Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows

TL;DR

This work addresses robust and efficient quantum state property learning from noisy near-term devices by introducing robust shallow shadows (RSS). RSS combines shallow, locally scrambled circuits with Bayesian noise learning and tensor-network post-processing to produce unbiased estimates for a broad class of observables while reducing sample complexity, even in the presence of realistic noise. The approach is validated experimentally on an IBM superconducting processor, demonstrating accurate predictions of fidelity, Pauli observables, and entanglement-related metrics, with depth-dependent tradeoffs and substantial improvements over traditional shallow or Pauli-only shadow strategies. Theoretical bounds, Bayesian calibration, and MPO/MPS representations together provide a scalable framework for reliable quantum state characterization on current hardware with practical implications for quantum information processing and quantum simulation.

Abstract

Extracting information efficiently from quantum systems is a major component of quantum information processing tasks. Randomized measurements, or classical shadows, enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Prepending a shallow random quantum circuit before measurements maintains this experimental friendliness, but also has favorable sample complexities for observables beyond low-weight Paulis, including high-weight Paulis and global low-rank properties such as fidelity. However, in realistic scenarios, quantum noise accumulated with each additional layer of the shallow circuit biases the results. To address these challenges, we propose the \emph{robust shallow shadows protocol}. Our protocol uses Bayesian inference to learn the experimentally relevant noise model and mitigate it in postprocessing. This mitigation introduces a bias-variance trade-off: correcting for noise-induced bias comes at the cost of a larger estimator variance. Despite this increased variance, as we demonstrate on a superconducting quantum processor, our protocol correctly recovers state properties such as expectation values, fidelity, and entanglement entropy, while maintaining a lower sample complexity compared to the random single qubit measurement scheme. We also theoretically analyze the effects of noise on sample complexity and show how the optimal choice of the shallow shadow depth varies with noise strength. This combined theoretical and experimental analysis positions the robust shallow shadow protocol as a scalable, robust, and sample-efficient protocol for characterizing quantum states on current quantum computing platforms.

Figures (13)

• Figure 1: A schematic overview of the robust shallow shadow protocol. In (a), we show an example of our randomized measurement scheme for a shallow circuit with $d=2$, which is a brickwork circuit comprised of twirled two-qubit gates. As shown in (b), these twirled gates are CNOT gates sandwiched by single-qubit random Cliffords. Our noise model is the sparse Pauli–Lindblad model berg2022probabilistic, which captures realistic noise effects such as qubit cross-talk. Upon twirling via single-qubit random Clifford gates, the effective noise channel simplifies from a full Pauli-Lindblad map (which has 9 two-body terms on each edge and 3 one-body terms for each node) to the one illustrated in (c), which has only one parameters for each edge and one parameter for each node. The left half of (a) shows the dataset collection process for both calibration and application states, and the right half shows our data postprocessing method. We use a Bayesian inference algorithm to estimate the noise parameters $\lambda$ of the quantum device, and use this to error mitigate our estimates of many different observables, ranging from fidelity to entanglement entropy.
• Figure 2: Sample complexity in robust shallow shadows (RSS) for Pauli observables with contiguous support of size $k$. In (a), we show a conceptual illustration of the three physical phenomena influencing sample complexity in the context of a classical random walk model: 1) operator spreading with a 'butterfly velocity' $v_B$; 2) particle density relaxation; and 3) noise-induced damping with rate $e^{-\lambda}$. The hatched region represents the space-time domain where operator spreading and relaxation occur: as time progresses (vertically), the initial operator of size $k$ (black dots) spreads ballistically (diagonal boundaries) while simultaneously relaxing to equilibrium density (vertical blue line). These phenomena collectively determine the sample complexity required for accurate observable estimation. In (b), we illustrate the qualitative impact of noise on the sample complexity upper bound, illustrating that increased noise levels lead to a slight increase in the sample complexity upper bound and a reduction in the optimal circuit depth. This reduction is approximately linear in the noise strength $\lambda$, with a proportionality coefficient $\alpha$. This is an example of the trade-offs involved in designing a noise-robust protocol. In (c), we illustrate the RMS error of reconstructed Paulis inferred using $10^4$ different circuits with $100$ shots each, as a function of the support size $k$. The dashed lines show the RMS error associated with direct calibration, while the solid lines correspond to the Bayesian learning protocol. The results demonstrate that Bayesian learning achieves comparable or better accuracy than direct calibration across all circuit depths $d$, while requiring significantly fewer calibration shots. This improved efficiency is particularly evident for larger support sizes, where the Bayesian approach maintains stable error rates even as $k$ increases.
• Figure 3: We apply RSS to predict multiple quantities including fidelity, Pauli observables, and subsystem purities. The hatched bars indicate recovered values without error mitigation, while the solid bars use error mitigation. In the top panel, we infer the fidelity of the experimentally prepared plus state $\ket{+}^{\otimes 18}$ and cluster state $\ket{\phi}$ with respect to the ideal (i.e., perfectly prepared) state. For the plus state, predictions agree with fiducial values obtained via direct fidelity estimation (i.e., measurement in the $X$ basis), showcasing the effectiveness of RSS in error mitigation. In contrast, predictions without error mitigation exhibit a decline in fidelity as circuit depth increases, underscoring the impact of noise. The lower left panel displays the predicted expectation values of Pauli observables, where RSS predictions maintain consistency and, for the plus state, agree with fiducial values. The lower right panel shows different subsystem purity predictions for a cluster state, illustrating how purity values are contingent upon the number of cuts within a subsystem. For instance, a subsystem in the bulk (formed by two cuts) has theoretical purity 0.25, whereas a boundary subsystem, with one cut, has a theoretical purity 0.5.
• Figure 4: Since the number of samples required to achieve a certain statistical error is proportional to the variance (square of the standard deviation) of the estimator, we use the standard deviation to indicate sample complexity. (a) The standard deviation of fidelity predictions decreases with increasing circuit depth, indicating reduced sample complexity. (b) The standard deviation for estimating Pauli operator expectations is plotted as a function of the Pauli weight $k$. We observe excellent agreement between experimental data (solid dots with error bars) and theoretical predictions (solid lines). Notably, shallow circuits ($d=2,4$) exhibit favorable sample complexity scaling for higher-weight Pauli operators, outperforming the $d=0$ scaling (proportional to $3^k$), as well as the theoretical upper bound of $2.28^k$.
• Figure 5: Prediction of subsystem purity in the AKLT resource state using RSS. In (a), we demonstrate how the RSS method can use a single dataset to concurrently predict the purity of all subsystems up to two qubits within AKLT resource states. We show theoretical predictions (left), experimental results with error mitigation (center), and the residual difference between mitigated and unmitigated results (right). The residual plot reveals that error mitigation systematically increases the predicted purities, bringing them closer to theoretical values, with the strongest corrections appearing in the three distinct AKLT clusters. The values at $(i,j)$ represent the purity of the reduced density matrix $\Tr(\rho_{ij}^2)$. The AKLT resource state has three clusters, each representing a smaller AKLT state with two spin-1 particles before fusion measurement; experimental predictions clearly show this pattern as well, and closely align with theoretical predictions. In (b), we show a schematic of the AKLT resource state before fusion measurements are applied to prepare the AKLT state.

Theorems & Definitions (14)

• Theorem 1: Sample complexity and optimal circuit depth, informal
• Theorem 2: Informal, Learning noisy Pauli weights
• Theorem 3: Informal, Noise-robust shallow shadow on time-independent noise
• Theorem 4: Learning noisy Pauli weights
• proof
• Lemma 5: Twirling of time-independent Markovian noise SymmetrizedNoisePhysRevA.94.052325
• Theorem 6: Noise-robust shallow shadow on time-independent Markovian noise
• proof
• Lemma 7: Twirling of time-dependent Markovian noise nonMarkovianRC
• Theorem 8: Noise-robust shallow shadow on time-dependent Markovian noise