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Physical Review A

Quantum algorithm for the microcanonical thermal pure quantum state method

Kaito Mizukami, Akihisa Koga

Published Jul 6, 2023• 1 citation
Publisher

Abstract

We propose a quantum algorithm to investigate finite-temperature properties of quantum many-body systems by implementing the microcanonical thermal pure quantum (TPQ) states [S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012)] on a quantum computer. In the formalism, the TPQ state is constructed by applying the power of a rescaled Hamiltonian to a random state. We clarify that, using techniques based on quantum signal processing, the power operation can be efficiently realized when the ground-state energy of a quantum system of interest is given with some precision. Our algorithm has an advantage in calculating thermodynamic quantities at low temperatures.

DOI
10.1103/physreva.108.012404

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We propose a quantum algorithm to investigate finite-temperature properties of quantum many-body systems by implementing the microcanonical thermal pure quantum (TPQ) states [S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012)] on a quantum computer. In the formalism, the TPQ state is constructed by applying the power of a rescaled Hamiltonian to a random state. We clarify that, using techniques based on quantum signal processing, the power operation can be efficiently realized when the ground-state energy of a quantum system of interest is given with some precision. Our algorithm has an advantage in calculating thermodynamic quantities at low temperatures.
We present a quantum algorithm for the microcanonical thermal pure quantum (TPQ) method, which has an advantage in evaluating thermodynamic quantities at finite temperatures, by combining with some recently developed techniques derived from quantum singular value transformation. When the ground energy of quantum systems has already been obtained precisely, the multiple products of the Hamiltonian are efficiently realized and the TPQ states at low temperatures are systematically constructed in quantum computations.
Abstract We present several features of a thermal pure quantum matrix product state (TPQ-MPS). TPQ-MPS can simulate finite temperature quantum many-body systems with a numerical cost comparable to the matrix product algorithm for the ground state in one-dimensional systems. Starting from a random matrix product state with auxiliary sites attached to the edges of the system, we successively operate the Hamiltonian with a shift-constant and obtain a series of TPQ-MPS that represents the microcanonical pure state at the highest temperature to the one at the lowest available temperature. How the parameters of the TPQ-MPS protocol influence the effective bond dimension χ of the matrix and the temperature distribution of these microcanonical states is examined.
We propose a method to calculate finite-temperature properties of a quantum many-body system for a microcanonical ensemble by introducing a pure quantum state named here an energy-filtered random-phase state, which is also a potentially promising application of near-term quantum computers. In our formalism, a microcanonical ensemble is specified by two parameters, i.e., the energy of the system and its associated energy window. Accordingly, the density of states is expressed as a sum of Gaussians centered at the target energy with its spread corresponding to the width of the energy window. We then show that the thermodynamic quantities such as entropy and temperature are calculated by evaluating the trace of the time-evolution operator and the trace of the time-evolution operator multiplied by the Hamiltonian of the system. We also describe how these traces can be evaluated using random diagonal-unitary circuits appropriate to quantum computation. The pure quantum state representing our microcanonical ensemble is related to a state of the form introduced by Wall and Neuhauser for the filter diagonalization method [M. R. Wall and D. Neuhauser, J. Chem. Phys. 102, 8011 (1995)], and therefore we refer to it as an energy-filtered random-phase state. The energy-filtered random-phase state is essentially a Fourier transform of a time-evolved state whose initial state is prepared as a random-phase state, and the cut-off time in the time-integral for the Fourier transform sets the inverse of the width of the energy window. The proposed method is demonstrated numerically by calculating thermodynamic quantities for the one-dimensional spin-1/2 Heisenberg model on small clusters up to 28 qubits, showing that the method is most efficient for the target energy around which the dense distribution of energy eigenstates is found.
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.
A thermal equilibrium state of a quantum many-body system can be represented by a typical pure state, which we call a thermal pure quantum (TPQ) state. We construct the canonical TPQ state, which corresponds to the canonical ensemble of the conventional statistical mechanics. It is related to the microcanonical TPQ state, which corresponds to the microcanonical ensemble, by simple analytic transformations. Both TPQ states give identical thermodynamic results, if both ensembles do, in the thermodynamic limit. The TPQ states corresponding to other ensembles can also be constructed. We have thus established the TPQ formulation of statistical mechanics, according to which all quantities of statistical-mechanical interest are obtained from a single realization of any TPQ state. We also show that it has great advantages in practical applications. As an illustration, we study the spin-1/2 kagome Heisenberg antiferromagnet.