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Computer Physics Communications

Methods for simulating time correlation functions in quantum systems

Dave Thirumalai, B. J. Berne

Published Feb 1, 1991• 41 citations
Publisher
DOI
10.1016/0010-4655(91)90266-n

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We present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.
A major problem still confronting molecular simulations is how to determine time-correlation functions of many-body quantum systems. In this paper the results of the maximum entropy (ME) and singular value decomposition (SVD) analytic continuation methods for calculating real time quantum dynamics from path integral Monte Carlo calculations of imaginary time time-correlation functions are compared with analytical results for quantum mechanical vibrational relaxation processes. This system studied is an exactly solvable system: a harmonic oscillator bilinearly coupled to a harmonic bath. The ME and SVD methods are applied to exact imaginary-time correlation functions with various level of added random noise, and also to imaginary-time data from path integral Monte Carlo (PIMC) simulations. The information gathered in the present benchmark study is valuable for the application of the analytic continuation of PIMC data to complex systems.
The n-time correlation function is pivotal for establishing connections between theoretical predictions and experimental observations of a quantum system. Conventional methods for computing n-time correlation functions on quantum computers, such as the Hadamard test, generally require an ancilla qubit that controls the entire system-an approach that poses challenges for digital quantum devices with limited qubit connectivity, as well as for analog quantum platforms lacking controlled operations. Here, we introduce a method to compute n-time correlation functions using only unitary evolutions on the system of interest, thereby eliminating the need for ancillas and the control operations. This approach substantially relaxes hardware connectivity requirements for digital processors and enables more practical measurements of n-time correlation functions on analog platforms. We demonstrate our protocol on IBM quantum hardware up to 12 qubits to measure the single-particle spectrum of the Schwinger model and the out-of-time-order correlator in the transverse-field Ising model. In the demonstration, we further introduce an error mitigation procedure based on signal processing that integrates signal filtering and correlation analysis, and successfully reproduces the noiseless simulation results from the noisy hardware. Our Letter highlights a route to exploring complex quantum many-body correlation functions in practice, even in the presence of realistic hardware limitations and noise.
Though quantum computing naturally offers an advantage for simulations of real-time quantum systems, implementing Imaginary-Time Evolution (ITE) is comparatively more difficult. Nevertheless, quantum implementations of ITE are useful both in cases where classical implementations have associated sign problems, and also as an exact method of preparing eigenstates on quantum computers. In this work we present an algorithm to obtain ITE of a generic quantum Hamiltonian by performing analytic continuation of measured real-time correlation functions. We present simulations and demonstrations on IBM quantum hardware of 1D Fokker-Planck equations for classical diffusion process and imaginary-time evolution of integrated correlation functions in 1D quantum mechanical scattering as examples to demonstrate the effectiveness of the method.