Post-selected Classical Query Complexity

TL;DR

This paper characterizes classical query algorithms with post-selection using nonnegative rational functions, revealing exponential separations from quantum algorithms and establishing connections to communication complexity and non-deterministic models.

Contribution

It introduces a novel characterization of post-selected classical query complexity via nonnegative rational functions, and demonstrates significant separations from quantum complexity for symmetric functions.

Findings

Post-selected classical query complexity equals the minimal degree of nonnegative rational approximations.

An exponential separation exists between post-selected classical and quantum query complexities for the Majority function.

Zero-error post-selected algorithms are equivalent to non-deterministic classical algorithms, requiring more queries than bounded-error versions.

Abstract

We study classical query algorithms with post-selection, and find that they are closely connected to rational functions with nonnegative coefficients. We show that the post-selected classical query complexity of a Boolean function is equal to the minimal degree of a rational function with nonnegative coefficients that approximates it (up to a factor of two). For post-selected quantum query algorithms, a similar relationship was shown by Mahadev and de Wolf, where the rational approximations are allowed to have negative coefficients. Using our characterisation, we find an exponentially large separation between post-selected classical query complexity and post-selected quantum query complexity, by proving a lower bound on the degree of rational approximations (with nonnegative coefficients) to the Majority function. This lower bound can be generalised to arbitrary symmetric functions, and allows us to find an unbounded separation between non-deterministic quantum and post-selected classical query complexity. All lower bounds carry over into the communication complexity setting. We show that the zero-error variants of post-selected query algorithms are equivalent to non-deterministic classical query algorithms, which in turn are characterised by nonnegative polynomials, and for some problems require exponentially more queries to the input than their bounded-error counterparts. Finally, we describe a post-selected query algorithm for approximating the Majority function, and an efficient query algorithm for approximate counting.