The Adversary Bound Revisited: From Optimal Query Algorithms to Optimal Control

TL;DR

This paper revisits the adversary bound in quantum computing, presenting a more general and accessible formulation that applies to optimal control problems and simplifies the understanding and implementation of quantum algorithms.

Contribution

It introduces a new perspective on the adversary bound using reduced density matrices, broadening its applicability beyond query problems to optimal control scenarios.

Findings

The new formulation simplifies understanding of the adversary bound.

The approach applies to a broader class of problems, including optimal control.

It improves algorithm efficiency by removing certain runtime factors.

Abstract

This note complements the paper "One-Way Ticket to Las Vegas and the Quantum Adversary" (arxiv:2301.02003). I develop the ideas behind the adversary bound - universal algorithm duality therein in a different form, using the same perspective as Barnum-Saks-Szegedy in which query algorithms are defined as sequences of feasible reduced density matrices rather than sequences of unitaries. This form may be faster to understand for a general quantum information audience: It avoids defining the "unidirectional relative $\gamma_{2}$-bound" and relating it to query algorithms explicitly. This proof is also more general because the lower bound (and universal query algorithm) apply to a class of optimal control problems rather than just query problems. That is in addition to the advantages to be discussed in Belovs-Yolcu, namely the more elementary algorithm and correctness proof that avoids phase estimation and spectral analysis, allows for limited treatment of noise, and removes another $\Theta(\log(1/\epsilon))$ factor from the runtime compared to the previous discrete-time algorithm.