Fermionic Partial Tomography via Classical Shadows

Journal Article · 09 September 2021 · Physical Review Letters

DOI: https://doi.org/10.1103/physrevlett.127.110504 · OSTI ID:1963683

Zhao, Andrew  ^[1]; Rubin, Nicholas C. ^[2]; Miyake, Akimasa ^[1]

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1. University of New Mexico, Albuquerque, NM (United States)
2. Google Research, Mountain View, CA (United States)

Here we propose a tomographic protocol for estimating any k-body reduced density matrix (k-RDM) of an n-mode fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting. Our sampling protocol uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all k-RDM elements to additive precision ϵ requires on the order of ($$^{n}_{k}$$)k^3/2log(n)/ϵ² repeated state preparations, which is optimal up to the logarithmic factor. Furthermore, numerical calculations show that our protocol offers a substantial improvement in constant overheads for k ≥ 2, as compared to prior deterministic strategies. We also adapt our method to particle-number symmetry, wherein the additional circuit depth may be halved at the cost of roughly 2–5 times more repetitions.

• Research Organization: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States); University of New Mexico, Albuquerque, NM (United States)
• Sponsoring Organization: USDOE Office of Science (SC); National Science Foundation (NSF)
• Grant/Contract Number: AC02-05CH11231
• OSTI ID: 1963683
• Journal Information: Physical Review Letters, Journal Name: Physical Review Letters Journal Issue: 11 Vol. 127; ISSN 0031-9007
• Publisher: American Physical Society (APS)Copyright Statement
• Country of Publication: United States
• Language: English