Abstract : Eigenvalue problems for matrices and spectral theory for unitary operators and self-adjoint operators are important research area of quantum walks. In fact, spectra of time-evolution operators of quantum walks determine the dynamics of quantum walkers. In this minisymposium, we are going to have eight talks on the scattering and the spectral theory for quantum walks as well as eigenvalue problems on quantum walks on finite graphs and related topics. Some of them are going to introduce studies of quantum walks in view of theoretical physics and laser engineering.
[01759] The Ihara expression of graph zeta functions
Format : Talk at Waseda University
Author(s) :
Ayaka Ishikawa (Yokohama National University)
Abstract : Konno and Sato showed that the Grover walk corresponds to the Sato zeta function. They also gave the characteristic polynomial of the transition matrix of the Grover walk, using the Ihara expression of the Sato zeta function.
We define the graph zeta function related to the Szegedy walk on a finite graph and give the Ihara expression. The Ihara expression will extend Konno-Sato's result to the Szegedy walk.
[02778] Topological stability of quantum walks and related
Format : Talk at Waseda University
Author(s) :
Chris Bourne (Nagoya University)
Abstract : Quantum walks with additional symmetries may possess topologically protected bound states, meaning these bound states are robust against small perturbations and changes. I will give a gentle mathematical introduction to this phenomenon, which borrows ideas from so-called topological phases of matter. I will also explain how the existence of topological bound states can be detected using a winding number-like formula.
[03296] Spectral mapping theorem for quantum walks on graphs
Format : Talk at Waseda University
Author(s) :
Kei Saito (Kanagawa University)
Abstract : Quantum walks (QWs) are well known as a quantum version of random walks, and their time evolution is described by a unitary operator. Segawa, Suzuki(2019) show that the spectrum of the QW is given by that of a self-adjoint operator on the underlying graph. In this talk we reconsider such a spectral mapping theorem from a different perspective and present a relation between the spectrum of QWs and the weighted line matrix.
[05096] The Segawa-Suzuki spectral mapping theorem, revisited
Format : Talk at Waseda University
Author(s) :
Yohei Tanaka (Gakushuin University)
Abstract : The Segawa-Suzuki spectral mapping theorem for chiral unitaries is particularly useful when studying spectral properties of chiral symmetric quantum walks. For example, it states that topologically protected bound states can be characterised by elements of the so-called birth and inherited eigenspaces. The purpose of this talk is to show that this characterisation has a yet another interpretation in terms of the real part of a chiral unitary.
Abstract : We study long time behavior of "open" quantum walks by introducing resonances and the resonance expansion. Eigenvalues on the unit circle characterize the localization of quantum walkers. However, an open quantum walk does not have eigenvalues but may have resonances inside the unit circle. Then the decay rate of the quantum walker in any bounded region is described by the modulus of resonances. The eigenstate expansion is generalized to the resonance expansion.
[04207] Spectral scattering theory for quantum walks
Format : Talk at Waseda University
Author(s) :
Akito Suzuki (Shinshu university)
Abstract : Spectral scattering theory works to understand the dynamics of quantum walks and obtain the limit distribution of the asymptotic velocity of the walker, which gives a quantum version of the central limit theorem. In this talk, I would like to talk about the recent development of spectral scattering theory for one-dimensional quantum walks.
[04051] Quantum Walk-Based Maze-Solving with Absorbing Holes
Format : Talk at Waseda University
Author(s) :
Leo Matsuoka (Hiroshima Institute of Technology)
Kenta Yuki (Freelancer)
Hynek Lavicka (STTech GmbH)
Etsuo Segawa (Yokohama National University)
Abstract : We propose a strategy for finding the shortest path on a bipartite graph maze using a discrete-time quantum walk with absorbing holes. Our numerical analysis shows that the chain of maximum trapped densities detects the shortest paths in most cases. Furthermore, we discuss the speed of the algorithm and propose a strategy for accelerating it using numerical analysis. Our results offer a potential model for autonomous maze-solving optimization by harnessing natural phenomena.
[03689] Distinguishability and Complexity in Non-Unitary Boson Sampling Dynamics
Format : Online Talk on Zoom
Author(s) :
Ken Mochizuki (RIKEN)
Abstract : We show that quantum walks of many photons are closely related to the boson sampling problem and computational complexity. In addition, we consider non-unitary quantum walks, which correspond to photonic dynamics in open quantum systems with post selection. We clarify that the distribution of photons can approach that of distinguishable particles in the long time limit, which makes the non-unitary boson sampling problem easy.
Ken Mochizuki and Ryusuke Hamazaki, Physical Review Research 5 013177 (2023).
