arXiv subjects

David Rosenbaum

Publications and source records attributed to David Rosenbaum.

3 recordsLinked to original sources

arXiv2013

Breaking the n^(log n) Barrier for Solvable-Group Isomorphism

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G and H are isomorphic. The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the special cases of p-groups and solvable groups --- ever since the n^(log n + O(1)) generator-enumeration algorithm. In this work, we present the first significant improvement over n^(log n) by showing that group isomorphism is n^((1 / 2) log_p n + O(1)) Turing reducible to composition-series isomorphism where p is the smallest prime dividing the order of the group. Combining our reduction with an n^(O(p / log p)) algorithm for p-group composition-series isomorphism, we obtain an n^((1 / 2) log n + O(1)) algorithm for p-group isomorphism. We then generalize our techniques from p-groups using Sylow bases to derive an n^((1 / 2) log n + O(log n / log log n)) algorithm for solvable-group isomorphism. Finally, we relate group isomorphism to the collision problem which allows us replace the 1 / 2 in the exponents with 1 / 4 using randomized algorithms and 1 / 6 using quantum algorithms.

cs.DS↗

arXiv2013

Optimal Quantum Circuits for Nearest-Neighbor Architectures

We show that the depth of quantum circuits in the realistic architecture where a classical controller determines which local interactions to apply on the kD grid Z^k where k >= 2 is the same (up to a constant factor) as in the standard model where arbitrary interactions are allowed. This allows minimum-depth circuits (up to a constant factor) for the nearest-neighbor architecture to be obtained from minimum-depth circuits in the standard abstract model. Our work therefore justifies the standard assumption that interactions can be performed between arbitrary pairs of qubits. In particular, our results imply that Shor's algorithm, controlled operations and fanouts can be implemented in constant depth, polynomial size and polynomial width in this architecture. We also present optimal non-adaptive quantum circuits for controlled operations and fanouts on a kD grid. These circuits have depth Theta(n^(1 / k)), size Theta(n) and width Theta(n). Our lower bound also applies to a more general class of operations.

quant-ph↗

arXiv2011

Quantum Algorithms for Tree Isomorphism and State Symmetrization

The graph isomorphism problem is theoretically interesting and also has many practical applications. The best known classical algorithms for graph isomorphism all run in time super-polynomial in the size of the graph in the worst case. An interesting open problem is whether quantum computers can solve the graph isomorphism problem in polynomial time. In this paper, an algorithm is shown which can decide if two rooted trees are isomorphic in polynomial time. Although this problem is easy to solve efficiently on a classical computer, the techniques developed may be useful as a basis for quantum algorithms for deciding isomorphism of more interesting types of graphs. The related problem of quantum state symmetrization is also studied. A polynomial time algorithm for the problem of symmetrizing a set of orthonormal states over an arbitrary permutation group is shown.

quant-ph↗