发表机构
Hetao Institute of Mathematics and Interdisciplinary Sciences; Beijing Institute of Mathematical Sciences and Applications（河套数学与交叉科学研究所; 北京应用数学与物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种单一固定浅电路作为量子分析器，取代经典影子协议中的随机测量设置，实现任意n量子比特态的最小信息完备测量，降低控制开销并保持解析可逆性。
AI 中文摘要
经典影子从可重复使用的经典记录中提取量子性质，这些记录通常通过随机测量设置获得。虽然单个设置可能是浅的，但在它们之间切换会引入超出常规度量的控制、校准和重构成本。在这里，我们为每个$n$构造一个单一固定的浅量子分析器，其玻恩结果取代外部采样的设置，作为可重复使用的影子快照的标签。该分析器将一个新制备的$n$量子比特基准寄存器$A$与对$A$和未知系统$S$的并行贝尔读出相结合，每个副本产生一个$2n$比特记录。同一电路实现了一个秩一的最小信息完备测量：一个固定设置取代了通常用于完全重构的$3^n$个局部泡利设置，同时保留了最小$d^2$个结果，其中$d=2^n$。其泡利对角框架允许解析求逆。对于固定的厄米可观测量，哈尔平均条件方差具有维度无关的系数，而状态均匀系数是维度相关的。对于$n\ge3$，最坏状态泡利方差在轴向扇区保持有界，在混合扇区按$\Theta(d)$缩放。系统接触前的基准制备使用$n-1$个任意两量子比特门，除$A$外无工作量子比特，并在全连接连通性下具有对数深度。未知系统经历一个平行的系统-辅助纠缠层，随后是局部哈达玛门和读出。测量辅助制备在宣布路由中使用$n+1$个额外量子比特实现$O(1)$自适应量子深度，或使用$O(n\log n)$个辅助量子比特实现确定性完成。这些权衡表明，通常逐次提供的部分测量设置随机性和控制复杂性可以编译到固定的可重用分析器中。
英文摘要
Classical shadows extract quantum properties from reusable classical records, typically obtained through randomized measurement settings. Although individual settings may be shallow, switching among them introduces control, calibration, and reconfiguration costs beyond conventional metrics. Here we construct, for every $n$, a single fixed shallow quantum analyzer whose Born outcomes replace externally sampled settings as labels for reusable shadow snapshots. The analyzer combines a freshly prepared $n$-qubit fiducial register $A$ with parallel Bell readout of $A$ and the unknown system $S$, producing one $2n$-bit record per copy. The same circuit realizes a rank-one minimal informationally complete measurement: one fixed setting replaces the $3^n$ local-Pauli settings conventionally used for complete reconstruction while retaining the minimum $d^2$ outcomes with $d=2^n$. Its Pauli-diagonal frame admits an analytic inverse. For fixed Hermitian observables, the Haar-averaged conditional variance has a dimension-independent coefficient, whereas the state-uniform coefficient is dimension dependent. For $n\ge3$, worst-state Pauli variances remain bounded in the axial sector and scale as $Θ(d)$ in the mixed sector. Fiducial preparation before system contact uses $n-1$ arbitrary two-qubit gates, no work qubits beyond $A$, and logarithmic depth under all-to-all connectivity. The unknown system undergoes one parallel system-ancilla entangling layer followed by local Hadamards and readout. Measurement-assisted preparation achieves $O(1)$ adaptive quantum depth using $n+1$ extra qubits in the heralded route, or $O(n\log n)$ auxiliaries for deterministic completion. These tradeoffs show that part of the measurement-setting randomness and control complexity normally supplied shot by shot can instead be compiled into a fixed reusable analyzer.
