A compressed classical description of quantum states
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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 ψ consists of its inner products with O(√(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 best known upper bound, due to Raz. We obtain an exponential improvement over Raz's protocol in terms of computational efficiency.
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