Topological obstructions to implementing quantum if-clause
release_fyw4jiymbbfw7oxwj6dxcf2cx4
by
Zuzana Gavorová, Matan Seidel, Yonathan Touati
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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