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 ${U}_{d}^{\ensuremath{-}1}$, whose failure probability decays exponentially in $k$. The protocol employs an adaptive strategy, proven necessary for the exponential performance. It requires that $k\ensuremath{\ge}d\ensuremath{-}1$, proven necessary for the exact implementation of ${U}_{d}^{\ensuremath{-}1}$ with quantum circuits. Moreover, even when quantum circuits with indefinite causal order are allowed, $k\ensuremath{\ge}d\ensuremath{-}1$ 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 $k\ensuremath{\le}3$ when $d=2$ and $k\ensuremath{\le}2$ 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.

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

Recent developments surrounding resource theories have shown that any quantum state or measurement resource, with respect to a convex (and compact) set of resourceless objects, provides an advantage in a tailored subchannel or state discrimination task, respectively. Here we show that an analogous, more general result is also true in the case of dynamical quantum resources, i.e., channels and instruments. In the scenario we consider, the tasks associated to a resource are input-output games. The advantage a resource provides in these games is naturally quantified by a generalized robustness measure. We illustrate our approach by applying it to a broad collection of examples, including classical and measure-and-prepare channels, measurement and channel incompatibility, local operations assisted by classical communication, and steering, as well as discussing its applicability to other resources in, e.g., quantum thermodynamics. We finish by showing that our approach generalizes to higher-order dynamics where it can be used, for example, to witness causal properties of supermaps.

Quantification of quantum dynamics with input-output games

Quantitative studies of irreversibility in statistical mechanics often involve the consideration of a reverse process, whose definition has been the object of many discussions, in particular for quantum mechanical systems. Here we show that the reverse channel very naturally arises from Bayesian retrodiction, in both classical and quantum theories. Previous paradigmatic results, such as Jarzynski's equality, Crooks' fluctuation theorem, and Tasaki's two-measurement fluctuation theorem for closed driven quantum systems, are all shown to be consistent with retrodictive arguments. Also, various corrections that were introduced to deal with nonequilibrium steady states or open quantum systems are justified on general grounds as remnants of Bayesian retrodiction. More generally, with the reverse process constructed on consistent logical inference, fluctuation relations acquire a much broader form and scope.

Fluctuation theorems from Bayesian retrodiction

This paper addresses the problem of designing universal quantum circuits to transform $k$ uses of a $d$-dimensional unitary input operation into a unitary output operation in a probabilistic heralded manner. Three classes of protocols are considered, parallel circuits, where the input operations can be performed simultaneously, adaptive circuits, where sequential uses of the input operations are allowed, and general protocols, where the use of the input operations may be performed without a definite causal order. For these three classes, we develop a systematic semidefinite programming approach that finds a circuit which obtains the desired transformation with the maximal success probability. We then analyze in detail three particular transformations: unitary transposition, unitary complex conjugation, and unitary inversion. For unitary transposition and unitary inverse, we prove that for any fixed dimension $d$, adaptive circuits have an exponential improvement in terms of uses $k$ when compared to parallel ones. For unitary complex conjugation and unitary inversion we prove that if the number of uses $k$ is strictly smaller than $d\ensuremath{-}1$, the probability of success is necessarily zero. We also discuss the advantage of indefinite causal order protocols over causal ones and introduce the concept of delayed input-state quantum circuits.

Probabilistic exact universal quantum circuits for transforming unitary operations

Similarly to quantum states, quantum operations can also be transformed by means of quantum superchannels, also known as process matrices. Quantum superchannels with multiple slots are deterministic transformations whichtake independent quantum operations as inputs. While they are enforced to respect the laws of quantum mechanics, the use of input operations may lack a definite causal order, and characterizations of general superchannels in terms of quantum objects with a physical implementation have been missing. In this paper, we provide a mathematical characterization for pure superchannels with two slots (also known as bipartite pure processes), which are superchannels preserving the reversibility of quantum operations. We show that the reversibility preserving condition restricts all pure superchannels with two slots to be either a quantum circuit only consisting of unitary operations or a coherent superposition of two unitary quantum circuits where the two input operations are differently ordered. The latter may be seen as a generalization of the quantum switch, allowing a physical interpretation for pure two-slot superchannels. An immediate corollary is that purifiable bipartite processes cannot violate device-independent causal inequalities.

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Consequences of preserving reversibility in quantum superchannels

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 $k\geq d-1$, proven necessary for exact implementation of $U_d^{-1}$ with quantum circuits. Moreover, even when quantum circuits with indefinite causal order are allowed, $k\geq d-1$ 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 $k\leq 3$ when $d=2$ and $k\leq 2$ 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 unknown quantum transformations: A universal protocol for inverting general unitary operations

We study equivalence determination of unitary operations, a task analogous to quantum state discrimination. The candidate states are replaced by unitary operations given as a quantum sample, i.e., a black-box device implementing a candidate unitary operation, and the discrimination target becomes another black-box. The task is an instance of higher-order quantum computation with the black-boxes as input. The optimal error probability is calculated by semidefinite programs. Arbitrary quantum operations applied between the black-boxes in a general protocol provide advantages over protocols restricted to parallelized use of the black-boxes. We provide a numerical proof of such an advantage. In contrast, a parallelized scheme is analytically shown to exhibit the optimal performance of general schemes for a particular number of quantum samples of the candidates. We find examples of finite-sample equivalence determination that achieve the same performance as when a classical description of the candidates are provided, although an exact classical description cannot be obtained from finite quantum samples.

Equivalence determination of unitary operations

We analyze a discrimination problem of a single-qubit unitary gate with two candidates, where the candidates are not provided with their classical description, but their quantum sample is. More precisely, there are three unitary quantum gates---one target and one sample for each of the two candidates---whose classical description is unknown except for their dimension. The target gate is chosen equally among the candidates. We obtain the optimal protocol that maximizes the expected success probability, assuming the Haar distribution for the candidates. This problem is originally introduced in Ref. [5] which provides a protocol achieving 7/8 in the expected success probability based on the ``unitary comparison'' protocol of Ref. [6]. The optimality of the protocol has been an open question since then. We prove the optimality of the comparison protocol, implying that only one of the two samples (one for each candidate) is needed to achieve an optimal discrimination. The optimization includes protocols outside the scope of quantum testers due to the dynamic ordering of the sample and target gates within a given protocol.

Atsushi Shimbo

Papers

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

Probabilistic exact universal quantum circuits for transforming unitary operations

Reversing unknown quantum transformations: A universal protocol for inverting general unitary operations

Equivalence determination of unitary operations

Optimal quantum discrimination of single-qubit unitary gates between two candidates