Quantum random walk finds application in efficient quantum algorithms as well as in quantum network theory. Here we study the mixing time of a discrete quantum walk over a square lattice in the presence of dynamic percolation and decoherence. We consider bit-flip and phase damping noise, and evaluate the instantaneous mixing time for both the cases. Using numerical analysis we show that, in the case of phase damping noise, the probability distribution of walker's position is sufficiently close to the uniform distribution after infinite time. However, during the action of bit-flip noise, even after infinite time the total variational distance between the two probability distributions is large enough.
Quantum random walk finds application in efficient quantum algorithms as well as in quantum network theory. Here we study the mixing time of a discrete quantum walk over a square lattice in the presence of dynamic percolation and decoherence. We consider bit-flip and phase damping noise, and evaluate the instantaneous mixing time for both the cases. Using numerical analysis we show that, in the case of phase damping noise, the probability distribution of walker's position is sufficiently close to the uniform distribution after infinite time. However, during the action of bit-flip noise, even after infinite time the total variational distance between the two probability distributions is large enough.
This study investigates the crossover dynamics of a one-dimensional discrete-time quantum walk (DTQW) in noisy environments using the density matrix formalism and Kraus operator representation of quantum operations. Decoherence is modeled by bit-flip, dephasing, and bit-phase noise channels, while dissipation is represented by generalized amplitude damping channels. The analysis of crossover dynamics involves examining the probability distribution, scaling exponent of variance, coherence measures, and entropy metrics such as Shannon and von Neumann entropy. The DTQW produces a symmetrical probability distribution of displacement under decoherence, with distinct features in scaling exponents of mean squared displacement or variance. Under dissipation, DTQW produces a non-symmetric probability distribution and spreads faster than classical walk under extreme decoherence. These suggest a broad class of crossover transport behavior, which can be classified as pure quantum walk, quantum-like walk, semi-classical-like walk, and classical-like walk. Remarkably, maximum Shannon entropy is achieved at lower degrees of decoherence. This study also analyzes the effect of temporal disorder in the coin operator modeled using fractional Gaussian noise (fGn). The crossover dynamic occurs in the form of weak localization, followed by enhanced entanglement and information backflow. Notably, the correlation of fGn is transferred to the coin's degree of entanglement and transpires in the two-point correlation of the von Neumann entropy fluctuation. These findings can be useful for implementing DTQW-based protocols in quantum error correction, quantum cryptography, and quantum memory on the noisy intermediate-scale quantum (NISQ) platform.
Following Brun's formalism for decoherent Discrete Time Quantum Walk (DTQW), a fully analytical formulation of first moment of DTQW has been presented when the quantum walker is subjected to a bit-flip noise and the initial state of the walker is considered to be delocalized up to three lattice sites in 1D. It is shown that when the initial state of the walker is also modulated by an initial phase, particular choice of initial phase angle can nullify the effect of the nonlocality in first moment in short time limit.
For a discrete time quantum walk (QW) on the N-cycle, allowing for decoherence on the coin, we derive a number of new results, including an explicit formula for the position probability distribution. For a QW of this type, we show that the mixing behavior tends, in the long-run, to a uniform distribution, regardless of the initial state of the system and irrespective of the parity of the number of nodes N. These results confirm the findings of previous authors who arrived at similar conclusions through extensive numerical simulations. In particular, we infer that the mixing time M(ϵ)ˉ for the time-everaged probability distribution is of order no greater than O(N2/ϵ).
In this thesis we study the propagation behavior of quantum particles which perform discrete-time quantum walks on spatial lattices.By definition, quantum walks are discrete-time evolutions such that a single time step couples only neighboring lattice sites.Additionally, we assume in this thesis that the dynamical law commutes with all translations of the spatial lattice, that is we consider translationally invariant quantum walks.The results presented in the following are separated into two parts: The first part is devoted to the study of single-particle quantum walks, and in the second part we investigate quantum walks of several particles.Single-particle quantum walks: Our aim is to study the asymptotic behavior of particles undergoing general translationally invariant quantum walks.Employing perturbation theory, we establish a direct method for computing the asymptotic distribution of the particle's position rescaled by an appropriate time-dependent factor.If the time evolution is implemented by a unitary operator commuting with all lattice translations and with empty pure point spectrum, we find linear scaling of the asymptotic position distribution.In this case, the asymptotic distribution of position divided by time is determined by the analog of the group velocity.If, on the other hand, the quantum walk is subject to decoherence, we expect that it behaves like a classical random walk.We provide explicit criteria under which this intuitive statement is true, that is, the asymptotic scaling of the particle's position distribution is that of a diffusion process.We find that the asymptotic distribution of position divided by the square root of time is a mixture of Gaussians in this case.However, we also provide examples of decoherent quantum walks which violate these criteria, and exhibit linear scaling of the asymptotic position distribution. Many-particle quantum walks:We investigate unitary quantum walks of two particles with collisional interactions.That is, the particles perform independent single-particle quantum walks if they are at distinct lattice sites, and an interaction takes place if both particles occupy the same lattice site.We prove that such quantum walks generically exhibit a molecular binding effect between the particles.More precisely, for a measurement of the particles' positions we have a bound on the probability to obtain a certain result which decays exponentially with the particles' distance.As a preparatory step for this analysis, we consider single-particle quantum walks with point defects, and provide a method to compute their eigenvalues in the spectral gap of the quantum walk without defect.We further generalize our model to arbitrary numbers of particles and study the behavior of local excitations in many-particle quantum walks.
The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(nlogn). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally nonuniform) distribution $\ensuremath{\pi}(x)$ in time O(n), and the stationary distribution is determined by the initial state of the walk. An explicit expression for $\ensuremath{\pi}(x)$ is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p\ensuremath{\approx}0.1$. A similar effect was previously reported for a quantum walk on an N-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps in obtaining---and preserving---a uniform distribution over the 2n sites of the hypercube in the shortest possible time.
We investigate one-dimensional discrete-time quantum walk on the line where the links between neighboring sites are randomly broken. Two link-broken ways, static percolation and dynamical percolation, are considered. The former means that the broken links are fixed in position space at each time step, while the latter is that broken links are varied with time step. Our attention focuses on the effects of these disorders on two physical quantities, the probability distribution and the entanglement between the coin degree of freedom and position degree of freedom. Choosing Hadamard coin operator and assuming the walker to start from the position eigenstate|0〉and attach itself to a coherent coin state 1/√2 (|↑〉+ i|↓〉), we give the statistical average results after making numerical calculations many times. The choices of coin operator and initial state, resulting in a symmetric probability distribution about origin in the ideal case, is helpful in comparing with different cases in different disorder strengths. It is shown that the probability distribution of static percolation quantum walk can change from a coherent behavior at short time to Anderson localization at longer time, while the dynamical percolation quantum walk can change to a classical diffusive behavior. With the decrease of the percolation probability, these transitions become faster. The entanglement for ideal case without disorder reaches a constant value after a short time evolution. The static percolation makes the entanglement less than that of ideal case and fluctuate irregularly around a certain value. The situation is very different for the dynamical percolation:the entanglement increases smoothly with the time step and can exceed the constant value in the ideal case at some time. Both of entanglements for two types of percolations decrease with reducing percolation probability. As a striking characteristic, the entanglement in dynamical case can tend to maximum regardless of percolation probability in long time limit, while the static case cannot. In the model for our study, the randomized unitary operations, induced by the static and dynamical percolations, can lead to some noticeable effects on the transport and entanglement of discrete time quantum walk. The results about the interplay between disorder and entanglement not only assist quantum information processing, but also give more options to further explore and understand disorder physical processes in nature.