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：如何公平地量化具有内在结构（如对称性或编码）的量子态的复杂度，是量子技术基准测试中的一个核心挑战。本文引入了参考依赖复杂度（RCC），这是一种根据可用的量子操作进行校准的信息论度量。其核心思想是利用量子相对熵来量化一个量子态与其“结构化真空”——即其受限子空间内的最大熵态——的偏离程度，从而仅对产生非平凡信息的过程进行“计价”。我们的核心成果是一个关键定理，它严格证明了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，以及一个精确的物理修正项（用于解释谱分布的非均匀性）构成。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成为“可审计的”。这项工作为量子技术提供了一把“标尺”，它既是结构公平的，又能实现跨平台比较，从而在过程的计算成本与最终态的结构化信息之间建立了一个严格可验证的约束关系。