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Andris Ambainis

Publications and source records attributed to Andris Ambainis.

At least 109 records · Page 6Linked to original sources
arXiv2005

Quantum Algorithms for Matching and Network Flows

We present quantum algorithms for the following graph problems: finding a maximal bipartite matching in time O(n sqrt{m+n} log n), finding a maximal non-bipartite matching in time O(n^2 (sqrt{m/n} + log n) log n), and finding a maximal flow in an integer network in time O(min(n^{7/6} sqrt m * U^{1/3}, sqrt{n U} m) log n), where n is the number of vertices, m is the number of edges, and U <= n^{1/4} is an upper bound on the capacity of an edge.

quant-ph↗
arXiv2005

A new quantum lower bound method, with an application to strong direct product theorem for quantum search

We present a new method for proving lower bounds on quantum query algorithms. The new method is an extension of adversary method, by analyzing the eigenspace structure of the problem. Using the new method, we prove a strong direct product theorem for quantum search. This result was previously proven by Klauck, Spalek and de Wolf (quant-ph/0402123) using polynomials method. No proof using adversary method was known before.

quant-ph↗
arXiv2005

Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range

We give a general method for proving quantum lower bounds for problems with small range. Namely, we show that, for any symmetric problem defined on functions $f:\{1, ..., N\}\to\{1, ..., M\}$, its polynomial degree is the same for all $M\geq N$. Therefore, if we have a quantum lower bound for some (possibly, quite large) range $M$ which is shown using polynomials method, we immediately get the same lower bound for all ranges $M\geq N$. In particular, we get $Ω(N^{1/3})$ and $Ω(N^{2/3})$ quantum lower bounds for collision and element distinctness with small range.

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arXiv2005

Quantum search algorithms

We review some of quantum algorithms for search problems: Grover's search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks.

quant-ph↗
arXiv2005

Probabilistic and Team PFIN-type Learning: General Properties

We consider the probability hierarchy for Popperian FINite learning and study the general properties of this hierarchy. We prove that the probability hierarchy is decidable, i.e. there exists an algorithm that receives p_1 and p_2 and answers whether PFIN-type learning with the probability of success p_1 is equivalent to PFIN-type learning with the probability of success p_2. To prove our result, we analyze the topological structure of the probability hierarchy. We prove that it is well-ordered in descending ordering and order-equivalent to ordinal epsilon_0. This shows that the structure of the hierarchy is very complicated. Using similar methods, we also prove that, for PFIN-type learning, team learning and probabilistic learning are of the same power.

cs.LG↗
arXiv2005

Quantum Search of Spatial Regions

Can Grover's algorithm speed up search of a physical region - for example a 2-D grid of size sqrt(n) by sqrt(n)? The problem is that sqrt(n) time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a d-dimensional hypercube in time O(sqrt n) for d at least 3, or O((sqrt n)(log n)^(3/2)) for d=2. More generally, we introduce a model of quantum query complexity on graphs, motivated by fundamental physical limits on information storage, particularly the holographic principle from black hole thermodynamics. Our results in this model include almost-tight upper and lower bounds for many search tasks; a generalized algorithm that works for any graph with good expansion properties, not just hypercubes; and relationships among several notions of `locality' for unitary matrices acting on graphs. As an application of our results, we give an O(sqrt(n))-qubit communication protocol for the disjointness problem, which improves an upper bound of Hoyer and de Wolf and matches a lower bound of Razborov.

quant-ph↗
arXiv2004

Polynomial degree vs. quantum query complexity

The degree of a polynomial representing (or approximating) a function f is a lower bound for the number of quantum queries needed to compute f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity Ω(M^{1.321...}). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a new, more general version of quantum adversary method.

quant-ph↗
arXiv2004

Multiparty Quantum Coin Flipping

We investigate coin-flipping protocols for multiple parties in a quantum broadcast setting: (1) We propose and motivate a definition for quantum broadcast. Our model of quantum broadcast channel is new. (2) We discovered that quantum broadcast is essentially a combination of pairwise quantum channels and a classical broadcast channel. This is a somewhat surprising conclusion, but helps us in both our lower and upper bounds. (3) We provide tight upper and lower bounds on the optimal bias epsilon of a coin which can be flipped by k parties of which exactly g parties are honest: for any 1 <= g <= k, epsilon = 1/2 - Theta(g/k). Thus, as long as a constant fraction of the players are honest, they can prevent the coin from being fixed with at least a constant probability. This result stands in sharp contrast with the classical setting, where no non-trivial coin-flipping is possible when g <= k/2.

quant-ph↗
arXiv2004

Small Pseudo-Random Families of Matrices: Derandomizing Approximate Quantum Encryption

A quantum encryption scheme (also called private quantum channel, or state randomization protocol) is a one-time pad for quantum messages. If two parties share a classical random string, one of them can transmit a quantum state to the other so that an eavesdropper gets little or no information about the state being transmitted. Perfect encryption schemes leak no information at all about the message. Approximate encryption schemes leak a non-zero (though small) amount of information but require a shorter shared random key. Approximate schemes with short keys have been shown to have a number of applications in quantum cryptography and information theory. This paper provides the first deterministic, polynomial-time constructions of quantum approximate encryption schemes with short keys. Previous constructions (quant-ph/0307104) are probabilistic--that is, they show that if the operators used for encryption are chosen at random, then with high probability the resulting protocol will be a secure encryption scheme. Moreover, the resulting protocol descriptions are exponentially long. Our protocols use keys of the same length as (or better length than) the probabilistic constructions; to encrypt $n$ qubits approximately, one needs $n+o(n)$ bits of shared key. An additional contribution of this paper is a connection between classical combinatorial derandomization and constructions of pseudo-random matrix families in a continuous space.

quant-ph↗
arXiv2004

Quantum Identification of Boolean Oracles

The oracle identification problem (OIP) is, given a set $S$ of $M$ Boolean oracles out of $2^{N}$ ones, to determine which oracle in $S$ is the current black-box oracle. We can exploit the information that candidates of the current oracle is restricted to $S$. The OIP contains several concrete problems such as the original Grover search and the Bernstein-Vazirani problem. Our interest is in the quantum query complexity, for which we present several upper and lower bounds. They are quite general and mostly optimal: (i) The query complexity of OIP is $O(\sqrt{N\log M \log N}\log\log M)$ for {\it any} $S$ such that $M = |S| > N$, which is better than the obvious bound $N$ if $M < 2^{N/\log^{3}N}$. (ii) It is $O(\sqrt{N})$ for {\it any} $S$ if $|S| = N$, which includes the upper bound for the Grover search as a special case. (iii) For a wide range of oracles ($|S| = N$) such as random oracles and balanced oracles, the query complexity is $Θ(\sqrt{N/K})$, where $K$ is a simple parameter determined by $S$.

quant-ph↗
arXiv2004

Coins Make Quantum Walks Faster

We show how to search N items arranged on a $\sqrt{N}\times\sqrt{N}$ grid in time $O(\sqrt N \log N)$, using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time $Ω(N)$ to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of $O(\sqrt{N})$ and give several extensions of quantum walk search algorithms for general graphs. The coin-flip operation needs to be chosen judiciously: we show that another ``natural'' choice of coin gives a walk that takes $Ω(N)$ steps. We also show that in 2 dimensions it is sufficient to have a two-dimensional coin-space to achieve the time $O(\sqrt{N} \log N)$.

quant-ph↗
arXiv2003

Cryptographic Randomized Response Techniques

We develop cryptographically secure techniques to guarantee unconditional privacy for respondents to polls. Our constructions are efficient and practical, and are shown not to allow cheating respondents to affect the ``tally'' by more than their own vote -- which will be given the exact same weight as that of other respondents. We demonstrate solutions to this problem based on both traditional cryptographic techniques and quantum cryptography.

cs.CC↗
arXiv2003

Distributed construction of quantum fingerprints

Quantum fingerprints are useful quantum encodings introduced by Buhrman, Cleve, Watrous, and de Wolf (Physical Review Letters, Volume 87, Number 16, Article 167902, 2001; quant-ph/0102001) in obtaining an efficient quantum communication protocol. We design a protocol for constructing the fingerprint in a distributed scenario. As an application, this protocol gives rise to a communication protocol more efficient than the best known classical protocol for a communication problem.

quant-ph↗
arXiv2003

Quantum random walks with decoherent coins

The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.

quant-ph↗
arXiv2002

Quantum Walks driven by many coins

Quantum random walks have been much studied recently, largely due to their highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum random walk on the line: the use of multiple quantum ``coins'' in order to diminish the effects of interference between paths. We find solutions to this system in terms of the single coin random walk, and compare the asymptotic limit of these solutions to numerical simulations. We find exact analytical expressions for the time-dependence of the first two moments, and show that in the long time limit the ``quantum mechanical'' behavior of the one-coin walk persists. We further show that this is generic for a very broad class of possible walks, and that this behavior disappears only in the limit of a new coin for every step of the walk.

quant-ph↗
arXiv2002

The quantum to classical transition for random walks

We look at two possible routes to classical behavior for the discrete quantum random walk on the line: decoherence in the quantum ``coin'' which drives the walk, or the use of higher-dimensional coins to dilute the effects of interference. We use the position variance as an indicator of classical behavior, and find analytical expressions for this in the long-time limit; we see that the multicoin walk retains the ``quantum'' quadratic growth of the variance except in the limit of a new coin for every step, while the walk with decoherence exhibits ``classical'' linear growth of the variance even for weak decoherence.

quant-ph↗
arXiv2002

Towards the Classical Communication Complexity of Entanglement Distillation Protocols with Incomplete Information

Quantum entanglement distillation protocols are LOCC protocols between Alice and Bob that convert imperfect EPR pairs, or, in general, partially entangled bipartite states into perfect or near-perfect EPR pairs. The classical communication complexity of these protocols is the minimal amount of classical communication needed for the conversion. In this paper, we focus on the communication complexity of protocols that operate with incomplete information, i.e., where the inputs are mixed states and/or prepared adversarially. We study 3 models of imperfect EPR pairs. In the measure-r model, r out of n EPR pairs are measured by an adversary; in the depolarization model, Bob's share of qubits underwent a depolarization channel; in the fidelity model, the only information Alice and Bob possess is the fidelity of the shared state. For the measure-r model and the depolarization model, we prove tight and almost-tight bounds on the outcome of LOCC protocols that don't use communication. For the fidelity model, we prove a lower bound on the communication complexity that matches the upper bound given by Ambainis, Smith, and Yang [ASY02].

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