Reversing Unknown Quantum Transformations: Universal Quantum Circuit for Inverting General Unitary Operations

Abstract

Given a quantum gate implementing a $d$-dimensional unitary operation $U_d$, without any specific description but $d$, and permitted to use $k$ times, we present a universal probabilistic heralded quantum circuit that implements the exact inverse $U_d^{-1}$, whose failure probability decays, exponentially in $k$. The protocol employs an adaptive strategy, proven necessary for the exponential performance. It requires $k\ge d-1$, proven necessary for exact implementation of $U_d^{-1}$ with quantum circuits. Moreover, even when quantum circuits with indefinite causal order are allowed, $k\ge d-1$ uses are required. We then present a finite set of linear and positive semidefinite constraints characterizing universal unitary inversion protocols and formulate a convex optimization problem whose solution is the maximum success probability for given $k$ and $d$. The optimal values are computed using semidefinite programming solvers for $k\le 3$ when $d=2$ and $k\le 2$ for $d=3$. With this numerical approach we show for the first time that indefinite causal order circuits provide an advantage over causally ordered ones in a task involving multiple uses of the same unitary operation.

arXiv commentCloser to the published version. This version slightly improves the text and provides more details on the proof of Theorem 2. Part of the technical methods are presented at arXiv:1909.01366. Matlab code to accompany this article can be found at https://github.com/mtcq/unitary_inverse

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