Search results for "computational complexity."
showing 10 items of 245 documents
The Descriptive Complexity Approach to LOGCFL
1998
Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free proj…
Superiority of exact quantum automata for promise problems
2011
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.
On Block Sensitivity and Fractional Block Sensitivity
2018
We investigate the relation between the block sensitivity bs(f) and fractional block sensitivity fbs(f) complexity measures of Boolean functions. While it is known that fbs(f) = O(bs(f)2), the best known separation achieves $${\rm{fbs}}\left( f \right) = \left( {{{\left( {3\sqrt 2 } \right)}^{ - 1}} + o\left( 1 \right)} \right){\rm{bs}}{\left( f \right)^{3/2}}$$ . We improve the constant factor and show a family of functions that give fbs(f) = (6−1/2 − o(1)) bs(f)3/2.
All Classical Adversary Methods Are Equivalent for Total Functions
2017
We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions and are equal to the fractional block sensitivity fbs( f ). That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. This equivalence also implies that for total functions, the relational adversary is equivalent to a simpler lower bound, which we call rank-1 relational adversary. For partial functions, we show unbounded separations between fbs( f ) and other adversary bounds, as well as between the adversary bounds themselves. We also show that, for partial functions, fractional block sensitivity canno…
Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations
2010
We present two new quantum algorithms. Our first algorithm is a generalization of amplitude amplification to the case when parts of the quantum algorithm that is being amplified stop at different times. Our second algorithm uses the first algorithm to improve the running time of Harrow et al. algorithm for solving systems of linear equations from O(kappa^2 log N) to O(kappa log^3 kappa log N) where \kappa is the condition number of the system of equations.
Classical automata on promise problems
2015
Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Secon…
Exact quantum algorithms have advantage for almost all Boolean functions
2014
It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that ${AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.
Quantum lower bound for inverting a permutation with advice
2014
Given a random permutation $f: [N] \to [N]$ as a black box and $y \in [N]$, we want to output $x = f^{-1}(y)$. Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but \emph{not} on the input $y$. Classically, there is a data structure of size $\tilde{O}(S)$ and an algorithm that with the help of the data structure, given $f(x)$, can invert $f$ in time $\tilde{O}(T)$, for every choice of parameters $S$, $T$, such that $S\cdot T \ge N$. We prove a quantum lower bound of $T^2\cdot S \ge \tilde{\Omega}(\epsilon N)$ for quantum algorithms that invert a random permutation $f$ on an $\epsilon$ fraction of…
Classical and Quantum Annealing in the Median of Three Satisfiability
2011
We determine the classical and quantum complexities of a specific ensemble of three-satisfiability problems with a unique satisfying assignment for up to N = 100 and 80 variables, respectively. In the classical limit, we employ generalized ensemble techniques and measure the time that a Markovian Monte Carlo process spends in searching classical ground states. In the quantum limit, we determine the maximum finite correlation length along a quantum adiabatic trajectory determined by the linear sweep of the adiabatic control parameter in the Hamiltonian composed of the problem Hamiltonian and the constant transverse field Hamiltonian. In the median of our ensemble, both complexities diverge e…
Search by quantum walks on two-dimensional grid without amplitude amplification
2011
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 \Theta(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 \Theta(1). This all…