Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states

Coles, Patrick J.; Cerezo, M.; Cincio, Łukasz

doi:10.1103/physreva.100.022103

Cited by 33 publications

(15 citation statements)

References 18 publications

(36 reference statements)

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“…which is operationally related to the fidelity of a process [15] and can be shown in certain cases to be closely related to the trace distance [4]. For computational efficiency, then, we use a Hilbert-Schmidt cost function…”

Section: Protocol Descriptionmentioning

confidence: 99%

Stochastic search for approximate compilation of unitaries

Shaffer¹

2021

Preprint

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Compilation of unitaries into a sequence of physical quantum gates is a critical prerequisite for execution of quantum algorithms. This work introduces STOQ, a stochastic search protocol for approximate unitary compilation into a sequence of gates from an arbitrary gate alphabet. We demonstrate STOQ by comparing its performance to existing product-formula compilation techniques for time-evolution unitaries on system sizes up to eight qubits. The compilations generated by STOQ are less accurate than those from product-formula techniques, but they are similar in runtime and traverse significantly different paths in state space. We also use STOQ to generate compilations of randomly-generated unitaries, and we observe its ability to generate approximately-equivalent compilations of unitaries corresponding to shallow random circuits. Finally, we discuss the applicability of STOQ to tasks such as characterization of nearterm quantum devices.

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Section: Protocol Descriptionmentioning

confidence: 99%

Stochastic search for approximate compilation of unitaries

Shaffer¹

2021

Preprint

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“…The Hilbert-Schmidt distance is dimension-dependent, and is related to the trace distance by the following [19]…”

Section: Preliminariesmentioning

confidence: 99%

Efficient Algorithms for Causal Order Discovery in Quantum Networks

Bai,

Wu,

Zhu

et al. 2020

Preprint

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Given black-box access to the input and output systems, we develop the first efficient quantum causal order discovery algorithm with polynomial query complexity with respect to the number of systems. We model the causal order with quantum combs, and our algorithm outputs the order of inputs and outputs that the given process is compatible with. Our algorithm searches for the last input and the last output in the causal order, removes them, and iteratively repeats the above procedure until we get the order of all inputs and outputs. Our method guarantees a polynomial running time for quantum combs with a low Kraus rank, namely processes with low noise and little information loss. For special cases where the causal order can be inferred from local observations, we also propose algorithms that have lower query complexity and only require local state preparation and local measurements. Our algorithms will provide efficient ways to detect and optimize available transmission paths in quantum communication networks, as well as methods to verify quantum circuits and to discover the latent structure of multipartite quantum systems.

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“…It not only touches upon some of the fundamental issues in quantum mechanics, but is also crucial to various applications in quantum information processing devices, such as quantum computers, teleporters, cloners, etc [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. One of the important aspects in this context concerns with various distance measures between quantum states [27][28][29][30][31][32][33][34][35][36][50][51][52]. A very important example of practical applicability of these distance measures is in quantifying the accuracy of a signal transmission in quantum communication, wherein one measures the distance between the transmitted and received states [49].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, a strong bound between trace distance and Hilbert-Schmidt distance is now known due to Ref. [51].…”

Section: Introductionmentioning

confidence: 99%

Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance

Kumar

2020

Phys. Rev. A

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Hilbert-Schmidt distance is one of the prominent distance measures in quantum information theory which finds applications in diverse problems, such as construction of entanglement witnesses, quantum algorithms in machine learning, and quantum state tomography. In this work, we calculate exact and compact results for the mean square Hilbert-Schmidt distance between a random density matrix and a fixed density matrix, and also between two random density matrices. In the course of derivation, we also obtain corresponding exact results for the distance between a Wishart matrix and a fixed Hermitian matrix, and two Wishart matrices. We verify all our analytical results using Monte Carlo simulations. Finally, we apply our results to investigate the Hilbert-Schmidt distance between reduced density matrices generated using coupled kicked tops.

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Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states

Cited by 33 publications

References 18 publications

Stochastic search for approximate compilation of unitaries

Stochastic search for approximate compilation of unitaries

Efficient Algorithms for Causal Order Discovery in Quantum Networks

Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance

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