Quantum Algorithms for a Set of Group Theoretic Problems

Fenner, Stephen A.; Zhang, Yong

doi:10.1007/11560586_18

Cited by 6 publications

(6 citation statements)

References 18 publications

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“…In quantum computing, black-box groups were previously investigated in the context of quantum algorithms, both in the Abelian [35,36,92] and the non-Abelian group setting [93,73,81,94,95,80,96,97]. Except for a few exceptions (cf.…”

Section: Relationship To Previous Workmentioning

confidence: 99%

The computational power of normalizer circuits over black-box groups

Bermejo-Vega,

Lin,

Nest

2014

Preprint

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This work presents a precise connection between Clifford circuits, Shor's factoring algorithm and several other famous quantum algorithms with exponential quantum speed-ups for solving Abelian hidden subgroup problems. We show that all these different forms of quantum computation belong to a common new restricted model of quantum operations that we call black-box normalizer circuits. To define these, we extend the previous model of normalizer circuits [1-3], which are built of quantum Fourier transforms, group automorphism and quadratic phase gates associated with an Abelian group G. In [1-3], the group is always given in an explicitly decomposed form. In our model, we remove this assumption and allow G to be a black-box group [4]. While standard normalizer circuits were shown to be efficiently classically simulable [1-3], we find that normalizer circuits are powerful enough to factorize and solve classically-hard problems in the black-box setting. We further set upper limits to their computational power by showing that decomposing finite Abelian groups is complete for the associated complexity class. In particular, solving this problem renders black-box normalizer circuits efficiently classically simulable by exploiting the generalized stabilizer formalism in [1][2][3]. Lastly, we employ our connection to draw a few practical implications for quantum algorithm design: namely, we give a no-go theorem for finding new quantum algorithms with black-box normalizer circuits, a universality result for low-depth normalizer circuits, and identify two other complete problems.

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Section: Relationship To Previous Workmentioning

confidence: 99%

The computational power of normalizer circuits over black-box groups

Bermejo-Vega,

Lin,

Nest

2014

Preprint

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show abstract

“…Most famous are Shor's factoring algorithm [3] and Grover's search algorithm [4]. There has also been extensive work on using a quantum computer to simulate quantum physics [5,6,7,8,9], an ongoing exploration of adiabatic algorithms [10,11,12], plus the discovery of quantum algorithms for differential equations [13], finding eigenvalues [14,15], numerical integration [16] and various problems in group theory [17,18,19] and knot theory [20,21].…”

Section: Introductionmentioning

confidence: 99%

Long-range coupling and scalable architecture for superconducting flux qubits

et al. 2007

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Constructing a fault-tolerant quantum computer is a daunting task. Given any design, it is possible to determine the maximum error rate of each type of component that can be tolerated while still permitting arbitrarily large-scale quantum computation. It is an underappreciated fact that including an appropriately designed mechanism enabling long-range qubit coupling or transport substantially increases the maximum tolerable error rates of all components. With this thought in mind, we take the superconducting flux qubit coupling mechanism described in [1] and extend it to allow approximately 500 MHz coupling of square flux qubits, 50 µm a side, at a distance of up to several mm. This mechanism is then used as the basis of two scalable architectures for flux qubits taking into account crosstalk and fault-tolerant considerations such as permitting a universal set of logical gates, parallelism, measurement and initialization, and data mobility.

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“…Mais especificamente, mostramos que DHSP, o problema de determinar se o subgrupo oculto possui ordem 1 em grupos de caixapreta (ver [Sdroievski 2018, Seção 2.4.6]) está na classe SZK. Para obter esse resultado, mostramos uma prova de conhecimento zero estatístico para o problema, baseada no resultado de [Fenner and Zhang 2005]. Além disso, mostramos que PDHSP, o mesmo problema para grupos simétricos, está na classe NISZK.…”

Section: Figura 4 Resumo Dos Resultados Obtidos Na Dissertaçãounclassified

Conhecimento Zero Estatístico e Reduções Eficientes para o Problema MKTP

Sdroievski

2019

Anais Do Concurso De Teses E Dissertações Da SBC (CTD-SBC)

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Este artigo e um resumo da dissertação de mestrado de mesmo título, na qual estudamos a complexidade de problemas computacionais candidatos a NP-intermediários. Nesse processo, conectamos a classe SZK, relacionada a provas de conhecimento zero, com o problema computacional MKTP, relacionado a teoria algorítmica da informação. Como contribuições originais, destacamos reduções aleatorizadas do PROBLEMA DO SUBGRUPO OCULTO (HSP) e outros problemas em teoria computacional de grupos para MKTP, e provas de conhecimento zero estat´ıstico para versões de decisão do problema HSP.

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Quantum Algorithms for a Set of Group Theoretic Problems

Cited by 6 publications

References 18 publications

The computational power of normalizer circuits over black-box groups

The computational power of normalizer circuits over black-box groups

Long-range coupling and scalable architecture for superconducting flux qubits

Conhecimento Zero Estatístico e Reduções Eficientes para o Problema MKTP

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