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We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an n-qubit channel E and an observable O, we aim to learn the mapping ρ↦Tr(OE[ρ]) to within a small error for most ρ sampled from a distribution D. Previously, Huang, Chen, and Preskill proved a surprising result that even if E is arbitrary, this task can be solved in time roughly nO(log(1/ϵ)), where ϵ is the target prediction error. However, their guarantee applied only to input distributions D invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states ρ. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution D, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.
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Pauli error estimation via Population Recovery
Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an n -qubit channel to precision ϵ in ℓ ∞ using just O ( 1 / ϵ 2 ) log ( n / ϵ ) applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an O ( 1 / ϵ ) factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability ≤ 1 / 4 .We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability 1 − η . In the regime of small η we extend our algorithm to achieve multiplicative precision 1 ± ϵ (i.e., additive precision ϵ η ) using just O ( 1 ϵ 2 η ) log ( n / ϵ ) applications of the channel.
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- Award ID(s):
- 1909310
- PAR ID:
- 10309487
- Date Published:
- Journal Name:
- Quantum
- Volume:
- 5
- ISSN:
- 2521-327X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.22331/q-2021-09-23-549
