Quantum-classical hybrid algorithm for computing imaginary-time response functions on Noisy Intermediate Scale Quantum devices

We propose a quantum-classical hybrid algorithm for computing imaginary-time response functions on Noisy Intermediate Scale Quantum (NISQ) devices. We benchmark this algorithm by performing calculations based on a quantum embedding theory.
The quantum embedding theory is an approximate theory that transforms a whole system into a quantum impurity model in which correlated electrons (impurity sites) are embedded in the environment (bath) that represents the remaining part. In particular, the dynamical mean-field theory (DMFT) is widely used in the field of condensed matter physics [1,2]. The biggest bottleneck in those calculations is solving the "quantum impurity model" numerically, i.e., computing the Green's function. Its computational complexity generally scales exponentially with the number of correlated orbitals for classical algorithms. In recent years, theoretical proposals have been made to compute the Green's function of a quantum impurity model in polynomial time by using a quantum computer [3,4,5]. In previous studies, the methods to compute Green's function using quantum computers have been mainly for real-time Green's functions [3,4]. On the other hand, the imaginary-time formulation allows for discretizing the bath with fewer bath sites. Therefore, there is a need for an efficient method to compute the imaginary-time Green's function on NISQ devices. However, the computation of the imaginary-time Green's function on quantum computers whose gates are unitary operators is a non-trivial task.
In this study, we extend the Variational Quantum Simulation (VQS) [6], which has been used for ground-state search, to the computation of the imaginary-time Green's function. Using a quantum circuit simulator, we perform test calculations of the algorithm for a dimer model and the single-band quantum impurity model obtained by DMFT. Based on the results, we discuss the computational resources and numerical stability of the algorithm on NISQ devices and compare it with other theoretical proposals [4, 5].

[1] G.Kotliar et al., Rev. Mod. Phys. 78, 865 (2006). [2] A.Georges et al., Rev. Mod. Phys. 68, 13 (1996).
[3] B.Bauer et al., Phys. Rev. X 6, 3 031045 (2016). [4] I.Rungger et al., arXiv:1910.04735v2 [quant-ph].
[5] H.Chen et al., arXiv :2105.01703v2 [quant-ph]. [6] S.McArdle et al., npj Quantum Inf. 5, 75 (2019).

[email protected] (Rihito Sakurai)