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arXiv:2007.03440 · Physical Review A

Experimentally undoing an unknown single-qubit unitary

Qin Feng, Tianfeng Feng, Yuling Tian, Maolin Luo, Xiaoqi Zhou

Published Jul 2, 2020• 2 citations
arXivHTMLPDF

Abstract

Undoing a unitary operation, i.e., reversing its action, is the task of canceling the effects of a unitary evolution on a quantum system, and it may be easily achieved when the unitary is known. Given a unitary operation without any specific description, however, it is a hard and challenging task to realize the inverse operation. Recently, a universal quantum circuit has been proposed [Phys. Rev. Lett. 123, 210502 (2019)] to undo an arbitrary unknown $d$-dimensional unitary $U$ by implementing its inverse with a certain probability. In this paper, we report the experimental reversal of three single-qubit unitaries $(d=2)$ by linear optical elements. The experimental results prove the feasibility of the reversing scheme, showing that the average fidelity of inverse unitaries is $F=0.9767\ifmmode\pm\else\textpm\fi{}0.0048$, in close agreement with the theoretical prediction.

DOI
10.1103/physreva.102.012602

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Undoing a unitary operation, i.e., reversing its action, is the task of canceling the effects of a unitary evolution on a quantum system, and it may be easily achieved when the unitary is known. Given a unitary operation without any specific description, however, it is a hard and challenging task to realize the inverse operation. Recently, a universal quantum circuit has been proposed [Phys. Rev. Lett. 123, 210502 (2019)] to undo an arbitrary unknown ddd-dimensional unitary UUU by implementing its inverse with a certain probability. In this paper, we report the experimental reversal of three single-qubit unitaries (d=2)(d=2)(d=2) by linear optical elements. The experimental results prove the feasibility of the reversing scheme, showing that the average fidelity of inverse unitaries is F=0.9767\ifmmode±\else\textpm\fi0.0048F=0.9767\ifmmode\pm\else\textpm\fi{}0.0048F=0.9767\ifmmode±\else\textpm\fi0.0048, in close agreement with the theoretical prediction.
We report a deterministic and exact protocol to reverse any unknown qubit-unitary operation, which simulates the time inversion of a closed qubit system. To avoid known no-go results on universal deterministic exact unitary inversion, we consider the most general class of protocols transforming unknown unitary operations within the quantum circuit model, where the input unitary operation is called multiple times in sequence and fixed quantum circuits are inserted between the calls. In the proposed protocol, the input qubit-unitary operation is called 4 times to achieve the inverse operation, and the output state in an auxiliary system can be reused as a catalyst state in another run of the unitary inversion. We also present the simplification of the semidefinite programming for searching the optimal deterministic unitary inversion protocol for an arbitrary dimension presented by M. T. Quintino and D. Ebler [Quantum 6, 679 (2022)2521-327X10.22331/q-2022-03-31-679]. We show a method to reduce the large search space representing all possible protocols, which provides a useful tool for analyzing higher-order quantum transformations for unitary operations.
Given a quantum gate implementing a ddd-dimensional unitary operation UdU_dUd​, without any specific description but ddd, and permitted to use kkk times, we present a universal probabilistic heralded quantum circuit that implements the exact inverse Ud−1U_d^{-1}Ud−1​, whose failure probability decays, exponentially in kkk. The protocol employs an adaptive strategy, proven necessary for the exponential performance. It requires k≥d−1k\geq d-1k≥d−1, proven necessary for exact implementation of Ud−1U_d^{-1}Ud−1​ with quantum circuits. Moreover, even when quantum circuits with indefinite causal order are allowed, k≥d−1k\geq d-1k≥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 kkk and ddd. The optimal values are computed using semidefinite programming solvers for k≤3k\leq 3k≤3 when d=2d=2d=2 and k≤2k\leq 2k≤2 for d=3d=3d=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.
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.
Reversing an unknown quantum evolution is of central importance to quantum information processing and fundamental physics, yet it remains a formidable challenge as conventional methods necessitate an infinite number of queries to fully characterize the quantum process. Here we introduce the Quantum Unitary Reversal Algorithm (QURA), a deterministic and exact approach to universally reverse arbitrary unknown unitary transformations using O(d2)\mathcal{O}(d^2)O(d2) calls of the unitary, where ddd is the system dimension. Our quantum algorithm resolves a fundamental problem of time-reversal simulations for closed quantum systems by confirming the feasibility of reversing any unitary evolution without knowing the exact process. The algorithm also provides the construction of a key oracle for unitary inversion in many quantum algorithm frameworks, such as quantum singular value transformation. It notably reveals a sharp boundary between the quantum and classical computing realms and unveils a quadratic quantum advantage in computational complexity for this foundational task.
In quantum mechanics, the unitary nature of time evolution makes it intrinsically reversible, given control over the system in question. Remarkably, there have been several recent demonstrations of protocols for reverting unknown unitaries in scenarios where even the interactions with the target system are unknown. These protocols are limited by their probabilistic nature, raising the fundamental question of whether time-reversal could be performed deterministically. Here we show that quantum physics indeed allows for this by exploiting the non-commuting nature of quantum operators, and demonstrate a recursive protocol for two-level quantum systems with an arbitrarily high probability of success. Using a photonic platform, we achieve an average rewinding fidelity of over 95%. Our protocol, requiring no knowledge of the quantum process to be rewound, is optimal in its running time, and brings quantum rewinding into a regime of practical relevance.