1,354,323 research outputs found
Optimal quantum query bounds for almost all Boolean functions
No full textWe 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 (A. Ambainis, 1999), and shows that van Dam's oracle interrogation (W. van Dam, 1998) 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
Polynomials, Quantum Query Complexity, and Grothendieck's Inequality
No full textWe show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function f is computable by a 1-query quantum algorithm with error bounded by epsilon<1/2 iff f can be approximated by a degree-2 polynomial with error bounded by epsilon'<1/2. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis [Aaronson/Ambainis, STOC 2015]. The proof uses Grothendieck's inequality to relate two matrix norms, with one norm corresponding to polynomial approximations and the other norm corresponding to quantum algorithms.
We also show two results for polynomials of higher degree. First, there is a total Boolean function which requires ~Omega(n) quantum queries but can be represented by a block-multilinear polynomial of degree ~O(sqrt(n)). Thus, in the general case (for an arbitrary number of queries), block-multilinear polynomials are not equivalent to quantum algorithms.
Second, for any constant degree k, the two notions of approximation by a polynomial (the standard and the block-multilinear) are equivalent. As a consequence, we solve an open problem from [Aaronson/Ambainis, STOC 2015], showing that one can estimate the value of any bounded degree-k polynomial p:{0,1}^n -> [-1,1] with O(n^{1-1/(2k)) queries
On the power of Ambainis lower bounds
No full textAbstractThe polynomial method and the Ambainis lower bound (or Alb, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of Alb's. We first use known Alb's to derive Ω(n1.5) lower bounds for BIPARTITENESS, BIPARTITENESS MATCHING and GRAPH MATCHING, in which the lower bound for BIPARTITENESS improves the previous Ω(n) one. We then show that all the three known Ambainis lower bounds have a limitation Nmin{C0(f),C1(f)}, where C0(f) and C1(f) are the 0- and 1-certificate complexities, respectively. This implies that for many problems such as TRIANGLE, k-CLIQUE, BIPARTITENESS and BIPARTITE/GRAPH MATCHING which draw wide interest and whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis techniques. Another consequence is that all the Ambainis lower bounds are not tight. For total functions, this upper bound for Alb's can be further improved to min{C0(f)C1(f),N·CI(f)}, where CI(f) is the size of max intersection of a 0- and a 1-certificate set. Again this implies that Alb's cannot improve the best known lower bound for some specific problems such as AND-OR TREE, whose precise quantum query complexity is still open. Finally, we generalize the three known Alb's and give a new Alb style lower bound method, which may be easier to use for some problems
Variable time amplitude amplification and quantum algorithms for linear algebra problems
No full textQuantum amplitude amplification is a method of increasing a success probability of an algorithm from a small epsilon>0 to Theta(1) with less repetitions than classically. In this paper, we generalize quantum amplitude amplification to the case when parts of the algorithm that is being amplified stop at different times.
We then apply the new variable time amplitude amplification to give two new quantum algorithms for linear algebra problems. Our first algorithm is an improvement of Harrow et al. algorithm for solving systems of linear equations. We improve the running time of the algorithm from O(k^2 log N) to O(k log^3 k log N) where k is the condition number of the system of equations. Our second algorithm tests whether a matrix A is singular or far from singular, faster then the previously known algorithms
Quantum search with variable times
No full textSince Grover's seminal work, quantum search has been studied in
great detail. In the usual search problem, we have a collection of
items and we would like to find .
We consider a new variant of this problem in which evaluating
for different may take a different number of time steps.
Let be the number of time steps required to evaluate .
If the numbers are known in advance, we give an algorithm
that solves the problem in O(sqrt{t_1^2+t_2^2+ldots+t_n^2)
steps. This is optimal, as we also show a matching lower bound.
The case, when are not known in advance, can be solved with a
polylogarithmic overhead. We also give an application of our new
search algorithm to computing read-once functions
Ambainis-Freivalds’ Algorithm for Measure-Once Automata
No full textAn algorithm given by Ambainis and Freivalds [1] constructs a quantum finite automaton (QFA) withO(logp) states recognizing the language L p = a i |i is divisible by p with probability 1 − ε, for any ε > 0 and arbitrary prime p. In [4] we gave examples showing that the algorithm is applicable also to quantum automata of very limited size. However, the Ambainis-Freivalds algoritm is tailored to constructing a measure-many QFA (defined by Kondacs and Watrous [2]), which cannot be implemented on existing quantum computers. In this paper we modify the algorithm to construct ameasure-once QFA of Moore and Crutchfield [3] and give examples of parameters for this automaton. We show for the language L p that a measure-once QFA can be twice as space efficient as measure-many QFA’s.</p
Quantum Algorithms for Computational Geometry Problems
No full textWe study quantum algorithms for problems in computational geometry, such as Point-On-3-Lines problem. In this problem, we are given a set of lines and we are asked to find a point that lies on at least 3 of these lines. Point-On-3-Lines and many other computational geometry problems are known to be 3Sum-Hard. That is, solving them classically requires time Ω(n^{2-o(1)}), unless there is faster algorithm for the well known 3Sum problem (in which we are given a set S of n integers and have to determine if there are a, b, c ∈ S such that a + b + c = 0). Quantumly, 3Sum can be solved in time O(n log n) using Grover’s quantum search algorithm. This leads to a question: can we solve Point-On-3-Lines and other 3Sum-Hard problems in O(n^c) time quantumly, for c<2? We answer this question affirmatively, by constructing a quantum algorithm that solves Point-On-3-Lines in time O(n^{1 + o(1)}). The algorithm combines recursive use of amplitude amplification with geometrical ideas. We show that the same ideas give O(n^{1 + o(1)}) time algorithm for many 3Sum-Hard geometrical problems
The Power and the Limits of Quantum Automata and Search Algorithms
No full textANOTĀCIJA
Kvantu skaitļošana ir nozare, kas pēta uz kvantu mehānikas likumiem balstīto
skaitļošanas modeļu īpašības. Disertācija ir veltīta kvantu skaitļošanas
algoritmiskiem aspektiem. Piedāvāti rezultāti trijos virzienos:
Kvantu galīgi automāti
Analizēta stāvokļu efektivitāte kvantu vienvirziena galīgam automātam.
Uzlabota labāka zināmā eksponenciālā atšķirība [AF98] starp
kvantu un klasiskajiem galīgajiem automātiem.
Grovera algoritma analīze
Pētīta Grovera algoritma noturība pret kļūdām. Vispārināts [RS08]
loģisko kļūdu modelis un piedāvāti vairāki jauni rezultāti.
Kvantu klejošana
Pētīta meklēšana 2D režģī izmantojot kvantu klejošanu. Paātrināts
[AKR05] kvantu klejošanas meklēšanas algoritms.
Atslēgas vārdi: Kvantu galīgi automāti, eksponenciālā atšķirība, Grovera
algoritms, noturība pret kļūdām, kvantu klejošana
LITERATŪRA
[AF98] A. Ambainis, R. Freivalds.
1-way quantum finite automata: strengths, weaknesses and generalizations.
Proceedings of the 39th IEEE Conference on Foundations of
Computer Science, 332-341, 1998.
arXiv:quant-ph/9802062v3
[AKR05] A. Ambainis, J. Kempe, A. Rivosh.
Coins make quantum walks faster.
Proceedings of SODA’05, 1099-1108, 2005.
[RS08] O. Regev, L. Schiff. Impossibility of a Quantum Speed-up with
a Faulty Oracle.
Proceedings of ICALP’2008, Lecture Notes in Computer Science,
5125:773-781, 2008.ABSTRACT
Quantum computation is the eld that investigates properties of models of
computation based on the laws of the quantum mechanics. The thesis is ded-
icated to algorithmic aspects of quantum computation and provides results
in three directions:
Quantum nite automata
We study space-eciency of one-way quantum nite automata. We
improve best known exponential separation [AF98] between quantum
and classical one-way nite automata.
Analysis of Grover's algorithm
We study fault-tolerance of Grover's algorithm. We generalize the
model of logical faults by [RS08] and present several new results.
Quantum walks
We study search by quantum walks on two-dimensional grid. We im-
prove (speed-up) quantum walk search algorithm by [AKR05].
Keywords: Quantum nite automata, exponential separation, Grover's al-
gorithm, fault-tolerance, quantum walks
BIBLIOGRAPHY
[AF98] A. Ambainis, R. Freivalds.
1-way quantum nite automata: strengths, weaknesses and gen-
eralizations.
Proceedings of the 39th IEEE Conference on Foundations of
Computer Science, 332-341, 1998.
arXiv:quant-ph/9802062v3
[AKR05] A. Ambainis, J. Kempe, A. Rivosh.
Coins make quantum walks faster.
Proceedings of SODA'05, 1099-1108, 2005.
[RS08] O. Regev, L. Schi. Impossibility of a Quantum Speed-up with
a Faulty Oracle.
Proceedings of ICALP'2008, Lecture Notes in Computer
Science, 5125:773-781, 2008
An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree
No full textWhile it is known that there is at most a polynomial separation between
quantum query complexity and the polynomial degree for total functions, the
precise relationship between the two is not clear for partial functions.
In this paper, we demonstrate an exponential separation between exact
polynomial degree and approximate quantum query complexity for a partial
Boolean function. For an unbounded alphabet size, we have a constant versus
polynomial separation.Comment: 13 pages, minimal changes with v
Provable Advantage for Quantum Strategies in Random Symmetric XOR Games
No full textNon-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 Theta((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 Theta(sqrt(ln(n))) for almost any symmetric XOR game
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