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arXiv:2305.19322 · Quantum

Robust Extraction of Thermal Observables from State Sampling and Real-Time Dynamics on Quantum Computers

Khaldoon Ghanem, Alexander Schuckert, Henrik Dreyer

Published Nov 3, 2023• 10 citations
arXivHTMLPDF

Abstract

Simulating properties of quantum materials is one of the most promising applications of quantum computation, both near- and long-term. While real-time dynamics can be straightforwardly implemented, the finite temperature ensemble involves non-unitary operators that render an implementation on a near-term quantum computer extremely challenging. Recently, Lu, Bañuls and Cirac \cite{Lu2021} suggested a "time-series quantum Monte Carlo method" which circumvents this problem by extracting finite temperature properties from real-time simulations via Wick's rotation and Monte Carlo sampling of easily preparable states. In this paper, we address the challenges associated with the practical applications of this method, using the two-dimensional transverse field Ising model as a testbed. We demonstrate that estimating Boltzmann weights via Wick's rotation is very sensitive to time-domain truncation and statistical shot noise. To alleviate this problem, we introduce a technique that imposes constraints on the density of states, most notably its non-negativity, and show that this way, we can reliably extract Boltzmann weights from noisy time series. In addition, we show how to reduce the statistical errors of Monte Carlo sampling via a reweighted version of the Wolff cluster algorithm. Our work enables the implementation of the time-series algorithm on present-day quantum computers to study finite temperature properties of many-body quantum systems.

DOI
10.22331/q-2023-11-03-1163

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Simulating properties of quantum materials is one of the most promising applications of quantum computation, both near- and long-term. While real-time dynamics can be straightforwardly implemented, the finite temperature ensemble involves non-unitary operators that render an implementation on a near-term quantum computer extremely challenging. Recently, Lu, Bañuls and Cirac \cite{Lu2021} suggested a "time-series quantum Monte Carlo method" which circumvents this problem by extracting finite temperature properties from real-time simulations via Wick's rotation and Monte Carlo sampling of easily preparable states. In this paper, we address the challenges associated with the practical applications of this method, using the two-dimensional transverse field Ising model as a testbed. We demonstrate that estimating Boltzmann weights via Wick's rotation is very sensitive to time-domain truncation and statistical shot noise. To alleviate this problem, we introduce a technique that imposes constraints on the density of states, most notably its non-negativity, and show that this way, we can reliably extract Boltzmann weights from noisy time series. In addition, we show how to reduce the statistical errors of Monte Carlo sampling via a reweighted version of the Wolff cluster algorithm. Our work enables the implementation of the time-series algorithm on present-day quantum computers to study finite temperature properties of many-body quantum systems.
Simulating thermal-equilibrium properties at finite temperature is crucial for studying quantum many-body systems. Quantum computers are expected to enable us to simulate large systems at finite temperatures, overcoming challenges faced by classical computers, such as the sign problem of the quantum Monte Carlo technique. Conventional methods suitable for fault-tolerant quantum computing (FTQC) devices are designed for studying large-scale quantum many-body systems but require a large number of ancilla qubits and a deep quantum circuit with many basic gates, making them unsuitable for the early stage of the FTQC era, when the availability of qubits and quantum gates is limited. In this paper, we propose a method suitable for quantum devices in this early stage to calculate the thermal-equilibrium expectation value of an observable at finite temperatures. Our proposal, named the Markov chain Monte Carlo with sampled pairs of unitaries (MCMC-SPU) algorithm, involves sampling simple quantum circuits and generating the corresponding statistical ensembles. This approach addresses the issues of resource demand and the decay in probability associated with postselection of measurement outcomes on ancilla qubits. We validate our proposal with a numerical simulation of the one-dimensional transverse-field Ising model as an illustrative example.
The computation of thermal properties of quantum many-body systems is a central challenge in our understanding of quantum mechanics. We introduce the Quantum Finite Temperature Lanczos Method (QFTLM), which extends the finite-temperature Lanczos method to quantum computers by combining real-time quantum Krylov methods with efficient preparation of typical states for trace estimation. This approach enables the computation of thermal expectation values while avoiding the exponential scaling inherent to classical exact simulation techniques. Numerical experiments on the transverse-field Ising model show that QFTLM can reproduce thermal observables over a wide temperature range. We further analyze the influence of Krylov dimension, number of trace-estimator states, and Trotter error, and show that suitable regularization is essential for robustness in noisy settings. These results establish QFTLM as a promising framework for finite-temperature quantum simulation.
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Preparing finite-temperature states in quantum simulators of spin systems, such as trapped ions or Rydberg atoms in optical tweezers, is challenging due to their almost perfect isolation from the environment. Here, we show how finite-temperature observables can be obtained with an algorithm motivated from the Jarzynski equality and equivalent to the one in Lu et al., PRX Quantum 2, 020321 (2021). It consists of classical importance sampling of initial states and a measurement of the Loschmidt echo with a quantum simulator. We use the method as a quantum-inspired classical algorithm and simulate the protocol with matrix product states to analyze the requirements on a quantum simulator. This way, we show that a finite-temperature phase transition in the long-range transverse-field Ising model can be characterized in trapped ion quantum simulators. We propose a concrete measurement protocol for the Loschmidt echo and discuss the influence of measurement noise, dephasing, as well as state preparation and measurement errors. We argue that the algorithm is robust against those imperfections under realistic conditions.
Quantum Monte Carlo (QMC) methods are essential for the numerical study of large-scale quantum many-body systems, yet their utility has been significantly hampered by the difficulty in computing key quantities such as off-diagonal operators and entanglement. This Letter introduces Bell-QMC, a novel QMC framework leveraging Bell sampling, a two-copy measurement protocol in the transversal Bell basis. We demonstrate that Bell-QMC enables an efficient and unbiased estimation of both challenging classes of observables, offering a significant advantage over previous QMC approaches. Notably, the entanglement across all system partitions can be computed in a single Bell-QMC simulation. We implement this method within the stochastic series expansion, where we design an efficient update scheme for sampling the configurations in the Bell basis. We demonstrate our algorithm in the one-dimensional transverse-field Ising model and the two-dimensional Z 2 lattice gauge theory, extracting universal quantum features using only simple diagonal measurements. This Letter establishes Bell-QMC as a powerful framework that significantly expands the accessible quantum properties in QMC simulations, providing a substantial advantage over conventional QMC.
Finite-Temperature Dynamical Phase Diagram of the 2+12+12+1D Quantum Ising Model
Mapping finite-temperature dynamical phase diagrams of quantum many-body models is a necessary step towards establishing a framework of far-from-equilibrium quantum many-body universality. However, this is quite difficult due, in part, to the severe challenges in representing the volume-law entanglement that is generated under nonequilibrium dynamics at finite temperatures. Here, we address these challenges with an efficient equilibrium quantum Monte Carlo (QMC) framework for computing the finite-temperature dynamical phase diagram. Our method uses energy conservation and the self-thermalizing properties of ergodic quantum systems to determine observables at late times after a quantum quench. We use this technique to chart the dynamical phase diagram of the 2+12+12+1D quantum Ising model generated by quenches of the transverse field in initial thermal states. Our approach allows us to track the evolution of dynamical phases as a function of both the initial temperature and transverse field. Surprisingly, we identify quenches in the ordered phase that cool the system as well as an interval of initial temperatures where it is possible to quench from the paramagnetic (PM) to ferromagnetic (FM) phases. Our method gives access to dynamical properties without explicitly simulating unitary time evolution, and is immediately applicable to other lattice geometries and interacting many-body systems. Finally, we propose a quantum simulation experiment on state-of-the-art digital quantum hardware to directly probe the predicted dynamical phases and their real-time formation.