Reachability Constraints in Variational Quantum Circuits: Optimization within Polynomial Group Module

TL;DR

This paper establishes a necessary condition for variational quantum algorithms to reach the exact ground state, linking module weights to the ability to achieve exact solutions and highlighting classical simulability for certain problems.

Contribution

It introduces a fundamental reachability constraint based on module projections, providing insights into when quantum approaches can be classically simulated for exact solutions.

Findings

Matching module weights is necessary for exact ground state reachability.

Certain problems, like Max Cut, can be classically simulated with polynomial time per step.

The condition applies to matchgate circuits and classical bit string solutions.

Abstract

This work identifies a necessary condition for any variational quantum approach to reach the exact ground state. Briefly, the norms of the projections of the input and the ground state onto each group module must match, implying that module weights of the solution state have to be known in advance in order to reach the exact ground state. An exemplary case is provided by matchgate circuits applied to problems whose solutions are classical bit strings, since all computational basis states share the same module-wise weights. Combined with the known classical simulability of quantum circuits for which observables lie in a small linear subspace, this implies that certain problems admit a classical surrogate for exact solution with each step taking $O(n^5)$ time. The Maximum Cut problem serves as an illustrative example.