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arXiv:2404.07912 · Journal of High Energy Physics

Quantum computation in fermionic thermal field theories

Wenyang Qian, Bin Wu

Published Jul 18, 2024• 6 citations
arXivHTMLPDF

Abstract

A bstract Thermal properties of quantum fields at finite temperature are crucial to understanding strongly interacting matter and recent development in quantum computing has provided an alternative and promising avenue of study. In this work, we study thermal field theories involving only fermions using quantum algorithms. We first delve into the presentations of fermion fields via qubits on digital quantum computers alongside the quantum algorithms such as quantum imaginary time evolutions employed to evaluate thermal properties of generic quantum field theories. Specifically, we show numerical results such as the thermal distribution and the energy density of thermal field theories for Majorana fermions in 1+1 dimensions using quantum simulators. In addition to free field theory, we also study the effects of interactions resulting from coupling with a spatially homogeneous Majorana field. In both cases, we show analytically that thermal properties of the system can be described using phase-space distributions, and the quantum simulation results agree with analytical and semiclassical expectations. Our work is an important step to understand thermal fixed points, preparing for quantum simulation of thermalization in real time.

DOI
10.1007/jhep07(2024)166

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A bstract Thermal properties of quantum fields at finite temperature are crucial to understanding strongly interacting matter and recent development in quantum computing has provided an alternative and promising avenue of study. In this work, we study thermal field theories involving only fermions using quantum algorithms. We first delve into the presentations of fermion fields via qubits on digital quantum computers alongside the quantum algorithms such as quantum imaginary time evolutions employed to evaluate thermal properties of generic quantum field theories. Specifically, we show numerical results such as the thermal distribution and the energy density of thermal field theories for Majorana fermions in 1+1 dimensions using quantum simulators. In addition to free field theory, we also study the effects of interactions resulting from coupling with a spatially homogeneous Majorana field. In both cases, we show analytically that thermal properties of the system can be described using phase-space distributions, and the quantum simulation results agree with analytical and semiclassical expectations. Our work is an important step to understand thermal fixed points, preparing for quantum simulation of thermalization in real time.
We present our recent studies on thermal field theories using quantum algorithms. We first delve into the representation of quantum fields via qubits on general digital quantum computers alongside the quantum algorithms employed to evaluate thermal properties of generic quantum field theories. Then, we show our numerical results of thermal field theories in 1+1 dimensions using quantum simulators. Both fermion and scalar fields will be discussed. These studies aim to understand thermal fixed points for our forthcoming work on studying thermalization in quantum field theories in real time quantum simulation.
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.
Fermions are basic building blocks in the standard model. Interactions among these elementary particles determine how they assemble and consequently form various states of matter in our nature. Simulating fermionic degrees of freedom is also a central problem in condensed matter physics and quantum chemistry, which is crucial to understanding high-temperature superconductivity, quantum magnetism and molecular structure and functionality. However, simulating interacting fermions by classical computing generically face the minus sign problem, encountering the exponential computation complexity. Ultracold atoms provide an ideal experimental platform for quantum simulation of interacting fermions. This highly-controllable system enables the realizing of nontrivial fermionic models, by which the physical properties of the models can be obtained by measurements in experiment. This deepens our understanding of related physical mechanisms and helps determine the key parameters. In recent years, there have been versatile experimental studies on quantum ground state physics, finite temperature thermal equilibrium, and quantum many-body dynamics, in fermionic quantum simulation systems. Quantum simulation offers an access to the physical problems that are intractable on the classical computer, including studying macroscopic quantum phenomena and microscopic physical mechanisms, which demonstrates the quantum advantages of controllable quantum systems. This paper briefly introduces the model of interacting fermions describing the quantum states of matter in such a system. Then we discuss various states of matter, which can arise in interacting fermionic quantum systems, including Cooper pair superfluids and density-wave orders. These exotic quantum states play important roles in describing high-temperature superconductivity and quantum magnetism, but their simulations on the classical computers have exponentially computational cost. Related researches on quantum simulation of interacting fermions in determining the phase diagrams and equation of states reflect the quantum advantage of such systems.
This thesis describes several topics related to finite temperature studies of strongly correlated systems: finite temperature density matrix embedding theory (FT-DMET), finite temperature metal-insulator transition, and quantum algorithms including quantum imaginary time evolution (QITE), quantum Lanczos (QLanczos), and quantum minimally entangled typical thermal states (QMETTS) algorithms. While the absolute zero temperature is not reachable, studies of physical and chemical problems at finite temperatures, especially at low temperature, is essential for understanding the quantum behaviors of materials in realistic conditions. Here we define low temperature as the temperature regime where the quantum effect is not largely dissipated due to thermal fluctuation. Treatment of systems at low temperature is specially difficult compared to both high temperature - where classical approximation can be applied - and zero temperature where only the ground state is required to describe the system of interest. FT-DMET is a wavefunction-based embedding scheme which can handle finite temperature simulations of a variety of strongly correlated problems. The "high-level in low-level" framework enables FT-DMET to tackle large bulk sizes and capture the majority of the entanglement at the same time. FT-DMET formulations and implementation details for both model systems and ab initio problems are provided in Chapter 2 and Chapter 3. Metal-insulator transition is a common but important phase transition in many strongly correlated materials. The widely accepted scheme to distinguish an insulator from a metal is band structure theory based on a single-particle picture. However, insulating phases caused by disorder or strong correlation cannot be explained merely with the band structure. In Chapter 4, we demonstrate that electron locality/mobility is a more general criteria to detect metal-insulator transition. We further introduce complex polarization as the order parameter to reflect the electron locality/mobility and provide a formalism based on thermofield theory to evaluate the complex polarization at finite temperature. Quantum algorithms are designed to perform simulations on a quantum device. The infrastructure of a quantum processing unit (QPU) utilizes the superposition property of quantum bits (qubits), and thus can potentially outplay the classical simulations in computational scaling for certain problems. In Chapter 5, we introduce the QITE algorithm, which can be applied to quantum simulations of both ground state and finite temperature problems. We further introduce a subspace method, QLanczos algorithm, and a a finite temperature quantum algorithm, QMETTS, where QITE is used as a building block for the two algorithms. We demonstrate above quantum algorithms with simulations on both classical computers and quantum computers.
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.
Calculations at finite temperatures are fundamental in different scientific fields, from nuclear physics to condensed matter. Evolution in imaginary time is a prominent classical technique for preparing thermal states of quantum systems. We propose a new quantum algorithm that prepares thermal states based on the quantum imaginary time propagation method, using a diluted operator with ancilla qubits to overcome the non-unitarity nature of the imaginary time operator. The presented method is the first that allows us to obtain the correct thermal density matrix on a general quantum processor for a generic Hamiltonian. We prove its reliability in the actual quantum hardware computing thermal properties for two and three neutron systems.