circuit.

17. said hybrid computer system comprising: (i) assigning molecular orbitals that can occupy a core space, a virtual space, and an active space to an active space based on a molecular system; (ii) A matrix product state (MPS) description of the ground state |ψ for a given coupling dimension D using an algorithm that approximates the ground state of said Hamiltonian by using the MSRG algorithm. ０ 〉; and (iii) The MPS description |ψ on the quantum computer ０ 〉, and (iv) constructing a variational quantum circuit that describes the dynamic correlation by coupling active orbitals in the core space and the active space; (v) generating an output result by minimizing the ground state energy from the result of executing the quantum circuit; 11. The hybrid computer arrangement of claim 10 configured to:

18. 10. A non-transitory computer-readable storage medium comprising specific computer-readable instructions executable on data processing hardware, the specific computer-readable instructions, when executed using the data processing hardware, performing a method according to any one of claims 1 to 9.

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