Miklos Santha

Quantum Algorithms for the Triangle Problem

We present two new quantum algorithms that either find a triangle (a copy of <i>K</i><inf>3</inf>) in an undirected graph <i>G</i> on <i>n</i> nodes, or reject if <i>G</i> is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes <i>Õ</i>(<i>n</i>^10/7) queries. The second algorithm uses <i>Õ</i>(<i>n</i>^13/10) queries, and it is based on a new design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only <i>O</i>(log <i>n</i>) qubits in its quantum subroutines, whereas the second one uses <i>O</i>(<i>n</i>) qubits. The Triangle Problem was first treated in [12], where an algorithm with <i>O</i>(<i>n</i> + √n|<i>E</i>|) query complexity was presented (here |<i>E</i>| is the number of edges of <i>G</i>).

Hidden translation and orbit coset in quantum computing

We give efficient quantum algorithms for the problems of <sc>Hidden Translation</sc> and <sc>Hidden Subgroup</sc> in a large class of non-abelian groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently <sc>Hidden Translation</sc> in Z <inf>p</inf>ⁿ, whenever <i>p</i> is a fixed prime. For the induction step, we introduce the problem <sc>Orbit Coset</sc> generalizing both <sc>Hidden Translation</sc> and <sc>Hidden Subgroup</sc>, and prove a powerful self-reducibility result: <sc>Orbit Coset</sc> in a finite group <i>G</i> is reducible to <sc>Orbit Coset</sc> in <i>G/N</i> and subgroups of <i>N</i>, for any solvable normal subgroup <i>N</i> of <i>G</i>.

Oblivious transfers and intersecting codes

Assume A owns t secret k-bit strings. She is willing to disclose one of them to B, at his choosing, provided he does not learn anything about the other strings. Conversely, B does not want A to learn which secret he chose to learn. A protocol for the above task is said to implement one-out-of-t string oblivious transfer, denoted (/sup t//sub 1/)-OT/sup k//sub 2/. This primitive is particularly useful in a variety of cryptographic settings. An apparently simpler task corresponds to the case k=1 and t=2 of two 1-bit secrets: this is known as one-out-of-two bit oblivious transfer, denoted (/sup 2//sub 1/)-OT/sub 2/. We address the question of implementing (/sup t//sub 1/)-OT/sup k//sub 2/ assuming the existence of a (/sup 2//sub 1/)-OT/sub 2/. In particular, we prove that unconditionally secure (/sup 2//sub 1/)-OT/sup k//sub 2/ can be implemented from /spl Theta/(k) calls to (/sup 2//sub 1/)-OT/sub 2/. This is optimal up to a small multiplicative constant. Our solution is based on the notion of self-intersecting codes. Of independent interest, we give several efficient new constructions for such codes. Another contribution of this paper is a set of information-theoretic definitions for correctness and privacy of unconditionally secure oblivious transfer.

Quantum algorithms for element distinctness

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp (1998), and imply an O(N/sup 3/4/ log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with /spl Theta/(N log N) classical complexity. We also prove a lower bound of /spl Omega/(/spl radic/N) comparisons for this problem and derive bounds for a number of related problems.

Deciding bisimilarity isP-complete

In finite labelled transition systems the problems of deciding strong bisimilarity, observation equivalence and observation congruence areP-complete under many—oneNC-reducibility. As a consequence, algorithms for automated analysis of finite state systems based on bisimulation seem to be inherently sequential in the following sense: the design of anNC algorithm to solve any of these problems will require an algorithmic breakthrough, which is exceedingly hard to achieve.

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.

On the hitting times of quantum versus random walks

The hitting time of a classical random walk (Markov chain) is the time required to detect the presence of—or equivalently, to find—a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the detection and finding problems have the same time complexity. In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case.Then, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem (Tulsi A.: Phys. Rev. A 78:012310 2008) for the 2D grid. Extending his result, we show that we can find a unique marked element with constant probability and with the same complexity as detection for a large class of quantum walks—the quantum analogue of state-transitive reversible ergodic Markov chains.Further, we prove that for any reversible ergodic Markov chain P, the quantum hitting time of the quantum analogue of P has the same order as the square root of the classical hitting time of P. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. In doing so, we define a notion of reversibility for a broad class of quantum walks and show how to derive from any such quantum walk a classical analogue. For the special case of quantum walks built on reflections, we show that the hitting time of the classical analogue is exactly the square of the quantum walk.

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.

Improved quantum query algorithms for triangle finding and associativity testing

We show that the quantum query complexity of detecting if an n-vertex graph contains a triangle is