Abstract
Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are well-known in physics and play an important role in the theory of quantum error-correcting codes. Our algorithm is based on non-commutative Fourier analysis of coset states which are quantum states that arise from a given black-box function. We use Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations. Furthermore, we use a new technique of changing labels of irreducible representations to obtain low-dimensional irreducible representations in the decomposition process. A feature of the presented algorithm is that in each iteration of the algorithm the quantum computer operates on two coset states simultaneously. This is an improvement over the previously best known quantum algorithm for these groups which required four coset states.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ambainis, A., Emerson, J.: Quantum t-designs: t-wise independence in the quantum world. In: Proceedings of the 22nd Annual IEEE Conference on Computational Complexity, pp. 129–140 (2007); Also arxiv preprint quant-ph/0701126
Bacon, D.: How a Clebsch-Gordan transform helps to solve the Heisenberg hidden subgroup problem. Quantum Information and Computation 8(5), 438–467 (2008)
Bacon, D.: Simon’s algorithm, Clebsch-Gordan sieves, and hidden symmetries of multiple squares (2008); Arxiv preprint quant-ph/0808.0174
Bacon, D., Childs, A., van Dam, W.: From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 469–478 (2005); Also arxiv preprint quant-ph/0504083
Beth, T.: On the computational complexity of the general discrete Fourier transform. Theoretical Computer Science 51, 331–339 (1987)
Brassard, G., Høyer, P.: An exact polynomial–time algorithm for Simon’s problem. In: Proceedings of Fifth Israeli Symposium on Theory of Computing and Systems ISTCS, pp. 12–33. IEEE Computer Society Press, Los Alamitos (1997); Also arxiv preprint quant–ph/9704027
Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction and orthogonal geometry. Physical Review Letters 78(3), 405–408 (1997); Also arxiv preprint quant-ph/9605005
Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Transactions on Information Theory 44(4), 1369–1387 (1998); Also arxiv preprint quant-ph/9608006
Childs, A., Schulman, L.J., Vazirani, U.: Quantum algorithms for hidden nonlinear structures. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 395–404 (2007); Also preprint arxiv:0705.2784
van Dam, W., Hallgren, S., Ip, L.: Quantum algorithms for some hidden shift problems. In: Proceedings of the Symposium on Discrete Algorithms (SODA), pp. 489–498 (2003); Also arxiv preprint quant–ph/0211140
Ettinger, M., Høyer, P., Knill, E.: The quantum query complexity of the hidden subgroup problem is polynomial. Information Processing Letters 91(1), 43–48 (2004); Also arxiv preprint quant–ph/0401083
Friedl, K., Ivanyos, G., Magniez, F., Santha, M., Sen, P.: Hidden translation and orbit coset in quantum computing. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pp. 1–9 (2003); Also arxiv preprint quant–ph/0211091
Gottesman, D.: A class of quantum error-correcting codes saturating the quantum Hamming bound. Physical Review A 54(3), 1862–1868 (1996); Also arxiv preprint quant-ph/9604038
Grigni, M., Schulman, L., Vazirani, M., Vazirani, U.: Quantum mechanical algorithms for the nonabelian hidden subgroup problem. Combinatorica, 137–154 (2004)
Hallgren, S.: Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. In: Proceedings of the 34th Annual ACM Symposium on Theory of computing, pp. 653–658 (2002)
Hales, L., Hallgren, S.: An improved quantum Fourier transform algorithm and applications. In: Proc. of the 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), pp. 515–525. IEEE Computer Society, Los Alamitos (2000)
Hallgren, S., Moore, C., Rötteler, M., Russell, A., Sen, P.: Limitations of quantum coset states for graph isomorphism. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 604–617 (2006)
Høyer, P.: Efficient Quantum Transforms (February 1997); Arxiv preprint quant-ph/9702028
Hallgren, S., Russell, A., Ta-Shma, A.: The hidden subgroup problem and quantum computation using group representations. SIAM Journal on Computing 32(4), 916–934 (2003)
Huppert, B.: Endliche Gruppen, vol. 1. Springer, Heidelberg (1983)
Ivanyos, G., Sanselme, L., Santha, M.: An efficient algorithm for hidden subgroup problem in extraspecial groups. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 586–597. Springer, Heidelberg (2007)
Kitaev, A.Y.: Quantum computations: algorithms and error correction. Russian Math. Surveys 52(6), 1191–1249 (1997)
Kuperberg, G.: A subexponential-time quantum algorithm for the dihedral hidden subgroup problem. SIAM Journal on Computing 35(1), 170–188 (2005); Also arxiv preprint quant–ph/0302112
Mosca, M., Ekert, A.: The hidden subgroup problem and eigenvalue estimation on a quantum computer. In: Williams, C.P. (ed.) QCQC 1998. LNCS, vol. 1509, pp. 174–188. Springer, Heidelberg (1999)
Moore, C., Rockmore, D., Russell, A., Schulman, L.: The power of basis selection in Fourier sampling: hidden subgroup problems in affine groups. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1113–1122 (2004); Also arxiv preprint quant-ph/0503095
Mosca, M., Zalka, C.H.: Exact quantum Fourier transforms and discrete logarithm algorithms. International Journal of Quantum Information 2(1), 91–100 (2004); Also arxiv preprint quant–ph/0301093
Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Regev, R.: Quantum computation and lattice problems. SIAM Journal on Computing 33(3), 738–760 (2004)
Radhakrishnan, J., Rötteler, M., Sen, P.: On the power of random bases in fourier sampling: Hidden subgroup problem in the heisenberg group. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1399–1411. Springer, Heidelberg (2005)
Sen, P.: Random measurement bases, quantum state distinction and applications to the hidden subgroup problem. In: Proceedings of the 21st Annual IEEE Conference on Computational Complexity, pp. 274–287 (2006); Also arxiv preprint quant-ph/0512085
Serre, J.P.: Linear Representations of Finite Groups. Springer, Heidelberg (1977)
Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26(5), 1484–1509 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Krovi, H., Rötteler, M. (2008). An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups. In: Calmet, J., Geiselmann, W., Müller-Quade, J. (eds) Mathematical Methods in Computer Science. Lecture Notes in Computer Science, vol 5393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89994-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-540-89994-5_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89993-8
Online ISBN: 978-3-540-89994-5
eBook Packages: Computer ScienceComputer Science (R0)