We present an adiabatic quantum algorithm for the abstract problem of searching marked vertices in a graph, or spatial search. Given a random walk (or Markov chain) P on a graph with a set of unknown marked vertices, one can define a related absorbing walk P' where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian H(s) from the interpolated Markov chain P(s)=(1-s)P+sP' and use it in an adiabatic quantum algorithm to drive an initial superposition over all vertices to a superposition over marked vertices. The adiabatic condition implies that for any reversible Markov chain and any set of marked vertices, the running time of the adiabatic algorithm is given by the square root of the classical hitting time. This algorithm therefore demonstrates a novel connection between the adiabatic condition and the classical notion of hitting time of a random walk. It also significantly extends the scope of previous quantum algorithms for this problem, which could only obtain a full quadratic speed-up for state-transitive reversible Markov chains with a unique marked vertex.

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. The number of steps of the quantum walk is quadratically smaller than the classical hitting time of any reversible random walk P on the graph. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the walk P and the absorbing walk P′, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of the interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not state-transitive, and in the presence of multiple marked vertices. As a consequence we make a progress on an open problem related to the spatial search on the 2D-grid.

Understanding NP-complete problems is a central topic in computer science. This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps between the ground and excited states of the quantum computer's Hamiltonian. We show that the statistics of the gaps can be analyzed in a novel way, borrowed from the study of quantum disordered systems in statistical mechanics. It turns out that due to a phenomenon similar to Anderson localization, exponentially small gaps appear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. This implies that unfortunately, adiabatic quantum optimization fails: the system gets trapped in one of the numerous local minima.

A non-local box is an abstract device into which Alice and Bob input bits x and y respectively and receive outputs a and b respectively, where a, b are uniformly distributed and the parity of a+b equals xy. Such boxes have been central to the study of quantum or generalized non-locality as well as the simulation of non-signaling distributions. In this paper, we start by studying how many non-local boxes Alice and Bob need in order to compute a Boolean function f. We provide tight upper and lower bounds in terms of the communication complexity of the function both in the deterministic and randomized case. We then proceed to show that the study of non-local box complexity has interesting applications for classical computation as well. In particular, we look at secure function evaluation, and study the question posed by Beimel and Malkin of how many Oblivious Transfer calls Alice and Bob need in order to securely compute a function f. We show that this question is related to the non-local box complexity of the function and conclude by greatly improving their bounds. Finally, another consequence of our results is that traceless two-outcome measurements on maximally entangled states can be simulated with 3 non-local boxes, while no finite bound was previously known.

Adiabatic quantum optimization has attracted a lot of attention because small scale simulations gave hope that it would allow to solve NP-complete problems efficiently. Later, negative results proved the existence of specifically designed hard instances where adiabatic optimization requires exponential time. In spite of this, there was still hope that this would not happen for random instances of NP-complete problems. This is an important issue since random instances are a good model for hard instances that can not be solved by current classical solvers, for which an efficient quantum algorithm would therefore be desirable. Here, we will show that because of a phenomenon similar to Anderson localization, an exponentially small eigenvalue gap appears in the spectrum of the adiabatic Hamiltonian for large random instances, very close to the end of the algorithm. This implies that unfortunately, adiabatic quantum optimization also fails for these instances by getting stuck in a local minimum, unless the computation is exponentially long.

We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite mea- surements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a, b distributed according to some pre-specified joint distribution p(a, b|x, y). Our re- sults apply to any non-signaling distribution, that is, those where Alice’s marginal distribution does not depend on Bob’s input, and vice versa. By introducing a simple new technique based on affine combinations of lower- complexity distributions, we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. The lower bounds we obtain can be expressed as linear programs (or SDPs for quantum communication). We show that the dual formulations have a striking interpre- tation, since they coincide with maximum violations of Bell and Tsirelson in- equalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distri- bution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, informa- tion theoretic methods are necessary to prove strong lower bounds. Finally, we give an exponential upper bound on quantum and classical commu- nication complexity in the simultaneous messages model, for any non-signaling distribution.

Consider the following general communication problem: Alice and Bob have to simulate a probabilistic function p, that with every (x, y) ∈ X × Y associates a probability distribution on A × B. The two parties, upon receiving inputs x and y, need to output a ∈ A, b ∈ B in such a manner that the (a, b) pair is distributed according to p(x, y). They share randomness, and have access to a channel that allows two-way communication. Our main focus is an instance of the above problem coming from the well known EPR experiment in quantum physics. In this paper, we are concerned with the amount of communication required to simulate the EPR experiment when it is repeated in parallel a large number of times, giving rise to a notion of amortized communication complexity. In the 3-dimensional case, Toner and Bacon showed that this problem could be solved using on average 0.85 bits of communication per repetition. We show that their approach cannot go below 0.414 bits, and we give a fundamentally different technique, relying on the reverse Shannon theorem, which allows us to reduce the amortized communication to 0.28 bits for dimension 3, and 0.410 bits for arbitrary dimension. We also give a lower bound of 0.13 bits for this problem (valid for one-way protocols), and conjecture that this could be improved to match the upper bounds. In our investigation we find interesting connections to a number of different problems in communication complexity. The results contained herein are entirely classical and no knowledge of the quantum phenomenon is assumed.

A quantum walk, i.e., the quantum evolution of a particle on a graph, is termed scalar if the internal space of the moving particle (often called the coin) has a dimension one. Here, we study the existence of scalar quantum walks on Cayley graphs, which are built from the generators of a group. After deriving a necessary condition on these generators for the existence of a scalar quantum walk, we present a general method to express the evolution operator of the walk, assuming homogeneity of the evolution. We use this necessary condition and the subsequent constructive method to investigate the existence of scalar quantum walks on Cayley graphs of various groups presented with two or three generators. In this restricted framework, we classify all groups - in terms of relations between their generators - that admit scalar quantum walks, and we also derive the form of the most general evolution operator. Finally, we point out some interesting special cases, and extend our study to a few examples of Cayley graphs built with more than three generators.

We present a protocol to simulate the quantum correlations of an arbitrary bipartite state, when the parties perform a measurement according to two traceless binary observables. We show that log(d) bits of classical communication is enough on average, where d is the dimension of both systems. To obtain this result, we use the sampling approach for simulating the quantum correlations. We discuss how to use this method in the case of qudits.

We propose a new method for designing quantum search algorithms for finding a “marked” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [FOCS'04, pp 32-41] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator, which is used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [FOCS04, pp 22-31] and Szegedy [FOCS'04, pp 32-41]. Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chain. In addition, it is conceptually simple, avoids several technical difficulties in the previous analyses, and leads to improvements in various aspects of several algorithms based on quantum walk.

We exhibit measurements for optimal state estimation which have a finite number of outcomes. This is achieved by a connection between finite optimal measurements and Gauss quadratures. The example we consider to illustrate this connection is that of state estimation on N qubits, all in a same pure state. Extensions to state estimation of mixed states are also discussed.

Gaussian quantum channels have recently attracted a growing interest, since they may lead to a tractable approach to the generally hard problem of evaluating quantum channel capacities. However, the analysis performed so far has always been restricted to memoryless channels. Here, we consider the case of a bosonic Gaussian channel with memory, and show that the classical capacity can be significantly enhanced by employing entangled input symbols instead of product symbols.

We study the problem of simulating quantum correlations with the help of shared randomness, supplemented with additional resources. More specifically, we focus on the case of local projective measurements on a qubit pair in a maximally entangled state for which different protocols using different resources have been proposed. We introduce a generic method to derive these protocols, which clarifies the link between them and sheds new light on the problem.

It is known that quantum correlations exhibited by a maximally entangled qubit pair can be simulated with the help of shared randomness, supplemented with additional resources, such as communication, post-selection or non-local boxes. For instance, in the case of projective measurements, it is possible to solve this problem with protocols using one bit of communication or making one use of a non-local box. We show that this problem reduces to a distributed sampling problem. We give a new method to obtain samples from a biased distribution, starting with shared random variables following a uniform distribution, and use it to build distributed sampling protocols. This approach allows us to derive, in a simpler and unified way, many existing protocols for projective measurements, and extend them to positive operator value measurements. Moreover, this approach naturally leads to a local hidden variable model for Werner states.

The bosonic quantum channels have recently attracted a growing interest, motivated by the hope that they open a tractable approach to the generally hard problem of evaluating quantum channel capacities. These studies, however, have always been restricted to memoryless channels. Here, it is shown that the classical capacity of a bosonic Gaussian channel with memory can be significantly enhanced if entangled symbols are used instead of product symbols. For example, the capacity of a photonic channel with 70%-correlated thermal noise of one third the shot noise is enhanced by about 11% when using 3.8-dB entangled light with a modulation variance equal to the shot noise.

Besides the traditional circuit-based model of quantum computation, several quantum algorithms based on a continuous-time Hamiltonian evolution have recently been introduced, including for instance continuous-time quantum walk algorithms as well as adiabatic quantum algorithms. Unfortunately, very little is known today about the behavior of these Hamiltonian algorithms in the presence of noise. Here, we perform a fully analytical study of the resistance to noise of these algorithms using perturbation theory combined with a theoretical noise model based on random matrices drawn from the Gaussian orthogonal ensemble, whose elements vary in time and form a stationary random process.

The study of quantum computation has been motivated by the hope of finding efficient quantum algorithms for solving classically hard problems. In this context, quantum algorithms by local adiabatic evolution have been shown to solve an unstructured search problem with a quadratic speedup over a classical search, just as Grover's algorithm. In this paper, we study how the structure of the search problem may be exploited to further improve the efficiency of these quantum adiabatic algorithms. We show that by nesting a partial search over a reduced set of variables into a global search, it is possible to devise quantum adiabatic algorithms with a complexity that, although still exponential, grows with a reduced order in the problem size.

We analyze three different quantum search algorithms, namely, the traditional circuit-based Grover's algorithm, its continuous-time analog by Hamiltonian evolution, and the quantum search by local adiabatic evolution. We show that these algorithms are closely related in the sense that they all perform a rotation, at a constant angular velocity, from a uniform superposition of all states to the solution state. This makes it possible to implement the two Hamiltonian-evolution algorithms on a conventional quantum circuit, while keeping the quadratic speedup of Grover's original algorithm. It also clarifies the link between the adiabatic search algorithm and Grover's algorithm.

We derive both numerically and analytically Bell inequalities and quantum measurements that present enhanced resistance to detector inefficiency. In particular, we describe several Bell inequalities which appear to be optimal with respect to inefficient detectors for small dimensionality d = 2,3,4 and two or more measurement settings at each side. We also generalize the family of Bell inequalities described by Collins et al. [Physical Review Lett. 88, 040404 (2002)] to take into account the inefficiency of detectors. In addition, we consider the possibility for pairs of entangled particles to be produced with probability less than 1. We show that when the pair production probability is small, one should in general use different Bell inequalities than when the pair production probability is high.

The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover's problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order sqrt(N), where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. (e-print quant-ph/0001106), where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.

The resonating-group method allows treating reactions in a fully microscopic way. The non-local resonating-group equation can be accurately solved on a Lagrange mesh involving few mesh points. This mesh technique is combined with either the R-matrix method or the Hulthén-Kohn method. The forbidden states can be eliminated by a special treatment. The accuracy of the technique of solution is illustrated on a solvable non-local potential. Phase shifts for the α+n and α+p scatterings are calculated with both variants of the resonating-group method on a Lagrange mesh and a comparison is performed between them and the equivalent generator-coordinate method.