8 research outputs found
On the adiabatic condition and the quantum hitting time of Markov chains
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) on a graph with a set of unknown marked vertices, one can
define a related absorbing walk where outgoing transitions from marked
vertices are replaced by self-loops. We build a Hamiltonian from the
interpolated Markov chain 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.Comment: 22 page
Google in a Quantum Network
We introduce the characterization of a class of quantum PageRank algorithms
in a scenario in which some kind of quantum network is realizable out of the
current classical internet web, but no quantum computer is yet available. This
class represents a quantization of the PageRank protocol currently employed to
list web pages according to their importance. We have found an instance of this
class of quantum protocols that outperforms its classical counterpart and may
break the classical hierarchy of web pages depending on the topology of the
web.Comment: RevTeX 4 file, color figure
A Note on Quantum Phase Estimation
In this work, we study the phase estimation problem. We show an alternative,
simpler and self-contained proof of query lower bounds. Technically, compared
to the previous proofs [NW99, Bes05], our proof is considerably elementary.
Specifically, our proof consists of basic linear algebra without using the
knowledge of Boolean function analysis and adversary methods. Qualitatively,
our bound is tight in the low success probability regime and offers a more
fine-grained trade-off. In particular, we prove that for any , every algorithm requires at least queries to
obtain an -approximation for the phase with probability at least p.
However, the existing bounds hold only when . Quantitatively, our
bound is tight since it matches the well-known phase estimation algorithm of
Cleve, Ekert, Macchiavello, and Mosca [CEMM98] which requires
queries to obtain an -approximation with a constant probability.
Following the derivation of the lower bound in our framework, we give a new and
intuitive interpretation of the phase estimation algorithm of [CEMM98], which
might be of independent interest
Quantum walk approach to simulating parton showers
This paper presents a novel quantum walk approach to simulating parton showers on a quantum computer. We demonstrate that the quantum walk paradigm offers a natural and more efficient approach to simulating parton showers on quantum devices, with the emission probabilities implemented as the coin flip for the walker, and the particle emissions to either gluons or quark pairs corresponding to the movement of the walker in two dimensions. A quantum algorithm is proposed for a simplified, toy model of a 31-step, collinear parton shower, hence significantly increasing the number of steps of the parton shower that can be simulated compared to previous quantum algorithms. Furthermore, it scales efficiently: the number of possible shower steps increases exponentially with the number of qubits, and the circuit depth grows linearly with the number of steps. Reframing the parton shower in the context of a quantum walk therefore brings dramatic improvements, and is a step towards extending the current quantum algorithms to simulate more realistic parton showers
Discrete-time Semiclassical Szegedy Quantum Walks
Quantum walks are promising tools based on classical random walks, with
plenty of applications such as many variants of optimization. Here we introduce
the semiclassical walks in discrete time, which are algorithms that combines
classical and quantum dynamics. Specifically, a semiclassical walk can be
understood as a classical walk where the transition matrix encodes the quantum
evolution. We have applied this algorithm to Szegedy's quantum walk, which can
be applied to any arbitrary weighted graph. We first have solved the problem
analytically on regular 1D cycles to show the performance of the semiclassical
walks. Next, we have simulated our algorithm in a general inhomogeneous
symmetric graph, finding that the inhomogeneity drives a symmetry breaking on
the graph. Moreover, we show that this phenomenon is useful for the problem of
ranking nodes in symmetric graphs, where the classical PageRank fails. We have
demonstrated experimentally that the semiclassical walks can be applied on real
quantum computers using the platform IBM Quantum.Comment: RevTex 4.2, 16 pages, 11 color figure
Discrete Quantum Walks on Graphs and Digraphs
This thesis studies various models of discrete quantum walks on graphs and digraphs via a spectral approach.
A discrete quantum walk on a digraph is determined by a unitary matrix , which acts on complex functions of the arcs of . Generally speaking, is a product of two sparse unitary matrices, based on two direct-sum decompositions of the state space. Our goal is to relate properties of the walk to properties of , given some of these decompositions.
We start by exploring two models that involve coin operators, one due to Kendon, and the other due to Aharonov, Ambainis, Kempe, and Vazirani. While is not defined as a function in the adjacency matrix of the graph , we find exact spectral correspondence between and . This leads to characterization of rare phenomena, such as perfect state transfer and uniform average vertex mixing, in terms of the eigenvalues and eigenvectors of . We also construct infinite families of graphs and digraphs that admit the aforementioned phenomena.
The second part of this thesis analyzes abstract quantum walks, with no extra assumption on . We show that knowing the spectral decomposition of leads to better understanding of the time-averaged limit of the probability distribution. In particular, we derive three upper bounds on the mixing time, and characterize different forms of uniform limiting distribution, using the spectral information of .
Finally, we construct a new model of discrete quantum walks from orientable embeddings of graphs. We show that the behavior of this walk largely depends on the vertex-face incidence structure. Circular embeddings of regular graphs for which has few eigenvalues are characterized. For instance, if has exactly three eigenvalues, then the vertex-face incidence structure is a symmetric -design, and is the exponential of a scalar multiple of the skew-symmetric adjacency matrix of an oriented graph. We prove that, for every regular embedding of a complete graph, is the transition matrix of a continuous quantum walk on an oriented graph