Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer

Estimating Molecular Thermal Averages with the Quantum Equation of Motion and Informationally Complete Measurements

By leveraging the Variational Quantum Eigensolver (VQE), the "quantum equation of motion" (qEOM) method established itself as a promising tool for quantum chemistry on near-term quantum computers and has been used extensively to estimate molecular excited states. Here, we explore a novel application of this method, employing it to compute thermal averages of quantum systems, specifically molecules like ethylene and butadiene. A drawback of qEOM is that it requires measuring the expectation values of a large number of observables on the ground state of the system, and the number of necessary measurements can become a bottleneck of the method. In this work, we focus on measurements through informationally complete positive operator-valued measures (IC-POVMs) to achieve a reduction in the measurement overheads by estimating different observables of interest through the measurement of a single set of POVMs. We show with numerical simulations that the qEOM combined with IC-POVM measurements ensures satisfactory accuracy in the reconstruction of the thermal state with a reasonable number of shots.

A linear response framework for quantum simulation of bosonic and fermionic correlation functions

Response functions are a fundamental aspect of physics; they represent the link between experimental observations and the underlying quantum many-body state. However, this link is often under-appreciated, as the Lehmann formalism for obtaining response functions in linear response has no direct link to experiment. Within the context of quantum computing, and via a linear response framework, we restore this link by making the experiment an inextricable part of the quantum simulation. This method can be frequency- and momentum-selective, avoids limitations on operators that can be directly measured, and can be more efficient than competing methods. As prototypical examples of response functions, we demonstrate that both bosonic and fermionic Green's functions can be obtained, and apply these ideas to the study of a charge-density-wave material on the ibm_auckland superconducting quantum computer. The linear response method provides a robust framework for using quantum computers to study systems in physics and chemistry.

Fragmented imaginary-time evolution for early-stage quantum signal processors

Simulating quantum imaginary-time evolution (QITE) is a significant promise of quantum computation. However, the known algorithms are either probabilistic (repeat until success) with unpractically small success probabilities or coherent (quantum amplitude amplification) with circuit depths and ancillary-qubit numbers unrealistically large in the mid-term. Our main contribution is a new generation of deterministic, high-precision QITE algorithms that are significantly more amenable experimentally. A surprisingly simple idea is behind them: partitioning the evolution into a sequence of fragments that are run probabilistically. It causes a considerable reduction in wasted circuit depth every time a run fails. Remarkably, the resulting overall runtime is asymptotically better than in coherent approaches, and the hardware requirements are even milder than in probabilistic ones. Our findings are especially relevant for the early fault-tolerance stages of quantum hardware.

Recent Advances for Quantum Neural Networks in Generative Learning

Quantum computers are next-generation devices that hold promise to perform calculations beyond the reach of classical computers. A leading method towards achieving this goal is through quantum machine learning, especially quantum generative learning. Due to the intrinsic probabilistic nature of quantum mechanics, it is reasonable to postulate that quantum generative learning models (QGLMs) may surpass their classical counterparts. As such, QGLMs are receiving growing attention from the quantum physics and computer science communities, where various QGLMs that can be efficiently implemented on near-term quantum machines with potential computational advantages are proposed. In this paper, we review the current progress of QGLMs from the perspective of machine learning. Particularly, we interpret these QGLMs, covering quantum circuit Born machines, quantum generative adversarial networks, quantum Boltzmann machines, and quantum variational autoencoders, as the quantum extension of classical generative learning models. In this context, we explore their intrinsic relations and their fundamental differences. We further summarize the potential applications of QGLMs in both conventional machine learning tasks and quantum physics. Last, we discuss the challenges and further research directions for QGLMs.

Towards Quantum Computing Phase Diagrams of Gauge Theories with Thermal Pure Quantum States

The phase diagram of strong interactions in nature at finite temperature and chemical potential remains largely theoretically unexplored due to inadequacy of Monte-Carlo-based computational techniques in overcoming a sign problem. Quantum computing offers a sign-problem-free approach, but evaluating thermal expectation values is generally resource intensive on quantum computers. To facilitate thermodynamic studies of gauge theories, we propose a generalization of the thermal-pure-quantum-state formulation of statistical mechanics applied to constrained gauge-theory dynamics, and numerically demonstrate that the phase diagram of a simple low-dimensional gauge theory is robustly determined using this approach, including mapping a chiral phase transition in the model at finite temperature and chemical potential. Quantum algorithms, resource requirements, and algorithmic and hardware error analysis are further discussed to motivate future implementations. Thermal pure quantum states, therefore, may present a suitable candidate for efficient thermal simulations of gauge theories in the era of quantum computing.

Exploring finite temperature properties of materials with quantum computers

Thermal properties of nanomaterials are crucial to not only improving our fundamental understanding of condensed matter systems, but also to developing novel materials for applications spanning research and industry. Since quantum effects arise at the nano-scale, these systems are difficult to simulate on classical computers. Quantum computers can efficiently simulate quantum many-body systems, yet current quantum algorithms for calculating thermal properties of these systems incur significant computational costs in that they either prepare the full thermal state on the quantum computer, or they must sample a number of pure states from a distribution that grows with system size. Canonical thermal pure quantum (TPQ) states provide a promising path to estimating thermal properties of quantum materials as they neither require preparation of the full thermal state nor require a growing number of samples with system size. Here, we present an algorithm for preparing canonical TPQ states on quantum computers. We compare three different circuit implementations for the algorithm and demonstrate their capabilities in estimating thermal properties of quantum materials. Due to its increasing accuracy with system size and flexibility in implementation, we anticipate that this method will enable finite temperature explorations of relevant quantum materials on near-term quantum computers.

Computing Free Energies with Fluctuation Relations on Quantum Computers

As a central thermodynamic property, free energy enables the calculation of virtually any equilibrium property of a physical system, allowing for the construction of phase diagrams and predictions about transport, chemical reactions, and biological processes. Thus, methods for efficiently computing free energies, which in general is a difficult problem, are of great interest to broad areas of physics and the natural sciences. The majority of techniques for computing free energies target classical systems, leaving the computation of free energies in quantum systems less explored. Recently developed fluctuation relations enable the computation of free energy differences in quantum systems from an ensemble of dynamic simulations. While performing such simulations is exponentially hard on classical computers, quantum computers can efficiently simulate the dynamics of quantum systems. Here, we present an algorithm utilizing a fluctuation relation known as the Jarzynski equality to approximate free energy differences of quantum systems on a quantum computer. We discuss under which conditions our approximation becomes exact, and under which conditions it serves as a strict upper bound. Furthermore, we successfully demonstrate a proof of concept of our algorithm using the transverse field Ising model on a real quantum processor. As quantum hardware continues to improve, we anticipate that our algorithm will enable computation of free energy differences for a wide range of quantum systems useful across the natural sciences.

Many-body thermodynamics on quantum computers via partition function zeros

Partition functions are ubiquitous in physics: They are important in determining the thermodynamic properties of many-body systems and in understanding their phase transitions. As shown by Lee and Yang, analytically continuing the partition function to the complex plane allows us to obtain its zeros and thus the entire function. Moreover, the scaling and nature of these zeros can elucidate phase transitions. Here, we show how to find partition function zeros on noisy intermediate-scale trapped-ion quantum computers in a scalable manner, using the XXZ spin chain model as a prototype, and observe their transition from XY-like behavior to Ising-like behavior as a function of the anisotropy. While quantum computers cannot yet scale to the thermodynamic limit, our work provides a pathway to do so as hardware improves, allowing the future calculation of critical phenomena for systems beyond classical computing limits.

Continuous-Variable Assisted Thermal Quantum Simulation

Simulation of a quantum many-body system at finite temperatures is crucially important but quite challenging. Here we present an experimentally feasible quantum algorithm assisted with continuous variable for simulating quantum systems at finite temperatures. Our algorithm has a time complexity scaling polynomially with the inverse temperature and the desired accuracy. We demonstrate the quantum algorithm by simulating a finite temperature phase diagram of the quantum Ising and Kitaev models. It is found that the important crossover phase diagram of the Kitaev ring can be accurately simulated by a quantum computer with only a few qubits and thus the algorithm may be implementable on current quantum processors. We further propose a protocol with superconducting or trapped ion quantum computers.

Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits

We extend the ability of an unitary quantum circuit by interfacing it with a classical autoregressive neural network. The combined model parametrizes a variational density matrix as a classical mixture of quantum pure states, where the autoregressive network generates bitstring samples as input states to the quantum circuit. We devise an efficient variational algorithm to jointly optimize the classical neural network and the quantum circuit to solve quantum statistical mechanics problems. One can obtain thermal observables such as the variational free energy, entropy, and specific heat. As a byproduct, the algorithm also gives access to low energy excitation states. We demonstrate applications of the approach to thermal properties and excitation spectra of the quantum Ising model with resources that are feasible on near-term quantum computers.

Generation of thermofield double states and critical ground states with a quantum computer

Our experiment prepares two types of nontrivial quantum states on a trapped ion quantum computer: the thermofield double state of the transverse-field Ising model at arbitrary temperature and the quantum critical state of the zero-temperature model. We use techniques motivated by the quantum approximate optimization algorithm, and we implement a hybrid quantum–classical optimization loop to prepare the quantum critical state. Our results pave the way for exploring strongly correlated models at finite temperature and teleportation protocols inspired by black hole physics.

Finite-temperature phases of many-body quantum systems are fundamental to phenomena ranging from condensed-matter physics to cosmology, yet they are generally difficult to simulate. Using an ion trap quantum computer and protocols motivated by the quantum approximate optimization algorithm (QAOA), we generate nontrivial thermal quantum states of the transverse-field Ising model (TFIM) by preparing thermofield double states at a variety of temperatures. We also prepare the critical state of the TFIM at zero temperature using quantum–classical hybrid optimization. The entanglement structure of thermofield double and critical states plays a key role in the study of black holes, and our work simulates such nontrivial structures on a quantum computer. Moreover, we find that the variational quantum circuits exhibit noise thresholds above which the lowest-depth QAOA circuits provide the best results.

Clustering of Conditional Mutual Information for Quantum Gibbs States above a Threshold Temperature

We prove that the quantum Gibbs states of spin systems above a certain threshold temperature are approximate quantum Markov networks, meaning that the conditional mutual information decays rapidly with distance. We demonstrate the exponential decay for short-ranged interacting systems and power-law decay for long-ranged interacting systems. Consequently, we establish the efficiency of quantum Gibbs sampling algorithms, a strong version of the area law, the quasilocality of effective Hamiltonians on subsystems, a clustering theorem for mutual information, and a polynomial-time algorithm for classical Gibbs state simulations.

Variational Thermal Quantum Simulation via Thermofield Double States

We present a variational approach for quantum simulators to realize finite temperature Gibbs states by preparing thermofield double (TFD) states. Our protocol is motivated by the quantum approximate optimization algorithm and involves alternating time evolution between the Hamiltonian of interest and interactions which entangle the system and its auxiliary counterpart. As a simple example, we demonstrate that thermal states of the 1D classical Ising model at any temperature can be prepared with perfect fidelity using L/2 iterations, where L is system size. We also show that a free fermion TFD can be prepared with nearly optimal efficiency. Given the simplicity and efficiency of the protocol, our approach enables near-term quantum platforms to access finite temperature phenomena via preparation of thermofield double states.

Machine learning & artificial intelligence in the quantum domain: a review of recent progress

Quantum information technologies, on the one hand, and intelligent learning systems, on the other, are both emergent technologies that are likely to have a transformative impact on our society in the future. The respective underlying fields of basic research-quantum information versus machine learning (ML) and artificial intelligence (AI)-have their own specific questions and challenges, which have hitherto been investigated largely independently. However, in a growing body of recent work, researchers have been probing the question of the extent to which these fields can indeed learn and benefit from each other. Quantum ML explores the interaction between quantum computing and ML, investigating how results and techniques from one field can be used to solve the problems of the other. Recently we have witnessed significant breakthroughs in both directions of influence. For instance, quantum computing is finding a vital application in providing speed-ups for ML problems, critical in our 'big data' world. Conversely, ML already permeates many cutting-edge technologies and may become instrumental in advanced quantum technologies. Aside from quantum speed-up in data analysis, or classical ML optimization used in quantum experiments, quantum enhancements have also been (theoretically) demonstrated for interactive learning tasks, highlighting the potential of quantum-enhanced learning agents. Finally, works exploring the use of AI for the very design of quantum experiments and for performing parts of genuine research autonomously, have reported their first successes. Beyond the topics of mutual enhancement-exploring what ML/AI can do for quantum physics and vice versa-researchers have also broached the fundamental issue of quantum generalizations of learning and AI concepts. This deals with questions of the very meaning of learning and intelligence in a world that is fully described by quantum mechanics. In this review, we describe the main ideas, recent developments and progress in a broad spectrum of research investigating ML and AI in the quantum domain.

Generalized Grover's Algorithm for Multiple Phase Inversion States

Grover's algorithm is a quantum search algorithm that proceeds by repeated applications of the Grover operator and the Oracle until the state evolves to one of the target states. In the standard version of the algorithm, the Grover operator inverts the sign on only one state. Here we provide an exact solution to the problem of performing Grover's search where the Grover operator inverts the sign on N states. We show the underlying structure in terms of the eigenspectrum of the generalized Hamiltonian, and derive an appropriate initial state to perform the Grover evolution. This allows us to use the quantum phase estimation algorithm to solve the search problem in this generalized case, completely bypassing the Grover algorithm altogether. We obtain a time complexity of this case of sqrt[D/M^{α}], where D is the search space dimension, M is the number of target states, and α≈1, which is close to the optimal scaling.

Quantum machine learning: a classical perspective

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning (ML) techniques to impressive results in regression, classification, data generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets is motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed up classical ML algorithms. Here we review the literature in quantum ML and discuss perspectives for a mixed readership of classical ML and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in ML are identified as promising directions for the field. Practical questions, such as how to upload classical data into quantum form, will also be addressed.

Quantum Enhanced Inference in Markov Logic Networks

Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference in MLNs is probabilistic and it is often performed by approximate methods such as Markov chain Monte Carlo (MCMC) Gibbs sampling. An MLN has many regular, symmetric structures that can be exploited at both first-order level and in the generated Markov network. We analyze the graph structures that are produced by various lifting methods and investigate the extent to which quantum protocols can be used to speed up Gibbs sampling with state preparation and measurement schemes. We review different such approaches, discuss their advantages, theoretical limitations, and their appeal to implementations. We find that a straightforward application of a recent result yields exponential speedup compared to classical heuristics in approximate probabilistic inference, thereby demonstrating another example where advanced quantum resources can potentially prove useful in machine learning.

Quantum speedup of Monte Carlo methods

Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work, we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.

Experimental observation of Lee-Yang zeros

Lee-Yang zeros are points on the complex plane of physical parameters where the partition function of a system vanishes and hence the free energy diverges. Lee-Yang zeros are ubiquitous in many-body systems and fully characterize their thermodynamics. Notwithstanding their fundamental importance, Lee-Yang zeros have never been observed in experiments, due to the intrinsic difficulty that they would occur only at complex values of physical parameters, which are generally regarded as unphysical. Here we report the first observation of Lee-Yang zeros, by measuring quantum coherence of a probe spin coupled to an Ising-type spin bath. The quantum evolution of the probe spin introduces a complex phase factor and therefore effectively realizes an imaginary magnetic field. From the measured Lee-Yang zeros, we reconstructed the free energy of the spin bath and determined its phase transition temperature. This experiment opens up new opportunities of studying thermodynamics in the complex plane.

From transistor to trapped-ion computers for quantum chemistry

Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motional degrees of freedom of trapped ions for solving problems in quantum chemistry, including molecular electronic structure, molecular dynamics, and vibronic coupling. We focus on applications that go beyond the capacity of classical computers, but may be realizable on state-of-the-art trapped-ion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a near-future trapped-ion-based technology.

Solovay-Kitaev decomposition strategy for single-qubit channels

Inspired by the Solovay-Kitaev decomposition for approximating unitary operations as a sequence of operations selected from a universal quantum computing gate set, we introduce a method for approximating any single-qubit channel using single-qubit gates and the controlled-not (cnot). Our approach uses the decomposition of the single-qubit channel into a convex combination of "quasiextreme" channels. Previous techniques for simulating general single-qubit channels would require as many as 20 cnot gates, whereas ours only needs one, bringing it within the range of current experiments.

Thermalization in nature and on a quantum computer

In this work, we show how Gibbs or thermal states appear dynamically in closed quantum many-body systems, building on the program of dynamical typicality. We introduce a novel perturbation theorem for physically relevant weak system-bath couplings that is applicable even in the thermodynamic limit. We identify conditions under which thermalization happens and discuss the underlying physics. Based on these results, we also present a fully general quantum algorithm for preparing Gibbs states on a quantum computer with a certified runtime and error bound. This complements quantum Metropolis algorithms, which are expected to be efficient but have no known runtime estimates and only work for local Hamiltonians.

A quantum-quantum Metropolis algorithm

The classical Metropolis sampling method is a cornerstone of many statistical modeling applications that range from physics, chemistry, and biology to economics. This method is particularly suitable for sampling the thermal distributions of classical systems. The challenge of extending this method to the simulation of arbitrary quantum systems is that, in general, eigenstates of quantum Hamiltonians cannot be obtained efficiently with a classical computer. However, this challenge can be overcome by quantum computers. Here, we present a quantum algorithm which fully generalizes the classical Metropolis algorithm to the quantum domain. The meaning of quantum generalization is twofold: The proposed algorithm is not only applicable to both classical and quantum systems, but also offers a quantum speedup relative to the classical counterpart. Furthermore, unlike the classical method of quantum Monte Carlo, this quantum algorithm does not suffer from the negative-sign problem associated with fermionic systems. Applications of this algorithm include the study of low-temperature properties of quantum systems, such as the Hubbard model, and preparing the thermal states of sizable molecules to simulate, for example, chemical reactions at an arbitrary temperature.

Preparing thermal states of quantum systems by dimension reduction

We present an algorithm that prepares thermal Gibbs states of one dimensional quantum systems on a quantum computer without any memory overhead, and in a time significantly shorter than other known alternatives. Specifically, the time complexity is dominated by the quantity N(‖h‖/T), where N is the size of the system, ‖h‖ is a bound on the operator norm of the local terms of the Hamiltonian (coupling energy), and T is the temperature. Given other results on the complexity of thermalization, this overall scaling is likely optimal. For higher dimensions, our algorithm lowers the known scaling of the time complexity with the dimension of the system by one.

Quantum algorithm for exact Monte Carlo sampling

We build a quantum algorithm which uses the Grover quantum search procedure in order to sample the exact equilibrium distribution of a wide range of classical statistical mechanics systems. The algorithm is based on recently developed exact Monte Carlo sampling methods, and yields a polynomial gain compared to classical procedures.