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arXiv:2312.16680 · Quantum

PT -symmetric mapping of three states and its implementation on a cloud quantum processor

Yaroslav Balytskyi, Yevgen V. Kotukh, Gennady Khalimov, Sang‐Yoon Chang

Published Apr 18, 2025• 2 citations

Abstract

PT -symmetric systems have garnered significant attention due to their unconventional properties. Despite the growing interest, there remains an ongoing debate about whether these systems outperform their Hermitian counterparts in practical applications, and if so, by what metrics this performance should be measured. We developed PT -symmetric approach for mapping N=3 pure qubit states to address this, implemented it using the dilation method, and demonstrated it on a superconducting quantum processor from the IBM Quantum Experience. For the first time, we derived exact expressions for the population of the post-selected PT -symmetric subspace for both N=2 and N=3 states. When applied to the discrimination of N=2 pure states, our algorithm provides an equivalent result to the conventional unambiguous quantum state discrimination. For N=3 states, our approach introduces novel capabilities not available in traditional Hermitian systems, enabling the transformation of an arbitrary set of three pure quantum states into another, at the cost of introducing an inconclusive outcome. Our algorithm has the same error rate for the attack on the three-state QKD protocol as the conventional minimum error, maximum confidence, and maximum mutual information strategies. For post-selected quantum metrology, our results provide precise conditions where PT -symmetric quantum sensors outperform their Hermitian counterparts in terms of information-cost rate. Combined with punctuated unstructured quantum database search, our method significantly reduces the qubit readout requirements at the cost of adding an ancilla, while maintaining the same average number of oracle calls as the original punctuated Grover's algorithm. Our work opens new pathways for applying PT symmetry in quantum communications, computing, and cryptography.

DOI: 10.22331/q-2025-04-18-1716

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PT -symmetric systems have garnered significant attention due to their unconventional properties. Despite the growing interest, there remains an ongoing debate about whether these systems outperform their Hermitian counterparts in practical applications, and if so, by what metrics this performance should be measured. We developed PT -symmetric approach for mapping N=3 pure qubit states to address this, implemented it using the dilation method, and demonstrated it on a superconducting quantum processor from the IBM Quantum Experience. For the first time, we derived exact expressions for the population of the post-selected PT -symmetric subspace for both N=2 and N=3 states. When applied to the discrimination of N=2 pure states, our algorithm provides an equivalent result to the conventional unambiguous quantum state discrimination. For N=3 states, our approach introduces novel capabilities not available in traditional Hermitian systems, enabling the transformation of an arbitrary set of three pure quantum states into another, at the cost of introducing an inconclusive outcome. Our algorithm has the same error rate for the attack on the three-state QKD protocol as the conventional minimum error, maximum confidence, and maximum mutual information strategies. For post-selected quantum metrology, our results provide precise conditions where PT -symmetric quantum sensors outperform their Hermitian counterparts in terms of information-cost rate. Combined with punctuated unstructured quantum database search, our method significantly reduces the qubit readout requirements at the cost of adding an ancilla, while maintaining the same average number of oracle calls as the original punctuated Grover's algorithm. Our work opens new pathways for applying PT symmetry in quantum communications, computing, and cryptography.

Abstract We present a novel

\mathcal{PT}

-symmetric approach for discriminating three pure qubit states and its implementation by the dilation method. Our approach based on the dilation method manipulating the Hilbert space of the system by two-staged

\mathcal{PT}

-symmetric evolution expedites the three-state discrimination process at the cost of introducing an inconclusive outcome. This modification of the Hilbert space of the qubit makes it possible to eliminate one of three states in the limit

\alpha\rightarrow\pm\frac{\pi}{2}

in the vicinity of the exceptional point corresponding to coalescing eigenvectors and eigenvalues, effectively reducing the problem to two-state discrimination, or reducing the geometry to the mirror-symmetric states for which an efficient discrimination strategy is developed. To demonstrate the effectiveness of our approach, we implement it on a superconducting quantum processor provided by IBM Quantum Experience employing an ancilla qubit. We conclude our paper by identifying potential applications, including those in quantum communication and cryptography, to facilitate further research and development.

Nonorthogonal quantum state discrimination (QSD) plays an important role in quantum information and quantum communication. In addition, compared to Hermitian quantum systems, parity-time-(

\mathcal{PT}

-)symmetric non-Hermitian quantum systems exhibit novel phenomena and have attracted considerable attention. Here, we experimentally demonstrate QSD in a

\mathcal{PT}

-symmetric system (i.e.,

\mathcal{PT}

-symmetric QSD), by having quantum states evolve under a

\mathcal{PT}

-symmetric Hamiltonian in a lossy linear optical setup. We observe that two initially nonorthogonal states can rapidly evolve into orthogonal states and the required evolution time can even be vanishing provided the matrix elements of the Hamiltonian become sufficiently large. We also observe that the cost of such a discrimination is a dissipation of quantum states into the environment. Furthermore, by comparing

\mathcal{PT}

-symmetric QSD with optimal strategies in Hermitian systems, we find that at the critical value,

\mathcal{PT}

-symmetric QSD is equivalent to the optimal unambiguous state discrimination in Hermitian systems. We also extend the

\mathcal{PT}

-symmetric QSD to the case of discriminating three nonorthogonal states. The QSD in a

\mathcal{PT}

-symmetric system opens a new door for quantum state discrimination, which has important applications in quantum computing, quantum cryptography, and quantum communication.

Quantum sensing utilizing unique quantum properties of non-Hermitian systems to realize ultra-precision measurements has been attracting increasing attention. However, the debate on whether non-Hermitian systems are superior to Hermitian counterparts in sensing remains an open question. Here, we investigate the quantum information in PT-symmetric quantum sensing utilizing two experimental schemes based on the trapped-ion platform. It turns out that the existence of advantages of non-Hermitian quantum sensing heavily depends on additional information resources carried by the extra degrees of freedom introduced to construct PT-symmetric quantum sensors. Moreover, the practical application of non-Hermitian quantum sensing with superior performance is primarily restricted by the additional resource consumption accompanied by the post-selection. Our study provides theoretical references for the construction of non-Hermitian quantum sensors with superior performance and has potential applications in research fields of quantum precision measurement and quantum information processing.

As a fundamental and challenging problem in quantum information processing, quantum state discrimination (QSD) has recently been revisited from the perspective of non-Hermitian (NH) physics. The existing work about QSD in NH systems is mostly limited to PT-symmetric or pseudo-Hermitian systems with real spectra, while generic NH Hamiltonians possess complex spectra. To explore the underlying physics of quantum state discrimination in NH systems, we first demonstrate that the unambiguous discrimination of non-orthogonal quantum states can be realized with P-pseudo-Hermitian and PT-symmetric Hamiltonians in the broken phase and the required evolution time can be arbitrarily small. We further extend the regime to generic non-Hermitian Hamiltonians and demonstrate the feasibility of unambiguous discrimination. Under the same energy constraint, we establish the criteria for constructing P-pseudo-Hermitian Hamiltonians that enable unambiguous discrimination of quantum states with smaller angular separation or within shorter evolution time than any fixed PT-symmetric system. Furthermore, we also discuss the potential impact of exceptional points (EPs) on QSD. Our results suggest that the non-orthogonal eigenstates of NH Hamiltonians, rather than PT symmetry or pseudo-Hermitian symmetry, are fundamental to unambiguous discrimination and information flow in NH systems.

Quantum computing is on the cusp of transforming the way we tackle complex problems, and the Grover search algorithm exemplifying its potential to revolutionize the search for unstructured large datasets, offering remarkable speedups over classical methods. Here, we report results for the implementation and characterization of a three-qubit Grover search algorithm using the state-of-the-art scalable quantum computing technology of superconducting quantum architectures. To delve into the algorithm's scalability and performance metrics, our investigation spans the execution of the algorithm across all eight conceivable single-result oracles, alongside nine two-result oracles, employing IBM Quantum's 127-qubit quantum computers. Moreover, we conduct five quantum state tomography experiments to precisely gauge the behavior and efficiency of our implemented algorithm under diverse conditions - ranging from noisy, noise-free environments to the complexities of real-world quantum hardware. By connecting theoretical concepts with real-world experiments, this study not only shed light on the potential of Noisy Intermediate-Scale Quantum Computers in facilitating large-scale database searches but also offer valuable insights into the practical application of the Grover search algorithm in real-world quantum computing applications.

The objective of this paper is to explain and elucidate the formalism of PT quantum mechanics by applying it to a well-known problem in conventional Hermitian quantum mechanics, namely the problem of state discrimination. Suppose that a system is known to be in one of two quantum states, |ψ(1)> or |ψ(2)>. If these states are not orthogonal, then the requirement of unitarity forbids the possibility of discriminating between these two states with one measurement; that is, determining with one measurement what state the system is in. In conventional quantum mechanics, there is a strategy in which successful state discrimination can be achieved with a single measurement but only with a success probability p that is less than unity. In this paper, the state-discrimination problem is examined in the context of PT quantum mechanics and the approach is based on the fact that a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states. It is shown that it is always possible to choose this inner product so that the two states |ψ(1)> and |ψ(2)> are orthogonal. Using PT quantum mechanics, one cannot achieve a better state discrimination than in ordinary quantum mechanics, but one can instead perform a simulated quantum state discrimination, in which with a single measurement a perfect state discrimination is realized with probability p.