Counting anticommuting Pauli pairs in linear time

TL;DR

This paper introduces a linear-time algorithm for counting anticommuting Pauli pairs in large collections, significantly improving efficiency over traditional quadratic methods in the bounded locality regime.

Contribution

The authors develop a novel linear-time algorithm that counts anticommuting Pauli pairs by maintaining subpattern counts, optimizing quantum workflow analysis.

Findings

Algorithm operates in O(m) time for bounded locality cases.

Efficiently processes large collections of Pauli strings.

Provides exact counts of anticommuting pairs with improved speed.

Abstract

Many quantum computing workflows manipulate long lists of Pauli strings. A basic classical subroutine involves taking $m$ Pauli strings on $n$ qubits, each of weight bounded by a constant, to determine if they are pairwise commuting, identify any counterexamples, or calculate the exact number of anticommuting unordered pairs. The standard general-purpose route represents Pauli strings in binary symplectic form and checks pairs in $O(m^2)$ time. Here, we provide an $O(m)$ algorithm for the bounded locality regime. It maintains counts of all labeled subpatterns of previously inserted strings and answers each new string query by a subset zeta identity. Our algorithm is particularly useful for processing large collections of Pauli strings within the bounded locality regime.