Hamiltonian-Driven Shadow Tomography of Quantum States

Abstract

Classical shadow tomography provides an efficient method for predicting functions of an unknown quantum state from a few measurements of the state. It relies on a unitary channel that efficiently scrambles the quantum information of the state to the measurement basis. Facing the challenge of realizing deep unitary circuits on near-term quantum devices, we explore the scenario in which the unitary channel can be shallow and is generated by a quantum chaotic Hamiltonian via time evolution. We provide an unbiased estimator of the density matrix for all ranges of the evolution time. We analyze the sample complexity of the Hamiltonian-driven shadow tomography. For Pauli observables, we find that it can be more efficient than the unitary-2-design-based shadow tomography in a sequence of intermediate time windows that range from an order-1 scrambling time to a time scale of $D^{1/6}$, given the Hilbert space dimension $D$. In particular, the efficiency of predicting diagonal Pauli observables is improved by a factor of $D$ without sacrificing the efficiency of predicting off-diagonal Pauli observables.

arXiv comment4+epsilon pages, 2 figures, with appendix. Add detailed discussion and numerical evidence in the new version. Add and modify some references

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Cited by in corpus (12)

• The Variational Quantum Eigensolver: a review of methods and best practices
• Provably efficient machine learning for quantum many-body problems
• Classical Shadows With Noise
• Classical Shadow Tomography with Locally Scrambled Quantum Dynamics
• Shallow shadows: Expectation estimation using low-depth random Clifford circuits
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• Optimising shadow tomography with generalised measurements
• Classical Shadows for Quantum Process Tomography on Near-term Quantum Computers
• Classical shadows with Pauli-invariant unitary ensembles
• Performance analysis of multi-shot shadow estimation
• Measuring energy by measuring any other observable