Symmetry-assisted adversaries for quantum state generation

TL;DR

This paper introduces a symmetry-assisted quantum adversary method to establish lower bounds on quantum state generation problems, advancing understanding of quantum query complexity and addressing open questions in the field.

Contribution

It generalizes adversary methods for quantum state generation and leverages problem symmetries to simplify bound calculations, providing new lower bounds and theoretical insights.

Findings

Proves an (\u221a N) lower bound for Index Erasure, matching known upper bounds.

Shows the multiplicative adversary method is stronger than the additive one for any problem.

Establishes a strong direct product theorem for quantum state generation.

Abstract

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the {\sc Graph Isomorphism} problem. We show that for the related problem of {\sc Index Erasure} our method leads to a lower bound of $\Omega(\sqrt N)$ which matches an upper bound obtained via reduction to quantum search on $N$ elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is based on two ideas: (i) on the one hand we generalize the known additive and multiplicative adversary methods to the case of quantum state generation, (ii) on the other hand we show how the symmetries of the underlying problem can be leveraged for the design of optimal adversary matrices and dramatically simplify the computation of adversary bounds. Taken together, these two ideas give the new result for {\sc Index Erasure} by using the representation theory of the symmetric group. Also, the method can lead to lower bounds even for small success probability, contrary to the standard adversary method. Furthermore, we answer an open question due to \v{S}palek [CCC'08] by showing that the multiplicative version of the adversary method is stronger than the additive one for any problem. Finally, we prove that the multiplicative bound satisfies a strong direct product theorem, extending a result by \v{S}palek to quantum state generation problems.