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arXiv:2206.05302 · PRX Quantum

Predicting Gibbs-State Expectation Values with Pure Thermal Shadows

Luuk Coopmans, Yuta Kikuchi, Marcello Benedetti

Published Jan 13, 2023• 38 citations
arXivHTMLPDF

Abstract

The preparation and computation of many properties of quantum Gibbs states is essential for algorithms such as quantum semidefinite programming and quantum Boltzmann machines.We propose a quantum algorithm that can predict M linear functions of an arbitrary Gibbs state with only O(log M ) experimental measurements.Our main insight is that for sufficiently large systems we do not need to prepare the nqubit mixed Gibbs state explicitly but, instead, we can evolve a random n-qubit pure state in imaginary time.The result then follows by constructing classical shadows of these random pure states.We propose a quantum circuit that implements this algorithm by using quantum signal processing for the imaginary time evolution.We numerically verify the efficiency of the algorithm by simulating the circuit for a tenspin-1/2 XXZ-Heisenberg model.In addition, we show that the algorithm can be successfully employed as a subroutine for training an eight-qubit fully connected quantum Boltzmann machine.

DOI
10.1103/prxquantum.4.010305

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The preparation and computation of many properties of quantum Gibbs states is essential for algorithms such as quantum semidefinite programming and quantum Boltzmann machines.We propose a quantum algorithm that can predict M linear functions of an arbitrary Gibbs state with only O(log M ) experimental measurements.Our main insight is that for sufficiently large systems we do not need to prepare the nqubit mixed Gibbs state explicitly but, instead, we can evolve a random n-qubit pure state in imaginary time.The result then follows by constructing classical shadows of these random pure states.We propose a quantum circuit that implements this algorithm by using quantum signal processing for the imaginary time evolution.We numerically verify the efficiency of the algorithm by simulating the circuit for a tenspin-1/2 XXZ-Heisenberg model.In addition, we show that the algorithm can be successfully employed as a subroutine for training an eight-qubit fully connected quantum Boltzmann machine.
Calculating the properties of Gibbs states is an important task in Quantum Chemistry and Quantum Machine Learning. Previous work has proposed a quantum algorithm which predicts Gibbs state expectation values forMMMobservables from only logMMMmeasurements, by combining classical shadows and quantum signal processing for a new estimator called Pure Thermal Shadows. In this work, we perform resource analysis for the circuits used in this algorithm, finding that quantum signal processing contributes most significantly to gate count and depth as system size increases. The implementation we use for this also features an improvement to the algorithm in the form of more efficient random unitary generation steps. Moreover, given the ramifications of the resource analysis, we argue that its potential utility could be constrained to Fault Tolerant devices sampling from the Gibbs state of a large, cool system.
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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.
Abstract This work presents a novel realization approach to quantum Boltzmann machines (QBMs). The preparation of the required Gibbs states, as well as the evaluation of the loss function’s analytic gradient, is based on variational quantum imaginary time evolution, a technique that is typically used for ground-state computation. In contrast to existing methods, this implementation facilitates near-term compatible QBM training with gradients of the actual loss function for arbitrary parameterized Hamiltonians which do not necessarily have to be fully visible but may also include hidden units. The variational Gibbs state approximation is demonstrated with numerical simulations and experiments run on real quantum hardware provided by IBM Quantum. Furthermore, we illustrate the application of this variational QBM approach to generative and discriminative learning tasks using numerical simulation.
Gibbs state preparation has emerged as a potential avenue for quantum advantage in the study of physical systems. Although there is known advantage for computing local properties of worst-case Gibbs states, recent results suggest a different picture for average-case local systems: at temperatures where their Gibbs states can be efficiently prepared quantumly, quasi-polynomial time classical algorithms can also compute local expectation values. Yet this leaves open the possibility of an advantage under a stronger notion of simulation: sampling. Gibbs sampling underlies Boltzmann machine-based quantum learning algorithms and quantum supremacy proposals, but its classical average-case complexity for physically relevant models remains unresolved. Here, we give strong evidence against an exponential quantum advantage in Gibbs sampling for an ensemble of local systems known as the quantum ppp-spin model. We describe a classical algorithm for sampling from the Gibbs state to any polynomially small error in total variation distance. Our algorithm is based on a meta-algorithm known as algorithmic stochastic localization (ASL). Our main technical result is a generalized convergence theorem for ASL which reduces approximate sampling from any distribution on the hypercube to estimating the mean of a related "tilted distribution" to sufficiently low additive error. We then extend known quasi-polynomial time algorithms for computing local expectation values of a related disordered model to compute these "tilted means" for the quantum ppp-spin model. Finally, we give a non-rigorous physics argument that this mean-estimation algorithm works down to the spin glass transition temperature. As efficient quantum algorithms also fail at this transition, this suggests they cannot Gibbs sample the quantum ppp-spin model at temperatures below that for which classical algorithms are also (almost) efficient.
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.