Representation of the Fermionic Boundary Operator

TL;DR

This paper presents an efficient quantum circuit for representing the boundary operator in computational geometry, leveraging fermionic operators to achieve exact implementation with minimal depth and errors.

Contribution

It introduces a novel fermionic operator-based representation of the boundary operator and constructs an $O(n)$-depth quantum circuit for exact implementation without approximation errors.

Findings

Exact quantum circuit for boundary operator using fermionic operators

Reduced circuit depth of $O(n)$ for implementation

Elimination of Trotterization and Taylor series errors

Abstract

The boundary operator is a linear operator that acts on a collection of high-dimensional binary points (simplices) and maps them to their boundaries. This boundary map is one of the key components in numerous applications, including differential equations, machine learning, computational geometry, machine vision and control systems. We consider the problem of representing the full boundary operator on a quantum computer. We first prove that the boundary operator has a special structure in the form of a complete sum of fermionic creation and annihilation operators. We then use the fact that these operators pairwise anticommute to produce an $O(n)$-depth circuit that exactly implements the boundary operator without any Trotterization or Taylor series approximation errors. Having fewer errors reduces the number of shots required to obtain desired accuracies.