Performance analysis of multi-shot shadow estimation

Abstract

Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for $K$ times after the same unitary evolution, and repeats this procedure for $M$ rounds of random sampled unitary. As a result, there are $MK$ times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable $O$. We find that in addition to the shadow-norm $\Vert O{\Vert }_{\mathrm{shadow}}$ introduced in [Huang et.al.~Nat.~Phys.~2020\cite{huang2020predicting}], the variance is also related to another norm, and we denote it as the cross-shadow-norm $\Vert O{\Vert }_{\mathrm{Xshadow}}$. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of $\Vert O{\Vert }_{\mathrm{Xshadow}}$. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.

Discussions on measuring a collection of observables and details on numerical simulation are added

References in corpus (13)

• Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
• The randomized measurement toolbox
• Experimental single-setting quantum state tomography
• Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors
• Scalable and Flexible Classical Shadow Tomography with Tensor Networks
• The Clifford group fails gracefully to be a unitary 4-design
• Optimising shadow tomography with generalised measurements
• Measuring Arbitrary Physical Properties in Analog Quantum Simulation
• Detecting entanglement in quantum many-body systems via permutation moments
• Classical shadows of fermions with particle number symmetry
• A hybrid framework for estimating nonlinear functions of quantum states
• Hardware-efficient learning of quantum many-body states
• Precision Bounds on Continuous-Variable State Tomography using Classical Shadows

Cited by in corpus (17)

• Shallow shadows: Expectation estimation using low-depth random Clifford circuits
• Experimental property-reconstruction in a photonic quantum extreme learning machine
• Thrifty shadow estimation: re-using quantum circuits and bounding tails
• Error-mitigated fermionic classical shadows on noisy quantum devices
• Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows
• Entanglement accelerates quantum simulation
• Machine learning on quantum experimental data toward solving quantum many-body problems
• Evaluating a quantum-classical quantum Monte Carlo algorithm with Matchgate shadows
• Tailored and Externally Corrected Coupled Cluster with Quantum Inputs
• Prog-QAOA: Framework for resource-efficient quantum optimization through classical programs
• On the connection between least squares, regularization, and classical shadows
• Holographic Classical Shadow Tomography
• Nearly query-optimal classical shadow estimation of unitary channels
• Duality theory for Clifford tensor powers
• Entropy density benchmarking of near-term quantum circuits
• Optimizing Circuit Reusing and its Application in Randomized Benchmarking
• Optimal randomized measurements for a family of non-linear quantum properties