Quantum Simulations Based on Parameterized Circuit of an Antisymmetric Matrix

TL;DR

This paper introduces a parameterized quantum circuit framework for approximating the exponential of an antisymmetric matrix, leveraging a special matrix structure and quantum Fourier transform properties to efficiently estimate eigenvalues and matrix exponentials.

Contribution

It presents a novel circuit design based on a uniform antisymmetric matrix with $\

Findings

The circuit efficiently estimates $e^A$ for antisymmetric matrices.

It can approximate eigenvalues of $A$ using $O(n^2)$ gates.

The eigenspectrum is constructed via rotation $Z$ gates and phase shifts.

Abstract

Given an antisymmetric matrix $A$ or the unitary matrix $U_A = e^A$-or an oracle whose answers can be used to infer information about $A$-in this paper we present a parameterized circuit framework that can be used to approximate a quantum circuit for $e^A$. We design the circuit based on a uniform antisymmetric matrix with $\{\pm 1\}$ elements, which has an eigenbasis that is a phase-shifted version of the quantum Fourier transform, and its eigenspectrum can be constructed by using rotation $Z$ gates. Therefore, we show that it can be used to directly estimate $e^A$ and its quantum circuit representation. Since the circuit is based on $O(n^2)$ quantum gates, which form the eigendecomposition of $e^A$ with separate building blocks, it can also be used to approximate the eigenvalues of $A$.