Closed-form analytic expressions for shadow estimation with brickwork circuits

Abstract

Properties of quantum systems can be estimated using classical shadows, which implement measurements based on random ensembles of unitaries. Originally derived for global Clifford unitaries and products of single-qubit Clifford gates, practical implementations are limited to the latter scheme for moderate numbers of qubits. Beyond local gates, the accurate implementation of very short random circuits with two-local gates is still experimentally feasible and, therefore, interesting for implementing measurements in near-term applications. In this work, we derive closed-form analytical expressions for shadow estimation using brickwork circuits with two layers of parallel two-local Haar-random (or Clifford) unitaries. Besides the construction of the classical shadow, our results give rise to sample-complexity guarantees for estimating Pauli observables. We then compare the performance of shadow estimation with brickwork circuits to the established approach using local Clifford unitaries and find improved sample complexity in the estimation of observables supported on sufficiently many qubits.

arXiv comment15+12 pages, several figures; v2: small improvements and new examples. Close to published version

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Cited by in corpus (16)

• Shallow shadows: Expectation estimation using low-depth random Clifford circuits
• Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix
• Enhanced observable estimation through classical optimization of informationally over-complete measurement data -- beyond classical shadows
• Classical shadows based on locally-entangled measurements
• Randomness-enhanced expressivity of quantum neural networks
• Many-body entropies and entanglement from polynomially-many local measurements
• Stability of classical shadows under gate-dependent noise
• Efficient Classical Shadow Tomography through Many-body Localization Dynamics
• Robust ultra-shallow shadows
• Approximate inverse measurement channel for shallow shadows
• Dual-unitary shadow tomography
• Nearly query-optimal classical shadow estimation of unitary channels
• Low variance estimations of many observables with tensor networks and informationally-complete measurements
• Holographic Classical Shadow Tomography
• Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States
• Generalized group designs: constructing novel unitary 2-, 3- and 4-designs