Topological obstructions to implementing quantum if-clause release_fyw4jiymbbfw7oxwj6dxcf2cx4

by Zuzana Gavorová, Matan Seidel, Yonathan Touati

Released as a article .

2022  

Abstract

Some tasks are impossible in a quantum circuit, even though their classical versions are easy in a classical circuit. An example with far-reaching consequences is cloning. Another task commonly used in classical computation is the if-clause. Its quantum version applies an unknown n-qubit unitary U∈ U(2^n) if and only if a control qubit is 1. We identify it with control_ϕ(U)=|0><0|⊗ I + e^i ϕ(U)|1><1|⊗ U, for some real function ϕ. To implement this operator, one query to the oracle U suffices in linear optics, but is not enough in a quantum circuit [Araújo et al., New J. Phys., 16(9):093026, 2014]. We extend this difference in query complexity to beyond exponential in n: Even with any number of queries to U and U^†, a quantum circuit with a success/fail measurement cannot implement control_ϕ(U) with a nonzero probability of success for all U∈ U(2^n) - not even approximately. The impossibility extends to process matrices, quantum circuits with relaxed causality. Our method regards a quantum circuit as a continuous function and uses topological arguments. Compared to the polynomial method [Beals et al., JACM, 48(4):778-797, 2001], it excludes quantum circuits of any query complexity. Our result does not contradict process tomography. We show directly why process tomography fails at if-clause, and suggest relaxations to random if-clause or entangled if-clause - their optimal query complexity remains open.
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Date   2022-04-18
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arXiv  2011.10031v2
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