The Hidden Subgroup Problem and Quantum Computation Using Group Representations

Hallgren, Sean; Russell, Alexander; Ta-Shma, Amnon

doi:10.1137/s009753970139450x

Cited by 99 publications

(77 citation statements)

References 23 publications

(24 reference statements)

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“…[HRTS03,GSVV04] show that O(log |G|) weak Fourier samplings suffice to reconstruct the normal core c(H) of the hidden subgroup H, where c(H) is the largest normal subgroup of G contained in H. Adapting their arguments, we prove the following result. …”

Section: Hidden Subgroup Problem and Fourier Samplingmentioning

confidence: 83%

“…Hallgren, Russell and Ta-Shma [HRTS03] showed that polynomially many iterations of weak Fourier sampling give enough information to reconstruct normal hidden subgroups. More generally, they show that the normal core c(H) of the hidden subgroup H (i. e. the largest normal subgroup of G contained in H) can be reconstructed via the weak method.…”

Section: Introductionmentioning

confidence: 99%

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On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group

Radhakrishnan

Rötteler²,

Sen³

2005

Lecture Notes in Computer Science

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The hidden subgroup problem (HSP) provides a unified framework to study problems of grouptheoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an efficient solution in the abelian case, not much is known for general non-abelian groups. Recently, some authors raised the question as to whether post-processing the Fourier spectrum by measuring in a random orthonormal basis helps for solving the HSP. Several negative results on the shortcomings of this random strong method are known. In this paper however, we show that the random strong method can be quite powerful under certain conditions on the group G. We define a parameter r(G) for a group G and show that O((log |G|/r(G))2 ) iterations of the random strong method give enough classical information to identify a hidden subgroup in G. We illustrate the power of the random strong method via a concrete example of the HSP over finite Heisenberg groups. We show that r(G) = Ω(1) for these groups; hence the HSP can be solved using polynomially many random strong Fourier samplings followed by a possibly exponential classical post-processing without further queries. The quantum part of our algorithm consists of a polynomial computation followed by measuring in a random orthonormal basis. This gives the first example of a group where random representation bases do help in solving the HSP and for which no explicit representation bases are known that solve the problem with (log G) O(1) Fourier samplings. As an interesting by-product of our work, we get an algorithm for solving the state identification problem for a set of nearly orthogonal pure quantum states.

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Section: Hidden Subgroup Problem and Fourier Samplingmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group

Radhakrishnan

Rötteler²,

Sen³

2005

Lecture Notes in Computer Science

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“…In this setting, it is possible to formulate the GI problems in terms as in (4). The formula can be translated into the following potential Hamiltonian: …”

Section: Adiabatic Quantum Walkmentioning

confidence: 99%

“…GI possesses peculiar features that make it an interesting candidate for an efficient quantum algorithm. In fact it is in NP but is not believed to be NP-Complete: like factoring, it belongs to the NP-Intermediate family [2] and is representative of the (non-Abelian) hidden subgroup problem family [3][4][5]. The best classical general algorithm solves GI for graphs of n vertices in time O(c √ n log n ), were c is a constant.…”

Section: Introductionmentioning

confidence: 99%

A quantum-walk-inspired adiabatic algorithm for solving graph isomorphism problems

Tamascelli¹,

Zanetti²

2014

J. Phys. A: Math. Theor.

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We present a 2-local quantum algorithm for graph isomorphism GI based on an adiabatic protocol. By exploiting continuous-time quantum-walks, we are able to avoid a mere diffusion over all possible configurations and to significantly reduce the dimensionality of the visited space. Within this restricted space, the graph isomorphism problem can be translated into the search of a satisfying assignment to a 2-SAT formula without resorting to perturbation gadgets or projective techniques. We present an analysis of the execution time of the algorithm on small instances of the graph isomorphism problem and discuss the issue of an implementation of the proposed adiabatic scheme on current quantum computing hardware.

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“…• Since the subgroups j Stab are not normal subgroups of N S , it follows from the work of Hallgren et al [55,56] that the standard non-abelian hidden subgroup algorithm will find the largest normal subgroup of N S lying in Moreover, one can not hope to use this QHS approach to Grover's algorithm to find a faster quantum algorithm. For Zalka [77] has shown that Grover's algorithm is optimal.…”

Section: Iia10 Is Grover's Algorithm a Qhs Algorithm?mentioning

confidence: 99%

Search for New Quantum Algorithms

Lomonaco¹,

Kauffman²

2006

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching data sources, Gathering and maintaining the data needed, and completing and reviewing the collection of information. SPONSOR/MONITOR'S ACRONYM(S) 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)DARPA AFRL-IF-RS-TR-2006-180 DISTRIBUTION AVAILABILITY STATEMENTApproved for public release; distribution unlimited. PA# 06-356 SUPPLEMENTARY NOTES ABSTRACTThe first objective of this effort, searching for new quantum algorithms, created six new quantum hidden subgroup algorithms. The second objective, improving the theoretical understanding of existing quantum algorithms, produced three new systematic procedures. Also, application of combinatorial group theory led to substantial progress in the understanding and analysis of nonabelian quantum hidden subgroup algorithms. Additionally, methods and techniques of quantum topology have been used to obtain new results in quantum computing including discovery of a relationship between quantum entanglement and topological linking. The last objective, analyzing issues associated with algorithm implementation proposed distributed quantum computing (DQC) as a fast track to scalable quantum computing with technology available within the next five years. A universal set of DQC primitives has been created and used to transform the quantum Fourier transform and the Shor algorithm into DQC. The additional computational overhead needed for DQC algorithms is insignificant and DQC is found to simplify the decoherence problem. SUBJECT TERMSquantum algorithms, quantum hidden subgroup algorithms, hidden subgroups, distributed quantum computation, distributed algorithms, Shor's algorithm, Grover's algorithm, non-abelian quantum hidden subgroup algorithms, decoherence, quantum computer architectures, topological quantum computation, quantum gates, anyonic computation SECURITY

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The Hidden Subgroup Problem and Quantum Computation Using Group Representations

Cited by 99 publications

References 23 publications

On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group

On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group

A quantum-walk-inspired adiabatic algorithm for solving graph isomorphism problems

Search for New Quantum Algorithms

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