The Symmetric Group Defies Strong Fourier Sampling

The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states

Quantum algorithms for problems in number theory, algebraic geometry, and group theory

Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.Comment: 20 pages, lecture notes for 2010 Summer School on Diversities in Quantum Computation/Information at Kinki Universit

The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer

A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and `hidden' or `unknown' subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian `hidden subgroup problem' can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or `flying qubits', instead of entire registers of control qubits.Comment: 16 pages, 3 figures, LaTeX2e, to appear in Proceedings of the 1st NASA International Conference on Quantum Computing and Quantum Communication (Springer-Verlag

Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.Comment: 10 page

Revisiting Shor's quantum algorithm for computing general discrete logarithms

We heuristically demonstrate that Shor's algorithm for computing general discrete logarithms, modified to allow the semi-classical Fourier transform to be used with control qubit recycling, achieves a success probability of approximately 60% to 82% in a single run. By slightly increasing the number of group operations that are evaluated quantumly, and by performing a limited search in the classical post-processing, we furthermore show how the algorithm can be modified to achieve a success probability exceeding 99% in a single run. We provide concrete heuristic estimates of the success probability of the modified algorithm, as a function of the group order, the size of the search space in the classical post-processing, and the additional number of group operations evaluated quantumly. In analogy with our earlier works, we show how the modified quantum algorithm may be simulated classically when the logarithm and group order are both known. Furthermore, we show how slightly better tradeoffs may be achieved, compared to our earlier works, if the group order is known when computing the logarithm.Comment: The pre-print has been extended to show how slightly better tradeoffs may be achieved, compared to our earlier works, if the group order is known. A minor issue with an integration limit, that lead us to give a rough success probability estimate of 60% to 70%, as opposed to 60% to 82%, has been corrected. The heuristic and results reported in the original pre-print are otherwise unaffecte

New Developments in Quantum Algorithms

In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201