# Quantum distinguishing complexity, zero-error algorithms, and statistical zero knowledge

## TL;DR

This paper introduces quantum distinguishing complexity, a new measure for quantum query algorithms, establishing relationships among various quantum complexities and implications for communication complexity lifting theorems.

## Contribution

It defines quantum distinguishing complexity and relates it to zero-error, bounded-error, and zero-knowledge quantum complexities, improving known bounds and providing new insights.

## Key findings

- Q_0(f)=O~(Q(f)^5) for all total functions f
- Q D(f) lower bounds QSZK(f) and upper bounds adversary bound
- Implications for lifting theorems in communication complexity

## Abstract

We define a new query measure we call quantum distinguishing complexity, denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, which must output a state close to |0> on a 0-input and a state close to |1> on a 1-input, a "quantum distinguishing algorithm" can output any state, as long as the output states for any 0-input and 1-input are distinguishable. Using this measure, we establish a new relationship in query complexity: For all total functions f, Q_0(f)=O~(Q(f)^5), where Q_0(f) and Q(f) denote the zero-error and bounded-error quantum query complexity of f respectively, improving on the previously known sixth power relationship. We also define a query measure based on quantum statistical zero-knowledge proofs, QSZK(f), which is at most Q(f). We show that QD(f) in fact lower bounds QSZK(f) and not just Q(f). QD(f) also upper bounds the (positive-weights) adversary bound, which yields the following relationships for all f: Q(f) >= QSZK(f) >= QS(f) = Omega(Adv(f)). This sheds some light on why the adversary bound proves suboptimal bounds for problems like Collision and Set Equality, which have low QSZK complexity. Lastly, we show implications for lifting theorems in communication complexity. We show that a general lifting theorem for either zero-error quantum query complexity or for QSZK would imply a general lifting theorem for bounded-error quantum query complexity.