Structure-Fair Quantum Circuit Complexity: An Auditable Information-Theoretic Lower Bound

TL;DR

The paper introduces Reference-Contingent Complexity (RCC) to quantify the cost of creating non-trivial quantum information on top of intrinsic structure, formalizing it as $C_R(\rho)=\frac{1}{\log \Gamma_R}D(\rho||\sigma_R)$ relative to a structured vacuum $\sigma_R$. It proves RCC provides a theorem-level lower bound for universal circuit complexity, with a leading linear term, a universal logarithmic correction, and a spectral-skew correction $\Delta_{\mathrm{spec}}(\rho)$; a smoothed form connects RCC to one-shot hypothesis testing via $D_H^\eta$, enabling auditable experimental estimation. The work also develops three operational principles (hypothesis testing, projective witnesses, dephasing bounds) to extract RCC from data, and presents three application interfaces—dynamical performance benchmarking, complexity-geometry calibration, and thermodynamic constraints—thereby linking information, dynamics, and geometry. An observational window further resolves degeneracies for pure states, introducing windowed RCC $C_{R,\Xi}$ and a workflow for experimentally squeezing the complexity spectrum. The framework is positioned as a structurally fair benchmark that complements existing complexity measures and opens directions toward unitary channels, open systems, and holographic dualities, with a concrete path toward auditable, cross-platform complexity comparisons.

Abstract

Quantifying the complexity of quantum states that possess intrinsic structure, such as symmetry or encoding, in a fair manner constitutes a core challenge in the benchmarking of quantum technologies. This paper introduces the Reference-Contingent Complexity (RCC), an information-theoretic measure calibrated by the available quantum operations. The core idea is to leverage the quantum relative entropy to quantify the deviation of a quantum state from its "structured vacuum"-namely, the maximum entropy state within its constrained subspace-thereby only pricing the process of creating non-trivial information. Our central result is a key theorem that rigorously proves the RCC serves as a lower bound for the complexity of any universal quantum circuit. This lower bound is comprised of a linear dominant term, a universal logarithmic correction, and a precise physical correction term that accounts for non-uniformity in the spectral distribution. Crucially, we establish a set of operational protocols, grounded in tasks like quantum hypothesis testing, which make this theoretical lower bound experimentally "auditable." This work provides a "ruler" for quantum technology that is structure-fair and enables cross-platform comparison, thereby establishing a strictly verifiable constraint between the computational cost of the process and the structured information of the final state.