Quantum Computation and Quadratically Signed Weight Enumerators

TL;DR

This paper establishes a deep connection between quantum computation and classical probabilistic computation through the lens of estimating quadratically signed weight enumerators, revealing new complexity class relationships.

Contribution

It proves quantum computation is equivalent to classical probabilistic computation with an oracle for these sums, introducing a canonical BQP-complete problem and new complexity insights.

Findings

Quantum computation is polynomially equivalent to classical probabilistic computation with an oracle.

Estimation of quadratically signed weight enumerators defines a canonical BQP-complete problem.

Problems involving these sums help explore relationships between complexity classes and quantum computing.

Abstract

We prove that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating the value of simple sums, quadratically signed weight enumerators. The problem of estimating these sums can be cast in terms of promise problems and has two interesting variants. An oracle for the unconstrained variant may be more powerful than quantum computation, while an oracle for a more constrained variant is efficiently solvable in the one-bit model of quantum computation. Thus, problems involving estimation of quadratically signed weight enumerators yield problems in BQP (bounded error quantum polynomial time) that are distinct from the ones studied so far, include a canonical BQP complete problem, and can be used to define and study complexity classes and their relationships to quantum computation.