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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… (More)

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G ∼ = H. For several decades, the n log p n+O(1) generator-enumeration bound (where p is the smallest prime dividing the order of the group) has been the best worst-case result for general groups. In this work, we show the first… (More)

In this work, we introduce bidirectional collision detection — a new algorithmic tool that applies to the collision problems that arise in many isomorphism problems. For the group isomorphism problem, we show that bidirectional collision detection yields a deterministic n (1/2) log n+O(1) time algorithm whereas previously the n log n+O(1)… (More)

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G ∼ = H. 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. Following a framework due to… (More)

We consider the isomorphism problem for groups specified by their multiplication tables. Until recently, the best published bound for the worst-case was achieved by the n log p n+O(1) generator-enumeration algorithm. In previous work with Fabian Wagner, we showed an n (1/2) log p n+O(log n/ log log n) time algorithm for testing isomorphism of p-groups by… (More)

We consider a generalization of the standard oracle model in which the oracle acts on the target with a permutation which is selected according to internal random coins. We show new exponential quantum speedups which may be obtained over classical algorithms in this oracle model. Even stronger, we describe several problems which are impossible to solve… (More)

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… (More)

This paper presents a new quantum array that can be used to control a single-qudit hermitian operator for an odd radix r > 2 by n controls using Θ n log 2 r+2 single-qudit controlled gates with one control and no an-cilla qudits. This quantum array is more practical than existing quantum arrays of the same complexity because it does not require the use of… (More)

In this paper, we prove results on the relationship between the complexity of the group and color isomorphism problems. The difficulty of color isomorphism problems is known to be closely linked to the the composition factors of the permutation group involved. Previous works are primarily concerned with applying color isomorphism to bounded degree graph… (More)