The Quantum Complexity of Computing Schatten p-norms
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.
Schatten p-norm
quantum complexity theory
complexity theory
one clean qubit model
Theory of computation~Quantum complexity theory
Theory of computation~Complexity classes
4:1-4:20
Regular Paper
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.
https://arxiv.org/abs/1706.09279
Chris
Cade
Chris Cade
School of Mathematics, University of Bristol, UK
Ashley
Montanaro
Ashley Montanaro
School of Mathematics, University of Bristol, UK
10.4230/LIPIcs.TQC.2018.4
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Chris Cade and Ashley Montanaro
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