Systems and methods for gibbs state generation in quantum circuits

Abstract

There is provided a computer-implemented hybrid variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian.

Description

SYSTEMS AND METHODS FOR GIBBS STATE GENERATION IN QUANTUM CIRCUITSFIEED OF THE INVENTION

[0001] The present invention relates to systems and hybrid variational methods for generating Gibbs states using quantum circuits, such as parametrised quantum circuits, and further for training parameterised quantum circuits to generate Gibbs states for use in Quantum Boltzmann Machines.BACKGROUND OF THE INVENTION

[0002] Quantum machine learning combines principles of quantum computing with machine learning techniques to solve problems more efficiently, and to solve different kinds of problems involving complex and quantum data patterns. An example of this is a class of energy-based models known as Quantum Boltzmann Machines (QBMs). Classical Boltzmann Machines are a type of machine learning model which consist of a neural network with visible units / neurons representing input data, hidden units / neurons capturing dependencies, and with connections - weights - between them. The input data is modelled using a scalar energy function defined over the input space. The model assigns low energy to input data -likely configurations - and high energy to unlikely configurations. The aim is to iteratively learn an energy function that correctly represents the probability distribution of the input data, where the lower energy states correspond to higher probability. Once trained, the model can be used in a generative manner to generate new data samples similar to the training data through the capturing of the underlying distribution.

[0003] QBMs are an adaption of Classical Boltzmann Machines to the quantum computing framework. Instead of the classical energy function the quantum model uses a Hermitian operator, a parametrized Hamiltonian. Here, hidden and visible units / neurons are represented by Qubits / Pauli matrices. The finite temperature state of the parametrized Hamiltonian is represented by a density matrix called the quantum Gibbs state. The aim of the model is to find the quantum Gibbs state that represents an input probability distribution. Compared to classical models, the quantum framework allows the use of quantum structures which are potentially inaccessible classically. Similarly, once trained, the model can be used in a generative manner to generate new data samples similar to the training data through the capturing of the underlying distribution.

[0004] However, training QBMs is complex, can be highly inefficient requiring many computational processing steps, and can be difficult to perform accurately.

[0005] In particular, a crucial and challenging aspect in the training of QBMs is the stage of Quantum Gibbs (thermal) state generation of the model Hamiltonian, and similarly the selection of the Hamiltonian model to use. A particular approach to finding the quantum Gibbs state in QBMs is that of Variational QBMs, where variational methods are used to approximate and optimise quantum Gibbs states and energies. In particular, this involves the use of an ansatz which is a parametric form of the quantum state chosen to approximate the true quantum state, and further involves optimising to adjust the parameters of the ansatz to minimise a cost function related to learning the desired / target quantum Gibbs state. This is often performed in a hybrid quantum and classical manner, with quantum computers used to generate the quantum Gibbs state as well as the Gibbs free energy which is the cost function to be minimised, and with classical algorithms used to handle the optimisation of the ansatz parameters. The ansatz is often a parameterised quantum circuit approximating the Gibbs state, and hence this parameterised quantum circuit is used to generate the Gibbs states.

[0006] Examples of existing variational techniques for quantum Gibbs state generation include:- Variational Quantum Imaginary Time Evolution (VarQITE) [Zoufal, C., Lucchi, A. & Woemer, S. Variational quantum Boltzmann machines. Quantum Mach. Intell. 3, 7 (2021) https: / / doi.orz / 10.1007 / s42484-020-00033-7. and arXiv:2007.00876 https: / / doi.or / 10.48550 / arXiv.2007.00876}Beta-Variational Quantum Eigensolver [Onno Huijgen et al 2024 Mach. Learn.: Sci. Technol. 5 025017]Truncated Taylor Series approximation [arXiv:2005.08797, https: / / doi.org / 10.48550 / arXiv.2005.08797]; and Variational Quantum Thermalizer (VQT) [arXiv: 1910.02071, https: / / doi.org / 10.48550 / arXiv.1910.02071]

[0007] However, these methods suffer from a number of challenges. Whilst these methods attempt to efficiently approximate the Gibbs state, nevertheless the generation of the Gibbs state is computationally challenging, requiring many processing steps. Further, the stability of the Gibbs state is affected by noise and decoherence, for instance from temperature. Indeed, the Hamiltonian can exist in different quantum phases based on its parameters and temperature, but the existing variational methods used for thermal state generation, for instance as used in QBMs, such as the above-mentioned variational quantumimaginary time evolution and beta- variational quantum eigensolvers do not account for these variations, leading to suboptimal performance across different regimes.

[0008] Certain aspects of the present disclosure and their embodiments may provide solutions to these or other challenges. For instance, certain aspects and embodiments may provide systems and methods for training QBMs capable of improved accuracy, speed and efficiency. For instance, certain aspects and embodiments may provide systems and hybrid variational methods for generating Gibbs states in parameterised circuits with improved accuracy, speed and efficiency.STATEMENT OF THE INVENTION

[0009] Aspects of the invention are defined by the accompanying claims. Advantageous optional features are defined in the dependent claims.

[0010] In accordance with an aspect there is provided a computer-implemented hybrid variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.

[0011] Various aspects and embodiments of the invention are described without limitation below, with reference to the figures.BRIEF DESCRIPTION OF THE DRAWINGS

[0012] There now follows, by way of example only, a detailed description of preferred embodiments of the present invention, with reference to the figures identified below.Figure 1 illustrates a method;Figure 2 illustrates a process;Figure 3 illustrates an example circuit configuration;Figure 4 illustrates an example circuit configuration;Figure 5 illustrates an example circuit configuration;Figure 6 illustrates a method;Figure 7 illustrates a process;Figure 8 is a table of results;Figure 9 is a table of results;Figure 10 is a table of results;Figure 11 is a table of results;Figure 12 is a table of results and graphs;Figure 13 illustrates an apparatus;Figure 14 illustrates an apparatus;Figure 15 illustrates an example circuit configuration.DETAILED DESCRIPTION

[0013] In the following description, functionally similar parts carry the same reference numerals between figures. The following sets forth specific details, such as particular aspects, embodiments or examples for purposes of explanation and not limitation. It will be appreciated by one skilled in the art that other examples may be employed apart from these specific details. Aspects and embodiments of the invention are now described,without limitation and by way of example only, with reference to the accompanying drawings.

[0014] Aspects and embodiments may provide improved methods and systems for performing hybrid variational methods of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian.

[0015] Aspects and embodiments may provide improved methods and systems for training a variational quantum Boltzmann machine.

[0016] Aspects and embodiments may provide a computer program which, when run on a hybrid quantum-classical computer, causes the computer to carry out a variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian.

[0017] Aspects and embodiments may provide an improved information processing apparatus comprising a memory and a processor connected to the memory, wherein the processor is configured to perform a variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian.

[0018] Indeed, there are many state of the art methods of performing Gibbs statee- / ?H _approximation and generation, where the Gibbs state p =where H is the Hamiltonian of the system and / 3 = = is the inverse temperature. However, each of these state of the artmethods suffers from distinct disadvantages.

[0019] Aspects and embodiments of the present application have identified distinct disadvantages in the state of the art methods of Gibbs State generation in general, regardless of applied context.

[0020] In particular, aspects and embodiments of the present application have identified that existing variational methods are incapable of generating the Gibbs state in certain parameter regimes of Hamiltonian, thus leading to suboptimal performance, especially when applied in a QBM (training) workflow. In particular, state of the art methods do not accurately or otherwise account for the fact that the Hamiltonian can exist in different quantum phases based on its parameters and temperature. Indeed, the existing variational methods used for thermal state generation, for instance as used in QBMs, such as the above-mentioned variational quantum imaginary time evolution and beta-variational quantum eigensolvers do not account for these variations, leading to suboptimal performance across different regimes.

[0021] Aspects and embodiments of the present application may advantageously overcome these disadvantages by providing a collective Gibbs state generation technique which works with good fidelity for Hamiltonians where some of the existing methods struggle. In particular, aspects and embodiments may provide collective optimisation over a range of different values for the parameters of the associated Hamiltonian, and hence thereby creating a generalised single optimisation. In particular, aspects and embodiments may provide a pre-trained parameterised circuit for a set and range of parameters of Hamiltonian, and thereby accurately generate the quantum Gibbs state across and within a particular parameter regime. Advantageously, after training, individual optimisation for every single parameter of the Hamiltonian is thereby no longer required. Hence aspects and embodiments may advantageously provide a single parameterised circuit which has been trained to learn and be optimised to generate quantum Gibbs state in all parameter regimes of the Hamiltonian. Hence aspects and embodiments may advantageously provide good quality and accurate Gibbs state generation irrespective of the parameter regime of Hamiltonian, and in particular to accurately and efficiently generate Gibbs states corresponding to any parameters of the Hamiltonian within the training set region, independent of any regimes / phases of Hamiltonian. In particular, aspects and embodiments may advantageously be more accurate in Gibbs state generation than state of the art methods such as VarQITE.

[0022] Accordingly, aspects and embodiments of the present invention may provide advantageously improved methods and systems for Gibbs state generation.

[0023] Aspects and embodiments of the present application have also identified that the existing variational methods of Gibbs state generation are disadvantageous^ only able to handle optimisation of single set of Hamiltonian parameters at a time. When such methods are used in the state of the art QBM workflows, nested loops are required within the QBM for Gibbs state generation, and these nested loops are required to perform optimisation of the parameters of the quantum circuit for each iterative QBM training loop when attempting to converge on the Gibbs state. Hence these makes the training process complex, inefficient, inaccurate and slow (see for instance [Zoufal, C., Lucchi, A. & Woemer, S. Variational quantum Boltzmann machines. Quantum Mach. Intell. 3, 7 (2021) htps: / / doi.org / 10.1007 / s42484-020-00Q33-7] and [Onno Huijgen el al 2024 Mach. Learn. : Sci. Technol. 5025017, 10.1088 / 2632-2153 / ad370f]).

[0024] In particular, as computing the Gibbs State is challenging, state of the art methods for variational QBM rely on and require nested loops to approximate the Gibbsstate. In particular, for each training epoch (outer loop) of variational QBM, nested loops are used to generate an approximation of the Gibbs state through defining a parameterised quantum circuit (ansatz) and using classical optimisation to find the optimal parameters that minimize a loss function, which is the Gibbs free energy. For each iteration of the outer loop, the inner loop on the quantum circuit is run multiple times to repeatedly measure the exact / approximate Gibbs free energy, which is optimised using a classical optimiser to adjust the parameters of the quantum circuit. Once trained, the quantum Gibbs state distribution is measured from the ansatz. Hence the nested loop structure allows the quantum system to improve its approximation of the Gibbs state over iterations. Once an approximate Gibbs state is generated by this inner loop for a Hamiltonian with particular coefficients, the loss function of the QBM (such as KL divergence) is updated using classical methods in the outer loop to match an input target distribution by adjusting the Hamiltonian coefficients. This step of approximating the Gibbs state with an inner loop and updating the loss function of QBM in the outer loop is repeated until QBM loss function reaches convergence. In conventional state of the art methods, this nested (inner) loop structure is essential to generate accurate Gibbs states. However, it is computationally highly inefficient, requiring many processing resources and taking a long time.

[0025] The idea of meta-leaming (learning to learn) in classical machine learning refers to training an outer (or upper / meta) algorithm that updates the inner learning algorithm (used for solving task such as image classification, defined by a dataset and objective) such that the model it learns improves an outer objective such as generalization performance or learning speed of the inner algorithm [https: / / arxiv.org / abs / 2004.05439]. This meta-leaming is also used in Quantum Machine learning for tasks such as generalization of ground states of parametrized Hamiltonian (Meta-VQE) [https: / / arxiv.org / abs / 2009.13545], training algorithms that does not rely on gradient computation [https: / / arxiv.org / pdf / 2304.07442] and training classical neural networks to assist in the quantum learning process [https: / / arxiv.org / pdf / 1907.05415; htps: / / link.springer.com / article / I0.1007 / s42484-020-00022 -w]. However, aspects and embodiments of the present application employ meta-leaming techniques for approximating the finite temperature state of a parametrized Hamiltonian which is not known in literature. In particular, according to aspects and embodiments of the present application, meta-leaming is introduced from a collective optimisation perspective. Unlike the state of the art approaches to training where the Gibbs state corresponding to a particular parameter value of Hamiltonian is approximated byminimizing the Gibbs free energy, aspects and embodiments consider a set of parameters of the Hamiltonian and construct a global cost function obtained by summing the truncated Gibbs free energy of all these Hamiltonians. Further, aspects and embodiments provide for optimising over this cost function to give an optimal ansatz that can give the Gibbs state corresponding to a Hamiltonian parameter which is not in the trained set of Hamiltonian parameters. It is noted that herein and in the art, the words ‘ansatz’ and ‘parametrised quantum circuit (PQC)’ may be used synonymously.

[0026] Aspects and embodiments of the present application may advantageously overcome the failings of the state of the art in a number of ways. In particular, when applied in the QBM workflows, aspects and embodiments of the present application may advantageously provide a pre-trained parameterised circuit capable of accurate and efficient Gibbs state generation and which avoids the need for the nested loop evaluation as in state of the art methods. Hence aspects and embodiments of the present application may provide for a QBM workflow which does not require nested loops, and is simpler and more efficient. Similarly, aspects and embodiments may provide for a QBM workflow which has reduced runtime and increased training efficiency with fewer processing steps. Hence aspects and embodiments may provide for reduced training cost. Aspects and embodiments may advantageously provide QBM workflows with improved precision Gibbs state generation and improved convergence loss functions as compared to state of the art methods.

[0027] Aspects and embodiments of the present application may advantageously provide variational QBMs of enhanced efficiency and performance, making advanced quantum computing techniques more accessible and cost-effective.

[0028] In particular, aspects and embodiments of the present application may avoid the nested loop evaluation for Gibbs state generation in variational QBM training, and in particular, avoid the extra optimisation and subsequent gradient computation of quantum circuits required beyond the full QBM training for each epoch. This can reduce the training time and complexity for QBM training especially where the Hamiltonian considered is complex in spin terms or number of qubits.

[0029] Figure 1 is a diagram illustrating a method according to an aspect. In particular, the method may be a computer-implemented hybrid variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, and may be performed on a hybrid quantum and classical computer as will be described further below.

[0030] Step Sllcomprises defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters.

[0031] Step S13 comprises initialising a set of tunable parameters of the parameterised quantum circuit.

[0032] Step S15 comprises performing collective optimisation of the tunable parameters with respect to the parameterised Hamiltonian, comprising for instance steps S17 to S19.

[0033] Step S17 comprises determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters: measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit; and evaluating a cost function based on the Gibbs state approximation.

[0034] Step S19 comprises evaluating a global cost function, the global cost function comprising the sum of the set of evaluated cost functions, wherein: if the global cost function is not minimised, updating the tunable parameters based on minimising the global cost function and iteratively performing the collective optimisation; and if the global cost function is minimised, defining the tunable parameters as trained parameters.

[0035] Any of the aspects and embodiments described herein, and any of steps of Figure 1, may be performed by an apparatus as described with reference to Figures 13 and 14 below, which concern a hybrid quantum and classical computer arrangement.

[0036] The Hamiltonian parameters may be the coupling constant J and the external magnetic field h.

[0037] The range of values for the parameters may be -2 to 2.

[0038] The set of evaluated cost functions may be stored in a variable such as a tensor.

[0039] The cost function and / or the global cost function may use the Gibbs free energy.

[0040] The cost function and / or the global cost function may use the Gibbs free energy with a truncated entropy term:wherein:h ”_1 is a vector comprising the parameters of Hamiltonian H(h ”_1);P = l / k_B T is inverse temperature;co ^,(|) 0 are the tuneable parameters of the parameterised quantum circuit; p(co ^,(|) 0 "*) is the Gibbs state approximation from the parameterised quantum circuit; and(l / )Tr(p(co ^,(|) 0 ”)A2 )-( 1 / ) is first order truncated Von-Neumann entropy associated with the Gibbs state approximation.

[0041] The updating the tuneable parameters based on minimising the global cost function may comprise using adaptive movement estimation (ADAM) optimiser.

[0042] The parameterised quantum circuit configuration may comprise: an encoding layer encoding the input set of Hamiltonian parameters and initialising the tuneable parameters of the parameterised quantum circuit configuration.

[0043] The encoding layer may comprise a Hardware Efficient Ansatz circuit.

[0044] The parameterised quantum circuit configuration may further comprise: a processing layer, the processing layer further comprises a Hamiltonian Variational Ansatz, wherein the gates of the Hamiltonian Variational Ansatz are parameterised based on the defined sets of Hamiltonian parameters.

[0045] The defined sets of Hamiltonian parameters may comprise the field or interaction terms of the Hamiltonian.

[0046] The encoding layer may comprise a neural network.

[0047] The neural network may be used to encode the input set of Hamiltonian parameters and output the tuneable parameters of the parameterised quantum circuit configuration.

[0048] According to an aspect there may be provided a computer-implemented method of training a variational quantum Boltzmann machine comprising:selecting a Hamiltonian comprising initial Hamiltonian parameters; measuring a distribution of Gibbs states using a pre-trained parameterised quantum circuit configuration trained in accordance with any manner as described herein, wherein the tuneable parameters are the trained parameters;adjusting the initial Hamiltonian parameters based on minimising a cost function, comprising:evaluating the cost function, comprising comparing the distribution of Gibbs states with a target distribution of measurements from a target dataset;generating updated Hamiltonian parameters, comprising adjusting the initial Hamiltonian parameters in a direction that reduces the cost function.

[0049] The updated Hamiltonian parameters may be defined as the trained Hamiltonian parameters if the cost function is minimised.

[0050] The pre-trained parameterised quantum circuit configuration may be selected from a plurality of pre-trained parameterised quantum circuit configurations trained in accordance with any manner as described herein, the selecting based on the complexity and number of parameters of the selected Hamiltonian comprising initial Hamiltonian parameters.

[0051] According to an aspect there is provided a method of using a trained variational quantum Boltzmann machine trained in accordance with any method as described herein, wherein it is used as part of a generative Al process to generate new data.

[0052] According to an aspect there is provided a computer program which, when run on a hybrid quantum-classical computer, causes the computer to carry out a variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.

[0053] According to an aspect there is provided an information processing apparatus comprising a memory and a processor connected to the memory, wherein the processor is configured to perform a variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.

[0054] Figure 2 illustrates specific embodiments of the method of Figure 1. In particular, features and steps of Figure 2 are annotated with references to the steps Sil toS19 from Figure 1, where this annotation indicates optional specific features which may form specific embodiments of the associated steps from Figure 1.

[0055] In particular, the method may be performed on a hybrid classical and quantum computer, as will be outlined below. In particular, aspects of the method may be performed on using ansatz quantum circuits, as will be described further below, in particular on a parameterised quantum ansatz circuit for using variational methods.

[0056] In particular, in a specific embodiment, step Sil comprises selecting a parameterised Hamiltonian in accordance with Step S 111. In the aspects and embodiments of the present application, the Hamiltonian chosen can take many forms and many levels of complexity, and is for instance guided by the specific context: for instance in the example of using QBMs to learn a target dataset, the parameterised Hamiltonian and the corresponding parameters will be sampled from a distribution such that it can model target data distribution.

[0057] In accordance with step Sil, the method involves at step 112 defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters. As noted above, varying Hamiltonians of different complexities can be chosen depending on the context. Hence any number of appropriate parameters may be chosen, and over any range of appropriate values. In a specific embodiment, the parameters chosen may for instance be two parameters, and these may be the coupling constant J and the external magnetic field h. The range for these values may be chosen appropriately, and in a specific embodiment may for instance be -2 to 2 for each parameter. Hence accordingly, a set of Hamiltonian parameters can be defined, depicted in Figure 2 as h ^_n vectors.

[0058] Hence, in accordance with step Sil the defining of a set of parameters and ranges for the parametrised Hamiltonian advantageously allows the system to cover a variety of different regimes of the Hamiltonian - hence covering different physical behaviours, and probably different quantum phases - such that the method may cover many different combinations and regimes of the Hamiltonian system through exploring these varying parameters in this way.

[0059] Step S13 comprises initialising a set of tuneable parameters of the parameterised quantum circuit. In a specific embodiment, step S13 comprises setting the tuneable parameters of the parameterised quantum circuit configuration at initial values, which may be random values. The parameterised quantum circuit configuration is chosen to be capable of performing Gibbs State generation, and the tuneable parameters of theparameterised quantum circuit configuration are chosen accordingly, and hence the number of different tuneable parameters for any of the parameterised quantum circuits as described herein may vary accordingly to different use cases such as different complexities of Hamiltonian. The tuneable parameters define the behaviour of the circuit and are associated with quantum gates that act on the qubits in the circuit. In a specific embodiment, the tuneable and / or trainable parameters are given as for example a> <p 9 but can take any values and any number of values, for instance in accordance with the complexity of the Hamiltonian being modelled. These will be described further below. It is these parameters that are to being optimised in methods and systems in accordance with the present disclosure, in particular advantageously being subject to collective optimisation over the range of values for the Hamiltonian parameters and hence in a manner that can advantageously accommodate and generalise the Gibbs State across the range of Hamiltonian parameters and regimes. Specific features and embodiments of the parameterised quantum circuit configuration will be described further below.

[0060] The Gibbs state pGibbs is a thermal state that describes the probability distribution of the system's energy levels at thermal equilibrium. It is defined as:Where:- H is the Hamiltonian of the systemP is the inverse temperature, 1 / kBT, where kB is the Boltzmann constant and T is the temperature- the bottom term normalizes the density matrix p Gibbs

[0061] Any appropriate value of can be chosen. In aspects and embodiments of the present application, p is chosen to be 1, which was advantageously determined to be neither too high nor too low, and to be a representatively useful value for performing accurate calculations. Other ranges of P, such as for instance 0.5 to 1 and 1 to 2 are also envisaged, where the higher the P value the closer the system approximates the classical non-quantum regime.

[0062] Merely for visual representation, Figure 2 includes a second conceptual box indicating ‘tuneable parameters a> <p 0, which is labelled both S13 and SI 9. This box is included for improved clarity of the flow diagram such that ‘updating the parameters’ forms a conceptual iterative loop - the parameters are set initially with a particular value inaccordance with S13, and are iteratively optimised throughout the collective optimisation process to be updated in accordance with step SI 9.

[0063] Step Sil and step S13 can be done in any order, or can be done simultaneously or in parallel. Steps Sil and S13 may be performed on a classical computer.

[0064] In step S15, the method comprises performing collective optimisation of the tunable parameters with respect to the parameterised Hamiltonian. Step S15 may comprise Steps S17 and SI 9, and specific embodiments within that.

[0065] Step S17 comprises determining a set of evaluated cost functions, comprising for each set of Hamiltonian parameters: measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit; and evaluating a cost function based on the Gibbs state approximation. The steps of S17 are performed iteratively for each set of parameters of the Hamiltonian.

[0066] In a specific embodiment step S17 comprises storing the cost function, and for instance storing the cost function in a variable such as a tensor, for instance initialized as a zero tensor in accordance with step S 171. This tensor stores the cost function associated with each set of parameters of the Hamiltonian, as will be described further below, and is iteratively updated during the processing. This step may be performed on a classical computer.

[0067] Step S17 comprises measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting a first set of Hamiltonian parameters into the parameterised quantum circuit configuration from the set of Hamiltonian parameters as previously defined across the range of parameter values. In a specific embodiment, this step may comprise step SI 72, comprising inputting the set of Hamiltonian parameters into a particular chosen parameterised circuit configuration, selected from a range of options of ansatz circuits as will be described further in specific embodiments below, for instance with reference to Figures 3 to 5. For instance, the type of parameterised quantum circuit configuration chosen may advantageously be selected based on the complexity of the parameterised Hamiltonian being considered. Regardless of the specific chosen parameterised quantum circuit, an (approximated) Gibbs state is generated and measured using the current tunable parameters of the parameterised quantum computer. These steps are performed on a quantum computer.

[0068] Step S17 comprises evaluating a cost function based on the Gibbs State approximation. In a specific embodiment, S17 comprises step SI 73 comprising evaluating the cost function for each h and adding this to the tensor. In a specific embodiment step S174 comprises iteratively repeating the steps S171 to 173 to process each of the sets of the Hamiltonian parameters and to thereby determine an evaluated cost function for each set of Hamiltonian parameters.

[0069] In specific embodiments, the cost function uses the Gibbs free energy. In particular, evaluating the cost function based on the Gibbs state approximation may comprise either computational evaluation of the Gibbs free energy, else may comprise measuring the Gibbs free energy. For instance, if the system size is small it is possible to generate the Gibbs state approximation from the parameterised quantum circuit configuration and to subsequently evaluate the Gibbs free energy on a classical computer using analytic expression - however when system size increases the associated size of Gibbs state approximation increases and becomes impossible to store in a classical computer, and hence it is necessary to evaluate the Gibbs free energy using quantum circuits by approximating the Gibbs free energy and measuring the resultant terms on the quantum circuit. Accordingly, in all specific embodiments using the Gibbs free energy, what is determined is either the exact or the approximate Gibbs free energy, as appropriate based on the system size. A specific embodiment of evaluating the Gibbs free energy using parameterised quantum circuits by approximating the Gibbs free energy and measuring the resultant terms on the quantum circuit is shown in Figure 15.

[0070] In particular, in a further specific embodiment the cost function uses the Gibbs free energy in the following form:<>Where:- h ”_1 is the vector comprising the parameters of Hamiltonian H(h ”_1);P = l / k_B T is the inverse temperature;co ^,(|) 0 are the tuneable parameters of the parameterised quantum circuit; p(co ^,(|) . () ) is the approximation to Gibbs state measured after tracing out ancillas from the parameterised quantum circuit; andp(co ",<|) ", 0 ") log (oi ",<|) ", 0 ") is the Von-Neumann entropy associated with the approximated Gibbs state.

[0071] In further specific embodiments, the cost function uses an approximation of the Gibbs Free Energy which includes a truncation of the entropy term, in particular a first order truncation using a Taylor series expansion of the entropy term for small qubit Hamiltonians. This advantageously allows faster and more efficient processing as the evaluation of the entropy term can prove difficult when performed using quantum circuit measurements. For instance, Hamiltonians with a small number of qubits, for example 2 to 3 qubits, and / or for example the Hamiltonian may only involve simple interactions (e.g., nearest-neighbour or pairwise interactions) and low-complexity terms, the truncated Gibbs free energy in the form of the cost function comprises:Where:- h "_1 is the vector comprising the parameters of Hamiltonian H(h "_1);P = l / k_B T is the inverse temperature;co ",<|> 0 " are the tuneable parameters of the parameterised quantum circuit; p(co ",(]) ", 0 ") is the approximation to Gibbs state measured after tracing out ancillas from the parameterised quantum circuit; and(l / )Tr(p(co ",<|) ", 0 ")A2 )-(l / ) is the first order truncated Von-Neumann entropy associated with the approximated Gibbs state.

[0072] It is noted that whilst all the above equations for the Gibbs free energy and the truncated Gibbs free energy are all shown with three tuneable parameters, this is merely an example representation. Any number of tuneable parameters may be used, and this may change for instance based on the complexity of the Hamiltonian in consideration, with fewer or more tuneable parameters of the ansatz quantum circuit being appropriate for different use cases. Regardless of the specific number of tuneable parameters, the form of all the above equations would be the same with only the number of tuneable parameters changing. A specific embodiment of evaluating the truncated Gibbs free energy in the above form using parameterised quantum circuits is shown in Figure 15.

[0073] Hence, in accordance with any described embodiment, a set of evaluated cost functions for each of the sets of Hamiltonian parameters is determined. In the specificembodiment of steps S171 to 174 this set of evaluated cost functions will be stored in as a variable, such as the tensor.

[0074] Regardless of the specific features of step SI 7, the iterative evaluation of the cost function for all of the set of Hamiltonian parameters occurs before proceeding to step S19.

[0075] Step S19 comprises evaluating a global cost function, the global cost function comprising the sum of the set of evaluated cost functions, wherein: if the global cost function is not minimised, updating the tunable parameters based on minimising the global cost function and iteratively performing the collective optimisation; and if the global cost function is minimised, defining the tunable parameters as trained parameters.

[0076] In a specific embodiment, step S19 comprises determining and evaluating a global cost function in accordance with step S 191. The global cost function may be a sum of the evaluated cost functions of step SI 7, and may be taken from the evaluated cost functions stored in the tensor in the specific embodiment of steps S171 to S174. For instance, this may comprise a summation of the Gibbs free energy of the Hamiltonian corresponding to each parameter, which are scalars and can be added with conventional summation. The global cost function thereby represents the cost function across each of the sets of Hamiltonians, and hence contributes to a collective optimisation process.

[0077] In specific embodiments, the individual as well as the global cost function uses the Gibbs free energy. In particular, in a further specific embodiment the cost function uses the Gibbs free energy in the following form:<>Where:- h "_1 is the vector comprising the parameters of Hamiltonian H(h "_1);P = l / k_B T is the inverse temperature;co ",<|) 0 " are the tuneable parameters of the parameterised quantum circuit; p(co ",(]) ", 0 ") is the approximation to Gibbs state measured after tracing out ancillas from the parameterised quantum circuit; andp(co ",(]) ", 0 ") log (co ",(|) ", 0 ") is the Von-Neumann entropy associated with the approximated Gibbs state.

[0078] In further specific embodiments, as previously defined, the individual as well as the global cost function uses an approximation of the Gibbs Free Energy is used which includes a truncation of the entropy term, in particular a first order truncation using a Taylor series expansion of the entropy term for small qubit Hamiltonians. This advantageously allows faster and more efficient processing as the evaluation of the entropy term can prove difficult when performed using quantum circuit measurements. The truncated Gibbs free energy in the form of the cost function comprises:Where:- h "_1 is the vector comprising the parameters of Hamiltonian H(h "_1);P = l / k_B T is the inverse temperature;co ",<|) 0 " are the tuneable parameters of the parameterised quantum circuit; p(co ",(]) ", 0 ") is the approximation to Gibbs state measured after tracing out ancillas from the parameterised quantum circuit; and(l / )Tr(p(co ",<|) ", 0 ")A2 )-(l / ) is the first order truncated Von-Neumann entropy associated with the approximated Gibbs state.

[0079] It is noted that whilst all the above equations for the Gibbs free energy and the truncated Gibbs free energy are all shown with three tuneable parameters, this is merely an example representation. Any number of tuneable parameters may be used, and this may change for instance based on the complexity of the Hamiltonian in consideration, with fewer or more tuneable parameters of the ansatz quantum circuit being appropriate for different use cases. Regardless of the specific number of tuneable parameters, the form of all the above equations would be the same with only the number of tuneable parameters changing. For instance, as will be seen with reference to Figure 3, only two variables are used in this instance. A specific embodiment of evaluating the truncated Gibbs free energy in the above form using parameterised quantum circuits is shown in Figure 15.

[0080] Hence in a specific embodiment the global cost function is obtained by the summation over truncated Gibbs free energy over all h.

[0081] Each of these modified cost functions are particularly advantageous in providing accurate and yet efficient calculation of the Gibbs free energy.

[0082] Following step S 191, the global cost function is assessed to determine whether it is minimised or converged. In accordance with step SI 92, in the instance that the global cost function is not minimised, the method proceeds to optimise the parameters and subsequently updates the parameters of the quantum ansatz circuit in accordance with step SI 93. The optimisation of the parameters is in the direction required to minimize the global cost function. This may be determined in any appropriate manner, using any appropriate optimisation algorithm or technique. For instance, in a specific embodiment the optimisation is performed using an Adaptive Moment Estimation (ADAM) optimisation algorithm, which for instance performs optimisation based on both the first moment (mean) and second moment (variance) of the gradient. Accordingly, the collective optimisation process S15 iteratively repeats and begins again from step S17 with the newly updated parameters of the quantum ansatz circuit. At an iteration, the global cost function will be determined to be minimized in step SI 94, wherein at which point it is determined that the values for the parameters of the parameterised quantum circuit configuration are thereby trained.

[0083] Step S19 in accordance with any embodiment may be performed using a classical computer.

[0084] Hence, in accordance with any aspect or embodiment as described herein, it is advantageously possible to determine the Gibbs free energy profile and generate corresponding Gibbs state which is accurately and efficiently generalised across the full range and regime of the Hamiltonian under consideration, rather than being limited to a single set of Hamiltonian parameters and parameter values as in the state of the art. In other words, it is advantageously possible to determine a generalised ansatz for the Gibbs state, defined through the optimised parameters of the parameterised circuit, where this generalized ansatz can generate accurate Gibbs states across the full range and regimes of the Hamiltonian. In particular, here the parameters of the parameterised quantum circuit configuration have been advantageously accurately and efficiently optimised to a single configuration which is advantageously able to generate accurate Gibbs states across the full range and regimes of the Hamiltonian chosen. As will be described further below, this pre-trained parameterised quantum circuit configuration with the fixed optimised parameters can be advantageously used in for instance training a QBM to model a target data set distribution with improved speed, accuracy and computational efficiency, and furthermore to be able to do so across a wide range of Hamiltonian regimes.

[0085] Further, it is envisaged that a collection of different pre-trained parameterised quantum circuits may have been pre-trained, each having fixed trained parameters having been optimised over different complexities of Hamiltonian, for instance over different set of ranges of parameter values and over different parameters. Accordingly, any aspect and embodiment as described may be repeated or performed for different chosen sets of Hamiltonian parameters over different ranges for the Hamiltonian parameter values. Accordingly, in aspects and embodiments, a particular optimised parameterised pre-trained parameterised circuit configuration - for instance having any of the processing layers as described herein - may be selected to be tailored to a particular target data distribution and / or Hamiltonian being considered, thereby improving the accuracy and performance and precision of the applicability of the different Gibbs state generations for use in a wide array of different circumstances and applications.

[0086] Figure 3 illustrates a specific embodiment of the parameterised quantum circuit configuration setup for use in any aspect and embodiment described herein, for instance for use as the chosen ansatz circuit configuration in step S17 and in particular in step S172. It is noted that any appropriate ansatz circuit can be used, for instance any appropriate near-term quantum hardware, or any appropriate hardware efficient ansatz circuit (such as Kandala, A., Mezzacapo, A., Temme, K. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242-246 (2017). https: / / doi.org / 10.1038 / nature23879), such as for instance an SU2 hardware efficient ansatz (such as Kandala, A., Mezzacapo, A., Temme, K. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242-246 (2017). https: / / doi.org / 10.1038 / nature23879) .

[0087] In particular, Figure 3 shows an example of a parameterised quantum circuit configuration in the form of a hardware efficient ansatz (such as Kandala, A., Mezzacapo, A., Temme, K. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242-246 (2017). https: / / doi.org / 10.1038 / nature23879). As previously described, the complexity of the chosen Hamiltonian can vary and be chosen appropriately due to context. In particular, the following specific embodiments will be disclosed in the context of a spin Hamiltonian with two parameters: H(lN )=H(h_l,h_2). However, this is merely representative and other Hamiltonians with different parameters and numbers of parameters are envisioned.

[0088] Similarly, the number of ancilla qubit layers can take any appropriate value. Merely as a representative example, the following specific embodiments in Figures 3 to 5 show the number of ancillas being chosen to be the same as the number of system qubits as this was determined to work well in performing accurate calculations. However, it is noted for instance that for Hamiltonians of lower complexity, for instance having one parameter rather than two, fewer ancillas may be used.

[0089] Regardless, the parameterised quantum circuit configuration embodies the tuneable parameters as previously described, wherein the tuneable parameters define the behaviour of the circuit and are associated with quantum gates that act on the qubits in the circuit. The ansatz quantum circuit may comprise an encoding layer and a processing layer. In specific embodiments the encoding layer and the processing layer may be the same component or may be different components, as described further below. The encoding layer encodes / embeds the input data which are the parameters of the Hamiltonian and, along with the processing layer(s) when present, sets the tuneable parameters of the ansatz quantum circuit, and hence encodes / embeds input data into a quantum state by initializing the qubits according to input data values by parametrising quantum gates (i.e. rotation gates). The encoding and optional processing layers perform quantum operations by applying the quantum gates to perform quantum transformations of the quantum state in a way that solves a specific problem or optimises a function, with the tuneable parameters adjusted during the algorithm’s optimisation phase. In other words, the encoding layer initialises the quantum state by mapping the input data onto the qubits, and along with the processing layer performs the main quantum computation by applying the variational gates to manipulate the qubits. Hence in the present application, the encoding layer encodes the sets of parameters of the Hamiltonians, and the tuneable parameters are embodied in the encoding layer components and processing layer components when present. Accordingly, in specific embodiments the encoding layer performs encoding the input set of Hamiltonian parameters and initialising the tuneable parameters of the parameterised quantum circuit configuration.

[0090] Hence, in a specific embodiment as depicted in Figure 3, a single circuit component comprises the encoding layer, for instance in the specific example depicted this is a hardware efficient ansatz, showing two ancilla qubits 31 and two system qubits 33, where the qubits are labelled q_n denoting n different ‘qubit wires’. The quantum gates 35 are depicted on each qubit, with the following variables:m’s and ’s are tuneable parameters1 is the number of the layerh have the parameters of the Hamiltonian.R is the rotation applied to the qubit, which are tuned during the variational optimisation process, where RZ is the qubit rotation around the z-axis and RY is the rotation around the y-axisHence the total tuneable parameters in the parameterised quantum circuit configuration is: 4 x (number of system qubits + number of ancillas) x 1 x size of h .

[0091] As noted above, the specific embodiment described is in the context of a spin Hamiltonian with two parameters: H(h )=H(h_l,h_2), where the is the set of Hamiltonian parameters as previously described in the embodiments of S 11. Hence in the present embodiment the encoding layer encodes the sets of parameters of the Hamiltonian and initialises the tuneable parameters of the hardware efficient ansatz. Hence in the step S17 of any embodiment as previously described, the encoding layer encodes each set h l, h_2 to h_n in turn, one per iteration of the loop of SI 7, until each of the sets of parameters for the Hamiltonian have been iteratively processed by step SI 7, and at each stage uses the first randomly initialised set of the tuneable parameters of the hardware efficient ansatz in accordance with the set of Hamiltonian parameters being encoded in that iteration.

[0092] Hence in the specific embodiment of Figure 3, the hardware efficient ansatz is an encoding layer performing iterative encoding of the sets of Hamiltonian parameters in accordance with any previously described embodiment of step Sil, with the tuneable parameters m and (p from any described embodiment of step S13 and tuned / trained in accordance with any described embodiment of steps S15 and SI 9. Further, the hardware efficient ansatz 30 that forms the encoding layer is then also used to measure / sample the (variational) Gibbs state and the exact / approximate Gibbs free energy as defined in any embodiment of step S17, as indicated in Figure 3 in the ‘ p_variational measurement’ 37 from the system qubits 33.

[0093] In particular, the specific embodiment of Figure 3 is useful for relatively simple Hamiltonian’s with a small number of parameters.

[0094] With reference to Figures 8 to 12, ‘Meta-VQT vl’ is a specific example of an embodiment of Figure 3.

[0095] Figure 4 illustrates a specific embodiment of the parameterised quantum circuit configuration setup for use in any aspect and embodiment described herein, forinstance for use in step S17 and in particular the chosen ansatz circuit configuration in step S172.

[0096] In particular, Figure 4 illustrates a specific embodiment of the parameterised quantum circuit configuration comprising the specific embodiment of Figure 3, and additionally a Hamiltonian Variational Ansatz (HVA) 40 (see for instance Roeland Wiersema et.al, Exploring Entanglement and Optimisation within the Hamiltonian Variational Ansatz, PRX Quantum 1, 020319, htps: / / ioumals.aps.Org / prxquantum / abstract / 10.l 103 / PRX Quantum.1 ,020319). In the specific embodiment of Figure 4, the encoding layer comprises the hardware efficient ansatz of Figure 3, and the processing layer comprises the HVA. In other words, processing as previously described is performed in both the hardware efficient ansatz (encoding layer) and the HVA (processing layer) in combination.

[0097] In particular, in instances in which the chosen Hamiltonian is more complex (i.e. more parameters etc.), it becomes more difficult and inefficient for the hardware efficient ansatz to accurately and effectively simulate the Gibbs state approximation corresponding to the chosen Hamiltonian. Accordingly, in instances in which more complicated Hamiltonians are chosen, to more accurately simulate a correspondingly complex Gibbs state approximation for the complex Hamiltonian, this specific embodiment additionally includes the special type of ansatz known as the HVA, which is well suited to deal with Hamiltonian’s with more complex terms.

[0098] For instance, merely for illustrative purposes, an example HVA 42 with five qubits is depicted for a spin Hamiltonian in accordance with the Ising model:With the conventional spin interaction term and magnetic field term. The example HVA 42 is depicted merely to illustrate the functionality of HVAs, and it is noted that in embodiments of the present application the HVA would be chosen accordingly to be compatible with the Hamiltonian considered and the number of qubits. In particular, in embodiments the hardware efficient ansatz and the HVA include the same gates, however in the HVA gates we implement the operations in the terms of the complexity of the Hamiltonian chosen. In particular, each of the gates 43 of the HVA are a combination of the fundamental gates 35 defined in the hardware efficient ansatz.

[0099] HVAs have gates which are parameterised to be similar to the Hamiltonian they are modelling. Accordingly, in the example HVA 42, the gates 43 are parameterised to be similar to the Ising model Hamiltonian. Accordingly:9 s are tuneable parameters1 is the number of processing layers- XX taking the form: e — i(T iX(TX'+1is the parametrized two qubit gate> • z- Z taking the form: e i is the parametrized single qubit gate.Hence the total tuneable parameters in the HVA processing layer is : 2 x (number of system qubits + number of ancillas) x 1.

[0100] Regardless of the specific Hamiltonian being considered, the HVA is specifically chosen and configured with the gates 43 parameterised in terms similar to the chosen Hamiltonian. Hence advantageously this assists the parameterised quantum circuit configuration comprising the hardware efficient ansatz 30 and the HVA 40 to more accurately approximate the full complexity of the Gibbs state approximation of the chosen Hamiltonian.

[0101] Hence accordingly, regardless of the specific Hamiltonian and parameters chosen, the set of Hamiltonian parameters as previously described in the embodiments of step S 11 are encoded into the encoding layer and the tuneable parameters of the hardware efficient ansatz are initialised. Hence in the step S17 of any embodiment as previously described, the encoding layer encodes each set h l, h_2 to h_n in turn, one per iteration of the loop of SI 7, until each of the sets of parameters for the Hamiltonian have been iteratively processed by step S17, and at each stage uses the first initialised tuneable parameters of the hardware efficient ansatz and the HVA in accordance with the set of Hamiltonian parameters being encoded in that iteration.

[0102] Hence in the specific embodiment of Figure 4, the hardware efficient ansatz is an encoding layer performing iterative encoding of the sets of Hamiltonian parameters in accordance with any previously described embodiment of step Sil, with the tuneable parameters m and from any described embodiment of step S13 and tuned / trained in accordance with any described embodiment of steps S15 and SI 9. Further, the hardware efficient ansatz 30 also forms part of the processing layer performing processing, the output of which is input into the HVA 40. The HVA 40 forms another part of the processing layer with the tuneable parameter 9 from any described embodiment of step S13 and tuned / trained in accordance with any described embodiment of steps S15 and S19, and can thereby be usedto measure / sample the (variational) Gibbs state and the exact / approximate Gibbs free energy as defined in any embodiment of step SI 7, as indicated in Figure 4 in the ' p_v ar national measurement’ 47 from the system qubits.

[0103] Hence the specific embodiment of Figure 4 is useful for encoding and processing more complex Hamiltonians, and wherein the HVA can be specially chosen and configured to be specifically tailored to the particular Hamiltonian chosen through the parametrisation of the gates in terms the same as or similar to the chosen Hamiltonian terms. For instance, in a specific embodiment the gates of the Hamiltonian Variational Ansatz are parameterised based on the defined sets of Hamiltonian parameters, and for instance optionally wherein the defined sets of Hamiltonian parameters comprises the field or interaction terms of the Hamiltonian.

[0104] With reference to Figures 8 to 12, ‘Meta-VQT v2’ is a specific example of an embodiment of Figure 4.

[0105] Figure 5 illustrates a specific embodiment of the parameterised quantum circuit configuration for use in any aspect and embodiment described herein, for instance for use in step S17 and in particular the chosen ansatz circuit configuration in step S172.

[0106] In particular, Figure 5 illustrates a specific embodiment of the parameterised quantum circuit configuration wherein the encoding layer comprises a neural network 50, and wherein the processing layer comprises a hardware efficient ansatz 30 and optionally an HVA 40. In a specific embodiment, the neural network 50 may be a feed forward neural network. However, any appropriate neural network may be used. The hardware efficient ansatz 30 and the HVA 40 are the same or similar to those as previously described with reference to Figures 3 and 4, with the exception of the differences noted below. The processing provided by the neural network 50 is performed on a classical computer. Hence in this embodiment the wording ‘parameterised quantum circuit’ is used conceptually to include the neural network, although in terms of the processing performed by the neural network this is performed on a classical computer. Hence this is a specific embodiment of the conceptual processing unit represented by the step S17 and in particular a specific embodiment of the ‘chosen ansatz circuit configuration’ defined in step SI 72, where the configuration includes the neural network 50 in combination with quantum circuitry such as the hardware efficient ansatz and optionally the HVA.

[0107] In particular, the neural network 50 is configured such that the parameters of the Hamiltonian, i.e. from each particular set of parameters as defined in embodiments of stepSil, can be encoded into the neural network 50 along with the tuneable parameters of the associated hardware efficient ansatz 30 and the optional HVA 40, i.e. along with the parameters (p of the hardware efficient ansatz 30 and the 9 of the optional HVA 40. Following training, the neural network 50 will be configured to take as input the sets of parameters for the Hamiltonian as defined in any embodiment of step Sil, and will output optimised tuneable parameters to be used to set the gates of the hardware efficient ansatz 30 and optionally for the HVA 40 where this is used. Hence in contrast to previous embodiments described with reference to Figures 3 and 4, in the specific embodiment of Figure 5 the sets of parameters of the Hamiltonian are encoded in the neural network 50 rather than in the gates of the hardware efficient ansatz. Hence in the specific embodiment of Figure 5, the output from the neural network is the initialized tuneable parameter < > of the hardware efficient ansatz 30 and 9 of HVA 40 where this is used.

[0108] As can be seen, the gate functions 35 are advantageously simpler here than in the specific embodiments described with reference to Figures 3 and 4 as the neural network 50 advantageously allows for not having encode the Hamiltonian parameters in the gates of the hardware efficient ansatz 30 and / or HVA 40. Indeed, the gate functions 35 do not require the Hamiltonian vector h, and because this specific embodiment does not encode the Hamiltonian parameters directly into the gates 35 as in the specific embodiments described with reference to Figures 3 and 4, there is further also no need for a> as here the Hamiltonian parameters are encoded into the neural network 50 allowing the simplified use of only a single parameter < > in place of the previous combination of M and .

[0109] Hence, in accordance with specific embodiments, the tuneable parameters of the parameterised quantum circuit may be obtained advantageously efficiently and accurately as outputs from the neural network 50.

[0110] Hence in the present embodiment, in the step S17 of any embodiment as previously described, the encoding layer in the form of the neural network 50 encodes each set h_ , h^2 to h_n in turn, one per iteration of the loop of SI 7, until each of the sets of parameters for the Hamiltonian have been iteratively processed by step S17.[oni] Hence in the specific embodiment of Figure 5, the hardware efficient ansatz 30 forms part of the processing layer, with the tuneable parameters and 9 from any described embodiment of step S13 and tuned / trained in accordance with any described embodiment of steps S15 and S19 being defined by the output of the neural network 50. Hence in the specific embodiments including the neural network 50, it is the parameters andweights of the neural network 50 which are tuned / trained in accordance with any described embodiment of steps S15 and S19.

[0112] In a first specific embodiment, the hardware efficient ansatz 30 is the only processing layer, and there is no HVA. In a second specific embodiment as depicted in Figure 5 the processing layer further also comprises the HVA 40, which as described with reference to Figure 4 has parametrised gates defined in accordance with the complexity of the chosen Hamiltonian. Hence regardless of which first or specific embodiment is being considered, the processing layer performs processing and can thereby be used to measure / sample the (variational) Gibbs state and the exact / approximate Gibbs free energy as defined in any embodiment of step SI 7, as indicated in Figure 5 in the ‘ p_variational measurement’ 57.

[0113] Hence advantageously the neural network 50 can deal with even more advanced and complex versions of the Hamiltonian and associated parameters, and further can do so in a manner with improved accuracy and efficiency as compared to exclusively quantum circuit based methods such as the hardware efficient ansatz and the HVA. In particular, the use of the neural network 50 advantageously reduces the number of quantum resources required by moving some of the computational processing of encoding away from the quantum based hardware efficient ansatz and HVA. Further, the use of the neural network 50 provides faster optimisation of the trainable / tunable parameters of the parameterised quantum circuit configuration as it uses fewer quantum resources such as the gates, and further because the level of complexity represented by the neural network 50 is much higher, allowing for instance non-linear connections which cannot be done in the quantum based hardware efficient ansatz and HVA components. Hence, as the complexity of the chosen Hamiltonian varies in different use cases, or with reference to different data distributions of target data sets. In particular, it is noted that in the specific embodiments described in relation to Figure 5, the HVA 40 is not required as the configuration generates high quality and accurate Gibbs states without it. In particular this is because the neural network 50 is advantageously able to handle a large amount of complexity and provides a large amount of calculating power through its complex operations, thereby relieving the quantum circuit components of this burden. However, the inclusion of the HVA 40 may even further improve the accuracy and precision of Gibbs state generation.

[0114] Any of the specific parameters shown in gates 35 of either of Figures 3 or 5 are merely representative and any other appropriate parameterising is envisaged.

[0115] With reference to Figures 8 to 12, ‘NN-Meta-VQT vl’ is a specific example of an embodiment of Figure 5 in which the HVA is not used, and ‘NN-Meta-VQT v2’ is a specific example of an embodiment of Figure 5 in which the HVA is used.

[0116] Accordingly, Figures 3 to 5 and the associated descriptions define specific embodiments of the ‘chosen ansatz circuit configuration’ as defined in step SI 72 of Figure 2, and which therefore can be used in any specific embodiment of step S17 as described herein. In particular, the choice of which of the specific embodiments and combinations of features which make up parameterised circuit can be made based on the complexity of the Hamiltonian chosen, for instance the Hamiltonian chosen to represent a particular target data set. The specific embodiments associated with and described in reference to Figure 5 represent the most powerful and most accurate methods, capable of dealing with more complicated Hamiltonians in a fast, accurate and efficient manner.

[0117] Hence each of the specific embodiments of Figures 3 to 5 when incorporated into embodiments of steps Sil to S19 - such as for instance when incorporated into the specific embodiment of Figure 2 - represent an advantageous approach to pre-training a single parameterised quantum circuit with a set of trained parameters. In particular, following step SI 9, and in particular step SI 94, the single parameterised quantum circuit configuration is thereby advantageously pre-trained with trained parameters such that it is advantageously generalised and optimised to generate Gibbs states across a range of parameter values of the Hamiltonian - as defined by the range of values for the parameters in the sets of Hamiltonian parameters - and hence such that it may thereby accurately and efficiently accommodate and account for Hamiltonians across a range of different regimes, phases and parameter values, in a manner that state of the art methods are incapable of.

[0118] Accordingly, the aspects and embodiments described with reference to Figures 2 to 5 provide an improved method of Gibbs state generation, capable of faster, more accurate and more efficient (i.e. fewer processing steps) generation of Gibbs states regardless of further specific application.

[0119] Further, it is envisaged that a collection of different pre-trained parameterised quantum circuits may have been pre-trained, each having fixed trained parameters having been optimised over different complexities of Hamiltonian, for instance over different set of ranges of parameter values and over different parameters. Accordingly, in aspects and embodiments, a particular optimised parameterised pre-trained parameterised circuit configuration - for instance having any of the encoding and processing layers as previouslydescribed - may be selected to be tailored to a particular target data distribution and / or Hamiltonian being considered, thereby improving the accuracy and performance and precision of the applicability of the different Gibbs state generations for use in a wide array of different circumstances and applications.

[0120] As previously outlined, a specific application may be the use of such pre-trained parameterised circuit configurations in QBM training and use, wherein a QBM is being trained to learn the distribution of a particular target data set. Here, the use of aspects and embodiments of the present application within the training flow of a QBM provides a number of distinct advantages in providing faster, more accurate and more efficient QBM training, in avoiding the disadvantageous^ time consuming nested loop processing required in the Gibbs state generation stage of conventional approaches, and in providing a Gibbs state generation capable of accounting for a wide range of regimes and phases of the Hamiltonian.

[0121] Figure 15 is a diagram illustrating an example of how the truncated Gibbs free energy as described in any aspect or embodiment herein may be measured using a parameterised quantum circuit in accordance with aspects and embodiments herein. Merely as a representative example, the parameterised quantum circuits shown are the ones shown in Figure 4, however any parameterised quantum circuit as described herein may be used, for instance the parameterised quantum circuit of Figure 3 or 5 may also be used.

[0122] In particular, Figure 15 shows configurations of the parameterised quantum circuit comprising the hardware efficient ansatz 30 and the HVA 40, wherein the parameterised quantum circuit is configured to measure the truncated Gibbs free energy as provided herein. Accordingly, the gates of the parameterised quantum circuit may appropriately configured to measure the resultant terms of the truncated Gibbs free energy, where the first term is: Tr(jo(m, < >,) and the second term is: Tr 0) . Asshown in Figure 15, a parameterised quantum circuit may first be configured to perform evaluation of the first term of the truncated Gibbs free energy, and the same or a different parameterised quantum circuit may be configured to perform evaluation of the second term of the truncated Gibbs free energy. The evaluation of the first and second terms may be performed either sequentially or simultaneously or at substantially the same time.

[0123] For performing evaluation of the second term, two parameterised quantum circuits will need to be configured, and will for instance include Hadamard gates for performing operations on a qubit to create a superposition state and may for instance include a controlled swap gate (i.e. Fredkin gate) which may for instance be used for statecomparison. Accordingly, as shown, one ancilla can be used to measure the p2term, with the measurement of the expectation value of the Pauli operator ozof the measurement being equivalent to measuring the p2.

[0124] Accordingly, Figure 15 shows the evaluation of each of the first and second terms of the truncated Gibbs free energy - which as previously described may be used as a cost function in the collective optimization step S15 to S19 as described in any manner herein. In particular, Figure 15 may be considered as an illustrative example of measuring the approximate Gibbs free energy for small systems using a quantum computer, for instance as previously mentioned for use with Hamiltonians with a small number of qubits, for example 2 to 3 qubits, and / or for example the Hamiltonian may only involve simple interactions (e.g. nearest-neighbour or pairwise interactions) and low-complexity terms.

[0125] With reference to the specific embodiments of the parameterised quantum circuit as shown in Figures 3 to 5, it is noted that different numbers of tuneable parameters are depicted. As previously described, these are merely representative examples and any number of suitable tuneable parameters may be used as previously described, for instance for use with different complexity Hamiltonians, and an appropriate Gibbs free energy or truncated Gibbs free energy may therefore be used in line with the equations as previously described but having different numbers of tuneable parameters, for instance in each specific instance to match the ansatz quantum circuit being used and hence.

[0126] Figure 6 is a diagram illustrating a method according to an aspect. In particular, the method may be a computer implemented method of training a variational quantum Boltzmann machine, and may be performed on a hybrid quantum and classical computer as will be described further below.

[0127] Step S21 comprises selecting a Hamiltonian comprising initial Hamiltonian parameters.

[0128] Step S23 comprises measuring a distribution of Gibbs states using a pretrained parameterised quantum circuit configuration trained in accordance with any aspect or embodiment as described herein, wherein the tuneable parameters are the trained parameters.

[0129] Step S25 comprises adjusting the initial Hamiltonian parameters based on minimising a cost function, comprising: evaluating the cost function, comprising comparing the distribution of Gibbs states with a target distribution of measurements from a target dataset; and generating updated Hamiltonian parameters, comprising adjusting the initial Hamiltonian parameters in a direction that reduces the cost function.

[0130] Any of the aspects and embodiments described herein, and any of steps of Figure 6, may be performed by an apparatus as described with reference to Figures 13 and 14 below, which concern a hybrid quantum and classical computer arrangement.

[0131] Figure 7 illustrates specific embodiments of the method of Figure 6. In particular, Figure 7 illustrates a specific embodiment of a training loop for a QBM.

[0132] In a specific embodiment of step S21, in accordance with step S211 a Hamiltonian is chosen with reference to a target dataset which the QBM is being trained to learn. Hence the Hamiltonian may be of varying complexity depending on context, having different numbers of parameters etc. The Hamiltonian has initial Hamiltonian parameters which are to be trained during the QBM training, and these are initialized with particular values. This may be random, else may be guided by consideration of the target dataset distribution. The distribution of the target dataset to be learned is referred to as Pdata- For instance, an example of a simple target dataset is the probability distribution of a bell state, however any target dataset can be used.

[0133] In a specific embodiment of step S23, in step S231 the Gibbs state for the Hamiltonian is set and measured. In particular, it is at this stage that state of the art systems would be required to perform a nested loop operation to iteratively train and optimise the parameters of a quantum circuit, leading to disadvantageous slow, inefficient and inaccurate Gibbs state generation. By contrast, in aspects and embodiments of the present application the measuring of the Gibbs state is performed using any chosen parameterised quantum circuit configuration as previously described, for instance as outlined in and with reference to Figures 2 to 5, wherein the chosen parameterised quantum circuit configuration is pretrained in accordance with any previously described method to have trained and fixed parameters for the components of the quantum circuit which have already been optimised (i.e. using collective optimisation) to generate a Gibbs state across a range of different parameter values, regimes and phases of the Hamiltonian. Accordingly, in step S232, a particular parameterised configuration is chosen to be used to generate the Gibbs states.

[0134] In particular, in accordance with any previously described aspect or embodiment, any of the specific embodiments of the parameterised quantum circuit configuration can be chosen - for instance any of the specific embodiments described with reference to Figures 2 to 5. Accordingly, a particular parameterised quantum circuit can be chosen based on the features, characteristics, complexity etc. of the associated Hamiltonian chosen to model the target data distribution . In particular, it is envisaged that a collection ofdifferent pre-trained parameterised quantum circuits may have been pre-trained, each having fixed trained parameters having been optimised over different complexities of Hamiltonian, for instance over different set of ranges of parameter values and over different parameters. Accordingly, in aspects and embodiments, a particular optimised parameterised pre-trained parameterised circuit configuration - for instance having any of the encoding and / or processing layers as previously described - may be selected to be tailored to the Hamiltonian being used to model the target data distribution, thereby improving the accuracy and performance and precision of the training of the QBM. Accordingly, a specific embodiment comprises selecting the pre-trained parameterised quantum circuit configuration from a plurality of pre -trained parameterised quantum circuit configurations trained in accordance with any previously described manner, the selecting based on the complexity and number of parameters of the selected Hamiltonian comprising initial Hamiltonian parameters.

[0135] Accordingly, regardless of specific parameterised quantum circuit configuration, instead of performing nested loops to optimise the quantum circuit before the Gibbs state can be generated, in aspects and embodiments of the present application a particular pre-trained parameterised quantum circuit is selected which already has pre-trained and fixed optimised parameters. Hence the Gibbs state can be immediately generated, and the distribution of Gibbs state measurements Pvariationai can be determined.

[0136] In a specific embodiment of step S25, in step S251 a cost function may be evaluated which evaluates the difference between between Paata and Pvariationai ■ Thecost function may be any appropriate cost function, for instance may be KL divergence or quantum relative entropy. In a further specific embodiment, a gradient may be evaluated, for instance comprising numerical differentiation. If the cost function is not minimized or converged, in step S252 the parameters of the Hamiltonian are optimised in a direction intended to minimise the cost function. The optimisation may be performed using any suitable optimiser, such as for instance an ADAM optimiser.

[0137] In step S27the training process for the QBM is completed when the cost function is minimised / converged, at which point the Hamiltonian parameters are deemed final and trained, and the QBM has been deemed to have learned the distribution of the target data set to a sufficient level of accuracy. Accordingly, with the final trained parameters for the Hamiltonian set, the QBM can thereby be subsequently used in a generative manner to take samples from the Gibbs state and to output newly generated data in a distribution similar or the same as the target data distribution. Accordingly in a specific embodiment, if the costfunction is minimised the updated Hamiltonian parameters are defined as trained Hamiltonian parameters.

[0138] Accordingly, QBMs trained in this manner are trained advantageously quickly, accurately and efficiently. In particular, QBMs can be trained in a manner that avoids the use of nested loops in the Gibbs state generation stage, which are otherwise required at each and every iteration of the updated Hamiltonian parameters at stage S23.

[0139] Further, it has been shown experimentally that generating Gibbs states in accordance with any of the aspects and embodiments described herein provides accurate and high precision Gibbs states for all Hamiltonians within the range of the values of the sets of Hamiltonian parameters on which the pre-trained parameterised quantum circuit configurations were trained, even if the values of the parameters of the Hamiltonian in question were not specifically within the sets of values for the parameters of the Hamiltonian during the training phase of the parameterised quantum circuit configuration. Further, it has also been shown experimentally that generating Gibbs states in accordance with any of the aspects and embodiments described herein provides accurate and high precision Gibbs states for Hamiltonians with parameters outside of the training range of the sets of value of the Hamiltonian, however with the further distance from the training range the lower the accuracy.

[0140] Accordingly, aspects and embodiments may advantageously provide for the generation of good quality Gibbs state irrespective of parameter regime of Hamiltonian.

[0141] Aspects and embodiments of the present application may advantageously provide a pre-trained parameterised quantum circuit configuration optimised and generalised for a set of parameters of Hamiltonian so as to accurately generate the quantum Gibbs state within a particular parameter regime. Accordingly, after training, individual optimisation for every single parameter is not required as is performed in for instance the nested loop optimisations of the state of the art.

[0142] Aspects and embodiments have been shown to outperform in terms of accuracy and efficiency existing methods such as VARQITE.

[0143] Aspects and embodiments of the present application may allow for the choice from a range of different parameterised quantum circuit configurations and for parameterised quantum circuit configurations trained over different ranges of sets of values of the Hamiltonian, and thereby advantageously provide methods capable of learning and generating quantum Gibbs state in all parameter regimes of the Hamiltonian.

[0144] Aspects and embodiments of the present application may advantageously incorporate the pre-trained parameterised quantum circuit configuration into QBM training and thereby avoid the nested loop used within the conventional training of QBM, advantageously decreasing the runtime and increasing the training efficiency of QBMs.

[0145] In aspects and embodiments of the present application, the precision of Gibbs state generation as well as the convergence of the loss function during QBM training is found to be better and more accurate than one existing methods for a given input data distribution and Hamiltonian.

[0146] Accordingly, with the final trained parameters for the Hamiltonian set, the QBM can thereby be subsequently used in a generative manner to take samples from the Gibbs state and to output newly generated data in a distribution similar or the same as the target data distribution. Hence according to a specific aspect, the trained variational quantum Boltzmann machine arrived at in accordance with any manner described herein may be used as or part of a generative Al system or method to generate new data.

[0147] In particular, the pre-trained parameterised quantum circuit configurations and the trained QBMs as provided for by methods in accordance with aspects and embodiments of the present application may advantageously provide for generative Gibbs state systems and methods which may be advantageously applied in a number of different fields and applications to generate new data.

[0148] In particular, relevant and related field of generative creation in which aspects and embodiments may be applied include broad quantum generative machine learning, generative systems based on energy / Hamiltonian based models, and variational hybrid quantum thermal state (Gibbs state) generations.

[0149] Specific applications in which aspects and embodiments may be applied to generate new data based on Gibbs State generation include for instance simulated quantum Gibbs states, accurate predictions of molecular structures and reaction rates, solving combinatorial optimisation problems, generative Quantum Machine Learning Models such as Quantum Boltzmann Machine that are superior to classical counterparts, solving semi-definite programs, quantum key distribution, quantum metrology.

[0150] Further specific applications in which aspects and embodiments may be applied to generate new data based on Gibbs State generation include for instance designing quantum counterparts of existing efficient classical Boltzmann machine (BM) trainingalgorithms, speech recognition, data reduction, feature extraction, classification, collaborative filtering, and travelling salesman problems, etc.

[0151] Figures 8 illustrates a table of results of generate Gibbs states using Meta-VQT vl and v2 and NN-Meta-VQT vl and v2 in accordance with aspects and embodiments of the present application as compared to the state of the art method VARQITE.

[0152] In particular, the test parameters were as follows:- Hamiltonian considered :Parameters : (J, h.)- Number of system qubits : 2, / ? = 1

[0153] Different parameter regimes for this Hamiltonian were identified by determining the non-analytic boundaries where the first and / or second order derivative of Gibbs free energy with respect to h and / or J diverges. Accordingly, the following was determined:- Analytic Regionso Regime 1 : h = [-1, -2], / = [1,2]o Regime 2 : h = [1,2], / = [1,2]o Regime 5 : h = [—0.5, 0.5], h = [—2, —1]Signatures of non-analytic boundarieso Regime 3 : J = [—2, —1], h = [1,2]o Regime 4 : J = [—2, —1], h = [—2, —1]o Regime 6 : J = [0.5, 1.5], h = [-0.1, 0.1]

[0154] Figure 8 considers the full regime and determines how the finite temperature state is approximated.

[0155] With reference to Figure 8:Meta-VQT vl includes 4 layers of hardware efficient ansatz (SU2) and the runtime was 40 minutes;Meta-VQT v2 includes 4 layers of hardware efficient ansatz (SU2) and 1 layer of Hamiltonian variational ansatz, and the runtime was 91 minutes;- NN-Meta-VQT vl includes 4 layers of hardware efficient ansatz (SU2) and the runtime was 36 minutes;- NN-Meta-VQT v2 includes 4 layers of hardware efficient ansatz (SU2) ) and 1 layer of Hamiltonian variational ansatz, and the runtime was 86 minutes.Each of the configurations used 2 qubits and 2 ancillas and performed 600 iterations

[0156] As can be seen, Meta-VQT v2 including the additional Hamiltonian ansatz to generate the states provides even further precision over Meta-VQT vl with the hardware efficient quantum circuit layers only. Further, NN-Meta VQT vl (classical neural network + quantum circuit layers) generates good quality Gibbs states even without the Hamiltonian variational ansatz. Indeed, precision of the generated Gibbs state with both Meta-VQT v2 (with HVA) and NN-Meta VQT vl (without HVA) is better than the VarQITE algorithm. The comparison can be made as NN-Meta VQT vl and Meta VQT vl have the same ansatz and depth as that of VarQITE.

[0157] Figures 9 and 10 illustrate tables of results of generate Gibbs states using NN-Meta-VQT vl in accordance with aspects and embodiments of the present application as compared to the state of the art method VARQITE.

[0158] In particular, the test parameters were again as follows:- Hamiltonian considered :Parameters : (J, h.)- Number of system qubits : 2, / ? = 1

[0159] Different parameter regimes for this Hamiltonian were identified by determining the non-analytic boundaries where the first and / or second order derivative of Gibbs free energy with respect to h and / or J diverges. Accordingly, the following was determined:- Analytic Regionso Regime 1 : h = [-1, -2], / = [1,2]o Regime 2 : h = [1,2], / = [1,2]o Regime 5 : h = [—0.5, 0.5], h = [—2, —1]Signatures of non-analytic boundarieso Regime 3 : J = [—2, —1], h = [1,2]o Regime 4 : J = [—2, —1], h = [—2, —1]o Regime 6 : J = [0.5, 1.5], h = [-0.1,0.!]

[0160] Figures 9 and 10 separately considers different regimes and determines how the finite temperature state is approximated. The ansatz chosen for NN-Meta VQT vl and the existing method VARQITE is same and a direct comparison can be done.

[0161] Figure 9 shows the parameter regimes of the complex Hamiltonian considered, and Figure 10 shows differing complexities of the 2 -qubit Hamiltonian. NN-Meta-VQT vl includes 4 layers of hardware efficient ansatz (SU2). NN-Meta VQT vl is advantageous over the existing method of VarQITE for generation of Gibbs state in all of these domains.

[0162] As can be seen in Figure 9:In Regime 6, performance of NN-Meta VQT vl and VarQITE is good with VarQITE having lower average trace distance.In Regimes 1, 2 and 5, NN-Meta VQT vl generates Gibbs states which are close to the exact ones. By contrast, the VarQITE generates states which are different from the exact Gibbs state and a few are very different from the exact one.In Regimes 4 and 3, compared to all regimes here the VarQITE is unable to generate any states which are closer / similar to the exact Gibbs state. By contrast, NN-Meta- VQT vl is able to generate good quality Gibbs states.

[0163] Overall, the performance of NN-Meta-VQT vl is advantageously accurate and independent of parameter regimes, whereas VarQITE performance is dependent on the parameter regime.

[0164] As can be seen in Figure 10, the average trace distance of the NN-Meta-VQT is advantageously independent of the complexity of the Hamiltonian, whereas for VARQITE it is highly dependent. Indeed, it is demonstrated that the NN-Meta-VQT vl based QBM being better than VARQITE based QBM is dependent upon the complexity of the Hamiltonian considered and the nature of the input distribution trained. If the Hamiltonian considered is simple and if it has only one commuting block of operators, then the performance of VARQITE is better or comparable with NN-Meta-VQT vl as shown. However, advantageously when more than one commuting blocks are present, NN-Meta-VQT vl generates Gibbs state more efficiently than the VARQITE. In summary, the performance of VARQITE is dependent on the complexity of the Hamiltonian while the performance of NN-Meta-VQT vl is advantageously independent of the complexity of Hamiltonian.

[0165] Figure 11 illustrates a table of results of generate Gibbs states using NN-Meta-VQT vl in accordance with aspects and embodiments of the present application as compared to the state of the art method VARQITE, where different inverse temperature values ft for the Gibbs state are used over the full regime of the Hamiltonian.

[0166] As can be seen, for both ft values the efficiency of NN-Meta-VQT vl is better or comparable to the VARQITE, but the accuracy is always higher.

[0167] Figure 12 illustrates a table of results relating to incorporating NN-Meta VQT vl in a QBM training workflow to learn an input probability distribution, and comparing this with the VARQITE based QBM, and shows associated plots for each of NN-Meta-VQT vl and VARQITE with KL divergence and trace distance against epochs.

[0168] In particular:- Number of layers for both methods : 4 layers of hardware efficient ansatz (SU2) - Number of qubits : 2Hamiltonian:Chosen example input data distribution for QBM training: [0.045, 0.16, 0.16, 0.62]

[0169] As can be seen, the time taken for training a NN-Meta-VQT based QBM is almost 50 times less than the VARQITE based QBM where the depth of VarQITE is chosen such that it gives similar precision of generated Gibbs state as that of NN-Meta-VQT vl. Further, the loss function of the QBM also converges better for NN-Meta-VQT vl based QBM compared to the VarQITE based QBM. This demonstrates a NN-Meta-VQT vl based QBM performing better than an existing VarQITE-based QBM.

[0170] For a complex Hamiltonian (with all possible spin interaction and field terms) and a random input distribution, the VarQITE based QBM loss function (KL divergence) takes on an average 1200 minutes to converge to a value close to zero with an average trace distance (between generated and exact Gibbs state) of 0.21. On the other hand, the time taken for NN-Meta-VQT vl based QBM training using this meta-ansatz takes only 25 minutes 10s on average, and generates Gibbs state with average trace distance of 0.135. Accordingly, this demonstrates numerical evidence where it is advantageously shown that both the quality of generated Gibbs states as well as the runtime considerably better for NN-Meta-VQT vl based QBM than the VarQITE based QBM.

[0171] Further, the VarQITE requires a depth 32 SU2 hardware efficient ansatz to achieve this average trace distance, while the NN-Meta-VQT vl generates better quality Gibbs states with only a depth 4 SU2 hardware efficient ansatz.Example Computer System implementation

[0172] Figure 13 is a block diagram of an information processing apparatus 10 or a computing device 10, such as a data storage server, which embodies the present invention, and which may be used to implement some or all of the operations of a method embodying the present invention, and perform some or all of the tasks of apparatus of an embodiment. The computing device 10 may be used to implement any of the method steps described above and / or any processes described above.

[0173] The computing device 10 comprises a processor 993 and memory 994. Optionally, the computing device also includes a network interface 997 for communication with other such computing devices. Optionally, the computing device also includes one or more input mechanisms such as keyboard and mouse 996, and a display unit such as one or more monitors 995. These elements may facilitate user interaction. The components are connectable to one another via a bus 992.

[0174] The memory 994 may include a computer readable medium, which term may refer to a single medium or multiple media (e.g., a centralized or distributed database and / or associated caches and servers) configured to carry computer-executable instructions. Computer-executable instructions may include, for example, instructions and data accessible by and causing a computer (e.g., one or more processors) to perform one or more functions or operations. For example, the computer-executable instructions may include those instructions for implementing a method disclosed herein, or any method steps disclosed herein, and / or any processes described above. Thus, the term “computer-readable storage medium” may also include any medium that is capable of storing, encoding or carrying a set of instructions for execution by the machine and that cause the machine to perform any one or more of the method steps of the present disclosure. The term “computer-readable storage medium” may accordingly be taken to include, but not be limited to, solid-state memories, optical media and magnetic media. By way of example, and not limitation, such computer-readable media may include non-transitory computer-readable storage media, including Random Access Memory (RAM), Read-Only Memory (ROM), Electrically Erasable Programmable Read-Only Memory (EEPROM), Compact Disc Read-Only Memory (CD-ROM) or other optical diskstorage, magnetic disk storage or other magnetic storage devices, flash memory devices (e.g., solid state memory devices).

[0175] The processor 993 is configured to control the computing device and execute processing operations, for example executing computer program code stored in the memory 994 to implement any of the method steps described herein. The memory 994 stores data being read and written by the processor 993 and may store training data and / or network weights and / or patches and / or updated patches and / or embeddings and / or vectors and / or graphs and / or representations and / or difference amounts and / or equations and / or other data, described above, and / or programs for executing any of the method steps and / or processes described above. As referred to herein, a processor may include one or more general-purpose processing devices such as a microprocessor, central processing unit, or the like. The processor may include a complex instruction set computing (CISC) microprocessor, reduced instruction set computing (RISC) microprocessor, very long instruction word (VLIW) microprocessor, or a processor implementing other instruction sets or processors implementing a combination of instruction sets. The processor may also include one or more special-purpose processing devices such as an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a digital signal processor (DSP), network processor, or the like. In one or more embodiments, a processor is configured to execute instructions for performing the operations and operations discussed herein. The processor 993 may be considered to comprise any of the modules described above. Any operations described as being implemented by a module may be implemented as a method by a computer and e.g. by the processor 993.

[0176] Optionally, the apparatus 10 includes a display unit 995 which may display a representation of data stored by the computing device.

[0177] The network interface (network I / F) 997 may be connected to a network, such as the Internet, and is connectable to other such computing devices via the network. The network I / F 997 may control data input / output from / to other apparatus via the network. Other peripheral devices such as microphone, speakers, printer, power supply unit, fan, case, scanner, trackerball etc. may be included in the computing device.

[0178] Methods embodying the present invention may be carried out on a computing device / apparatus 10 such as that illustrated in Figure 13. Such a computing device need not have every component illustrated in Figure 13, and may be composed of a subset of those components. For example, the apparatus 10 may comprise the processor 993 and the memory994 connected to the processor 993. Or the apparatus 10 may comprise the processor 993, the memory 994 connected to the processor 993, and the display 995. A method embodying the present invention may be carried out by a single computing device in communication with one or more data storage servers via a network. The computing device may be a data storage itself storing at least a portion of the data.

[0179] The processor 993 may comprise at least one “classical” processor and at least one quantum computing device / processor. The at least one “classical” processor may be configured to implement methods steps and / or operations of a method / computer program, corresponding to any of the method steps described above, for instance with reference to Figures 1 and 6, wherein when some steps / operations include “using” quantum models / agents and in those operations the “classical” processor controls the at least one quantum computing device / processor to implement the quantum agents and carry out the operations.

[0180] Rather than the processor 993 comprising at least one “classical” processor and at least one quantum computing device / processor, the processor may comprise or be the at least one “classical” processor and the computing device 10 may additionally comprise at least one quantum computing device / processor configured as described above. The at least one quantum computing device / processor may be provided separately from or inside the computing device 10 and may e.g. be connected to the computing device 10 via a bus.

[0181] Figure 14 is a schematic diagram illustrating a hybrid quantum-classical computing device 900 comprising a classical computer construction 901 and a quantum computer configuration 902. The classical computer construction 901 comprises a central processing unit (CPU) 930, a main storage 940, an auxiliary storage 950, and input / output device(s) 960. The quantum computer configuration 902 comprises a quantum arithmetic processor (which may be referred to as a quantum processor) 910. The quantum arithmetic processor 910 may comprise a qubit device 912 and a qubit control signal generator 914. The qubit device 912 is configured to represent the physical qubits where quantum information is stored and processed, and the qubit control signal generator 914 is configured to produce the signals required to manipulate those qubits and perform the operations in the quantum arithmetic processor 910. In particular the qubit device 910 is configured to hold and manipulate quantum states, and may be implemented using various technologies, such as for instance superconducting circuits, trapped ions, or photonic systems, and the qubit control signal generator 914 is configured to manipulate the qubits of the qubit device 912 byproducing control signals, such as for instance microwave pulses or laser beams etc. The computing device 900 comprises a bus 920 connecting the components described above. The auxiliary storage 950 may comprise any of an operating system (OS) 952, a quantum circuit program 954, and a quantum computational control program 956. The input / output device(s) 960 may comprise any of a display 962, a keyboard and / or mouse 964 and an external memory (memory card) 966.

[0182] The computing device 900 may be considered to correspond to the computing device 10 of Figure 13. Or the classical computer construction 901 may be considered to correspond to the computing device 10 and description of the computing device 10 may apply to the classical computer construction 901 (that is, the CPU 930 may correspond to the processor 993, the main storage 940 and auxiliary storage 950 may correspond to the memory 994, the input / output device(s) 960 may correspond to the display 995 and input 996, and the classical computer construction 901 may comprise an interface corresponding to interface 997). In other words, the CPU 930 may be configured to implement methods steps and / or operations of a method / computer program, corresponding to any of the method steps described above, for instance as described with reference to Figures 1 and 6, wherein some steps / operations include “using” quantum MU models / agents and in those operations the CPU 930 may be configured to control the quantum processor 910 and carry out the operations.

[0183] The computing devices 10 and 900 may be referred as information processing apparatuses, and may work in tandem with quantum processing performed in the quantum computer configuration 902 and with classical processing performed in the classical computer configuration 901. The classical computer configuration 901 may control and manage the operation and instruction of the system as a whole, and / or the quantum computer configuration 902 in particular.

[0184] A method embodying the present invention may be carried out by a plurality of computing devices operating in cooperation with one another, for instance a hybrid quantum-classical computer as defined above. One or more of the plurality of computing devices may be a data storage server storing at least a portion of the data.

[0185] The invention may be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations of them. The invention may be implemented as a computer program or computer program product, i.e., a computer program tangibly embodied in a non-transitory information carrier, e.g., in a machine-readable storagedevice, or in a propagated signal, for execution by, or to control the operation of, one or more hardware modules.

[0186] A computer program may be in the form of a stand-alone program, a computer program portion or more than one computer program and may be written in any form of programming language, including compiled or interpreted languages, and it may be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a data processing environment. A computer program may be deployed to be executed on one module or on multiple modules at one site or distributed across multiple sites and interconnected by a communication network.

[0187] Method steps of the invention may be performed by one or more programmable processors executing a computer program to perform functions of the invention by operating on input data and generating output. Apparatus of the invention may be implemented as programmed hardware or as special purpose logic circuitry, including e.g., an FPGA (field programmable gate array) or an ASIC (application-specific integrated circuit).

[0188] Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read-only memory or a random access memory or both. The essential elements of a computer are a processor for executing instructions coupled to one or more memory devices for storing instructions and data.

[0189] For the purposes of the present disclosure, the term “machine learning model” encompasses within its scope the following concepts:- machine Learning algorithms, comprising processes or instructions through which data may be used in a training process to generate a model artefact for performing a given task, or for representing a real world process or system;- the model artefact that is created by such a training process, and which comprises the computational architecture that performs the task; and- the process performed by the model artefact in order to complete the task.

[0190] References to “machine learning model”, “model”, model parameters”, “model information”, etc., may thus be understood as relating to any one or more of the above concepts encompassed within the scope of “Machine learning model”.The above-described embodiments of the present invention may advantageously be used independently of any other of the embodiments or in any feasible combination with one or more others of the embodiments.ALTERNATIVE EMBODIMENTS

[0191] The embodiments described above are illustrative of, rather than limiting to, the present invention. Alternative embodiments apparent on reading the above description may nevertheless fall within the scope of the invention.

[0192] Alternative statements of the invention are recited below as numbered clauses: Clause 1: A computer-implemented hybrid variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.Clause 2: The computer-implemented method of clause 1, wherein the Hamiltonian parameters are the coupling constant J and the external magnetic field h.Clause 3 : The computer-implemented method of clause 2, wherein the range of values for the Hamiltonian parameters is -2 to 2.Clause 4: The computer-implemented method of any preceding clause, further comprising storing the set of evaluated cost functions in a variable such as a tensor.Clause 5: The computer-implemented method of any preceding clause, wherein the cost function and / or the global cost function uses the Gibbs free energy.Clause 6: The computer-implemented method of clause 5, wherein the cost function and / or the global cost function uses the Gibbs free energy with a truncated entropy term:wherein:h ”_1 is a vector comprising the parameters of Hamiltonian H(h ”_1);P = l / k_B T is inverse temperature;co ^,(|) T O" are the tuneable parameters of the parameterised quantum circuit; p(co ",(|) ". 0 ") is the Gibbs state approximation from the parameterised quantum circuit; and(l / )Tr(p(co ",(|) ". 0 ")z2 )-( 1 / ) is first order truncated Von-Neumann entropy associated with the Gibbs state approximation.Clause 7: The computer-implemented method of any preceding clause, wherein the updating the tuneable parameters based on minimising the global cost function comprises using an adaptive movement estimation (ADAM) optimiser.Clause 8: The computer-implemented method of any preceding clause, wherein the parameterised quantum circuit configuration comprises:an encoding layer encoding the input set of Hamiltonian parameters and initialising the tuneable parameters of the parameterised quantum circuit configuration.Clause 9: The computer-implemented method of clause 8, wherein the encoding layer comprises a Hardware Efficient Ansatz circuit.Clause 10: The computer-implemented method of clause 8 or clause 9, wherein the parameterised quantum circuit configuration further comprises:a processing layer, the processing layer further comprises a Hamiltonian Variational Ansatz, wherein the gates of the Hamiltonian Variational Ansatz are parameterised based on the defined sets of Hamiltonian parameters.Clause 11: The computer-implemented method of clause 10, wherein the defined sets of Hamiltonian parameters comprises the field or interaction terms of the Hamiltonian.Clause 12: The computer-implemented method of any of clauses 10 or 11, wherein the encoding layer comprises a neural network.Clause 13: The computer-implemented method of clause 12, wherein the neural network is used to encode the input set of Hamiltonian parameters and output the tuneable parameters of the parameterised quantum circuit configuration.Clause 14: A computer-implemented method of training a variational quantum Boltzmann machine comprising:selecting a Hamiltonian comprising initial Hamiltonian parameters; measuring a distribution of Gibbs states using a pre-trained parameterised quantum circuit configuration trained in accordance with any preceding clause, wherein the tuneable parameters are the trained parameters;adjusting the initial Hamiltonian parameters based on minimising a cost function, comprising:evaluating the cost function, comprising comparing the distribution of Gibbs states with a target distribution of measurements from a target dataset;generating updated Hamiltonian parameters, comprising adjusting the initial Hamiltonian parameters in a direction that reduces the cost function.Clause 15: The computer implemented method of clause 14, wherein if the cost function is minimised, defining the updated Hamiltonian parameters as trained Hamiltonian parameters.Clause 16: The computer-implemented method of clause 15, further comprising selecting the pre-trained parameterised quantum circuit configuration from a plurality of pre-trained parameterised quantum circuit configurations trained in accordance with any of clauses 1 to 13.Clause 17: The computer-implemented method of clause 16, wherein the selecting is based on the complexity and number of parameters of the selected Hamiltonian comprising initial Hamiltonian parameters.Clause 18: A computer-implemented method of using a trained variational quantum Boltzmann machine, trained in accordance with the method of any of clauses 14 to 17, as part of a generative Al process.Clause 19: A computer program which, when run on a hybrid quantum-classical computer, causes the computer to carry out a variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.Clause 20: An information processing apparatus comprising a memory and a processor connected to the memory, wherein the processor is configured to perform a variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.

Claims

We Claim:

1. A computer-implemented hybrid variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.

2. The computer-implemented method of claim 1, wherein the Hamiltonian parameters are the coupling constant J and the external magnetic field h.

3. The computer-implemented method of claim 2, wherein the range of values for the Hamiltonian parameters is -2 to 2.

4. The computer-implemented method of any preceding claim, further comprising storing the set of evaluated cost functions in a variable such as a tensor.

5. The computer-implemented method of any preceding claim, wherein the cost function and / or the global cost function uses the Gibbs free energy.

6. The computer-implemented method of claim 5, wherein the cost function and / or the global cost function uses the Gibbs free energy with a truncated entropy term:wherein:h ”_1 is a vector comprising the parameters of Hamiltonian H(h ”_1);P = l / k_B T is inverse temperature;co ^,(|) T O" are the tuneable parameters of the parameterised quantum circuit; p(co ",(|) ". 0 ") is the Gibbs state approximation from the parameterised quantum circuit; and(l / )Tr(p(co ",(|) ". 0 ")z2 )-( 1 / ) is first order truncated Von-Neumann entropy associated with the Gibbs state approximation.

7. The computer-implemented method of any preceding claim, wherein the updating the tuneable parameters based on minimising the global cost function comprises using an adaptive movement estimation (ADAM) optimiser.

8. The computer-implemented method of any preceding claim, wherein the parameterised quantum circuit configuration comprises:an encoding layer encoding the input set of Hamiltonian parameters and initialising the tuneable parameters of the parameterised quantum circuit configuration.

9. The computer-implemented method of claim 8, wherein the encoding layer comprises a Hardware Efficient Ansatz circuit.

10. The computer-implemented method of claim 8 or claim 9, wherein the parameterised quantum circuit configuration further comprises:a processing layer, the processing layer further comprises a Hamiltonian Variational Ansatz, wherein the gates of the Hamiltonian Variational Ansatz are parameterised based on the defined sets of Hamiltonian parameters.

11. The computer-implemented method of claim 10, wherein the defined sets of Hamiltonian parameters comprises the field or interaction terms of the Hamiltonian.

12. The computer-implemented method of any of claims 10 or 11, wherein the encoding layer comprises a neural network.

13. The computer-implemented method of claim 12, wherein the neural network is used to encode the input set of Hamiltonian parameters and output the tuneable parameters of the parameterised quantum circuit configuration.

14. A computer-implemented method of training a variational quantum Boltzmann machine comprising:selecting a Hamiltonian comprising initial Hamiltonian parameters; measuring a distribution of Gibbs states using a pre-trained parameterised quantum circuit configuration trained in accordance with any preceding claim, wherein the tuneable parameters are the trained parameters;adjusting the initial Hamiltonian parameters based on minimising a cost function, comprising:evaluating the cost function, comprising comparing the distribution of Gibbs states with a target distribution of measurements from a target dataset;generating updated Hamiltonian parameters, comprising adjusting the initial Hamiltonian parameters in a direction that reduces the cost function.

15. The computer implemented method of claim 14, wherein if the cost function is minimised, defining the updated Hamiltonian parameters as trained Hamiltonian parameters.

16. The computer-implemented method of claim 15, further comprising selecting the pretrained parameterised quantum circuit configuration from a plurality of pre-trained parameterised quantum circuit configurations trained in accordance with any of claims 1 to 13.

17. The computer-implemented method of claim 16, wherein the selecting is based on the complexity and number of parameters of the selected Hamiltonian comprising initial Hamiltonian parameters.

18. A computer-implemented method of using a trained variational quantum Boltzmann machine, trained in accordance with the method of any of claims 14 to 17, as part of a generative Al process.

19. A computer program which, when run on a hybrid quantum-classical computer, causes the computer to carry out a variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.

20. An information processing apparatus comprising a memory and a processor connected to the memory, wherein the processor is configured to perform a variational method of training a parametrised quantum circuit configuration to perform Gibbs state generation of a parametrised Hamiltonian, comprising:defining sets of Hamiltonian parameters, wherein each set comprises Hamiltonian parameters with fixed values and the sets collectively define a range of values for each of the Hamiltonian parameters;initialising a set of tuneable parameters of the parameterised quantum circuit configuration;performing collective optimisation of the tuneable parameters with respect to the parameterised Hamiltonian, comprising:determining a set of evaluated cost functions comprising an evaluated cost function for each set of Hamiltonian parameters, comprising for each set of Hamiltonian parameters:measuring a Gibbs state approximation from the parameterised quantum circuit configuration, comprising inputting the set of Hamiltonian parameters into the parameterised quantum circuit configuration;evaluating a cost function based on the Gibbs state approximation; andevaluating a global cost function, the global cost function comprising a sum of the set of evaluated cost functions, wherein:if the global cost function is not minimised, updating the tuneable parameters based on minimising the global cost function and iteratively performing the collective optimisation; andif the global cost function is minimised, defining the tuneable parameters as trained parameters.

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