Search results for "Quantum complexity theory"
showing 10 items of 12 documents
Learning-Graph-Based Quantum Algorithm for k-distinctness
2012
We present a quantum algorithm solving the $k$-distinctness problem in $O(n^{1-2^{k-2}/(2^k-1)})$ queries with a bounded error. This improves the previous $O(n^{k/(k+1)})$-query algorithm by Ambainis. The construction uses a modified learning graph approach. Compared to the recent paper by Belovs and Lee arXiv:1108.3022, the algorithm doesn't require any prior information on the input, and the complexity analysis is much simpler. Additionally, we introduce an $O(\sqrt{n}\alpha^{1/6})$ algorithm for the graph collision problem where $\alpha$ is the independence number of the graph.
Span programs for functions with constant-sized 1-certificates
2012
Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constant-sized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the element distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph. We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit. We prove the power of our approach by designing a quantum algorithm for the triangle problem with query complexity O(n35/27) that is better than O(n13/10) of the best p…
Enlarging the gap between quantum and classical query complexity of multifunctions
2013
Quantum computing aims to use quantum mechanical effects for the efficient performance of computational tasks. A popular research direction is enlarging the gap between classical and quantum algorithm complexity of the same computational problem. We present new results in quantum query algorithm design for multivalued functions that allow to achieve a large quantum versus classical complexity separation. To compute a basic finite multifunction in a quantum model only one query is enough while classically three queries are required. Then, we present two generalizations and a modification of the original algorithm, and obtain the following complexity gaps: Q UD (M′) ≤ N versus C UD (M′) ≥ 3N,…
How Low Can Approximate Degree and Quantum Query Complexity Be for Total Boolean Functions?
2012
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.
Reordering Method and Hierarchies for Quantum and Classical Ordered Binary Decision Diagrams
2017
We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to “width” complexity. It is known that maximal complexity gap between deterministic and quantum model is exponential. But there are few examples of such functions. We present method (called “reordering”), which allows to build Boolean function g from Boolean Function f, such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is \(2^{\varOmega (n/log n)}\) and present quantum OBD…
On Physical Problems that are Slightly More Difficult than QMA
2013
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).
2014
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 9 th root of the classical randomized query complexity. This resolves a conjecture of Watrous from 2002. Second, inspired by recent work of O’Donnell et al. and Dinur et al., we conjecture that every bounded low-degree polynomial has a “highly influential” …
Quantum Attacks on Classical Proof Systems - The Hardness of Quantum Rewinding
2014
Quantum zero-knowledge proofs and quantum proofs of knowledge are inherently difficult to analyze because their security analysis uses rewinding. Certain cases of quantum rewinding are handled by the results by Watrous (SIAM J Comput, 2009) and Unruh (Eurocrypt 2012), yet in general the problem remains elusive. We show that this is not only due to a lack of proof techniques: relative to an oracle, we show that classically secure proofs and proofs of knowledge are insecure in the quantum setting. More specifically, sigma-protocols, the Fiat-Shamir construction, and Fischlin's proof system are quantum insecure under assumptions that are sufficient for classical security. Additionally, we show…
Effects of Kolmogorov complexity present in inductive inference as well
1997
For all complexity measures in Kolmogorov complexity the effect discovered by P. Martin-Lof holds. For every infinite binary sequence there is a wide gap between the supremum and the infimum of the complexity of initial fragments of the sequence. It is assumed that that this inevitable gap is characteristic of Kolmogorov complexity, and it is caused by the highly abstract nature of the unrestricted Kolmogorov complexity.
Non-intersecting Complexity
2006
A new complexity measure for Boolean functions is introduced in this article. It has a link to the query algorithms: it stands between both polynomial degree and non-deterministic complexity on one hand and still is a lower bound for deterministic complexity. Some inequalities and counterexamples are presented and usage in symmetrisation polynomials is considered.