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
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.
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 construct exact deterministic circuits for transposing and inverting an unknown qutrit unitary
U∈SU(3) using only forward queries. Three queries prepare a catalyst independently of the input state. Four further queries implement transposition while returning the catalyst unchanged and uncorrelated with the output. Two uses within an antisymmetric encoding implement inversion with eight processing queries and return the same catalyst. Reusing the catalyst implements
m transpositions or inversions with
3+4m or
3+8m total queries, respectively. More generally, we prove that exact transposition and inversion of an unknown
U∈SU(d),
d≥2, require at least
d+1 and
d2−1 processing queries, respectively, in fixed causal order. These processing bounds hold for any program state prepared independently of the input state using finitely many forward queries, even if that state is consumed during processing. Both our catalytic qutrit algorithms attain their respective bounds. Moreover, exact qutrit transposition with four processing queries requires at least three preparation queries, and our construction attains this bound.
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) calls of the unitary, where
d 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.
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.