Preparing thermal states on a quantum computer can have a variety of applications, from simulating many-body quantum systems to training machine learning models. Variational circuits have been proposed for this task on near-term quantum computers, but several challenges remain, such as finding a scalable cost-function, avoiding the need of purification, and mitigating noise effects. We propose a new algorithm for thermal state preparation that tackles those three challenges by exploiting the noise of quantum circuits. We consider a variational architecture containing a depolarizing channel after each unitary layer, with the ability to directly control the level of noise. We derive a closed-form approximation for the free-energy of such circuit and use it as a cost function for our variational algorithm. By evaluating our method on a variety of Hamiltonians and system sizes, we find several systems for which the thermal state can be approximated with a high fidelity. However, we also show that the ability for our algorithm to learn the thermal state strongly depends on the temperature: while a high fidelity can be obtained for high and low temperatures, we identify a specific range for which the problem becomes more challenging. We hope that this first study on noise-assisted thermal state preparation will inspire future research on exploiting noise in variational algorithms.
Preparing thermal states on a quantum computer can have a variety of applications, from simulating many-body quantum systems to training machine learning models. Variational circuits have been proposed for this task on near-term quantum computers, but several challenges remain, such as finding a scalable cost-function, avoiding the need of purification, and mitigating noise effects. We propose a new algorithm for thermal state preparation that tackles those three challenges by exploiting the noise of quantum circuits. We consider a variational architecture containing a depolarizing channel after each unitary layer, with the ability to directly control the level of noise. We derive a closed-form approximation for the free-energy of such circuit and use it as a cost function for our variational algorithm. By evaluating our method on a variety of Hamiltonians and system sizes, we find several systems for which the thermal state can be approximated with a high fidelity. However, we also show that the ability for our algorithm to learn the thermal state strongly depends on the temperature: while a high fidelity can be obtained for high and low temperatures, we identify a specific range for which the problem becomes more challenging. We hope that this first study on noise-assisted thermal state preparation will inspire future research on exploiting noise in variational algorithms.
The preparation of Gibbs thermal states is an important task in quantum computation with applications in quantum simulation, quantum optimization, and quantum machine learning. However, many algorithms for preparing Gibbs states rely on quantum subroutines which are difficult to implement on near-term hardware. Here, we address this by (i) introducing an objective function that, unlike the free energy, is easily measured, and (ii) using dynamically generated, problem-tailored ansätze. This allows for arbitrarily accurate Gibbs state preparation using low-depth circuits. To verify the effectiveness of our approach, we numerically demonstrate that our algorithm can prepare high-fidelity Gibbs states across a broad range of temperatures and for a variety of Hamiltonians.
Preparation of Gibbs distributions is an important task for quantum computation. It is a necessary first step in some types of quantum simulations and further is essential for quantum algorithms such as quantum Boltzmann training. Despite this, most methods for preparing thermal states are impractical to implement on near-term quantum computers because of the memory overheads required. Here we present a variational approach to preparing Gibbs states that is based on minimizing the free energy of a quantum system. The key insight that makes this practical is the use of Fourier series approximations to the logarithm that allows the entropy component of the free-energy to be estimated through a sequence of simpler measurements that can be combined together using classical post processing. We further show that this approach is efficient for generating high-temperature Gibbs states, within constant error, if the initial guess for the variational parameters for the programmable quantum circuit are sufficiently close to a global optima. Finally, we examine the procedure numerically and show the viability of our approach for five-qubit Hamiltonians using Trotterized adiabatic state preparation as an ansatz.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
Nature is governed by precise physical laws, which can inspire the discovery of new computer-run simulation algorithms. Thermal states are the most ubiquitous for they are the equilibrium states of matter. Simulating thermal states of quantum matter has applications ranging from quantum machine learning to better understanding of high-temperature superconductivity and quantum chemistry. The computational complexity of this task is hopelessly hard for classical computers. The existing quantum algorithms come with caveats: most either require quantum phase estimation rendering them impractical for current noisy hardware, or are variational which face obstacles such as initialization, barren plateaus, and a general lack of provable guarantee. We provide two quantum algorithms with provable guarantees to prepare thermal states on (near-term) quantum computers that avoid these drawbacks. The first algorithm is inspired by the natural thermalization process where the ancilla qubits act as the infinite thermal bath. This algorithm can potentially run in polynomial time to sample thermal distributions of ergodic systems -- the vast class of physical systems that equilibrate in isolation with respect to local observables. The second algorithm works for any system and in general runs in exponential time. However, it requires significantly smaller quantum resources than previous such algorithms. In addition, we provide an error mitigation technique for both algorithms to fight back decoherence, which enables us to run our algorithms on the near-term quantum devices. To illustration, we simulate the thermal state of the hardcore Bose-Hubbard model on the latest generation of available quantum computers.
Traditional quantum algorithms with guaranteed performance generally require fully coherent quantum computers to operate, making error-correction a necessity. In contrast, variational quantum algorithms seek to remove this requirement by formulating the computation as the approximate optimization of a functional. This optimization is carried out in the space of heuristic functions defined by a parameterized (variational) quantum circuit. Taking advantage of the flexibility in the definition of these heuristics, we can expand the operability of variational algorithms within the regime of coherence times of noisy intermediate-scale quantum devices, avoiding the necessity of error-correction. In this dissertation, we present the application of the variational quantum computing approach to problems in quantum simulation, quantum state preparation, quantum error-correction, and generative modeling.\nIn the first part, we investigate the use of the variational quantum eigensolver (VQE) for simulating the ground state of fermionic systems. We start by studying the implementation of VQE using the unitary coupled cluster (UCC) ansatz and propose strategies to reduce its cost. We use these insights to carry out some of the first experimental quantum computations of molecular energies using simplified versions of this ansatz. We also propose an extension of UCC to study problems in condensed matter physics, along with a new low-depth circuit ansatz for preparing non-gaussian fermionic states on quantum computers. We show the potential of our approach to describe strongly correlated fermionic systems.\nIn the second part, we develop three new variational quantum algorithms for problems in quantum computing and machine learning, namely: 1) the quantum autoencoder, to compress ensembles of quantum states, 2) the variational quantum error corrector, to find device-tailored quantum encoding and recovery circuits for quantum error correction, and 3) the variational quantum generator, to generate classical probability distributions. These techniques offer efficient ways to design new quantum circuits for state preparation, to find more effective error-correcting codes and to perform generative modeling with quantum computers. Our work provides insights into the design of variational quantum algorithms and establishes practical guidelines to implement these methods on near-term quantum computers, as well as some future research directions for this field.
Thermal equilibrium states of many-body Hamiltonians are essential for probing quantum chaos, finite-temperature phases of matter, and training quantum machine learning models, yet generating large collections of such states across different Hamiltonians remains costly with existing methods. We introduce a powerful operation, the quantum thermal-drift channel, to construct a measurement-controlled sampling algorithm that autonomously generates thermal states together with their system Hamiltonians as labels for general physical models. We prove that our algorithm is efficient: the total gate count scales polynomially with system size and quadratically with inverse temperature, providing the first polynomial resource bound for random thermal state generation. We characterize the distribution of sampled Hamiltonians as a normal distribution reweighted by partition functions, which quantifies a trade-off between sampling accuracy and effective label range. Level-spacing statistics computed from sampled thermal states of a 2D transverse-field Ising model show a crossover to Wigner-Dyson universality, confirming that the sampler captures nontrivial chaotic correlations. Finally, a variational quantum classifier trained on the generated dataset achieves near-optimal accuracy in predicting Hamiltonian properties of unseen states. These results establish a scalable, quantum-native route for thermodynamic simulation and labeled quantum data generation in many-body systems.