BQP and PPAD

TL;DR

This paper explores the relationship between quantum complexity class BQP and the class PPAD, proposing a conjecture about their inclusion, analyzing it under oracle models, and discussing open problems in computational complexity.

Contribution

It introduces a conjecture that PPAD might be contained in BQP, provides a necessary and sufficient condition for this, and demonstrates the conjecture's limitations under oracle models.

Findings

PPAD is not contained in BQP under the oracle model.

A necessary and sufficient condition for PPAD ⊆ BQP is established.

Open problems and future directions in complexity theory are discussed.

Abstract

We initiate the study of the relationship between two complexity classes, BQP (Bounded-Error Quantum Polynomial-Time) and PPAD (Polynomial Parity Argument, Directed). We first give a conjecture that PPAD is contained in BQP, and show a necessary and sufficient condition for the conjecture to hold. Then we prove that the conjecture is not true under the oracle model. In the end, we raise some interesting open problems/future directions.