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arXiv:2009.04648

Many Body Thermodynamics on Quantum Computers via Partition Function Zeros

Akhil Francis, Daoquan Zhu, C. Huerta Alderete, Sonika Johri, Xiao Xiao, James K. Freericks, Christopher R. Monroe, Norbert M. Linke, Alexander F. Kemper

Published Sep 10, 2020• 1 citation
arXivHTMLPDF

Abstract

Interacting quantum systems illustrate complex phenomena including phase transitions to novel ordered phases. The universal nature of critical phenomena reduces their description to determining only the transition temperature and the critical exponents. Numerically calculating these results for systems in new universality classes is complicated due to critical slowing down, requiring increasing resources near the critical point. An alternative approach analytically continues the calculation onto the complex plane and determines the partition function via its zeros. Here we show how to robustly perform this analysis on noisy intermediate scale trapped ion quantum computers in a scalable manner, using the XXZ model as a prototype. We illustrate the 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 determination of critical phenomena for systems that cannot be solved otherwise.

DOI
10.48550/arxiv.2009.04648

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Interacting quantum systems illustrate complex phenomena including phase transitions to novel ordered phases. The universal nature of critical phenomena reduces their description to determining only the transition temperature and the critical exponents. Numerically calculating these results for systems in new universality classes is complicated due to critical slowing down, requiring increasing resources near the critical point. An alternative approach analytically continues the calculation onto the complex plane and determines the partition function via its zeros. Here we show how to robustly perform this analysis on noisy intermediate scale trapped ion quantum computers in a scalable manner, using the XXZ model as a prototype. We illustrate the 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 determination of critical phenomena for systems that cannot be solved otherwise.
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.
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.
By setting the inverse temperature β loose to occupy the complex plane, Fisher showed that the zeros of the complex partition function Z , if approaching the real β axis, reveal a thermodynamic phase transition. More recently, Fisher zeros were used to mark the dynamical phase transition in quench dynamics. It remains unclear, however, how Fisher zeros can be employed to better understand quantum phase transitions or the nonunitary dynamics of open quantum systems. Here we answer this question by a comprehensive analysis of the analytically continued one-dimensional transverse field Ising model. We exhaust all the Fisher zeros to show that in the thermodynamic limit they congregate into a remarkably simple pattern in the form of continuous open or closed lines. These Fisher lines evolve smoothly as the coupling constant is tuned, and a qualitative change identifies the quantum critical point. By exploiting the connection between Z and the thermofield double states, we obtain analytical expressions for the short- and long-time dynamics of the survival amplitude, including its scaling behavior at the quantum critical point. We point out Z can be realized and probed in monitored quantum circuits. The exact analytical results are corroborated by the numerical tensor renormalization group. We further show that similar patterns of Fisher zeros also emerge in other spin models. Therefore, the approach outlined may serve as a powerful tool for interacting quantum systems. Published by the American Physical Society 2024
While the zeros of complex partition functions, such as Lee-Yang zeros and Fisher zeros, have been pivotal in characterizing temperature-driven phase transitions, extending this concept to zero temperature remains an open question. In this work, we propose a solution to this issue by calculating the imaginary-temperature zeros (ITZs), which are defined as the roots of the imaginary-temperature partition function. We illustrate the analytical properties of ITZs in the transverse-field Ising chain, showing that the ITZs' distribution can distinguish between various phases and signify the critical exponents. Universal singular behaviors manifest in such quantities as the edge density of ITZs and the magnetization, with the scaling exponents remarkably differing from those in Lee-Yang theory. We further illuminate the consistency between ITZs and the zeros of the spectral form factor, which offers a practical path for the experimental detection of ITZs.
Concepts of the complex partition functions and the Fisher zeros provide intrinsic statistical mechanisms for finite temperature and real time dynamical phase transitions. We extend the utility of these complexifications to quantum phase transitions. We exactly identify different Fisher zeros on lines or closed curves and elucidate their correspondence with domain-wall excitations or confined mesons for the one-dimensional transverse field Ising model. The crossover behavior of the Fisher zeros provides a fascinating picture for criticality near the quantum phase transition, where the excitation energy scales are quantitatively determined. We further confirm our results by tensor network calculations and demonstrate a clear signal of deconfined meson excitations from the disruption of the closed zero curves. Our results unambiguously show significant features of Fisher zeros for a quantum phase transition and open up a new route to explore quantum criticality.
Phase transitions and critical phenomena are the most universal phenomena in nature.To understand the phase transitions and critical phenomena of a given system as a continuous function of temperature and to obtain the partition function zeros in the complex temperature plane indicating most effectively phase transitions and critical phenomena, we need to calculate the density of states.Currently, Wang-Landau Monte Carlo algorithm is one of the most efficient Monte Carlo methods to calculate the approximate density of states.Using Wang-Landau Monte Carlo algorithm, the density of states for the Ising model on L x L square lattices (L = 4 ~ 32) with periodic boundary conditions is obtained, and the partition function zeros of the Ising model are evaluated in the complex temperature plane.By examining the behavior of the first partition function zero (partition function zero closest to the positive real axis), phase transitions and critical phenomena can be much more accurately analyzed.The approximate first zeros of the Ising ferromagnet, obtained from Wang-Landau algorithm, are quite close to the exact ones, indicating that it is a reliable method for calculating the density of states and the first partition function zeros.