Creating cat states in one-dimensional quantum walks using delocalized initial states

We show that arbitrary quantum evolution, both unitary and non-unitary ones, can be efficiently realized via one-dimensional quantum walk. This is done based on the equivalence between quantum evolution and positive-operated valued measurements, and an algorithm of properly controlling one-dimensional discrete-time quantum walk to realize arbitrary positive-operated valued measurements. Furthermore, our method based on one-dimensional quantum walk can be easily generalized to other platforms Our method enriches technics for quantum information processes, and the applications of quantum walk.

One-dimensional quantum walks with a time and spin-dependent phase shift

In this article we investigate the effects of shifting position decoherence, arising from the incoherent tunneling effect in the experimental realization of the quantum walk, on the one-dimensional discrete time quantum walk. We show that in the regime of this type of noise the quantum behavior of the walker does not vanish, in contrast to the coin decoherence for which the walker undergoes a quantum-to-classical transition even for weak noise. In particular, we show that the quadratic dependence of the variance on the time and also the coin–position entanglement, i.e. two important quantum aspects of the coherent quantum walk, are preserved in the presence of tunneling decoherence. Furthermore, we present an explicit expression for the probability distribution of the decoherent one-dimensional quantum walk in terms of the corresponding coherent probabilities, and show that this type of decoherence smooths the probability distribution.

Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the Z2 \times Z2 topological invariant in a simple way that is consistent with a general definition for 1-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondance. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.

Quantum walks are known to propagate quadratically faster than their classical counterparts and are used to model dynamics in various quantum systems. The spread of the quantum walk in position space shows anomalous diffusion behavior. By controlling the action of quantum coin operation on the corresponding coin degree of freedom of the walker, one can demonstrate control over the diffusion behavior. In this work, we report different forms of coin operations on quantum walks exhibiting anomalous diffusion behavior. Homogeneous and accelerated quantum walks display superdiffusive behavior, whereas uncorrelated static and dynamic disorders in the evolution induce strong and weak localization of the particle indicating subdiffusive and normal diffusive behavior. The role played by the interference effects in the spreading of the walker has remained elusive and our aim in this work is to present the interplay between quantum coherence and mean squared displacement of the walker. We employ two reliable measures of coherence for conclusively establishing the role of quantum interference as the driving force behind the anomalous diffusive behavior in the dynamics of quantum walks.

We study the generation of hybrid entanglement in a one-dimensional quantum walk. In particular, we explore the preparation of maximally entangled states between position and spin degrees of freedom. We address it as an optimization problem, where the cost function is the Schmidt norm. We then benchmark the algorithm and compare the generation of entanglement between the Hadamard quantum walk, the random quantum walk and the optimal quantum walk. Finally, we discuss an experimental scheme with a photonic quantum walk in the orbital angular momentum of light. The experimental measurement of entanglement can be achieved with quantum state tomography.

Controllable simulation of topological phases and edge states with quantum walk

The staggered quantum walk is a type of discrete-time quantum walk model without a coin which can be generated on a graph using particular partitions of the graph nodes. We design Hamiltonians for potential realization of the staggered dynamics on a two-dimensional lattice composed of superconducting microwave resonators connected with tunable couplings. The naive generalization of the one-dimensional staggered dynamics generates two uncoupled one-dimensional quantum walks thus more complex partitions need to be employed. However, by analyzing the coherence of the dynamics, we show that the quantumness of the evolution corresponding to two independent one-dimensional quantum walks can be elevated to the level of a single two-dimensional quantum walk, only by modifying the boundary conditions. In fact, by changing the lattice boundary conditions (or topology), we explore the walk on different surfaces such as torus, Klein bottle, real projective plane and sphere. The coherence and the entropy reach different levels depending on the topology of the surface. We observe that the entropy captures similar information as coherence, thus we use it to explore the effects of boundaries on the dynamics of the continuous-time quantum walk and the classical random walk.

This study investigated the unitary equivalence classes of one-dimensional quantum walks with and without initial states. We determined the unitary equivalence classes of one-dimensional quantum walks, two-phase quantum walks with one defect, complete two-phase quantum walks, one-dimensional quantum walks with one defect and translation-invariant one-dimensional quantum walks.

Quantum walks are promising for information processing tasks because on regular graphs they spread quadratically faster than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, whereby the spread of the walker stays within a finite region even in the infinite time limit. It is therefore important to understand when we can expect a quantum walk to be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a generic one-dimensional quantum walk -- the split-step walk -- to different forms of static disorder. We find that introducing static, symmetry-preserving disorder in the parameters of the walk leads to Anderson localization. In the completely disordered limit, however, a delocalization sets in, and the walk spreads subdiffusively. Using an efficient numerical algorithm, we calculate the bulk topological invariants of the disordered walk, and interpret the disorder-induced Anderson localization and delocalization transitions using these invariants.

This study investigates unitary equivalent classes of one-dimensional quantum walks. We prove that one-dimensional quantum walks are unitary equivalent to quantum walks of Ambainis type and that translation-invariant one-dimensional quantum walks are Szegedy walks. We also present a necessary and sufficient condition for a one-dimensional quantum walk to be a Szegedy walk.

The dynamic behavior of a physical system often originates from its spectral properties. In open systems, where the effective non-Hermitian description enables a wealth of spectral structures in the complex plane, the concomitant dynamics are significantly enriched, whereas the identification and comprehension of the underlying connections are challenging. Here we experimentally demonstrate the correspondence between the transient self-acceleration of local excitations and the non-Hermitian spectral topology using lossy photonic quantum walks. Focusing first on one-dimensional quantum walks, we show that the measured short-time acceleration of the wave function is proportional to the area enclosed by the eigenspectrum. We then reveal a similar correspondence in two-dimension quantum walks, where the self-acceleration is proportional to the volume enclosed by the eigenspectrum in the complex parameter space. In both dimensions, the transient self-acceleration crosses over to a long-time behavior dominated by a constant flow at the drift velocity. Our results unveil the universal correspondence between spectral topology and transient dynamics, and offer a sensitive probe for phenomena in non-Hermitian systems that originate from spectral geometry.

Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a lossy environment may destroy these phases. We investigate the behaviour of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian respects exact {{mathcal {P}}}{{mathcal {T}}}-symmetry. Although the topological nature persists in two-dimensional quantum walks as well, the {{mathcal {P}}}{{mathcal {T}}}-symmetry has no role to play there. Furthermore, we observe topological phase transition in two-dimensional quantum walks that is induced by losses in the system.

Disentanglement and loss of quantum correlations due to one global collective noise effect are described for two-qubit Schrödinger cat and Werner states of a four level trapped ion quantum system. Once the Jaynes–Cummings ionic interactions are mapped onto a Dirac spinor structure, the elementary tools for computing quantum correlations of two-qubit ionic states are provided. With two-qubit quantum numbers related to the total angular momentum and to its projection onto the direction of an external magnetic field (which lifts the degeneracy of the ion’s internal levels), a complete analytical profile of entanglement for the Schrödinger cat and Werner states is obtained. Under vacuum noise (during spontaneous emission), the two-qubit entanglement in the Schrödinger cat states is shown to vanish asymptotically. Otherwise, the robustness of Werner states is concomitantly identified, with the entanglement content recovered by their noiseless-like evolution. Most importantly, our results point to a firstly reported sudden transition between classical and quantum decay regimes driven by a classical collective noise on the Schrödinger cat states, which has been quantified by the geometric discord.

We analyze the application of the history state formalism to quantum walks. The formalism allows one to describe the whole walk through a pure quantum history state, which can be derived from a timeless eigenvalue equation. It naturally leads to the notion of system-time entanglement of the walk, which can be considered as a measure of the number of orthogonal states visited in the walk. We then focus on one-dimensional discrete quantum walks, where it is shown that such entanglement is independent of the initial spin orientation for real Hadamard-type coin operators and real initial states (in the standard basis) with definite site parity. Moreover, in the case of an initially localized particle it can be identified with the entanglement of the unitary global operator that generates the whole history state, which is related to its entangling power and can be analytically evaluated. Besides, it is shown that the evolution of the spin subsystem can also be described through a spin history state with an extended clock. A connection between its average entanglement (over all initial states) and that of the operator generating this state is also derived. A quantum circuit for generating the quantum walk history state is provided as well.