Simulating Quantum Computations with Tutte Polynomials

TL;DR

This paper introduces a classical heuristic algorithm that computes quantum probability amplitudes by evaluating Tutte polynomials of graphic matroids, linking quantum circuit analysis with combinatorial graph invariants.

Contribution

It presents a novel method connecting quantum amplitudes to Tutte polynomial evaluations and offers an efficient classical algorithm for Clifford circuit amplitudes.

Findings

The algorithm effectively computes quantum amplitudes for certain circuits.

Experimental results show competitive performance on random quantum circuits.

Explicit formulas for Clifford circuit amplitudes are derived.

Abstract

We establish a classical heuristic algorithm for exactly computing quantum probability amplitudes. Our algorithm is based on mapping output probability amplitudes of quantum circuits to evaluations of the Tutte polynomial of graphic matroids. The algorithm evaluates the Tutte polynomial recursively using the deletion-contraction property while attempting to exploit structural properties of the matroid. We consider several variations of our algorithm and present experimental results comparing their performance on two classes of random quantum circuits. Further, we obtain an explicit form for Clifford circuit amplitudes in terms of matroid invariants and an alternative efficient classical algorithm for computing the output probability amplitudes of Clifford circuits.