Split-step Quantum Walk Research Articles

Absence of bound states for quantum walks and CMV matrices via reflections

We give a criterion based on reflection symmetries in the spirit of Jitomirskaya–Simon to show absence of point spectrum for (split-step) quantum walks and Cantero–Moral–Velázquez (CMV) matrices. To accomplish this, we use some ideas from a recent paper by the authors and collaborators to implement suitable reflection symmetries for such operators. We give several applications. For instance, we deduce arithmetic delocalization in the phase for the unitary almost-Mathieu operator and singular continuous spectrum for generic CMV matrices generated by the Thue–Morse subshift.

On rotated CMV operators and orthogonal polynomials on the unit circle

Split-step quantum walk operators can be expressed as a generalized version of CMV operators with complex transmission coefficients, which we call rotated CMV operators. Following the idea of Cantero, Moral and Velazquez's original construction of the original CMV operators from the theory of orthogonal polynomials on the unit circle (OPUC), we show that rotated CMV operators can be constructed similarly via a rotated version of OPUCs with respect to the same measure, and admit an analogous LM -factorization as the original CMV operators. We also develop the rotated second kind polynomials corresponding to the rotated OPUCs. We then use the LM -factorization of rotated alternate CMV operators to compute the Gesztesy–Zinchenko transfer matrices for rotated CMV operators.

Eigenvalues and threshold resonances of a two-dimensional split-step quantum walk with strong shift

Eigenvalues and threshold resonances of a two-dimensional split-step quantum walk with strong shift

Index Theory of Chiral Unitaries and Split-Step Quantum Walks

Building from work by Cedzich et al. and Suzuki et al., we consider topological and index-theoretic properties of chiral unitaries, which are an abstraction of the time evolution of a chiral-symmetric self-adjoint operator. Split-step quantum walks provide a rich class of examples. We use the index of a pair of projections and the Cayley transform to define topological indices for chiral unitaries on both Hilbert spaces and Hilbert $C^*$-modules. In the case of the discrete time evolution of a Hamiltonian-like operator, we relate the index for chiral unitaries to the index of the Hamiltonian. We also prove a double-sided winding number formula for anisotropic split-step quantum walks on Hilbert $C^*$-modules, extending a result by Matsuzawa.

The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition

It is recently shown that a split-step quantum walk possesses a chiral symmetry, and that a certain well-defined index can be naturally assigned to it. The index is a well-defined Fredholm index if and only if the associated unitary time-evolution operator has spectral gaps at both [Formula: see text] and [Formula: see text] In this paper, we extend the existing index formula for the Fredholm case to encompass the non-Fredholm case (i.e. gapless case). We make use of a natural extension of the Fredholm index to the non-Fredholm case, known as the Witten index. The aim of this paper is to fully classify the Witten index of the split-step quantum walk by employing the spectral shift function for a rank one perturbation of a fourth-order difference operator. It is also shown in this paper that the Witten index can take half-integer values in the non-Fredholm case.

Topological invariants in quantum walks

Discrete-time quantum walks (DTQWs) provide a convenient platform for a realization of many topological phases in noninteracting systems. They often offer more possibilities than systems with a static Hamiltonian. Nevertheless, researchers are still looking for DTQW symmetries protecting topological phases and for definitions of appropriate topological invariants. Although the majority of DTQW studies on this topic focus on the so-called split-step quantum walk, two distinct topological phases can be observed in more basic models. Here we infer topological properties of the basic DTQWs directly from the mapping of the Brillouin zone to the Bloch Hamiltonian. We show that for translation-symmetric systems they can be characterized by a homotopy relative to special points. We also propose a topological invariant corresponding to this concept. This invariant indicates the number of edge states at the interface between two distinct phases.

Optical realization of one-dimensional generalized split-step quantum walks

Quantum walks are more than tools for building quantum algorithms. They have been used effectively to model and simulate quantum dynamics in many complex physical processes. Particularly, a variant of discrete-time quantum walk known as split-step quantum walk is closely related to Dirac cellular automata and topological insulators, whose realizations rely on position-dependent control of evolution operators. Owing to the ease of manipulating multiple degrees of freedom of photons, we provide an optical setup of split-step operators which, in combination with position-dependent coin (PDC) operation, can accomplish a table-top setup of generalized split-step walks. Also, we propose an optical implementation for PDC operation that allows, for instance, realizing electric quantum walks, control localization dynamics, and emulate space-time curvature effects. In addition, we propose a setup to realize any t-step split-step quantum walk involving 2 J-plates, 2 variable waveplates, a half-waveplate, an optical switch, and an optical delay line.

Spectral analysis for a multi-dimensional split-step quantum walk with a defect

This paper studies the spectrum of a multi-dimensional split-step quantum walk with a defect that cannot be analyzed in previous papers by Fuda et al. (Quantum Inf Process 18:203–226, 2017; J Math Phys 59:082201, 2018). To this end, we have developed a new technique which allow us to use a spectral mapping theorem for the one-defect model. We also derive the time-averaged limit measure for one-dimensional case as an application of the spectral analysis.

Unitary equivalence classes of split-step quantum walks

In this study, we investigate the unitary equivalence of split-step quantum walks (SSQWs). Introduced by Kitagawa and generalized by Suzuki, SSQWs are two-state discrete-time quantum walks on a one-dimensional lattice. In this work, we define a new class of quantum walks that includes all SSQWs by making use of the rank conditions, which state that (1) the rank of operators representing the walk from vertex x to vertex $$x\pm 1$$ is less than or equal to 1 and that (2) the rank from vertex x to x (i.e., to itself) is less than or equal to 2. Initially, we abstractly define quantum walks in this class, then clarify by presenting the explicit form of such quantum walks. We also calculate their unitary equivalence classes and show that four real parameters are required for each vertex to parameterize the unitary equivalence classes. Further, we consider the unitary equivalence of Suzuki’s SSQWs and prove that all parameters of Suzuki’s SSQWs can be real. Finally, we discuss the chiral symmetry of SSQWs.

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

We investigate the role of a time and spin-dependent phase shift on the evolution of one-dimensional discrete-time quantum walks. By employing Floquet engineering a time and spin-dependent phase shift (ϕ) is stroboscopically imprinted onto the walker's wave function at the end of each step of the walk. For a quantum walk driven by the standard protocol and with rational values of the phase factor, i.e., ϕ/2π=p/q we show with our numerical simulations that revivals with equal periods occur in the probability distribution of the walk. In the split-step quantum walk we employ two different coin operators which are parametrized by a parameter δθ. By making use of δθ as a control parameter we demonstrate that for ϕ/2π=p/q the walk exhibits three major types of dynamics: ballistic spreading, revivals, and localization. For an irrational value of ϕ/2π our results show revivals with unpredictable periods, and the walker remains localized in a small region of the lattice. Furthermore, in view of an experimental realization we investigate the robustness of revivals against noise in the phase shift. Our results show that signatures of revivals persist for smaller values of the noise parameter while these vanish for larger values. Our work is important in the context of quantum computation and simulation with quantum walks where different types of the walk dynamics can be engineered by making use of the time and spin-dependent phases shift, and the coin parameter as control knobs.

Persistence of topological phases in non-Hermitian quantum walks

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.

Topological delocalization in the completely disordered two-dimensional quantum walk

We investigate numerically and theoretically the effect of spatial disorder on two-dimensional split-step discrete-time quantum walks with two internal "coin" states. Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts. We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization but to a diffusive spread instead. This is a delocalization, which happens because disorder places the quantum walk to a critical point between different anomalous Floquet-Anderson insulating topological phases. We base this explanation on the relationship of this general quantum walk to a simpler case more studied in the literature and for which disorder-induced delocalization of a topological origin has been observed. We review topological delocalization for the simpler quantum walk, using time evolution of the wave functions and level spacing statistics. We apply scattering theory to two-dimensional quantum walks and thus calculate the topological invariants of disordered quantum walks, substantiating the topological interpretation of the delocalization and finding signatures of the delocalization in the finite-size scaling of transmission. We show criticality of the Haar random quantum walk by calculating the critical exponent $\eta$ in three different ways and find $\eta$ $\approx$ 0.52 as in the integer quantum Hall effect. Our results showcase how theoretical ideas and numerical tools from solid-state physics can help us understand spatially random quantum walks.

An index theorem for split-step quantum walks

Split-step quantum walks are models of supersymmetric quantum walk, and thus their Witten indices can be defined. We prove that the Witten index of a split-step quantum walk coincides with the difference between the winding numbers of functions corresponding to the right-limit of coins and the left-limit of coins. As a corollary, we give an alternative derivation of the index formula for split-step quantum walks, which is recently obtained by Suzuki and Tanaka.

Measuring a dynamical topological order parameter in quantum walks

Quantum processes of inherent dynamical nature, such as quantum walks, defy a description in terms of an equilibrium statistical physics ensemble. Until now, identifying the general principles behind the underlying unitary quantum dynamics has remained a key challenge. Here, we show and experimentally observe that split-step quantum walks admit a characterization in terms of a dynamical topological order parameter (DTOP). This integer-quantized DTOP measures, at a given time, the winding of the geometric phase accumulated by the wavefunction during a quantum walk. We observe distinct dynamical regimes in our experimentally realized quantum walks, and each regime can be attributed to a qualitatively different temporal behavior of the DTOP. Upon identifying an equivalent many-body problem, we reveal an intriguing connection between the nonanalytic changes of the DTOP in quantum walks and the occurrence of dynamical quantum phase transitions.

The Witten index for 1D supersymmetric quantum walks with anisotropic coins

Chirally symmetric discrete-time quantum walks possess supersymmetry, and their Witten indices can be naturally defined. The Witten index gives a lower bound for the number of topologically protected bound states. The purpose of this paper is to give a complete classification of the Witten index associated with a one-dimensional split-step quantum walk. It turns out that the Witten index of this model exhibits striking similarity to the one associated with a Dirac particle in supersymmetric quantum mechanics.

Supersymmetry for chiral symmetric quantum walks

Quantum walks have attracted attention as a promising platform realizing topological phenomena, and many physicists have introduced various types of indices to characterize topologically protected bound states that are robust against perturbations. In this paper, we introduce an index from a supersymmetric point of view. This allows us to define indices for all chiral symmetric quantum walks such as multi-dimensional split-step quantum walks and quantum walks on graphs, for which there has been no index theory. Moreover, the index gives a lower bound on the number of bound states robust against compact perturbations. We also calculate the index for several concrete examples including the unitary transformation that appears in Grover’s search algorithm.

Eigenvalue measurement of topologically protected edge states in split-step quantum walks

We study topological phenomena of quantum walks by implementing a novel protocol that extends the range of accessible properties to the eigenvalues of the walk operator. To this end, we experimentally realise for the first time a split-step quantum walk with decoupling, which allows for investigating the effect of a bulk-boundary while realising only a single bulk configuration. The experimental platform is implemented with the well-established time-multiplexing architecture based on fibre-loops and coherent input states. The symmetry protected edge states are approximated with high similarities and we read-out the phase relative to a reference for all modes. In this way we observe eigenvalues which are distinguished by the presence or absence of sign flips between steps. Furthermore, the results show that investigating a bulk-boundary with a single bulk is experimentally feasible when decoupling the walk beforehand.

Simulating Dirac Hamiltonian in curved space-time by split-step quantum walk

Dirac particle represents a fundamental constituent of our nature. Simulation of Dirac particle dynamics by a controllable quantum system using quantum walks will allow us to investigate the non-classical nature of dynamics in its discrete form. In this work, starting from a modified version of one-spatial dimensional general inhomogeneous split-step discrete quantum walk we derive an effective Hamiltonian which mimics a single massive Dirac particle dynamics in curved (1 + 1) space-time dimension coupled to U(1) gauge potential—which is a forward step towards the simulation of the unification of electromagnetic and gravitational forces in lower dimension and at the single particle level. Implementation of this simulation scheme in simple qubit-system has been demonstrated. We show that the same Hamiltonian can represent (2 + 1) space-time dimensional Dirac particle dynamics when one of the spatial momenta remains fixed. We also discuss how we can include U(N) gauge potential in our scheme, in order to capture other fundamental force effects on the Dirac particle. The emergence of curvature in the two-particle split-step quantum walk has also been investigated while the particles are interacting through their entangled coin operation.

Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations

For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in the work of Higuchi et al. (e-print arXiv:1506.06457) [see also E. Segawa and A. Suzuki, Quantum Stud.: Math. Found. 3, 11 (2016)]. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there are no eigenvalues around ±1 in terms of a discriminant operator. We also provide a criterion for when eigenvalues ±1 exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations.

Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped Dirac equation

We study the dynamics of discrete-time quantum walk using quantum coin operations, $\hat{C}(\theta_1)$ and $\hat{C}(\theta_2)$ in time-dependent periodic sequence. For the two-period quantum walk with the parameters $\theta_1$ and $\theta_2$ in the coin operations we show that the standard deviation [$\sigma_{\theta_1, \theta_2} (t)$] is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations $\theta_1$ or $\theta_2$, $\sigma_{\theta_1, \theta_2}(t) = \min \{\sigma_{\theta_1}(t), \sigma_{\theta_2}(t) \}$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walks, we present the bounds on the dynamics of three- and higher period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using $\theta_1$ and $\theta_2$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\theta=0$, here we show the recovery of the massless Dirac equation with non-zero $\theta$ parameters contributing to the intriguing interference in the dynamics in a totally non-relativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.