Understanding Quantum Imaginary Time Evolution and its Variational form

TL;DR

The paper addresses the challenge of obtaining ground states for Ising-type Hamiltonians by examining Quantum Imaginary Time Evolution (QITE) and its variational form (varQITE). It details how QITE replaces nonunitary imaginary-time evolution with stepwise unitary updates $U_m=e^{-i A_m au}$, with $A_m$ expanded as Pauli strings and coefficients determined via a linear system, and discusses scaling through Trotterization and localized domains. It then presents varQITE, deriving parameter dynamics from the McLachlan variational principle, with $M_{ij}\dot{ heta}_j=-V_i$ and circuit-based methods to estimate $M$ and $V$ using the parameter-shift rule. The methods are demonstrated on the transverse-field Ising model, highlighting how domain size $D$ and ansatz expressivity affect convergence and fidelity, and revealing practical trade-offs between deterministic but costly QITE and hardware-friendly varQITE, including potential hybrid strategies. Overall, the work guides practical implementation of QITE and varQITE on near-term quantum devices and outlines avenues for combining their strengths to scalable ground-state computations.

Abstract

Many computationally hard problems can be encoded in quantum Hamiltonians. The solution to these problems is given by the ground states of these Hamiltonians. A state-of-the-art algorithm for finding the ground state of a Hamiltonian is the so-called Quantum Imaginary Time Evolution (QITE) which approximates imaginary time evolution by a unitary evolution that can be implemented in quantum hardware. In this paper, we review the original algorithm together with a comprehensive computer program, as well as, the variational version of it.

Figures (1)

• Figure 1: (Left) Energy of the state obtained with both algorithms and for different domain sizes $D$ of QITE. (Right) Fidelity between the exact ground state $\ket{\psi_{gs}}$ and the states obtained from the algorithms. The time step for QITE is $\Delta \tau=0.25$ and the time step for varQITE is $\Delta \tau=0.05$. The ground state $\ket{\psi_{gs}}$ and its energy $E_{gs}$ of the TFIM are calculated using Exact Digonalization.