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arXiv:quant-ph/0611022 · Physical Review A

Wigner formula of rotation matrices and quantum walks

Takahiro Miyazaki, Makoto Katori, Norio Konno

Published Jul 31, 2007• 38 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.76.012332

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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)(2j+1)(2j+1)-dimensional unitary representations of rotations with half-integers jjj's are useful to analyze the probability laws of quantum walks. For any value of half-integer jjj, 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)(2j+1)(2j+1) is even, the probability measure of limit distribution is given by a superposition of (2j+1)∕2(2j+1)∕2(2j+1)∕2 terms of scaled Konno's density functions, and if (2j+1)(2j+1)(2j+1) is odd, it is a superposition of jjj 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.
One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$ of all joint moments of two components of walker's pseudovelocity, Xt/t{X}_{t}/tXt​/t and Yt/t{Y}_{t}/tYt​/t, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.
The analysis of a physical problem simplifies considerably when one uses a suitable coordinate system. We apply this approach to the discrete-time quantum walks with coins given by 2j+12j+12j+1-dimensional Wigner rotation matrices (Wigner walks), a model which was introduced in Miyazaki et al. [Phys. Rev. A 76, 012332 (2007)]. First, we show that from the three parameters of the coin operator only one is physically relevant for the limit density of the Wigner walk. Next, we construct a suitable basis of the coin space in which the limit density of the Wigner walk acquires a much simpler form. This allows us to identify various dynamical regimes which are otherwise hidden in the standard basis description. As an example, we show that it is possible to find an initial state which reduces the number of peaks in the probability distribution from generic 2j+12j+12j+1 to a single one. Moreover, the models with integer jjj lead to the trapping effect. The derived formula for the trapping probability reveals that it can be highly asymmetric and it deviates from purely exponential decay. Explicit results are given up to dimension 5.
We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers j=1/2,1,3/2,...j=1/2, 1, 3/2, ...j=1/2,1,3/2,..., introduced by Miyazaki {\it et al.}, each of which the walker's wave function has 2j+12j+12j+1 components and hopping range at each time step is 2j2j2j. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konno's density function. Since Konno's density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the (2j+1)(2j+1)(2j+1)-component model can have 2j+12j+12j+1 pikes, when 2j+12j+12j+1 is even. When jjj becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the j→∞j \to \inftyj→∞ limit.
We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.
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∕21∕21∕2, in which each wave number kkk of the walker's wave function is mapped to a point q(k)\mathbf{q}(k)q(k) in the three-dimensional momentum space and q(k)\mathbf{q}(k)q(k) makes a planar orbit as kkk changes its value in $[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi})$. The integration over kkk 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 evolution of a walker in standard "Discrete-time Quantum Walk (DTQW)" is determined by coin and shift unitary operators. The conditional shift operator shifts the position of the walker to right or left by unit step size while the direction of motion is specified by the coin operator. This scenario can be generalized by choosing the step size randomly at each step in some specific interval. For example, the value of the roll of a dice can be used to specify the step size after throwing the coin. Let us call such a quantum walk "Discrete-time Random Step Quantum Walk (DTRSQW)". A completely random probability distribution is obtained whenever the walker follows the DTRSQW. We have also analyzed two more types of quantum walks, the "Discrete-time Un-biased Quantum Walk (DTUBQW)" and the "Discrete-time Biased Quantum Walk (DTBQW)". In the first type, the step size is kept different than unit size but the same for left and right shifts, whereas in the second type left and right shifts can also be different. The probability distribution in DTUBQW is found to follow a certain rule. The standard deviation (σσσ) of DTRSQW is higher than DTQW and hence DTRSQW spreads faster. The σσσ of DTUBQW shows sawtooth behavior with faster spread than DTQW for some specific values of rotation angles and steps.