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Sample-optimal tomography of quantum states

Authors:

Aram W. Harrow,

Nengkun YuAuthors Info & Claims

STOC '16: Proceedings of the forty-eighth annual ACM symposium on Theory of Computing

Pages 913 - 925

https://doi.org/10.1145/2897518.2897585

Published: 19 June 2016 Publication History

Abstract

It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. This is the quantum analogue of the problem of estimating a probability distribution given some number of samples.

Previously, it was known only that estimating states to error є in trace distance required O(dr²/є²) copies for a d-dimensional density matrix of rank r. Here, we give a measurement scheme (POVM) that uses O( (dr/ δ ) ln(d/δ) ) copies to estimate ρ to error δ in infidelity. This implies O( (dr / є²)· ln(d/є) ) copies suffice to achieve error є in trace distance. For fixed d, our measurement can be implemented on a quantum computer in time polynomial in n.

We also use the Holevo bound from quantum information theory to prove a lower bound of Ω(dr/є²)/ log(d/rє) copies needed to achieve error є in trace distance. This implies a lower bound Ω(dr/δ)/log(d/rδ) for the estimation error δ in infidelity. These match our upper bounds up to log factors.

Our techniques can also show an Ω(r²d/δ) lower bound for measurement strategies in which each copy is measured individually and then the outcomes are classically post-processed to produce an estimate. This matches the known achievability results and proves for the first time that such “product” measurements have asymptotically suboptimal scaling with d and r.

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Sample-optimal tomography of quantum states

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cover image ACM Conferences

STOC '16: Proceedings of the forty-eighth annual ACM symposium on Theory of Computing

June 2016

1141 pages

ISBN:9781450341325

DOI:10.1145/2897518

General Chair:
Daniel Wichs
Northeastern, USA
,
Program Chair:
Yishay Mansour
Tel Aviv

Copyright © 2016 ACM.

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Published: 19 June 2016

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June 19 - 21, 2016

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Grier DPashayan HSchaeffer L(2024)Sample-optimal classical shadows for pure statesQuantum10.22331/q-2024-06-17-13738(1373)Online publication date: 17-Jun-2024
https://doi.org/10.22331/q-2024-06-17-1373
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