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arXiv:2605.04981

Exact identification of unknown unitary processes

Santiago Llorens, Arnau Diebra, Michal Sedlák, Ramon Muñoz-Tapia

quant-phSubmitted May 6, 2026• Announced May 7, 2026• 0 citations
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Abstract

The accurate identification of faulty hardware is a fundamental requirement for reliable quantum information processing. We address this problem in a quantum setting, where a series of $n$ devices is intended to apply the same unitary operation, but $k$ malfunctioning devices among them apply a different, unknown unitary action. Under the assumption of complete ignorance regarding the specific unitary transformation applied, we model our hypotheses using representation-theoretic tools and study the zero-error protocol for identifying these faulty devices. We derive the optimal success probability for the single- and two-anomaly scenarios, demonstrating that it is independent of the total number of devices in the series. Furthermore, we present a simple protocol that makes use of ancillary systems that achieves this optimal limit. Notably, this protocol offers significant operational advantages, such as allowing us to test each device independently. Finally, we extend our analysis to the general scenario in which both the number of anomalies and the local dimension of the systems are arbitrary, evaluating our protocol's performance and conjecturing its global optimality in the general case.

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The accurate identification of faulty hardware is a fundamental requirement for reliable quantum information processing. We address this problem in a quantum setting, where a series of nnn devices is intended to apply the same unitary operation, but kkk malfunctioning devices among them apply a different, unknown unitary action. Under the assumption of complete ignorance regarding the specific unitary transformation applied, we model our hypotheses using representation-theoretic tools and study the zero-error protocol for identifying these faulty devices. We derive the optimal success probability for the single- and two-anomaly scenarios, demonstrating that it is independent of the total number of devices in the series. Furthermore, we present a simple protocol that makes use of ancillary systems that achieves this optimal limit. Notably, this protocol offers significant operational advantages, such as allowing us to test each device independently. Finally, we extend our analysis to the general scenario in which both the number of anomalies and the local dimension of the systems are arbitrary, evaluating our protocol's performance and conjecturing its global optimality in the general case.
We consider the problem of correctly identifying a malfunctioning quantum device that forms part of a network of N such devices, which can be considered as the quantum analog of classical anomaly detection. In the case where the devices in question are sources assumed to prepare identical quantum pure states, with the faulty source producing a different anomalous pure state, we show that the optimal probability of successful identification requires a global quantum measurement. We also put forth several local measurement strategies—both adaptive and nonadaptive—that achieve the same optimal probability of success in the limit where the number of devices to be checked is large. In the case where the faulty device performs a known unitary operation, we show that the use of entangled probes provides an improvement that even allows perfect identification for values of the unitary parameter that surpass a certain threshold. Finally, if the faulty device implements a known qubit channel, we find that the optimal probability for detecting the position of rank-one and rank-two Pauli channels can be achieved by product state inputs and separable measurements for any size of network, whereas for rank-three and general amplitude damping channels, optimal identification requires entanglement with N qubit ancillas. Published by the American Physical Society 2024
Suppose we want to identify an input state with one of two unknown reference states, where the input state is guaranteed to be equal to one of the reference states. We assume that no classical knowledge of the reference states is given, but a certain number of copies of them are available instead. Two reference states are independently and randomly chosen from the state space in a unitary invariant way. This is called the quantum state identification problem, and the task is to optimize the mean identification success probability. In this paper, we consider the case where each reference state is pure and bipartite, and generally entangled. The question is whether the maximum mean identification success probability can be attained by means of a local operations and classical communication (LOCC) measurement scheme. Two types of identification problem are considered when a single copy of each reference state is available. We show that a LOCC scheme attains the globally achievable identification probability in the minimum-error identification problem. In the unambiguous identification problem, however, the maximal success probability by means of LOCC is shown to be less than the globally achievable identification probability.
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.
Quantum noise constitutes a fundamental obstacle to realizing practical quantum technologies. To address the pivotal challenge of identifying quantum systems least affected by noise, we introduce the purest quantum state identification, which can be used to improve the accuracy of quantum computation and communication. We formulate a rigorous paradigm for identifying the purest quantum state among KKK unknown nnn-qubit quantum states using total NNN quantum state copies. For incoherent strategies, we derive the first adaptive algorithm achieving error probability exp⁡(−Ω(NH1log⁡(K)2n))\exp\left(- Ω\left(\frac{N H_1}{\log(K) 2^n }\right) \right)exp(−Ω(log(K)2nNH1​​)), fundamentally improving quantum property learning through measurement optimization. By developing a coherent measurement protocol with error bound exp⁡(−Ω(NH2log⁡(K)))\exp\left(- Ω\left(\frac{N H_2}{\log(K) }\right) \right)exp(−Ω(log(K)NH2​​)), we demonstrate a significant separation from incoherent strategies, formally quantifying the power of quantum memory and coherent measurement. Furthermore, we establish a lower bound by demonstrating that all strategies with fixed two-outcome incoherent POVM must suffer error probability exceeding $ \exp\left( - O\left(\frac{NH_1}{2^n}\right)\right)$. This research advances the characterization of quantum noise through efficient learning frameworks. Our results establish theoretical foundations for noise-adaptive quantum property learning while delivering practical protocols for enhancing the reliability of quantum hardware.
We consider the problem of reproducing one quantum measurement given the ability to perform another. We give a general framework and specific protocols for this problem. For example, we show how to use available “imperfect” devices a small number of times to implement a target measurement with average error that drops off exponentially with the number of imperfect measurements used. We hope that could be useful in near-term applications as a type of lightweight error mitigation of the measuring devices. As well as the view to practical applications, we consider the question from a general theoretical perspective in the most general setting where both the available and target measurements are arbitrary generalized quantum measurements. We show that this general problem in fact reduces to the ability to reproduce the statistics of (complete) von Neumann measurements, and that in the asymptotic limit of infinitely many uses of the available measurement, a simple protocol based upon “classical cloning” can perfectly achieve this task. We show that asymptotically any nontrivial quantum measurement can reproduce any other. We also study optimal protocols for a fixed number of uses of the available measurement. This includes, but is not limited to, improving both noisy and lossy quantum measurements. Furthermore, we show that, in a setting where we perform multiple measurements in parallel, we can achieve finite-rate measurement reproduction, by using block-coding techniques from classical information theory. Finally, we show that advantages can also be gained by making use of probabilistic protocols.
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.