Topological obstructions to implementing quantum ifclause
Abstract
Some tasks are impossible in a quantum circuit, even though their classical versions are easy in a classical circuit. An example with farreaching consequences is cloning. Another task commonly used in classical computation is the ifclause. Its quantum version applies an unknown $n$qubit unitary $U\in U(2^n)$ if and only if a control qubit is $1$. We identify it with $control_\phi(U)=\left0\right>\!\!\left<0\right\otimes I + e^{i \phi(U)}\left1\right>\!\!\left<1\right\otimes U$, for some real function $\phi$. 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 $U^\dagger$, a quantum circuit with a success/fail measurement cannot implement $control_\phi(U)$ with a nonzero probability of success for all $U\in U(2^n)$  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):778797, 2001], it excludes quantum circuits of any query complexity. Our result does not contradict process tomography. We show directly why process tomography fails at ifclause, and suggest relaxations to random ifclause or entangled ifclause  their optimal query complexity remains open.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 arXiv:
 arXiv:2011.10031
 Bibcode:
 2020arXiv201110031G
 Keywords:

 Quantum Physics
 EPrint:
 13 pages, 4 figures, 3 tables + Appendix: 24 pages, 6 figures. Changes in this version: main text completely rewritten  shortened, the rest moved to appendix