arXiv subjects

Miklos Santha

Publications and source records attributed to Miklos Santha.

At least 55 records · Page 3Linked to original sources

arXiv2015

Nonlocality and conflicting interest games

Nonlocality enables two parties to win specific games with probabilities strictly higher than allowed by any classical theory. Nevertheless, all known such examples consider games where the two parties have a common interest, since they jointly win or lose the game. The main question we ask here is whether the nonlocal feature of quantum mechanics can offer an advantage in a scenario where the two parties have conflicting interests. We answer this in the affirmative by presenting a simple conflicting interest game, where quantum strategies outperform classical ones. Moreover, we show that our game has a fair quantum equilibrium with higher payoffs for both players than in any fair classical equilibrium. Finally, we play the game using a commercial entangled photon source and demonstrate experimentally the quantum advantage.

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arXiv2014

An efficient quantum algorithm for finding hidden parabolic subgroups in the general linear group

In the theory of algebraic groups, parabolic subgroups form a crucial building block in the structural studies. In the case of general linear groups over a finite field $F_q$, given a sequence of positive integers $n_1, ..., n_k$, where $n=n_1+...+n_k$, a parabolic subgroup of parameter $(n_1, ..., n_k)$ in $GL_n(F_q)$ is a conjugate of the subgroup consisting of block lower triangular matrices where the $i$th block is of size $n_i$. Our main result is a quantum algorithm of time polynomial in $\log q$ and $n$ for solving the hidden subgroup problem in $GL_n(F_q)$, when the hidden subgroup is promised to be a parabolic subgroup. Our algorithm works with no prior knowledge of the parameter of the hidden parabolic subgroup. Prior to this work, such an efficient quantum algorithm was only known for the case $n=2$ (A. Denney, C. Moore, and A. Russell (2010), Quantum Inf. Comput., Vol. 10, pp. 282-291), and for minimal parabolic subgroups (Borel subgroups), for the case when $q$ is not much smaller than $n$ (G. Ivanyos: Quantum Inf. Comput., Vol. 12, pp. 661-669).

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arXiv2014

Generalized Wong sequences and their applications to Edmonds' problems

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n by n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matrices in B, symbolic determinant identity testing (SDIT) is the question to decide whether there exists a nonsingular matrix in B. The constructive versions of these problems are asking to find a matrix of maximum rank, respectively a nonsingular matrix, if there exists one. Our first algorithm solves the constructive SMR when B is spanned by unknown rank one matrices, answering an open question of Gurvits. Our second algorithm solves the constructive SDIT when B is spanned by triangularizable matrices, but the triangularization is not given explicitly. Both algorithms work over finite fields of size at least n+1 and over the rational numbers, and the first algorithm actually solves (the non-constructive) SMR independently from the field size. Our main tool to obtain these results is to generalize Wong sequences, a classical method to deal with pairs of matrices, to the case of pairs of matrix spaces.

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arXiv2013

Polynomial time quantum algorithms for certain bivariate hidden polynomial problems

We present a new method for solving the hidden polynomial graph problem (HPGP) which is a special case of the hidden polynomial problem (HPP). The new approach yields an efficient quantum algorithm for the bivariate HPGP even when the input consists of several level set superpositions, a more difficult version of the problem than the one where the input is given by an oracle. For constant degree, the algorithm is polylogarithmic in the size of the base field. We also apply the results to give an efficient quantum algorithm for the oracle version of the HPP for an interesting family of bivariate hidden functions. This family includes diagonal quadratic forms and elliptic curves.

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arXiv2013

Improved bounds for the randomized decision tree complexity of recursive majority

We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of $(1/2-δ) \cdot 2.57143^h$ for the two-sided-error randomized decision tree complexity of evaluating height $h$ formulae with error $δ\in [0,1/2)$. This improves the lower bound of $(1-2δ)(7/3)^h$ given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of $(1-2δ) \cdot 2.55^h$ given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most $(1.007) \cdot 2.64944^h$. The previous best known algorithm achieved complexity $(1.004) \cdot 2.65622^h$. The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel "interleaving" of two recursive algorithms.

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arXiv2012

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 $O(n^{9/7})$. This improves the previous best algorithm of Belovs making $O(n^{35/27})$ queries. For the problem of determining if an operation $\circ : S \times S \rightarrow S$ is associative, we give an algorithm making $O(|S|^{10/7})$ queries, the first improvement to the trivial $O(|S|^{3/2})$ application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work, the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.

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arXiv2012

A learning graph based quantum query algorithm for finding constant-size subgraphs

Let $H$ be a fixed $k$-vertex graph with $m$ edges and minimum degree $d >0$. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an $n$-vertex graph contains $H$ as a subgraph is $O(n^{2-2/k-t})$, where $ t = \max{\frac{k^2- 2(m+1)}{k(k+1)(m+1)}, \frac{2k - d - 3}{k(d+1)(m-d+2)}}$. The previous best algorithm of Magniez et al. had complexity $\widetilde O(n^{2-2/k})$.

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arXiv2012

New bounds on the classical and quantum communication complexity of some graph properties

We study the communication complexity of a number of graph properties where the edges of the graph $G$ are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are: * An Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication. * A deterministic upper bound of O(n^{3/2}log n) bits for deciding if a bipartite graph contains a perfect matching, and a quantum lower bound of Omega(n) for this problem. * A Theta(n^2) bound for the randomized communication complexity of deciding if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for the quantum communication complexity of this problem. The first two quantum lower bounds are obtained by exhibiting a reduction from the n-bit Inner Product problem to these graph problems, which solves an open question of Babai, Frankl and Simon. The third quantum lower bound comes from recent results about the quantum communication complexity of composed functions. We also obtain essentially tight bounds for the quantum communication complexity of a few other problems, such as deciding if G is triangle-free, or if G is bipartite, as well as computing the determinant of a distributed matrix.

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arXiv2012

Hidden Symmetry Subgroup Problems

We advocate a new approach of addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the Hidden Symmetry Subgroup Problem (HSSP), which is a generalization of the well-studied Hidden Subgroup Problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this connection, we obtain the first efficient quantum algorithm for the hidden polynomial problem for multivariate quadratic polynomials over fields of constant characteristic. We also apply the new methods to polynomial function graph problems and present an efficient quantum procedure for constant degree multivariate polynomials over any field. This result improves in several ways the currently known algorithms.

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arXiv2011

Search via Quantum Walk

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk á la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.

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arXiv2011

The complexity of approximate Nash equilibrium in congestion games with negative delays

We extend the study of the complexity of finding an $\eps$-approximate Nash equilibrium in congestion games from the case of positive delay functions to delays of arbitrary sign. We first prove that in symmetric games with $α$-bounded jump the $\eps$-Nash dynamic converges in polynomial time when all delay functions are negative, similarly to the case of positive delays. We then establish a hardness result for symmetric games with $α$-bounded jump and with arbitrary delay functions: in that case finding an $\eps$-Nash equilibrium becomes $\PLS$-complete.

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arXiv2011

On the power of a unique quantum witness

In a celebrated paper, Valiant and Vazirani raised the question of whether the difficulty of NP-complete problems was due to the wide variation of the number of witnesses of their instances. They gave a strong negative answer by showing that distinguishing between instances having zero or one witnesses is as hard as recognizing NP, under randomized reductions. We consider the same question in the quantum setting and investigate the possibility of reducing quantum witnesses in the context of the complexity class QMA, the quantum analogue of NP. The natural way to quantify the number of quantum witnesses is the dimension of the witness subspace W in some appropriate Hilbert space H. We present an efficient deterministic procedure that reduces any problem where the dimension d of W is bounded by a polynomial to a problem with a unique quantum witness. The main idea of our reduction is to consider the Alternating subspace of the d-th tensor power of H. Indeed, the intersection of this subspace with the d-th tensor power of W is one-dimensional, and therefore can play the role of the unique quantum witness.

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arXiv2010

Optimal Direct Sum Results for Deterministic and Randomized Decision Tree Complexity

A Direct Sum Theorem holds in a model of computation, when solving some k input instances together is k times as expensive as solving one. We show that Direct Sum Theorems hold in the models of deterministic and randomized decision trees for all relations. We also note that a near optimal Direct Sum Theorem holds for quantum decision trees for boolean functions.

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arXiv2008

Quantum walk based search algorithms

In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We present a rather detailed description of a somewhat simplified version of the MNRS algorithm. Finally, in the query complexity model, we show how quantum walks can be applied to the following search problems: Element Distinctness, Matrix Product Verification, Restricted Range Associativity, Triangle, and Group Commutativity.

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arXiv2008

On the hitting times of quantum versus random walks

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. 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. Finally, 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 for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems.

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arXiv2007

An efficient quantum algorithm for the hidden subgroup problem in nil-2 groups

In this paper we extend the algorithm for extraspecial groups in \cite{iss07}, and show that the hidden subgroup problem in nil-2 groups, that is in groups of nilpotency class at most 2, can be solved efficiently by a quantum procedure. The algorithm presented here has several additional features. It contains a powerful classical reduction for the hidden subgroup problem in nilpotent groups of constant nilpotency class to the specific case where the group is a $p$-group of exponent $p$ and the subgroup is either trivial or cyclic. This reduction might also be useful for dealing with groups of higher nilpotency class. The quantum part of the algorithm uses well chosen group actions based on some automorphisms of nil-2 groups. The right choice of the actions requires the solution of a system of quadratic and linear equations. The existence of a solution is guaranteed by the Chevalley-Warning theorem, and we prove that it can also be found efficiently.

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arXiv2007

An efficient quantum algorithm for the hidden subgroup problem in extraspecial groups

Extraspecial groups form a remarkable subclass of p-groups. They are also present in quantum information theory, in particular in quantum error correction. We give here a polynomial time quantum algorithm for finding hidden subgroups in extraspecial groups. Our approach is quite different from the recent algorithms presented in [17] and [2] for the Heisenberg group, the extraspecial p-group of size p3 and exponent p. Exploiting certain nice automorphisms of the extraspecial groups we define specific group actions which are used to reduce the problem to hidden subgroup instances in abelian groups that can be dealt with directly.

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arXiv2005

Quantum Algorithms for the Triangle Problem

We present two new quantum algorithms that either find a triangle (a copy of $K_{3}$) in an undirected graph $G$ on $n$ nodes, or reject if $G$ is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes $\tilde{O}(n^{10/7})$ queries. The second algorithm uses $\tilde{O}(n^{13/10})$ queries, and it is based on a design concept of Ambainis~\cite{amb04} that incorporates the benefits of quantum walks into Grover search~\cite{gro96}. The first algorithm uses only $O(\log n)$ qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in~\cite{bdhhmsw01}, where an algorithm with $O(n+\sqrt{nm})$ query complexity was presented, where $m$ is the number of edges of $G$.

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