Limit Distribution of Quantum Walks and Weyl Equation

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.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.

Apr 14, 20170 citations

Limit Theorems for Open Quantum Random Walks

Norio Konno, Hyun Jae Yoo

arXiv:1209.1419Journal of Statistical PhysicsDec 13, 201251 citations

Quantum walks and orbital states of a Weyl particle

Makoto Katori, Soichi Fujino, Norio Konno

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 $\mathbf{q}(k)$ 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.

arXiv:quant-ph/0503142Physical Review AJul 14, 200526 citations

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.