Quantum walks can find a marked element on any graph

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $M$ consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $HT(P,M)$ of any reversible random walk $P$ on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity $HT^+(\mathit{P,M})$ which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk $P$ and the absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk $P$ is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters $p_M$ (the probability of picking a marked vertex from the stationary distribution) and $HT^+(\mathit{P,M})$ are known.Comment: 50 page

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) $P$ 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.Comment: 22 page

Hitting time for the continuous quantum walk

We define the hitting (or absorbing) time for the case of continuous quantum walks by measuring the walk at random times, according to a Poisson process with measurement rate $\lambda$. From this definition we derive an explicit formula for the hitting time, and explore its dependence on the measurement rate. As the measurement rate goes to either 0 or infinity the hitting time diverges; the first divergence reflects the weakness of the measurement, while the second limit results from the Quantum Zeno effect. Continuous-time quantum walks, like discrete-time quantum walks but unlike classical random walks, can have infinite hitting times. We present several conditions for existence of infinite hitting times, and discuss the connection between infinite hitting times and graph symmetry.Comment: 12 pages, 1figur

Quantum walks with infinite hitting times

Hitting times are the average time it takes a walk to reach a given final vertex from a given starting vertex. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition, which for discrete time quantum walks is that the degeneracy of the evolution operator be greater than the degree of the graph. The set of initial states which give an infinite hitting time form a subspace. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. In the case of the discrete walk, if this condition is satisfied the walk will have infinite hitting times for any choice of a coin operator, and we give a class of graphs with infinite hitting times for any choice of coin. Hitting times are not very well-defined for continuous time quantum walks, but we show that the idea of infinite hitting-time walks naturally extends to the continuous time case as well.Comment: 28 pages, 3 figures in EPS forma

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

Non-Markovian dynamics for an open two-level system without rotating wave approximation: Indivisibility versus backflow of information

By use of the two measures presented recently, the indivisibility and the backflow of information, we study the non-Markovianity of the dynamics for a two-level system interacting with a zero-temperature structured environment without using rotating wave approximation (RWA). In the limit of weak coupling between the system and the reservoir, and by expanding the time-convolutionless (TCL) generator to the forth order with respect to the coupling strength, the time-local non-Markovian master equation for the reduced state of the system is derived. Under the secular approximation, the exact analytic solution is obtained and the sufficient and necessary conditions for the indivisibility and the backflow of information for the system dynamics are presented. In the more general case, we investigate numerically the properties of the two measures for the case of Lorentzian reservoir. Our results show the importance of the counter-rotating terms to the short-time-scale non-Markovian behavior of the system dynamics, further expose the relations between the two measures and their rationality as non-Markovian measures. Finally, the complete positivity of the dynamics of the considered system is discussed

Common variable immunodeficiency complicated with hemolytic uremic syndrome

Common variable immunodeficiency is a primary immunodeficiency disease characterized by reduced serum immunoglobulins and heterogeneous clinical features. Recurrent pyogenic infections of upper and lower respiratory tracts are the main clinical manifestations of common variable immunodeficiency. Hemolytic uremic syndrome is a multisystemic disorder characterized by thrombocytopenia, microangiopathic hemolytic anemia, and organ ischemia due to platelet aggregation in the arterial microvasculature. This is one of the rare cases of patients diagnosed with common variable immunodeficiency, which was complicated by hemolytic uremic syndrome

Phenotypic characterisation of regulatory T cells in dogs reveals signature transcripts conserved in humans and mice

Regulatory T cells (Tregs) are a double-edged regulator of the immune system. Aberrations of Tregs correlate with pathogenesis of inflammatory, autoimmune and neoplastic disorders. Phenotypically and functionally distinct subsets of Tregs have been identified in humans and mice on the basis of their extensive portfolios of monoclonal antibodies (mAb) against Treg surface antigens. As an important veterinary species, dogs are increasingly recognised as an excellent model for many human diseases. However, insightful study of canine Tregs has been restrained by the limited availability of mAb. We therefore set out to characterise CD4+CD25high T cells isolated ex vivo from healthy dogs and showed that they possess a regulatory phenotype, function, and transcriptomic signature that resembles those of human and murine Tregs. By launching a cross-species comparison, we unveiled a conserved transcriptomic signature of Tregs and identified that transcript hip1 may have implications in Treg function

Subexponential rate versus distance with time-multiplexed quantum repeaters

Quantum communications capacity using direct transmission over length-L optical fiber scales as R∼e-αL, where α is the fiber's loss coefficient. The rate achieved using a linear chain of quantum repeaters equipped with quantum memories, probabilistic Bell state measurements (BSMs), and switches used for spatial multiplexing, but no quantum error correction, was shown to surpass the direct-transmission capacity. However, this rate still decays exponentially with the end-to-end distance, viz., R∼e-sαL, with s<1. We show that the introduction of temporal multiplexing - i.e., the ability to perform BSMs among qubits at a repeater node that were successfully entangled with qubits at distinct neighboring nodes at different time steps - leads to a subexponential rate-vs-distance scaling, i.e., R∼e-tαL, which is not attainable with just spatial or spectral multiplexing. We evaluate analytical upper and lower bounds to this rate and obtain the exact rate by numerically optimizing the time-multiplexing block length and the number of repeater nodes. We further demonstrate that incorporating losses in the optical switches used to implement time multiplexing degrades the rate-vs-distance performance, eventually falling back to exponential scaling for very lossy switches. We also examine models for quantum memory decoherence and describe optimal regimes of operation to preserve the desired boost from temporal multiplexing. Quantum memory decoherence is seen to be more detrimental to the repeater's performance over switching losses. © 2021 American Physical Society.Immediate accessThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]