Auxiliary-Free Replica Shadows: Efficient Estimation of Multiple Nonlinear Quantum Properties

TL;DR

Nonlinear properties of quantum states such as $\mathrm{tr}(O\rho^t)$ are notoriously costly to estimate with standard shadow methods due to exponential sampling requirements. The AFRS framework combines $t$ copies of $\rho$, a common random unitary $V$, and a joint entangling operation $\mathcal{R}$ to produce unbiased estimators of $\rho^t$ with variance bounds $\mathrm{Var}(\widehat{o_t}) \le \|O_0\|^2_{\mathrm{sh},\mathcal{E}} + \|O\|_\infty^2$, achieving exponential sampling improvement over the original shadow protocol. It further introduces Local-AFRS for constant-depth circuits when estimating local observables, enabling practical implementation on near-term devices, and shows that a collection of $L$ observables can be estimated with sample complexity $M t \sim \mathcal{O}( t \log L)$ under multiplexing. The work provides a path toward efficient estimation of nonlinear properties, with implications for quantum metrology, error mitigation, and many-body physics.

Abstract

Efficient estimation of nonlinear properties is a significant yet challenging task from quantum information processing to many-body physics. Current methodologies often suffer from an exponential sampling cost or require auxiliary qubits and deep quantum circuits. To address these limitations, we propose an efficient auxiliary-free replica shadow (AFRS) framework, which leverages the power of the joint entangling operation on a few input replicas while integrating the mindset of shadow estimation. We rigorously prove that AFRS can offer exponential improvements in estimation accuracy compared with the conventional shadow method, and facilitate the simultaneous estimation of various nonlinear properties, unlike the destructive swap test. Additionally, we introduce an advanced local-AFRS variant tailored to estimating local observables with constant-depth quantum circuits, significantly simplifying the experimental implementation. Our work paves the way for efficient and practical estimation of nonlinear properties on near-term quantum devices.

Figures (14)

• Figure 1: An illustration of the AFRS framework. Experimental stage (up): $t$ replicas of $\rho$ are first evolved by the same random unitary $V$, then entangled via a fine-tuned joint operation $\mathcal{R}$, and finally measured in the computational basis to generate outcome $\mathbf{x}$. Post-processing stage (down): The mapping strategy $\mathcal{G}_{\mathbf{x}\to\mathbf{b}}$ transforms $\mathbf{x}$ to $\mathbf{b}$, which along with the unitary $V$ and function $f(\mathbf{x})$ forms an estimator $\widehat{\rho^t}$ of $\rho^t$. Repeating the whole procedure $M$ times generates the shadow set $\{\widehat{\rho^t_{(i)}}\}^M_{i=1}$ for estimating nonlinear properties.
• Figure 2: Scaling of estimation error with respect to the qubit number for OS, AFRS and local-AFRS protocols. The processed state is a noisy $n$-qubit GHZ state, $\rho = 0.7\ket{\text{GHZ}}\bra{\text{GHZ}}+0.3{\mathbb{I}}_d/d$. In (a), $V\in \mathcal{E}_{\mathrm{LCl}}$ is used to estimate $o_2 =\tr(O\rho^2)$, with $O=Z_1Z_2$ being a local Pauli observable. In (b), $V\in\mathcal{E}_{\mathrm{Cl}}$ is used to estimate the fidelity $F_2 =\bra{\text{GHZ}}\rho^2\ket{\text{GHZ}}$ with $O = \ket{\text{GHZ}}\bra{\text{GHZ}}$. The estimation errors for different protocols are compared under the same sample number $N$ of $\rho$, typically with $N\ll d$ to show the asymptotic scaling. The parameter $\alpha$ denotes the fitting value of the slope with $\mathrm{Error}\sim \mathcal{O}(d^\alpha)$.
• Figure 3: (a) Quantum circuit of local-AFRS. (b) Left-view of panel (a) along the gray cross-section. Each circle denotes a qubit from one replica of $V(\rho)$. The dashed boxed indicate subsequent joint unitaries. For a local observable $O=O_A\otimes \mathbb{I}_{\bar{A}}$, the block $\mathcal{R}_{\bar{A}}$ can be substituted by the qubit-wise form $\bigotimes_{i\in \bar{A}} \mathcal{R}_{(i)}$. (c) Example of entangling unitary for odd $n$: $U_{\text{odd}} = \bigotimes_{j=1}^{\lfloor n/2 \rfloor}\mathcal{R}_{2j-1,2j} \otimes \mathcal{R}_n$ and $U_{\text{even}} = \mathcal{R}_1 \otimes\bigotimes_{j=1}^{{\lfloor n/2 \rfloor}}\mathcal{R}_{2j,2j+1}$.
• Figure 4: Circuit compilation of $\mathcal{R}$. (a) Compilation of $\mathcal{R}$ in the single-qubit case. (b) Simplification of the circuit by utilizing classical post-processing. (c) The generalization to the $n$-qubit case. Here, we take $n=4$, and assume that the parity check $c^1=0$, and $c^i=1$ for $i\in\{2,3,4\}$. For the first qubit pair, the quantum circuit is similar to the single-qubit case, but there is a joint unitary consisting of CNOTs among the last three qubits in the first replica. In addition, as in the single-qubit case shown in (b), there are some classical post-processings to calculate the final measurement labels $\mathbf{x}_1$ and $\mathbf{x}_2$.
• Figure 5: Estimation performance of OS, AFRS and local-AFRS protocols. The quantum state is a noisy $5$-qubit GHZ state $\rho = 0.7\ket{\text{GHZ}}\bra{\text{GHZ}}+0.3{\mathbb{I}}_d/d$. Local Clifford ensemble with $V\in\mathcal{E}_{\mathrm{LCl}}$ is utilized for evaluating $o_2 = \tr(O\rho^2)$ with the sample number $N=1000$ in (a), and $\langle O \rangle_{\mathrm{VD}}^{(t=2)}=\tr(O\rho^2)/\tr(\rho^2)$ in (b). The local observable in (b) is set as $O=Z_1 Z_2$, and the black dashed line represents the theoretical result of $\langle O \rangle_{\mathrm{VD}}^{(t=2)}=0.994$.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Corollary 1
• Lemma 1
• proof : Proof of Observation \ref{['obs:local']}
• Theorem S3
• Proposition S1
• proof : Proof of \ref{['th:NewEst']}
• Proposition S2
• proof : Proof of Corollary 1