Analytical Lower Bound on Query Complexity for Transformations of Unknown Unitary Operations

TL;DR

This paper establishes fundamental lower bounds on the number of queries needed to perform transformations like inversion and conjugation on unknown unitary operations, confirming the optimality of existing protocols and introducing a new differentiation-based framework.

Contribution

It provides analytical lower bounds for query complexity of unitary transformations, introduces a differentiation framework for such bounds, and proves the impossibility of catalytic protocols for complex conjugation.

Findings

Lower bound of d^2 queries for unitary inversion, matching existing protocols.

Introduction of a differentiation-based framework for deriving query bounds.

Proof that catalytic protocols cannot achieve complex conjugation of unitaries.

Abstract

Recent developments have revealed deterministic and exact protocols for performing complex conjugation, inversion, and transposition of a general $d$-dimensional unknown unitary operation using a finite number of queries to a black-box unitary operation. In this work, we establish analytical lower bounds for the query complexity of unitary inversion, transposition, and complex conjugation, which hold even if the input unitary is an unknown logarithmic-depth unitary. Specifically, our lower bound of $d^2$ for unitary inversion demonstrates the asymptotic optimality of the deterministic exact inversion protocol, which operates with $O(d^2)$ queries. We introduce a novel framework utilizing differentiation to derive these lower bounds on query complexity for general differentiable functions $f: \mathrm{SU}(d)\to \mathrm{SU}(d)$. As a corollary, we prove that a catalytic protocol -- a new concept recently noted in the study of exact unitary inversion -- is impossible for unitary complex conjugation. Furthermore, we extend our framework to the partially known setting, where the input unitary operation is promised to be within a subgroup of $\mathrm{SU}(d)$ and the probabilistic setting, where transformations succeed probabilistically.