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Quantum Information and Computation

Decomposing finite Abelian groups

Ka Chun Cheung, Michele Mosca

Published Oct 1, 2001• 16 citations
Publisher

Abstract

This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups into a product of cyclic groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann Hypothesis) also leads to an efficient algorithm for computing class numbers (known to be at least as difficult as factoring).

DOI
10.26421/qic1.3-2

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This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups into a product of cyclic groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann Hypothesis) also leads to an efficient algorithm for computing class numbers (known to be at least as difficult as factoring).
Decomposing finite Abelian black-box groups into cyclic factors is a basic problem in quantum computation. We give a quantum algorithm for decomposing finite Abelian black-box groups by adapting the quantum sampling and classical lattice-reduction method of Regev's factoring algorithm. The algorithm computes an invariant-factor decomposition and corresponding cyclic generators with high probability. For a group G of order at most 2^n with unique encodings and reversible group-operation cost T_op = Omega(sqrt(n)), our algorithm uses O(sqrt(n)) quantum circuits, each with O~(n T_op) gates and executed at most O(n) times. For comparison, we consider the Cheung-Mosca algorithm, which decomposes the Sylow p-subgroups separately. We also consider the common-modulus implementation of the extended Cheung-Mosca algorithm. Our algorithm reduces the sum of the circuit gate counts from O~(n^2 T_op) to O~(n^(3/2) T_op). The total quantum time bound reduces from O~(n^3 T_op) to O~(n^(5/2) T_op), and the quantum space bound reduces from O(n^2) to O(n) qubits. The classical computation uses polynomially many bit operations and group-operation queries. These improvements rely on two technical ingredients. We prove that the integer relation lattices used in our algorithm admit, with high probability, integral bases whose vectors have Euclidean norm at most exp(O(sqrt(n))). We preserve the circuit-size advantage of Regev's algorithm for group decomposition by incorporating all O(n) sampled generators while keeping the lattice-reduction dimension at O(sqrt(n)).
The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of abelian algorithms of the Shor/Simon genre. Our strategy is to make this class of algorithms as mathematically transparent as possible. By the phrase "mathematically transparent" we mean to expose, to bring to the surface, and to make explicit the concealed mathematical structures that are inherently and fundamentally a part of such algorithms. In so doing, we create symbolic abelian quantum hidden subgroup algorithms that are analogous to the those symbolic algorithms found within such software packages as Axiom, Cayley, Maple, Mathematica, and Magma. As a spin-off of this effort, we create three different generalizations of Shor's quantum factoring algorithm to free abelian groups of finite rank. We refer to these algorithms as wandering (or vintage Z_Q) Shor algorithms. They are essentially quantum algorithms on free abelian groups A of finite rank n which, with each iteration, first select a random cyclic direct summand Z of the group A and then apply one iteration of the standard Shor algorithm to produce a random character of the "approximating" finite group A=Z_Q, called the group probe. These characters are then in turn used to find either the order P of a maximal cyclic subgroup Z_P of the hidden quotient group H_phi, or the entire hidden quotient group H_phi. An integral part of these wandering quantum algorithms is the selection of a very special random transversal, which we refer to as a Shor transversal. The algorithmic time complexity of the first of these wandering Shor algorithms is found to be O(n^2(lgQ)^3(lglgQ)^(n+1)).
The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism.