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.

翻译：以公平方式量化具有内在结构（如对称性或编码）的量子态的复杂度，是量子技术基准测试的核心挑战。本文引入参考依赖复杂度（RCC），这是一种根据可用量子操作校准的信息论度量。其核心思想是利用量子相对熵来量化量子态与其“结构真空”——即其受限子空间内的最大熵态——的偏离，从而仅对产生非平凡信息的过程进行定价。我们的核心结果是一个关键定理，严格证明了RCC可作为任何通用量子电路复杂度的下界。该下界由一个线性主导项、一个普适的对数修正项，以及一个用于解释谱分布非均匀性的精确物理修正项组成。至关重要的是，我们建立了一套基于量子假设检验等任务的操作协议，使得这一理论下界在实验上具有“可审计性”。这项工作为量子技术提供了一把“标尺”，它具备结构公平性并支持跨平台比较，从而在过程计算成本与最终态的结构化信息之间建立了严格可验证的约束关系。