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Andris Ambainis

Publications and source records attributed to Andris Ambainis.

At least 73 records · Page 4Linked to original sources
arXiv2014

Quantum walk algorithm for element distinctness

We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N^{2/3}) query quantum algorithm. This improves the previous O(N^{3/4}) query quantum algorithm of Buhrman et.al. (quant-ph/0007016) and matches the lower bound by Shi (quant-ph/0112086). The algorithm also solves the generalization of element distinctness in which we have to find k equal items among N items. For this problem, we get an O(N^{k/(k+1)}) query quantum algorithm.

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arXiv2014

On physical problems that are slightly more difficult than QMA

We study the complexity of computational problems from quantum physics. Typically, they are studied using the complexity class QMA (quantum counterpart of NP) but some natural computational problems appear to be slightly harder than QMA. We introduce new complexity classes consisting of problems that are solvable with a small number of queries to a QMA oracle and use these complexity classes to quantify the complexity of several natural computational problems (for example, the complexity of estimating the spectral gap of a Hamiltonian).

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arXiv2014

The Need for Structure in Quantum Speedups

Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate. First, we show that for any problem that is invariant under permuting inputs and outputs (like the collision or the element distinctness problems), the quantum query complexity is at least the 7th root of the classical randomized query complexity. (An earlier version of this paper gave the 9th root.) This resolves a conjecture of Watrous from 2002. Second, inspired by recent work of O'Donnell et al. (2005) and Dinur et al. (2006), we conjecture that every bounded low-degree polynomial has a "highly influential" variable. Assuming this conjecture, we show that every T-query quantum algorithm can be simulated on most inputs by a poly(T)-query classical algorithm, and that one essentially cannot hope to prove P!=BQP relative to a random oracle.

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arXiv2013

Spatial Search on Grids with Minimum Memory

We study quantum algorithms for spatial search on finite dimensional grids. Patel et al. and Falk have proposed algorithms based on a quantum walk without a coin, with different operators applied at even and odd steps. Until now, such algorithms have been studied only using numerical simulations. In this paper, we present the first rigorous analysis for an algorithm of this type, showing that the optimal number of steps is $O(\sqrt{N\log N})$ and the success probability is $O(1/\log N)$, where $N$ is the number of vertices. This matches the performance achieved by algorithms that use other forms of quantum walks.

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arXiv2013

Weak Parity

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in the quantum case for any eps>0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.

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arXiv2013

New upper bound on block sensitivity and certificate complexity in terms of sensitivity

Sensitivity \cite{CD82,CDR86} and block sensitivity \cite{Nisan91} are two important complexity measures of Boolean functions. A longstanding open problem in decision tree complexity, the "Sensitivity versus Block Sensitivity" question, proposed by Nisan and Szegedy \cite{Nisan94} in 1992, is whether these two complexity measures are polynomially related, i.e., whether $bs(f)=O(s(f)^{O(1)})$. We prove an new upper bound on block sensitivity in terms of sensitivity: $bs(f) \leq 2^{s(f)-1} s(f)$. Previously, the best upper bound on block sensitivity was $bs(f) \leq (\frac{e}{\sqrt{2π}}) e^{s(f)} \sqrt{s(f)}$ by Kenyon and Kutin \cite{KK}. We also prove that if $\min\{s_0(f),s_1(f)\}$ is a constant, then sensitivity and block sensitivity are linearly related, i.e. $bs(f)=O(s(f))$.

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arXiv2013

Quantum algorithms for search with wildcards and combinatorial group testing

We consider two combinatorial problems. The first we call "search with wildcards": given an unknown n-bit string x, and the ability to check whether any subset of the bits of x is equal to a provided query string, the goal is to output x. We give a nearly optimal O(sqrt(n) log n) quantum query algorithm for search with wildcards, beating the classical lower bound of Omega(n) queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most k special items out of a set of n items, given the ability to make queries of the form "does the set S contain any special items?" for any subset S of the n items. We give a simple quantum algorithm which uses O(k log k) queries to solve this problem, as compared with the classical lower bound of Omega(k log(n/k)) queries.

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arXiv2013

Parameterized Quantum Query Complexity of Graph Collision

We present three new quantum algorithms in the quantum query model for \textsc{graph-collision} problem: \begin{itemize} \item an algorithm based on tree decomposition that uses $O\left(\sqrt{n}t^{\sfrac{1}{6}}\right)$ queries where $t$ is the treewidth of the graph; \item an algorithm constructed on a span program that improves a result by Gavinsky and Ito. The algorithm uses $O(\sqrt{n}+\sqrt{α^{**}})$ queries, where $α^{**}(G)$ is a graph parameter defined by \[α^{**}(G):=\min_{VC\text{-- vertex cover of}G}{\max_{\substack{I\subseteq VC\\I\text{-- independent set}}}{\sum_{v\in I}{°{v}}}};\] \item an algorithm for a subclass of circulant graphs that uses $O(\sqrt{n})$ queries. \end{itemize} We also present an example of a possibly difficult graph $G$ for which all the known graphs fail to solve graph collision in $O(\sqrt{n} \log^c n)$ queries.

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arXiv2013

How Low Can Approximate Degree and Quantum Query Complexity be for Total Boolean Functions?

It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Omega(log n), and that this bound is achieved for some functions. In this paper we study the case of approximate degree and bounded-error quantum query complexity. We show that for these measures the correct lower bound is Omega(log n / loglog n), and we exhibit quantum algorithms for two functions where this bound is achieved.

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arXiv2013

Provable Advantage for Quantum Strategies in Random Symmetric XOR Games

Non-local games are widely studied as a model to investigate the properties of quantum mechanics as opposed to classical mechanics. In this paper, we consider a subset of non-local games: symmetric XOR games of $n$ players with 0-1 valued questions. For this class of games, each player receives an input bit and responds with an output bit without communicating to the other players. The winning condition only depends on XOR of output bits and is constant w.r.t. permutation of players. We prove that for almost any $n$-player symmetric XOR game the entangled value of the game is $Θ(\frac{\sqrt{\ln{n}}}{n^{1/4}})$ adapting an old result by Salem and Zygmund on the asymptotics of random trigonometric polynomials. Consequently, we show that the classical-quantum gap is $Θ(\sqrt{\ln{n}})$ for almost any symmetric XOR game.

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arXiv2013

Exact quantum query complexity of EXACT and THRESHOLD

A quantum algorithm is exact if it always produces the correct answer, on any input. Coming up with exact quantum algorithms that substantially outperform the best classical algorithm has been a quite challenging task. In this paper, we present two new exact quantum algorithms for natural problems: 1) for the problem EXACT_k^n in which we have to determine whether the sequence of input bits x_1, ..., x_n contains exactly k values x_i=1; 2) for the problem THRESHOLD_k^n in which we have to determine if at least k of n input bits are equal to 1.

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arXiv2012

Optimal quantum query bounds for almost all Boolean functions

We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis, and shows that van Dam's oracle interrogation is essentially optimal for almost all functions. Our proof uses the fact that the acceptance probability of a T-query algorithm can be written as the sum of squares of degree-T polynomials.

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arXiv2011

Quantum strategies are better than classical in almost any XOR game

We initiate a study of random instances of nonlocal games. We show that quantum strategies are better than classical for almost any 2-player XOR game. More precisely, for large n, the entangled value of a random 2-player XOR game with n questions to every player is at least 1.21... times the classical value, for 1-o(1) fraction of all 2-player XOR games.

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arXiv2011

Search by quantum walks on two-dimensional grid without amplitude amplification

We study search by quantum walk on a finite two dimensional grid. The algorithm of Ambainis, Kempe, Rivosh (quant-ph/0402107) takes O(\sqrt{N log N}) steps and finds a marked location with probability O(1/log N) for grid of size \sqrt{N} * \sqrt{N}. This probability is small, thus amplitude amplification is needed to achieve Θ(1) success probability. The amplitude amplification adds an additional O(\sqrt{log N}) factor to the number of steps, making it O(\sqrt{N} log N). In this paper, we show that despite a small probability to find a marked location, the probability to be within an O(\sqrt{N}) neighbourhood (at an O(\sqrt[4]{N}) distance) of the marked location is Θ(1). This allows to skip amplitude amplification step and leads to an O(\sqrt{log N}) speed-up. We describe the results of numerical experiments supporting this idea, and we prove this fact analytically.

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arXiv2011

Worst case analysis of non-local games

Non-local games are studied in quantum information because they provide a simple way for proving the difference between the classical world and the quantum world. A non-local game is a cooperative game played by 2 or more players against a referee. The players cannot communicate but may share common random bits or a common quantum state. A referee sends an input $x_i$ to the $i^{th}$ player who then responds by sending an answer $a_i$ to the referee. The players win if the answers $a_i$ satisfy a condition that may depend on the inputs $x_i$. Typically, non-local games are studied in a framework where the referee picks the inputs from a known probability distribution. We initiate the study of non-local games in a worst-case scenario when the referee's probability distribution is unknown and study several non-local games in this scenario.

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arXiv2011

Superiority of exact quantum automata for promise problems

In this note, we present an infinite family of promise problems which can be solved exactly by just tuning transition amplitudes of a two-state quantum finite automata operating in realtime mode, whereas the size of the corresponding classical automata grow without bound.

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arXiv2011

Quantum property testing for bounded-degree graphs

We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion, we also prove an Omega(N^(1/4)) quantum query lower bound, thus ruling out the possibility of an exponential quantum speedup. Our quantum algorithms follow from a combination of classical property testing techniques due to Goldreich and Ron, derandomization, and the quantum algorithm for element distinctness. The quantum lower bound is obtained by the polynomial method, using novel algebraic techniques and combinatorial analysis to accommodate the graph structure.

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