arXiv subjects

Renato Portugal

Publications and source records attributed to Renato Portugal.

At least 37 records · Page 2Linked to original sources

arXiv2020

Entropy production in the quantum walk

We explore the notion of generated entropy in open quantum systems. We focus on the study of the discrete-time quantum walk on the line, from the entropy production perspective. We argue that the evolution of the coin can be modeled as an open two-level system that exchanges energy with the lattice at some effective temperature that depends on the initial state. The entropy balance shows that there is a positive entropy production during the evolution, in accordance with the second law of thermodynamics.

quant-ph↗

arXiv2020

Discrete-Time Quantum Walks on Oriented Graphs

The interest in quantum walks has been steadily increasing during the last two decades. It is still worth to present new forms of quantum walks that might find practical applications and new physical behaviors. In this work, we define discrete-time quantum walks on arbitrary oriented graphs by partitioning a graph into tessellations, which is a collection of disjoint cliques that cover the vertex set. By using the adjacency matrices associated with the tessellations, we define local unitary operators, whose product is the evolution operator of our quantum walk model. We introduce a parameter, called alpha, that quantifies the amount of orientation. We show that the parameter alpha can be tuned in order to increase the amount of quantum walk-based transport on oriented graphs.

quant-ph↗

arXiv2020

Experimental Implementation of Quantum Walks on IBM Quantum Computers

The development of universal quantum computers has achieved remarkable success in recent years, culminating with the quantum supremacy reported by Google. Now is possible to implement short-depth quantum circuits with dozens of qubits and to obtain results with significant fidelity. Quantum walks are good candidates to be implemented on the available quantum computers. In this work, we implement discrete-time quantum walks with one and two interacting walkers on cycles, two-dimensional lattices, and complete graphs on IBM quantum computers. We are able to obtain meaningful results using the cycle, the two-dimensional lattice, and the complete graph with 16 nodes each, which require 4-qubit quantum circuits up to depth 100.

quant-ph↗

arXiv2020

Total tessellation cover and quantum walk

We propose the total staggered quantum walk model and the total tessellation cover of a graph. This model uses the concept of total tessellation cover to describe the motion of the walker who is allowed to hop both to vertices and edges of the graph, in contrast with previous models in which the walker hops either to vertices or edges. We establish bounds on $T_t(G)$, which is the smallest number of tessellations required in a total tessellation cover of $G$. We highlight two of these lower bounds $T_t(G) \geq ω(G)$ and $T_t(G)\geq is(G)+1$, where $ω(G)$ is the size of a maximum clique and $is(G)$ is the number of edges of a maximum induced star subgraph. Using these bounds, we define the good total tessellable graphs with either $T_t(G)=ω(G)$ or $T_t(G)=is(G)+1$. The $k$-total tessellability problem aims to decide whether a given graph $G$ has $T_t(G) \leq k$. We show that $k$-total tessellability is in $\mathcal{P}$ for good total tessellable graphs. We establish the $\mathcal{NP}$-completeness of the following problems when restricted to the following classes: ($is(G)+1$)-total tessellability for graphs with $ω(G) = 2$; $ω(G)$-total tessellability for graphs $G$ with $is(G)+1 = 3$; $k$-total tessellability for graphs $G$ with $\max\{ω(G), is(G)+1\}$ far from $k$; and $4$-total tessellability for graphs $G$ with $ω(G) = is(G)+1 = 4$. As a consequence, we establish hardness results for bipartite graphs, line graphs of triangle-free graphs, universal graphs, planar graphs, and $(2,1)$-chordal graphs.

cs.DM↗

arXiv2019

The Tessellation Cover Number of Good Tessellable Graphs

A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges, and the tessellation cover number, denoted by $T(G)$, is the size of a smallest tessellation cover. The \textsc{$t$-tessellability} problem aims to decide whether a graph $G$ has $T(G)\leq t$ and is $\mathcal{NP}$-complete for $t\geq 3$. Since the number of edges of a maximum induced star of $G$, denoted by $is(G)$, is a lower bound on $T(G)$, we define good tessellable graphs as the graphs~$G$ such that $T(G)=is(G)$. The \textsc{good tessellable recognition (gtr)} problem aims to decide whether $G$ is a good tessellable graph. We show that \textsc{gtr} is $\mathcal{NP}$-complete not only if $T(G)$ is known or $is(G)$ is fixed, but also when the gap between $T(G)$ and $is(G)$ is large. As a byproduct, we obtain graph classes that obey the corresponding computational complexity behaviors.

cs.CC↗

arXiv2019

Multiparticle quantum walk with a gas-like interaction

We analyze the dynamics of multiparticle discrete-time quantum walk on the two-dimensional lattice, with an interaction inspired on a classical model for gas collision, called HPP model. In this classical model, the direction of motion changes only when the particles collide head-on, preserving momentum and energy. In our quantum model, the dynamics is driven by the usual quantum-walk evolution operator if the particles are on different nodes, and is driven by the HPP rules if the particles are in the same node, linearly extended for superpositions. Using this new form of evolution operator, we numerically analyze three physical quantities for the two-walker case: The probability distribution of the position of one walker, the standard deviation of the position of one walker, and the entanglement between the walkers as a function of the number of steps. The numerical analysis implies that the entanglement between the walkers as a function of the number of steps initially increases and quickly tends to a constant value, which depends on the initial condition. We compare the results obtained using the HPP interaction with the equivalent ones using the phase interaction, which is based on an evolution operator that inverts the sign of the coin operator if the walkers are in the same position.

quant-ph↗

arXiv2018

Eigenbasis of the Evolution Operator of 2-Tessellable Quantum Walks

Staggered quantum walks on graphs are based on the concept of graph tessellation and generalize some well-known discrete-time quantum walk models. In this work, we address the class of 2-tessellable quantum walks with the goal of obtaining an eigenbasis of the evolution operator. By interpreting the evolution operator as a quantum Markov chain on an underlying multigraph, we define the concept of quantum detailed balance, which helps to obtain the eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of the double discriminant matrix of the quantum Markov chain. To obtain the remaining eigenvectors, we have to use the quantum detailed balance conditions. If the quantum Markov chain has a quantum detailed balance, there is an eigenvector for each fundamental cycle of the underlying multigraph. If the quantum Markov chain does not have a quantum detailed balance, we have to use two fundamental cycles linked by a path in order to find the remaining eigenvectors. We exemplify the process of obtaining the eigenbasis of the evolution operator using the kagome lattice (the line graph of the hexagonal lattice), which has symmetry properties that help in the calculation process.

quant-ph↗

arXiv2018

Complexity Bounds on Quantum Search Algorithms in finite-dimensional Networks

We establish a lower bound concerning the computational complexity of Grover's algorithms on fractal networks. This bound provides general predictions for the quantum advantage gained for searching unstructured lists. It yields a fundamental criterion, derived from quantum transport properties, for the improvement a quantum search algorithm achieves over the corresponding classical search in a network based solely on its spectral dimension, $d_{s}$. Our analysis employs recent advances in the interpretation of the venerable real-space renormalization group (RG) as applied to quantum walks. It clarifies the competition between Grover's abstract algorithm, i.e., a rotation in Hilbert space, and quantum transport in an actual geometry. The latter is characterized in terms of the quantum walk dimension $d_{w}^{Q}$ and the spatial (fractal) dimension $d_{f}$ that is summarized simply by the spectral dimension of the network. The analysis simultaneously determines the optimal time for a quantum measurement and the probability for successfully pin-pointing a marked element in the network. The RG further encompasses an optimization scheme devised by Tulsi that allows to tune this probability to certainty, leaving quantum transport as the only limiting process. It considers entire families of problems to be studied, thereby establishing large universality classes for quantum search, which we verify with extensive simulations. The methods we develop could point the way towards systematic studies of universality classes in computational complexity to enable modification and control of search behavior.

cond-mat.stat-mech↗

arXiv2018

Staggered Quantum Walk on Hexagonal Lattices

A discrete-time staggered quantum walk was recently introduced as a generalization that allows to unify other versions, such as the coined and Szegedy's walk. However, it also produces new forms of quantum walks not covered by previous versions. To explore their properties, we study here the staggered walk on a hexagonal lattice. Such a walk is defined using a set of overlapping tessellations that cover the graph edges, and each tessellation is a partition of the node set into cliques. The hexagonal lattice requires at least three tessellations. Each tessellation is associated with a local unitary operator and the product of the local operators defines the evolution operator of the staggered walk on the graph. After defining the evolution operator on the hexagonal lattice, we analyze the quantum walk dynamics with the focus on the position standard deviation and localization. We also obtain analytic results for the time complexity of spatial search algorithms with one marked node using cyclic boundary conditions.

quant-ph↗

arXiv2018

Element Distinctness Revisited

The element distinctness problem is the problem of determining whether the elements of a list are distinct, that is, if $x=(x_1,...,x_N)$ is a list with $N$ elements, we ask whether the elements of $x$ are distinct or not. The solution in a classical computer requires $N$ queries because it uses sorting to check whether there are equal elements. In the quantum case, it is possible to solve the problem in $O(N^{2/3})$ queries. There is an extension which asks whether there are $k$ colliding elements, known as element $k$-distinctness problem. This work obtains optimal values of two critical parameters of Ambainis' seminal quantum algorithm [SIAM J.~Comput., 37, 210-239, 2007]. The first critical parameter is the number of repetitions of the algorithm's main block, which inverts the phase of the marked elements and calls a subroutine. The second parameter is the number of quantum walk steps interlaced by oracle queries. We show that, when the optimal values of the parameters are used, the algorithm's success probability is $1-O(N^{1/(k+1)})$, quickly approaching 1. The specification of the exact running time and success probability is important in practical applications of this algorithm.

quant-ph↗

arXiv2018

Discretization of continuous-time quantum walks via the staggered model with Hamiltonians

We characterize a close connection between the continuous-time quantum-walk model and a discrete-time quantum-walk version, based on the staggered model with Hamiltonians in a class of Cayley graphs, which can be considered as a discretization of continuous-time quantum walks. This connection provides examples of perfect state transfer and instantaneous uniform mixing in the staggered model. On the other hand, we provide some more examples of perfect state transfer and instantaneous uniform mixing in the staggered model that cannot be reproduced by the continuous-time model.

quant-ph↗

arXiv2018

Quantum Walks via Quantum Cellular Automata

Very much as its classical counterpart, quantum cellular automata are expected to be a great tool for simulating complex quantum systems. Here we introduce a partitioned model of quantum cellular automata and show how it can simulate, with the same amount of resources (in terms of effective Hilbert space dimension), various models of quantum walks. All the algorithms developed within quantum walk models are thus directly inherited by the quantum cellular automata. The latter, however, has its structure based on local interactions between qubits, and as such it can be more suitable for present (and future) experimental implementations.

quant-ph↗

arXiv2017

Partition-based discrete-time quantum walks

We introduce a family of discrete-time quantum walks, called two-partition model, based on two equivalence-class partitions of the computational basis, which establish the notion of local dynamics. This family encompasses most versions of unitary discrete-time quantum walks driven by two local operators studied in literature, such as the coined model, Szegedy's model, and the 2-tessellable staggered model. We also analyze the connection of those models with the two-step coined model, which is driven by the square of the evolution operator of the standard discrete-time coined walk. We prove formally that the two-step coined model, an extension of Szegedy model for multigraphs, and the two-tessellable staggered model are unitarily equivalent. Then, selecting one specific model among those families is a matter of taste not generality.

math-ph↗

arXiv2017

Coined Quantum Walks as Quantum Markov Chains

We analyze the equivalence between discrete-time coined quantum walks and Szegedy's quantum walks. We characterize a class of flip-flop coined models with generalized Grover coin on a graph $Γ$ that can be directly converted into Szegedy's model on the subdivision graph of $Γ$ and we describe a method to convert one model into the other. This method improves previous results in literature that need to use the staggered model and the concept of line graph, which are avoided here.

quant-ph↗

arXiv2017

Quantum search on the two-dimensional lattice using the staggered model with Hamiltonians

Quantum search on the two-dimensional lattice with one marked vertex and cyclic boundary conditions is an important problem in the context of quantum algorithms with an interesting unfolding. It avails to test the ability of quantum walk models to provide efficient algorithms from the theoretical side and means to implement quantum walks in laboratories from the practical side. In this paper, we rigorously prove that the recent-proposed staggered quantum walk model provides an efficient quantum search on the two-dimensional lattice, if the reflection operators associated with the graph tessellations are used as Hamiltonians, which is an important theoretical result for validating the staggered model with Hamiltonians. Numerical results show that on the two-dimensional lattice staggered models without Hamiltonians are not as efficient as the one described in this paper and are, in fact, as slow as classical random-walk-based algorithms.

quant-ph↗

arXiv2017

Renormalization of the Unitary Evolution Equation for Coined Quantum Walks

We consider discrete-time evolution equations in which the stochastic operator of a classical random walk is replaced by a unitary operator. Such a problem has gained much attention as a framework for coined quantum walks that are essential for attaining the Grover limit for quantum search algorithms in physically realizable, low-dimensional geometries. In particular, we analyze the exact real-space renormalization group (RG) procedure recently introduced to study the scaling of quantum walks on fractal networks. While this procedure, when implemented numerically, was able to provide some deep insights into the relation between classical and quantum walks, its analytic basis has remained obscure. Our discussion here is laying the groundwork for a rigorous implementation of the RG for this important class of transport and algorithmic problems, although some instances remain unresolved. Specifically, we find that the RG fixed-point analysis of the classical walk, which typically focuses on the dominant Jacobian eigenvalue $λ_{1}$, with walk dimension $d_{w}^{RW}=\log_{2}λ_{1}$, needs to be extended to include the subdominant eigenvalue $λ_{2}$, such that the dimension of the quantum walk obtains $d_{w}^{QW}=\log_{2}\sqrt{λ_{1}λ_{2}}$. With that extension, we obtain analytically previously conjectured results for $d_{w}^{QW}$ of Grover walks on all but one of the fractal networks that have been considered.

quant-ph↗

arXiv2017

Staggered Quantum Walks with Hamiltonians

Quantum walks are recognizably useful for the development of new quantum algorithms, as well as for the investigation of several physical phenomena in quantum systems. Actual implementations of quantum walks face technological difficulties similar to the ones for quantum computers, though. Therefore, there is a strong motivation to develop new quantum-walk models which might be easier to implement. In this work, we present an extension of the staggered quantum walk model that is fitted for physical implementations in terms of time-independent Hamiltonians. We demonstrate that this class of quantum walk includes the entire class of staggered quantum walk model, Szegedy's model, and an important subset of the coined model.

quant-ph↗

arXiv2016

Staggered quantum walks with superconducting microwave resonators

The staggered quantum walk model on a graph is defined by an evolution operator that is the product of local operators related to two or more independent graph tessellations. A graph tessellation is a partition of the set of nodes that respects the neighborhood relation. Flip-flop coined quantum walks with the Hadamard or Grover coins can be expressed as staggered quantum walks by converting the coin degree of freedom into extra nodes in the graph. We propose an implementation of the staggered model with superconducting microwave resonators, where the required local operations are provided by the nearest neighbor interaction of the resonators coupled through superconducting quantum interference devices. The tunability of the interactions makes this system an excellent toolbox for this class of quantum walks. We focus on the one-dimensional case and discuss its generalization to a more general class known as triangle-free graphs.

quant-ph↗