Phase shadow: A noise-tolerant path to global quantum property estimation

Abstract

Measuring global quantum properties-such as the fidelity to complex multipartite states-is both an essential and experimentally challenging task. Classical shadow estimation offers favorable sample complexity, but typically relies on many-qubit circuits that are difficult to realize on current platforms. We propose the robust phase shadow scheme, a measurement framework based on random circuits with controlled-$Z$ as the unique entangling gate type, tailored to architectures such as trapped ions and neutral atoms. Leveraging tensor diagrammatic reasoning, we rigorously analyze the induced circuit ensemble and show that phase shadows match the performance of full Clifford-based ones. Importantly, our approach supports a noise-robust extension via purely classical post-processing, enabling reliable estimation under gate-dependent noise where existing techniques often fail. Additionally, by exploiting structural properties of random stabilizer states, we design an efficient post-processing algorithm that resolves a key computational bottleneck in previous shadow protocols. Our results enhance the practicality of shadow-based techniques, providing a robust and scalable route for estimating global properties in noisy quantum systems.

Figures (18)

• Figure 1: Outline of the (robust) phase shadow framework. The random phase circuit is in the form ${CZ}$-${S}$-${H}$ before the final computational basis measurement, where the layer of two-qubit ${CZ}$ gates may suffer gate-dependent noise. For both noiseless (\ref{['th:PshadowMain']}) and noisy scenarios (\ref{['th:PshadowNoise']}), via proper and efficient post-processing, our framework can return unbiased estimator of the off-diagonal part of $\rho$, say $\rho_f$, with favorable statistical performance.
• Figure 2: (a) An illustration of \ref{['eq:23moment']} of $m=2$ case. The union of the two permutation unitaries $\mathbb{I}_D^{\otimes 2}\cup \mathbb{S}_D=\mathbb{I}_D^{\otimes 2}+ \mathbb{S}_D-\Delta_D$, with the parity projector $\Delta_D$ being their intersection, say $\mathbb{I}_D^{\otimes 2}\cap \mathbb{S}_D=\Delta_D$. For $D=2$, the corresponding matrix representation is also shown. (b) A tensor diagram illustration of the (noiseless) measurement channel in \ref{['Eq:idealchannel']}. By tracing the first copy, one obtains the output proportional to $\mathbb{I}_D+\rho_f$. (c) In the presence of noise, the moment function expands from a compact form (left) to a sum of $3^n$ tensor-product terms (right), each built from qubit-level operators ${\Delta_2,(\mathbb{S}_2 - \Delta_2),(\mathbb{I}_2^{\otimes 2} - \Delta_2)}$, as shown by the colored blocks.
• Figure 3: (a) Variance of fidelity estimation for the GHZ* state (noiseless case), comparing Pauli, Clifford measurements, and PS huang2020predicting under different qubit number $n$ of the system. The GHZ* state is locally equivalent to the canonical GHZ state zhou2019detecting. (b) Fidelity estimation of the GHZ* state using RPS and PS measurements for $n \in \{25,35,45\}$ qubits under different noise level $p_e$ ($N = 50,\!000$ snapshots for each data point). (c) Cubic root of the post-processing time for fidelity estimation of three graph states using RPS ($n = 15$--$65$, $N = 10,\!000$). The details on the numerical simulations are left to Supplementary Note 8.
• Figure 4: (Left) Estimation bias and (right) logarithmic estimation variance for 10-qubit random stabilizer states, comparing generalized RPS (orange) and its extension to random Clifford circuits (green) with the Clifford RSE protocol (purple) chen2021robust. Each data point is computed using $N=10^4$ measurement snapshots and averaged over 100 random stabilizer states. The RSE protocol is calibrated using $10^6$ runs for each noise level. The details on the numerical simulations are left to Supplementary Note 10.
• Figure 5: A tensor diagrammatic illustration of \ref{['Eq:EMvarfinal']}. (a) Using the Principle of Inclusion-Exclusion, we transform $V_n(\pi_{(23)})\cup V_n(\pi_{(123)})\cup V_n(\pi_{(132)})$ to an alternating sum of Boolean tensors. We observe that the common elements of $V_n(\pi_{(123)})$ and $V_n(\pi_{(132)})$ are all contained in $V_n(\pi_{(23)})$. (b) The intersection like $V_n(\pi_{(123)})\cap V_n(\pi_{(23)})$ and $V_n(\pi_{(123)})\cap V_n(\pi_{(132)})$ equal to the Hadamard product of them, respectively. (c) The contribution of the tensor $V_n(\pi_{(123)})\cap V_n(\pi_{(23)})$ to the estimation variance. By reconfiguring the tensor network (untwist the blue line), we derive the simplification $\sum_{\mathbf{x}}\bra{\mathbf{x}}\rho\ket{\mathbf{x}}\bra{\mathbf{x}}O_f^2\ket{\mathbf{x}} = \tr(\rho_dO_f^2)$.

Theorems & Definitions (20)

• Proposition 1: 2nd and 3rd moments
• Theorem 1: Unbiased and efficient recovery with phase shadow estimation
• Proposition 2: Pauli decomposition
• Theorem 2: Guarantees of unbiased and efficient recovery under realistic noise
• Proposition 3: Efficient classical post-processing (informal)
• Theorem 3: Generalized RPS under gate-dependent Pauli noise
• Lemma 1: Time complexity
• Definition 1: Moment functions
• Lemma 2: Twirling channels
• proof