Coherence and Imaginarity as Resources in Quantum Circuit Complexity

TL;DR

This paper explores how quantum coherence and imaginarity resources influence quantum circuit complexity, establishing new bounds and relationships that deepen understanding of quantum resource roles in computation.

Contribution

It introduces tighter lower bounds on circuit cost using coherence and imaginarity measures, revealing scenarios where imaginarity offers advantages over coherence.

Findings

Lower bounds on circuit cost via Tsallis relative α entropy of coherence.

Explicit bounds of circuit cost for typical quantum gates.

Imaginarity provides nontrivial constraints even when coherence bounds are zero.

Abstract

Quantum circuit complexity quantifies the minimal number of gates needed to realize a unitary transformation and plays a central role in quantum computation. In this work, we investigate the complexity of quantum circuits through coherence and imaginarity resources. We establish a lower bound on the circuit cost by the Tsallis relative $\alpha$ entropy of cohering power, which is shown to be tighter than the one presented by Bu et al.[\textit{Communications in Mathematical Physics} 405, no. 7 (2024):161] under restrictive conditions. As a consequence, we obtain the relationships between the circuit cost and the coherence generating power via probabilistic average in terms of skew information/relative entropy, and present explicit bounds of the circuit cost for typical quantum gates. Moreover, we derive lower bounds on the circuit cost via the imaginaring power of the circuit, induced by the Tsallis relative $\alpha$ entropy and relative entropy. We demonstrate that imaginarity can yield nontrivial constraints on the circuit cost even when coherence-based lower bounds are zero (e.g., for the $T$ gate), which implies that imaginarity may provide advantages under certain circumstances compared with coherence. Our results may help better understand the connections between quantum resources and circuit complexity.