Equivalence, Identity, and Unitarity Checking in Black-Box Testing of Quantum Programs

Peixun Long, Jianjun Zhao

Tiancheng Jin, Bo Yang, Haibo Yu +1 more

Peixun Long, Jianjun Zhao

Quantum programs exhibit inherent non-deterministic behavior, which poses more significant challenges for error discovery compared to classical programs. While several testing methods have been proposed for quantum programs, they often overlook fundamental questions in black-box testing. In this paper, we bridge this gap by presenting three novel algorithms specifically designed to address the challenges of equivalence, identity, and unitarity checking in black-box testing of quantum programs. We also explore optimization techniques for these algorithms, including specialized versions for equivalence and unitarity checking, and provide valuable insights into parameter selection to maximize performance and effectiveness. To evaluate the effectiveness of our proposed methods, we conducted comprehensive experimental evaluations, which demonstrate that our methods can rigorously perform equivalence, identity, and unitarity checking, offering robust support for black-box testing of quantum programs.

Mingsheng Ying, Nengkun Yu, Yuan Feng +1 more

厚 新保

Quantum state discrimination is one of the most fundamental tasks in quantum information since it describes the read-out process of classical information from quantum states in quantum communications.In quantum state discrimination, a given test state is guaranteed to be identical to one of a set of candidate quantum states.The task is to determine the identity of the test state by performing an appropriate quantum measurement based on the classical descriptions of the candidate states.Quantum state discrimination also has several connections to the principles of quantum mechanics such as the no-cloning theorem and the no-signaling principle.One of fundamental properties of quantum mechanics is that the exact classical description of unknown quantum states cannot be retrieved from a single copy of the physical state.Quantum state discrimination in the case that the candidate states are given as two unknown physical states (reference states) but their classical descriptions are not given, has been investigated.The two reference states are labeled one and two, respectively.The task is to determine which of the two reference states is equivalent to the test state, whereas the descriptions of the states are not of interest.We call such a task as equivalence determination of quantum states.Discrimination of quantum operations has been also investigated and it has been shown that there is an intrinsic difference between discrimination of quantum state and discrimination of quantum operations.We analyze a unitary operation version of the equivalence determination task by using the concept of higher-order quantum computation to investigate properties of discrimination of other quantum objects.We consider three black-boxes (test box and reference boxes) implementing unknown unitary operations given as physical systems.The test box is guaranteed to implement one of the two unitary operations given by the reference boxes.Equivalence determination of unitary operations is a task to determine the reference box implementing the same unitary operation as the one by the test box.We assume that the unitary operations of the reference boxes are randomly sampled from the Haar measure of SU(2).We first consider the case that each of three black-boxes can be used only once both in parallel and ordered uses of the black-boxes.We show that the optimal success probability for equivalence determination is 7/8 by obtaining analytical solutions of the corresponding semidefinite programmings (SDPs) both in the parallel and ordered strategies.The optimal success probability can be achieved even without using one of the reference boxes.We also showed that entanglement of an initial state across the systems on which the reference boxes act and the test box acts is necessary to achieve the optimal success probability.We then consider the case that the multiple uses of the reference boxes are allowed.We consider the parallel strategies up to four uses of each of the reference boxes and obtain the optimal success probabilities by numerically solving the semidefinite programmings.We numerically show that the ordered strategies give improvement over the parallel strategies and the optimal success probability varies depending on the order of the black-boxes when one reference box is used once and the other reference box is used twice.This result indicates that an appropriate order of the black-boxes under the ordered strategies is necessary to obtain improvement over the parallel strategies. Chapter 1 Introduction Notations• H i , K j : Hilbert spaces labeled with i and j.• L(H): The set of linear operators on H.• L(H, K): The set of linear operators from H to K.• S(H) : The set of positive semidefinite linear operation of unit trace on H • Tr H : The partial trace over H • U † : The conjugate transpose of U • {Π i }: The positive operator-valued measure • { Π i }: The quantum tester • δ ij : The Kronecker delta, equal to 1 if i = j and 0 otherwise • dµ(•) : The Haar measure, the uniform distribution of the space of unitary operations • |I⟩⟩ : Unnormalized maximally entangled vector defined as |I⟩⟩ = i |i⟩|i⟩ • M : A quantum channel • M : Choi operator of a quantum channel M• U j : The irreducible subspace of the total angular momentum j.

Mingsheng Ying