Finite Imaginary-Time Evolution for Polynomial Unconstrained Binary Optimization

TL;DR

The paper introduces FinITE, a method for implementing finite imaginary-time evolution on quantum computers for polynomial unconstrained binary optimization problems, enabling ground-state preparation without quadratization.

Contribution

Develops FinITE, a finite-beta imaginary-time evolution technique using LCU for diagonal Pauli-Z Hamiltonians, avoiding quadratization and providing explicit fidelity thresholds.

Findings

Verified the identity on MaxCut and HUBO instances via statevector simulations.

Demonstrated finite-beta threshold prediction and amplitude amplification in shot-based simulations.

Established a closed-form relation between success probability and ground-state fidelity.

Abstract

Imaginary-time evolution is a standard primitive for ground-state preparation but is nonunitary, precluding direct quantum implementation. We develop Finite Imaginary-Time Evolution (FinITE), a finite-beta construction for diagonal Pauli-Z cost Hamiltonians arising from polynomial unconstrained binary optimization (PUBO) instances, including QUBO and HUBO cases. FinITE uses the linear-combination-of-unitaries (LCU) framework to implement a scaled imaginary-time propagator. The commuting Pauli-Z structure makes termwise block-encodings compose without product-formula error, and higher-order Pauli-Z terms are handled directly without quadratization. The structure yields an exact finite-beta identity between the LCU success probability and the ground-subspace fidelity. Combined with a gap-based fidelity lower bound, the identity yields a closed-form sufficient imaginary-time threshold beta-star for a chosen target fidelity. The threshold depends on estimates of the spectral gap and the initial ground-subspace overlap. Because the LCU success event is flagged by a known ancilla outcome, we integrate fixed-point amplitude amplification with an explicit query-complexity bound. Statevector simulations verify the identity on a five-vertex MaxCut (QUBO) and an eight-qubit cubic HUBO instance, and shot-based simulations on the MaxCut instance illustrate the predicted finite-beta threshold and amplification procedure.