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IET Quantum Communication

Quantum Schemes for Distinguishing Quantum Gates

Songsong Dai

Published Jan 1, 2025• 0 citations
Publisher

Abstract

ABSTRACT Discrimination of quantum gates is fundamental to quantum computation and information. This process involves determining whether an unknown unitary gate belongs to one of two sets of quantum gates, or . In this paper, we first introduce a scheme for distinguishing between two specific single‐qubit gates. Subsequently, we present a scheme for discriminating between two particular two‐qubit gates. Both of these discrimination schemes are exact quantum algorithms.

DOI
10.1049/qtc2.70009

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ABSTRACT Discrimination of quantum gates is fundamental to quantum computation and information. This process involves determining whether an unknown unitary gate belongs to one of two sets of quantum gates, or . In this paper, we first introduce a scheme for distinguishing between two specific single‐qubit gates. Subsequently, we present a scheme for discriminating between two particular two‐qubit gates. Both of these discrimination schemes are exact quantum algorithms.
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.
Abstract Distinguishing physical processes is one of the fundamental problems in quantum physics. Although distinguishability of quantum preparations and quantum channels have been studied considerably, distinguishability of quantum measurements remains largely unexplored. We investigate the problem of single-shot discrimination of quantum measurements using two strategies, one based on single quantum systems and the other one based on entangled quantum systems. First, we formally define both scenarios. We then construct sets of measurements (including non-projective) in arbitrary finite dimensions that are perfectly distinguishable within the second scenario using quantum entanglement, while not in the one based on single quantum systems. Furthermore, we show that any advantage in measurement discrimination tasks over single systems is a demonstration of Einstein–Podolsky–Rosen ‘quantum steering’. Alongside, we prove that all pure two-qubit entangled states provide an advantage in a measurement discrimination task over one-qubit systems.
In order to stabilize quantum algorithms against decoherence one has to fulfill two requirements. Firstly, one has to develop an appropriate quantum error correcting code. Secondly, one has to find a set of suitable unitary quantum transformations acting on the physical qubits which preserve the properties of this error correcting quantum code and which allow the implementation of a universal set of quantum gates. This is a challenging task in particular if we restrict ourselves to a limited class of two-particle interactions by which the physical qubits can be controlled. For the special cases of four and six physical qubits we discuss a set of such quantum gates which satisfy these two conditions for the recently developed detected-jump correcting quantum codes [1]. These quantum codes are capable of stabilizing distinguishable qubits against decoherence arising from spontaneous decay processes.
Quantum-state filtering is a variant of the unambiguous state discrimination problem: the states are grouped in sets and we want to determine to which particular set a given input state belongs. The simplest case, when the NNN given states are divided into two subsets and the first set consists of one state only while the second consists of all of the remaining states, is termed quantum-state filtering. We derived previously the optimal strategy for the case of NNN nonorthogonal states, ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$, for distinguishing $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. In a previous paper [Phys. Rev. Lett. 90, 257901 (2003)], we sketched an appplication of the results to probabilistic quantum algorithms. Here we fill the gaps and give the complete derivation of the probabilstic quantum algorithm that can optimally distinguish between two classes of Boolean functions, that of the balanced functions and that of the biased functions. The algorithm is probabilistic, it fails sometimes but when it does it lets us know that it did. Our approach can be considered as a generalization of the Deutsch-Jozsa algorithm that was developed for the discrimination of balanced and constant Boolean functions.
Dynamics of many-qubit systems, that may correspond to computational processing with quantum systems, can be efficiently and generally approximated by a sequence of two- and single-qubit gates. In practical applications, however, a quantum gate prepared as a unitary transformation may appear as a noisy channel and consequently may inhibit quantum advantages. In this work, we apply the scheme of channel discrimination to detect if a quantum gate that is actually realized is unitary or noisy. We show that a two-qubit unitary transformation and its noisy counterpart can be optimally discriminated by local resources, without the necessity of creating entanglement repeatedly. It is also shown that the scheme can be applied to estimation of the fraction of noise existing in quantum gates.
The problem of distinguishing two unitary transformations, or quantum gates, is analyzed and a function reflecting their statistical distinguishability is found. Given two unitary operations, U1 and U2, it is proved that there always exists a finite number N such that U(x in circle N)(1) and U(x in circle N)(2) are perfectly distinguishable, although they were not in the single-copy case. This result can be extended to any finite set of unitary transformations. Finally, a fidelity for one-qubit gates, which satisfies many useful properties from the point of view of quantum information theory, is presented.