Estimating the ground-state energy of Hamiltonians is a fundamental task for which it is believed that quantum computers can be helpful. Several approaches have been proposed toward this goal, including algorithms based on quantum phase estimation and hybrid quantum-classical optimizers involving parameterized quantum circuits, the latter falling under the umbrella of the variational quantum eigensolver. Here, we analyze the performance of quantum Boltzmann machines for this task, which is a less explored ansatz based on parameterized thermal states and which is not known to suffer from the barren-plateau problem. We delineate a hybrid quantum-classical algorithm for this task and rigorously prove that it converges to an $\varepsilon$-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in $\varepsilon^{-1}$, the number of parameters, and the norm of the Hamiltonian being optimized. Our algorithm estimates the gradient of the energy function efficiently by means of a quantum circuit construction that combines classical random sampling, Hamiltonian simulation, and the Hadamard test. Additionally, supporting our main claims are calculations of the gradient and Hessian of the energy function, as well as an upper bound on the matrix elements of the latter that is used in the convergence analysis.
Estimating the ground-state energy of Hamiltonians is a fundamental task for which it is believed that quantum computers can be helpful. Several approaches have been proposed toward this goal, including algorithms based on quantum phase estimation and hybrid quantum-classical optimizers involving parameterized quantum circuits, the latter falling under the umbrella of the variational quantum eigensolver. Here, we analyze the performance of quantum Boltzmann machines for this task, which is a less explored ansatz based on parameterized thermal states and which is not known to suffer from the barren-plateau problem. We delineate a hybrid quantum-classical algorithm for this task and rigorously prove that it converges to an ε-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in ε−1, the number of parameters, and the norm of the Hamiltonian being optimized. Our algorithm estimates the gradient of the energy function efficiently by means of a quantum circuit construction that combines classical random sampling, Hamiltonian simulation, and the Hadamard test. Additionally, supporting our main claims are calculations of the gradient and Hessian of the energy function, as well as an upper bound on the matrix elements of the latter that is used in the convergence analysis.
We introduce evolved quantum Boltzmann machines as a variational ansatz for quantum optimization and learning tasks. Given two parameterized Hamiltonians G(θ) and H(ϕ), an evolved quantum Boltzmann machine consists of preparing a thermal state of the first Hamiltonian G(θ) followed by unitary evolution according to the second Hamiltonian H(ϕ). Alternatively, one can think of it as first realizing imaginary time evolution according to G(θ) followed by real time evolution according to H(ϕ). After defining this ansatz, we provide analytical expressions for the gradient vector and illustrate their application in ground-state energy estimation and generative modeling, showing how the gradient for these tasks can be estimated by means of quantum algorithms that involve classical sampling, Hamiltonian simulation, and the Hadamard test. We also establish analytical expressions for the Fisher-Bures, Wigner-Yanase, and Kubo-Mori information matrix elements of evolved quantum Boltzmann machines, as well as quantum algorithms for estimating each of them, which leads to at least three different general natural gradient descent algorithms based on this ansatz. Along the way, we establish a broad generalization of the main result of [Luo, Proc. Am. Math. Soc. 132, 885 (2004)], proving that the Fisher-Bures and Wigner-Yanase information matrices of general parameterized families of states differ by no more than a factor of two in the matrix (Loewner) order, making them essentially interchangeable for training when using natural gradient descent.
Born-rule generative modeling, a central task in quantum machine learning, seeks to learn probability distributions that can be efficiently sampled by measuring complex quantum states. One hope is for quantum models to efficiently capture probability distributions that are difficult to learn and simulate by classical means alone. Quantum Boltzmann machines were proposed about one decade ago for this purpose, yet efficient training methods have remained elusive. In this paper, I overcome this obstacle by proposing a practical solution that trains quantum Boltzmann machines for Born-rule generative modeling. Two key ingredients in the proposal are the Donsker-Varadhan variational representation of the classical relative entropy and the quantum Boltzmann gradient estimator of [Patel et al., arXiv:2410.12935]. I present the main result for a more general ansatz known as an evolved quantum Boltzmann machine [Minervini et al., arXiv:2501.03367], which combines parameterized real- and imaginary-time evolution. I also show how to extend the findings to other distinguishability measures beyond relative entropy. Finally, I present four different hybrid quantum-classical algorithms for the minimax optimization underlying training, and I discuss their theoretical convergence guarantees.
Thermal states play a fundamental role in various areas of physics, and they are becoming increasingly important in quantum information science, with applications related to semidefinite programming, quantum Boltzmann machine learning, Hamiltonian learning, and the related task of estimating the parameters of a Hamiltonian. Here we establish formulas underlying the basic geometry of parameterized thermal states, and we delineate quantum algorithms for estimating the values of these formulas. More specifically, we establish formulas for the Fisher--Bures and Kubo--Mori information matrices of parameterized thermal states, and our quantum algorithms for estimating their matrix elements involve a combination of classical sampling, Hamiltonian simulation, and the Hadamard test. These results have applications in developing a natural gradient descent algorithm for quantum Boltzmann machine learning, which takes into account the geometry of thermal states, and in establishing fundamental limitations on the ability to estimate the parameters of a Hamiltonian, when given access to thermal-state samples. For the latter task, and for the special case of estimating a single parameter, we sketch an algorithm that realizes a measurement that is asymptotically optimal for the estimation task. We finally stress that the natural gradient descent algorithm developed here can be used for any machine learning problem that employs the quantum Boltzmann machine ansatz.
The preparation of quantum Gibbs states is a fundamental challenge in quantum computing, essential for applications ranging from modeling open quantum systems to quantum machine learning. Building on the Meta-Variational Quantum Eigensolver framework proposed by Cervera-Lierta et al. and a problem-driven ansatz design, we introduce two meta-learning algorithms: Meta-Variational Quantum Thermalizer (Meta-VQT) and Neural Network Meta-VQT (NN-Meta VQT) for efficient thermal state preparation of parametrized Hamiltonians on Noisy Intermediate-Scale Quantum (NISQ) devices. Meta-VQT utilizes a fully quantum ansatz, while NN Meta-VQT integrates a quantum--classical hybrid architecture. Both use collective optimization over a finite set of training parameter values to prepare approximate Gibbs states at held-out parameter values within the sampled parameter domain. We validate our methods on Transverse Field Ising Models (TFIM) with up to eight qubits and on a two-qubit Heisenberg model with all field terms. Within these finite-size benchmarks, the trained models accurately prepare thermal states at test points inside the training interval. For the six and eight qubit cases of TFIM, meta-learned parameters when combined with appropriately designed ansatz serve as warm-start initializations, significantly outperforming random initializations in the optimization tasks. Furthermore, a 3 qubit Kitaev ring example showcases our algorithm's effectiveness across finite-temperature crossover regimes. Finally, we apply our algorithms to train a Quantum Boltzmann Machine (QBM) on a 2-qubit Heisenberg model with all field terms, achieving enhanced training efficiency, improved Gibbs state accuracy, and a 30-fold runtime speedup over existing techniques such as variational quantum imaginary time (VarQITE)-based QBM highlighting the practicality of meta-algorithm-based QBMs.
Abstract Quantum Boltzmann machines (QBMs) are machine-learning models for both classical and quantum data. We give an operational definition of QBM learning in terms of the difference in expectation values between the model and target, taking into account the polynomial size of the data set. By using the relative entropy as a loss function, this problem can be solved without encountering barren plateaus. We prove that a solution can be obtained with stochastic gradient descent using at most a polynomial number of Gibbs states. We also prove that pre-training on a subset of the QBM parameters can only lower the sample complexity bounds. In particular, we give pre-training strategies based on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. We verify these models and our theoretical findings numerically on a quantum and a classical data set. Our results establish that QBMs are promising machine learning models.
In this thesis we have implemented three many-body methods in quantum mechanics on quantum computers, namely the Quantum Phase Estimation (QPE) algorithm, the Variational Quantum Eigensolver (VQE) algorithm and the Quantum Adiabatic Time Evolution (QATE) algorithm.We utilized these methods to approximate the ground state energy of the pairing model Hamiltonian.We provided a comparison between the ideal simulation of a quantum computer and a simulation including the noise model from some of IBM's actual quantum devices.For the VQE algortihm, we also run the method on the IBM Q London five-qubit device.As the time complexity (circuit depth) of a quantum algorithm (quantum circuit) is often the bottleneck on current quantum computers, we also introduced a circuit optimization scheme, called Recursive Circuit Optimization, which can be utilized to approximate the action of a quantum circuit by learning the parameters of a parametrized unitary operation and only evaluating small parts of the circuit at a time.We tested out the method by trying to optimize a random circuit, the QATE algorithm and the unitary coupled cluster doubles (UCCD) ansatz.Finally, we implemented a quantum circuit that allows us to encode a real data set into the amplitudes of a quantum state, and we utilized this circuit to implement several realizations of dense and recurrent neural networks on quantum computers.We proposed neural network layers that are consisting of hardware-and parameter-efficient parametrized quantum circuits, and we have made a Python framework to construct and train deep quantum neural networks.We then tested the neural networks by trying to learn a simple non-linear function as well as finding the smallest eigenvalue of the pairing Hamiltonian Full Configuration Interaction (FCI) matrix.We found that all the quantum many-body methods showed promise in providing an upper bound on the ground state energy for the ideal simulation of a quantum computer.When performing the simulation with noise and the execution on the real quantum device however, we found that only the VQE algorithm was able to yield reasonable results due to short circuit depth.Still, the success of the method was found to be dependent on the interaction strength g of the quantum system, as the approximation of the ground state energy started to deviate from the actual value as we increased g.For the recursive circuit optimization of a random circuit, we found that we were able to approximate the state given by the random circuit, and we learned that we should expect some error in the approximation due to finite measurements and the flexibility of the parametrized unitary.When utilizing the recursive optimization scheme to approximate the QATE algorithm, we found that although we were not able to exactly reproduce the action of the operator, we did provide an upper bound to the ground state energy close to the one given by the actual QATE algorithm.In the same time, the circuit depth of the algorithm was drastically decreased.When utilizing the scheme for the VQE algorithm, we got promising results while reducing the circuit depth of the UCCD ansatz.Over to the neural networks, we managed to approximate a non-linear function utilizing a hardware-efficient three layered network, with only three qubits in the intermediate layers.We showed that our proposed neural network are suitable for near-term quantum computers by successfully approximating the lowest eigenvalue of the FCI matrix when running a simulation with the noise model from the IBM Q London five-qubit computer.i I would first like to thank my supervisor, Morten Hjorth-Jensen, for being helpful and motivational throughout my time at the University of Oslo.Your door has always been open when help is needed, which has been a great relief in stressful times.I would also like to thank my friends and family for believing in me and providing emotional support.Your continuous interest in my studies and well-being has meant a lot to me.My next thanks goes to Heidi K. J.I highly appreciate your never-ending support.Finally I would like to thank the computational physics study group.You have been invaluable to me in both a social and educational manner.