Isometry operations encode the quantum information of the input system to a larger output system, while the corresponding decoding operation would be an inverse operation of the encoding isometry operation. Given an encoding operation as a black box from a d -dimensional system to a D -dimensional system, we propose a universal protocol for isometry inversion that constructs a decoder from multiple calls of the encoding operation. This is a probabilistic but exact protocol whose success probability is independent of D . For a qubit ( d=2 ) encoded in n qubits, our protocol achieves an exponential improvement over any tomography-based or unitary-embedding method, which cannot avoid D -dependence. We present a quantum operation that converts multiple parallel calls of any given isometry operation to random parallelized unitary operations, each of dimension d . Applied to our setup, it universally compresses the encoded quantum information to a D -independent space, while keeping the initial quantum information intact. This compressing operation is combined with a unitary inversion protocol to complete the isometry inversion. We also discover a fundamental difference between our isometry inversion protocol and the known unitary inversion protocols by analyzing isometry complex conjugation and isometry transposition. General protocols including indefinite causal order are searched using semidefinite programming for any improvement in the success probability over the parallel protocols. We find a sequential "success-or-draw" protocol of universal isometry inversion for d=2 and D=3 , thus whose success probability exponentially improves over parallel protocols in the number of calls of the input isometry operation for the said case.
Isometry operations encode the quantum information of the input system to a larger output system, while the corresponding decoding operation would be an inverse operation of the encoding isometry operation. Given an encoding operation as a black box from a d -dimensional system to a D -dimensional system, we propose a universal protocol for isometry inversion that constructs a decoder from multiple calls of the encoding operation. This is a probabilistic but exact protocol whose success probability is independent of D . For a qubit ( d=2 ) encoded in n qubits, our protocol achieves an exponential improvement over any tomography-based or unitary-embedding method, which cannot avoid D -dependence. We present a quantum operation that converts multiple parallel calls of any given isometry operation to random parallelized unitary operations, each of dimension d . Applied to our setup, it universally compresses the encoded quantum information to a D -independent space, while keeping the initial quantum information intact. This compressing operation is combined with a unitary inversion protocol to complete the isometry inversion. We also discover a fundamental difference between our isometry inversion protocol and the known unitary inversion protocols by analyzing isometry complex conjugation and isometry transposition. General protocols including indefinite causal order are searched using semidefinite programming for any improvement in the success probability over the parallel protocols. We find a sequential "success-or-draw" protocol of universal isometry inversion for d=2 and D=3 , thus whose success probability exponentially improves over parallel protocols in the number of calls of the input isometry operation for the said case.
Identification of possible transformations of quantum objects including quantum states and quantum operations is indispensable in developing quantum algorithms. Universal transformations, defined as input-independent transformations, appear in various quantum applications. Such is the case for universal transformations of unitary operations. However, extending these transformations to non-unitary operations is nontrivial and largely unresolved. Addressing this, we introduce isometry adjointation protocols that transform an input isometry operation into its adjoint operation, which include both unitary operation and quantum state transformations. The paper details the construction of parallel and sequential isometry adjointation protocols, derived from unitary inversion protocols using quantum combs and the (dual) Clebsch-Gordan transforms, and achieving optimal approximation error. This error is shown to be independent of the output dimension of the isometry operation. In particular, we explicitly obtain an asymptotically optimal parallel protocol achieving an approximation error ϵ=Θ(d2/n) , where d is the input dimension of the isometry operation and n is the number of calls of the isometry operation. The research also extends to isometry inversion and universal error detection, employing semidefinite programming to assess optimal performances. The findings suggest that the optimal performance of general protocols in isometry adjointation and universal error detection is not dependent on the output dimension, and that indefinite causal order protocols offer advantages over sequential ones in isometry inversion and universal error detection.
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.
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.
Access to the complex conjugate of an unknown quantum channel is a useful resource in quantum oracle problems, motivating the question of how such access can be simulated using only a limited number of calls to the original channel. We determine the optimal deterministic protocol for approximately implementing the complex conjugate isometry V from n uses of an unknown isometry channel V:Cd→CD. We derive a closed-form expression for the optimal fidelity and prove that a parallel protocol is optimal even among general quantum superchannels, including adaptive and indefinite-causal-order strategies. The formula implies a query complexity n=Θ(d[(D−d)/ε+1]) for achieving infidelity ε. We also present a circuit construction based on the quantum Schur transform and the dual Clebsch--Gordan transform, with circuit complexity O(poly(D,1/ε)). This task is extended to the multi-copy case V⊗n↦V⊗k. For fixed d<D and k, we show that the optimal fidelity for the multi-copy case is 1−kd(D−d)/n+o(n−1), and that this value is asymptotically attained by a parallel estimation-based protocol. Finally, combining the isometry protocol with random Stinespring dilations yields a protocol for complex conjugation of unknown rank-r quantum channels whose query complexity is optimal up to a constant factor if the Kraus rank r is constant.
Given a quantum gate implementing a d-dimensional unitary operation Ud, 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 Ud−1, whose failure probability decays, exponentially in k. The protocol employs an adaptive strategy, proven necessary for the exponential performance. It requires k≥d−1, proven necessary for exact implementation of Ud−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 programming 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.
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.