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arXiv:2008.09555 · Physical Review A

Probabilistic storage and retrieval of qubit phase gates

Michal Sedlák, Mário Ziman

Published Sep 17, 2020• 10 citations
arXivHTMLPDF

Abstract

Probabilistic storage and retrieval (PSR) of unitary quantum dynamics is possible with exponentially small failure probability with respect to the number of systems used as a quantum memory [M. Sedl\'ak, A. Bisio, and M. Ziman, Phys. Rev. Lett. 122, 170502 (2019)]. Here we study improvements due to a priori knowledge about the unitary transformation to be stored. In particular, we study $N\ensuremath{\rightarrow}1$ PSR of qubit phase gates, i.e., qubit rotations around the $Z$ axis with an unknown angle, and show that if we access the gate only $N$ times the optimal probability of perfect retrieving of its single use is $N/(N+1)$. We propose a quantum circuit realization for the optimal protocol and show that a programmable phase gate [G. Vidal, L. Masanes, and J. I. Cirac, Phys. Rev. Lett. 88, 047905 (2002)] can be turned into a $({2}^{k}\ensuremath{-}1)\ensuremath{\rightarrow}1$ optimal PSR of phase gates and requires only $k$ controlled-not gates, while having exponentially small failure probability in $k$.

DOI
10.1103/physreva.102.032618

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Probabilistic storage and retrieval (PSR) of unitary quantum dynamics is possible with exponentially small failure probability with respect to the number of systems used as a quantum memory [M. Sedl'ak, A. Bisio, and M. Ziman, Phys. Rev. Lett. 122, 170502 (2019)]. Here we study improvements due to a priori knowledge about the unitary transformation to be stored. In particular, we study N\ensuremath→1N\ensuremath{\rightarrow}1N\ensuremath→1 PSR of qubit phase gates, i.e., qubit rotations around the ZZZ axis with an unknown angle, and show that if we access the gate only NNN times the optimal probability of perfect retrieving of its single use is N/(N+1)N/(N+1)N/(N+1). We propose a quantum circuit realization for the optimal protocol and show that a programmable phase gate [G. Vidal, L. Masanes, and J. I. Cirac, Phys. Rev. Lett. 88, 047905 (2002)] can be turned into a (2k\ensuremath−1)\ensuremath→1({2}^{k}\ensuremath{-}1)\ensuremath{\rightarrow}1(2k\ensuremath−1)\ensuremath→1 optimal PSR of phase gates and requires only kkk controlled-not gates, while having exponentially small failure probability in kkk.
We address the question of quantum memory storage for quantum dynamics. In particular, we design an optimal protocol for N→1 probabilistic storage and retrieval of unitary channels on d-dimensional quantum systems. If we access the unknown unitary gate only N times, the optimal success probability of perfect single-use retrieval is N/(N-1+d^{2}). The derived size of the memory system exponentially improves the known upper bound on the size of the program register needed for probabilistic programmable quantum processors. Our results are closely related to probabilistic perfect alignment of reference frames and probabilistic port-based teleportation.
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We show that it is possible to reduce the number of two-qubit gates needed for the construction of an arbitrary controlled-unitary transformation by up to 2 times using a tunable controlled-phase gate. On the platform of linear optics, where two-qubit gates can only be achieved probabilistically, our method significantly reduces the amount of components and increases success probability of a two-qubit gate. The experimental implementation of our technique presented in this Letter for a controlled single-qubit unitary gate demonstrates that only one tunable controlled-phase gate is needed instead of two standard controlled-not gates. Thus, not only do we increase the success probability by about 1 order of magnitude (with the same resources), but also avoid the need for conducting quantum nondemolition measurement otherwise required to join two probabilistic gates. Subsequently, we generalize our method to a higher order, showing that n-times controlled gates can be optimized by replacing blocks of controlled-not gates with tunable controlled-phase gates.
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The purpose of unitary synthesis is to find a gate sequence that optimally approximates a target unitary transformation. A new synthesis approach, called probabilistic synthesis, has been introduced, and its superiority has been demonstrated over traditional deterministic approaches with respect to approximation error and gate length. However, the optimality of current probabilistic synthesis algorithms is unknown. We obtain the tight lower bound on the approximation error obtained by the optimal probabilistic synthesis, which guarantees the sub-optimality of current algorithms. We also show its tight upper bound, which improves and unifies current upper bounds depending on the class of target unitaries. These two bounds reveal the fundamental relationship of approximation error between probabilistic approximation and deterministic approximation of unitary transformations. From a computational point of view, we show that the optimal probability distribution can be computed by the semidefinite program (SDP) we construct. We also construct an efficient probabilistic synthesis algorithm for single-qubit unitaries, rigorously estimate its time complexity, and show that it reduces the approximation error quadratically compared with deterministic algorithms.
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 that k≥d-1, proven necessary for the exact implementation of U_{d}^{-1} with quantum circuits. Moreover, even when quantum circuits with indefinite causal order are allowed, k≥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 programing solvers for k≤3 when d=2 and k≤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.