A Compressed Classical Description of Quantum States

Authors David Gosset, John Smolin

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Author Details

David Gosset
  • IBM T. J. Watson Research Center, Yorktown Heights, USA
  • Department of Combinatorics and Optimization and Institute for Quantum Computing, University of Waterloo, Canada
John Smolin
  • IBM T. J. Watson Research Center, Yorktown Heights, USA


DG thanks Scott Aaronson, Sergey Bravyi, Aram Harrow, Shelby Kimmel, Yi-Kai Liu, Ashley Montanaro, and Oded Regev for helpful discussions. We thank Sergey Bravyi for his Clifford-manipulation code, and Robin Kothari for pointing out the relevance of the lower bound from Ref. [Gavinsky et al., 2007].

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David Gosset and John Smolin. A Compressed Classical Description of Quantum States. In 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 135, pp. 8:1-8:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We show how to approximately represent a quantum state using the square root of the usual amount of classical memory. The classical representation of an n-qubit state psi consists of its inner products with O(sqrt{2^n}) stabilizer states. A quantum state initially specified by its 2^n entries in the computational basis can be compressed to this form in time O(2^n poly(n)), and, subsequently, the compressed description can be used to additively approximate the expectation value of an arbitrary observable. Our compression scheme directly gives a new protocol for the vector in subspace problem with randomized one-way communication complexity that matches (up to polylogarithmic factors) the optimal upper bound, due to Raz. We obtain an exponential improvement over Raz’s protocol in terms of computational efficiency.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Quantum complexity theory
  • Quantum computation
  • Quantum communication complexity
  • Classical simulation


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