The time-evolution equation of a one-dimensional quantum walker is exactly mapped to the three-dimensional Weyl equation for a zero-mass particle with spin $1∕2$, in which each wave number $k$ of the walker's wave function is mapped to a point $\mathbf{q}(k)$ in the three-dimensional momentum space and $\mathbf{q}(k)$ makes a planar orbit as $k$ changes its value in $[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi})$. The integration over $k$ providing the real-space wave function for a quantum walker corresponds to considering an orbital state of a Weyl particle, which is defined as a superposition (curvilinear integration) of the energy-momentum eigenstates of a free Weyl equation along the orbit. Konno's novel distribution function of a quantum walker's pseudovelocities in the long-time limit is fully controlled by the shape of the orbit and how the orbit is embedded in the three-dimensional momentum space. The family of orbital states can be regarded as a geometrical representation of the unitary group U(2) and the present study will propose a new group-theoretical point of view for quantum-walk problems.
The time-evolution equation of a one-dimensional quantum walker is exactly mapped to the three-dimensional Weyl equation for a zero-mass particle with spin 1∕2, in which each wave number k of the walker's wave function is mapped to a point q(k) in the three-dimensional momentum space and q(k) makes a planar orbit as k changes its value in [\ensuremath−\ensuremathπ,\ensuremathπ). The integration over k providing the real-space wave function for a quantum walker corresponds to considering an orbital state of a Weyl particle, which is defined as a superposition (curvilinear integration) of the energy-momentum eigenstates of a free Weyl equation along the orbit. Konno's novel distribution function of a quantum walker's pseudovelocities in the long-time limit is fully controlled by the shape of the orbit and how the orbit is embedded in the three-dimensional momentum space. The family of orbital states can be regarded as a geometrical representation of the unitary group U(2) and the present study will propose a new group-theoretical point of view for quantum-walk problems.
Inspired by the covering-space method in path integrals on multiply connected spaces, we here present a universal formula of time-evolution kernels for continuous- and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Lambda}}$ is an arbitrary lattice and $\mathrm{\ensuremath{\Gamma}}$ is a discrete group whose action on $\mathrm{\ensuremath{\Lambda}}$ has no fixed points. We show that the time-evolution kernel on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$ can be written as a weighted sum of time-evolution kernels on $\mathrm{\ensuremath{\Lambda}}$, where the summation is over the orbit of initial point in $\mathrm{\ensuremath{\Lambda}}$ and weight factors are given by a one-dimensional unitary representation of $\mathrm{\ensuremath{\Gamma}}$. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulas of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on $\mathrm{\ensuremath{\Lambda}}/\mathrm{\ensuremath{\Gamma}}$, all of which are constructed in exactly the same way as for the time-evolution kernel.
Discrete-time quantum walks are considered a discretization of Dirac equations and the study for them has been getting attention since around 2000. In this paper, we focus on a quantum walk on a line related to a Weyl equation in a space. The quantum walker with two inner states, also interpreted as a quantum particle with the down-spin and the up-spin state in physics, spreads on the line and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.
Quantization of a random-walk model is performed by giving a qudit (a multicomponent wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of the walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine quantum coins. We show that Wigner's (2j+1)-dimensional unitary representations of rotations with half-integers j's are useful to analyze the probability laws of quantum walks. For any value of half-integer j, convergence of all moments of walker's pseudovelocity in the long-time limit is proved. It is generally shown for the present models that, if (2j+1) is even, the probability measure of limit distribution is given by a superposition of (2j+1)∕2 terms of scaled Konno's density functions, and if (2j+1) is odd, it is a superposition of j terms of scaled Konno's density functions and a Dirac's delta function at the origin. For the two-, three-, and four-component models, the probability densities of limit distributions are explicitly calculated and their dependence on the parameters of quantum coins and on the initial qudit of walker is completely determined. Comparison with computer simulation results is also shown.
A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac walks, provide a description of the free evolution of relativistic quantum fields in the small wave-vector regime. The clash between the intrinsic discreteness of quantum walks and the continuous symmetries of special relativity is resolved by giving a definition of change of inertial frame in terms of a change of values of the constants of motion, which leaves the walk operator unchanged. Starting from the family of 1 + 1 dimensional Dirac walks with all possible values of the mass parameter, we introduce a unique walk encompassing the latter as an extra degree of freedom, and we derive its group of changes of inertial frames. This symmetry group contains a non-linear realization of SO+(2,1)⋉R3; since one of the two space-like dimensions does not correspond to an actual spatial degree of freedom but rather the mass, we interpret it as a 2 + 1 dimensional de Sitter group. This group also contains a non-linear realization of the proper orthochronous Poincaré group SO+(1,1)⋉R2 in 1 + 1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry in the limit of small wave-vectors and masses. Surprisingly, for the Dirac walk with a fixed value of the mass parameter, the group of allowed changes of reference frame does not have a consistent interpretation in the limit of small wave-vectors.
The weak convergence theorems of the one- and two-dimensional simple quantum walks, SQW(d),d=1,2, show a striking contrast to the classical counterparts, the simple random walks, SRW(d). In the SRW(d), the distribution of position X(t) of the particle starting from the origin converges to the Gaussian distribution in the diffusion scaling limit, in which the time scale T and spatial scale L both go to infinity as the ratio L/T is kept finite. On the other hand, in the SQW(d), the ratio L/T is kept to define the pseudovelocity V(t)=X(t)/t, and then all joint moments of the components ${V}_{j}(t),1\ensuremath{\leqslant}j\ensuremath{\leqslant}d$, of V(t) converge in the $T=L\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. The limit distributions have novel structures such that they are inverted-bell shaped and their supports are bounded. In the present paper we claim that these properties of the SQW(d) can be explained by the theory of relativistic quantum mechanics. We show that the Dirac equation with a proper ultraviolet cutoff can provide a quantum walk model in three dimensions, where the walker has a four-component qubit. We clarify that the pseudovelocity V(t) of the quantum walker, which solves the Dirac equation, is identified with the relativistic velocity. Since the quantum walker should be a tardyon, not a tachyon, ∣V(t)∣<c, where c is the speed of light, and this restriction (the causality) is the origin of the finiteness of supports of the limit distributions universally found in quantum walk models. By reducing the number of components of momentum in the Dirac equation, we obtain the limit distributions of pseudovelocities for the lower dimensional quantum walks. We show that the obtained limit distributions for the one- and two-dimensional systems have common features with those of SQW(1) and SQW(2).
It is well known that 1+1-dimensional Weyl and Dirac equations in the relativistic quantum mechanics are related to some quantum walks. However, in 3+1-dimensions, discrete-time quantum walks corresponding to the above two equations have not been obtained. We show that the 3+1-dimensional Weyl and Dirac walkers cannot be obtained by generalizing the above well-known model to 3+1-dimensional cases. Apparently, these 3+1-dimensional extended models seem to be well defined, but they do not have unitary time evolution. Thus, the square of the magnitude of each spinor is not able to be any probability distribution. In addition, we present two Schrödinger walkers, and one of which has unitary time evolution. This unitary Schrödinger walker can move to all sites at each time step, unlike the existing models. Therefore, it is mentioned that adequate models of 3+1-dimensional Weyl and Dirac walkers will also be given like this unitary Schrödinger walker.