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arXiv:0905.0085

On Anomaly Identification and the Counterfeit Coin Problem

Eldin Wee Chuan Lim

Published May 1, 2009• 2 citations
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Abstract

We address a well-known problem in combinatorics involving the identification of counterfeit coins with a systematic approach. The methodology can be applied to cases where the total number of coins is exceedingly large such that brute force or enumerative comparisons become impractical. Based on the principle behind this approach, the minimum effort necessary for identification of the counterfeit coin can be determined and expressed as a simple relation. We further suggest a possible application of this methodology to error detection in quantum information processing.

DOI
10.48550/arxiv.0905.0085

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We address a well-known problem in combinatorics involving the identification of counterfeit coins with a systematic approach. The methodology can be applied to cases where the total number of coins is exceedingly large such that brute force or enumerative comparisons become impractical. Based on the principle behind this approach, the minimum effort necessary for identification of the counterfeit coin can be determined and expressed as a simple relation. We further suggest a possible application of this methodology to error detection in quantum information processing.
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
The problem of reliable identification of quantum hash states, which are images of quantum hash functions, is considered. It is shown that this can be done with a high probability for a linear number of O(n/(1−ϵ2))O(n/(1-\epsilon^{2}))O(n/(1−ϵ2)) experiments when hashing words of length nnn with ϵ\epsilonϵ -resistance to collisions. The results of a numerical experiment on the implementation of the procedure are presented, demonstrating the result of the above. We provide such an experiment to simulate quantum hashing for binary words of length 5.