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The Quantum Complexity of Computing Schatten p-norms

Authors Chris Cade, Ashley Montanaro

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ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Complexity classes
Keywords
  • Schatten p-norm
  • quantum complexity theory
  • complexity theory
  • one clean qubit model

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Abstract

We consider the quantum complexity of computing Schatten p-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating Tr(|A|^p) for a log-local n-qubit Hamiltonian A and p=poly(n), up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantity up to a somewhat higher level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by the quantum algorithm is substantially better than a natural classical algorithm for the problem. The same problem can be solved for arbitrary sparse matrices in BQP. One application of the algorithm is the approximate computation of the energy of a graph.

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Chris Cade and Ashley Montanaro. The Quantum Complexity of Computing Schatten p-norms. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.TQC.2018.4

Author Details

Chris Cade
  • School of Mathematics, University of Bristol, UK
Ashley Montanaro
  • School of Mathematics, University of Bristol, UK

Funding

CC was supported by the EPSRC. AM was supported by an EPSRC Early Career Fellowship (EP/L021005/1). No new data were created during this study.

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