Quantum states cannot be transmitted efficiently classically

Ashley Montanaro

doi:10.22331/q-2019-06-28-154

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Quantum states cannot be transmitted efficiently classically

2019-06-28

Ashley Montanaro *

DOI：10.22331/q-2019-06-28-154

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Abstract

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We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of n qubits (held by another), up to constant accuracy, must transmit at least Omega(2(n)) bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an is an element of-net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires Omega(N) classical queries for an input of size N. Second, a nonlocal task which can be solved using n Bell pairs, but for which any approximate classical solution must communicate Omega(2(n)) bits.

Keywords:

ONE-WAY COMMUNICATION

EXPONENTIAL SEPARATION

COMPLEXITY