Classical Shadows with Improved Median-of-Means Estimation

Abstract

The classical shadows protocol, introduced by Huang et al. [Nat. Phys. 16, 1050 (2020)], makes use of the median-of-means (MoM) estimator to efficiently estimate the expectation values of $M$ observables with failure probability $\delta$ using only $\mathcal{O}(\log (M/\delta ))$ measurements. In their analysis, Huang et al. used loose constants in their asymptotic performance bounds for simplicity. However, the specific values of these constants can significantly affect the number of shots used in practical implementations. To address this, we studied a modified MoM estimator proposed by Minsker [PMLR 195, 5925 (2023)] that uses optimal constants and involves a U-statistic over the data set. For efficient estimation, we implemented two types of incomplete U-statistics estimators, the first based on random sampling and the second based on cyclically permuted sampling. We compared the performance of the original and modified estimators when used with the classical shadows protocol with single-qubit Clifford unitaries (Pauli measurements) for an Ising spin chain, and global Clifford unitaries (Clifford measurements) for the Greenberger-Horne-Zeilinger (GHZ) state. While the original estimator outperformed the modified estimators for Pauli measurements, the modified estimators showed improved performance over the original estimator for Clifford measurements. Our findings highlight the importance of tailoring estimators to specific measurement settings to optimize the performance of the classical shadows protocol in practical applications.

15 pages, 13 figures

References in corpus (51)

• Efficient simulation of one-dimensional quantum many-body systems
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Mixed-state entanglement from local randomized measurements
• Provably efficient machine learning for quantum many-body problems
• Efficient estimation of Pauli observables by derandomization
• Sample-optimal tomography of quantum states
• Fermionic partial tomography via classical shadows
• Robust shadow estimation
• Perfect Sampling with Unitary Tensor Networks
• Classical Shadows With Noise
• Experimental Estimation of Quantum State Properties from Classical Shadows
• Classical Shadow Tomography with Locally Scrambled Quantum Dynamics
• Matchgate Shadows for Fermionic Quantum Simulation
• Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States
• Experimental quantum state measurement with classical shadows
• Shadows of quantum machine learning
• Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors
• Shallow shadows: Expectation estimation using low-depth random Clifford circuits
• Scalable and Flexible Classical Shadow Tomography with Tensor Networks
• Precision-guaranteed quantum tomography
• Quantum scrambling with classical shadows
• Classical Shadows for Quantum Process Tomography on Near-term Quantum Computers
• Operator relaxation and the optimal depth of classical shadows
• Classical shadows with Pauli-invariant unitary ensembles
• Improved machine learning algorithm for predicting ground state properties
• Quantum Error Mitigated Classical Shadows
• Classical shadow tomography for continuous variables quantum systems
• Thrifty shadow estimation: re-using quantum circuits and bounding tails
• Training variational quantum circuits with CoVaR: covariance root finding with classical shadows
• Estimating gate-set properties from random sequences
• Learnability transitions in monitored quantum dynamics via eavesdropper's classical shadows
• Shadow tomography on general measurement frames
• Error-mitigated fermionic classical shadows on noisy quantum devices
• Group-theoretic error mitigation enabled by classical shadows and symmetries
• Evaluating a quantum-classical quantum Monte Carlo algorithm with Matchgate shadows
• Classical shadows based on locally-entangled measurements
• Measurement optimization of variational quantum simulation by classical shadow and derandomization
• Sample-optimal classical shadows for pure states
• Filtering crosstalk from bath non-Markovianity via spacetime classical shadows
• Optimising quantum tomography via shadow inversion
• Efficient Local Classical Shadow Tomography with Number Conservation
• Stability of classical shadows under gate-dependent noise
• On Classical and Hybrid Shadows of Quantum States
• Efficient Classical Shadow Tomography through Many-body Localization Dynamics
• Qudit Shadow Estimation Based on the Clifford Group and the Power of a Single Magic Gate
• Decoherence through Ancilla Anyon Reservoirs
• Robust ultra-shallow shadows
• Classical shadows meet quantum optimal mass transport
• Quantum Circuit Cutting for Classical Shadows
• Biased Estimator Channels for Classical Shadows