Perfect initialization of a quantum computer of neutral atoms in an optical lattice of large lattice constant

We propose a scheme for the initialization of a quantum computer based on neutral atoms trapped in an optical lattice with large lattice constant. Our focus is the development of a compacting scheme to prepare a perfect optical lattice of simple orthorhombic structure with unit occupancy. Compacting is accomplished by sequential application of two types of operations: a flip operator that changes the internal state of the atoms, and a shift operator that moves them along the lattice principal axis. We propose physical mechanisms for realization of these operations and we study the effects of motional heating of the atoms. We carry out an analysis of the complexity of the compacting scheme and show that it scales linearly with the number of lattice sites per row of the lattice, thus showing good scaling behavior with the size of the quantum computer.Comment: 18 page

Adiabatic Quantum Computing with Phase Modulated Laser Pulses

Implementation of quantum logical gates for multilevel system is demonstrated through decoherence control under the quantum adiabatic method using simple phase modulated laser pulses. We make use of selective population inversion and Hamiltonian evolution with time to achieve such goals robustly instead of the standard unitary transformation language.Comment: 19 pages, 6 figures, submitted to JOP

Quantum walks on Cayley graphs

We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given, including a family of quantum walks on the hypercube involving a Clifford Algebra.Comment: J. Phys. A (accepted for publication

Bounds for mixing time of quantum walks on finite graphs

Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov, Ambainis, Kempe and Vazirani (2001) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speed-up in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e., walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.Comment: This is the journal version (except formatting); it is a significant revision of the previous version, in particular, it contains a new result about the convergence of quantum walks with decoherence; 16 page

Security of practical private randomness generation

Measurements on entangled quantum systems necessarily yield outcomes that are intrinsically unpredictable if they violate a Bell inequality. This property can be used to generate certified randomness in a device-independent way, i.e., without making detailed assumptions about the internal working of the quantum devices used to generate the random numbers. Furthermore these numbers are also private, i.e., they appear random not only to the user, but also to any adversary that might possess a perfect description of the devices. Since this process requires a small initial random seed, one usually speaks of device-independent randomness expansion. The purpose of this paper is twofold. First, we point out that in most real, practical situations, where the concept of device-independence is used as a protection against unintentional flaws or failures of the quantum apparatuses, it is sufficient to show that the generated string is random with respect to an adversary that holds only classical-side information, i.e., proving randomness against quantum-side information is not necessary. Furthermore, the initial random seed does not need to be private with respect to the adversary, provided that it is generated in a way that is independent from the measured systems. The devices, though, will generate cryptographically-secure randomness that cannot be predicted by the adversary and thus one can, given access to free public randomness, talk about private randomness generation. The theoretical tools to quantify the generated randomness according to these criteria were already introduced in [S. Pironio et al, Nature 464, 1021 (2010)], but the final results were improperly formulated. The second aim of this paper is to correct this inaccurate formulation and therefore lay out a precise theoretical framework for practical device-independent randomness expansion.Comment: 18 pages. v3: important changes: the present version focuses on security against classical side-information and a discussion about the significance of these results has been added. v4: minor changes. v5: small typos correcte

Improved Error-Scaling for Adiabatic Quantum State Transfer

We present a technique that dramatically improves the accuracy of adiabatic state transfer for a broad class of realistic Hamiltonians. For some systems, the total error scaling can be quadratically reduced at a fixed maximum transfer rate. These improvements rely only on the judicious choice of the total evolution time. Our technique is error-robust, and hence applicable to existing experiments utilizing adiabatic passage. We give two examples as proofs-of-principle, showing quadratic error reductions for an adiabatic search algorithm and a tunable two-qubit quantum logic gate.Comment: 10 Pages, 4 figures. Comments are welcome. Version substantially revised to generalize results to cases where several derivatives of the Hamiltonian are zero on the boundar

Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach

In this paper, we study the $k$-forest problem in the model of resource augmentation. In the $k$-forest problem, given an edge-weighted graph $G(V,E)$, a parameter $k$, and a set of $m$ demand pairs $\subseteq V \times V$, the objective is to construct a minimum-cost subgraph that connects at least $k$ demands. The problem is hard to approximate---the best-known approximation ratio is $O(\min\{\sqrt{n}, \sqrt{k}\})$. Furthermore, $k$-forest is as hard to approximate as the notoriously-hard densest $k$-subgraph problem. While the $k$-forest problem is hard to approximate in the worst-case, we show that with the use of resource augmentation, we can efficiently approximate it up to a constant factor. First, we restate the problem in terms of the number of demands that are {\em not} connected. In particular, the objective of the $k$-forest problem can be viewed as to remove at most $m-k$ demands and find a minimum-cost subgraph that connects the remaining demands. We use this perspective of the problem to explain the performance of our algorithm (in terms of the augmentation) in a more intuitive way. Specifically, we present a polynomial-time algorithm for the $k$-forest problem that, for every $\epsilon>0$, removes at most $m-k$ demands and has cost no more than $O(1/\epsilon^{2})$ times the cost of an optimal algorithm that removes at most $(1-\epsilon)(m-k)$ demands

Approximability of the Subset Sum Reconfiguration Problem

The subset sum problem is a well-known NP-complete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NP-hard, and is PSPACE-complete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in the reconfiguration. We show that this maximization problem admits a polynomial-time approximation scheme (PTAS), while the problem is APX-hard if we are given a conflict graph

Quantum magic rectangles: Characterization and application to certified randomness expansion

We study a generalization of the Mermin-Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these rectangular games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for $m \times n$ rectangular games of dimensions $m,n \geq 3$ there are quantum strategies that win with certainty, while for dimensions $1 \times n$ quantum strategies do not outperform classical strategies. The final case of dimensions $2 \times n$ is richer, and we give upper and lower bounds that both outperform the classical strategies. Finally, we apply our findings to quantum certified randomness expansion to find the noise tolerance and rates for all magic rectangle games. To do this, we use our previous results to obtain the winning probability of games with a distinguished input for which the devices give a deterministic outcome, and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput. 46, 1304 (2017)].Comment: 23 pages, 3 figures; published version with minor correction

Quantum transport on two-dimensional regular graphs

We study the quantum-mechanical transport on two-dimensional graphs by means of continuous-time quantum walks and analyse the effect of different boundary conditions (BCs). For periodic BCs in both directions, i.e., for tori, the problem can be treated in a large measure analytically. Some of these results carry over to graphs which obey open boundary conditions (OBCs), such as cylinders or rectangles. Under OBCs the long time transition probabilities (LPs) also display asymmetries for certain graphs, as a function of their particular sizes. Interestingly, these effects do not show up in the marginal distributions, obtained by summing the LPs along one direction.Comment: 22 pages, 11 figure, acceted for publication in J.Phys.