[Paper Review] On the support of the Grover walk on higher-dimensional lattices

Norio Konno, Sarato Takahashi|arXiv (Cornell University)|Jan 28, 2020

Quantum chaos and dynamical systems8 references4 citations

TL;DR

This paper determines the minimum support size for stationary measures of the Grover walk on d-dimensional lattices by solving the eigenvalue problem $U_G\Psi = \lambda\Psi$ for $|\lambda|=1$. For moving shifts, the minimum support is $2^d$ points; for flip-flop shifts, it is 4 points for $d \geq 2$, with no finite support in 1D. The results refine prior work using spectral and algebraic methods.

ABSTRACT

This paper presents the minimum supports of states for stationary measures of the Grover walk on the d-dimensional lattice by solving the corresponding eigenvalue problem. The numbers of the minimum supports for moving and flip-flop shifts are 2^d (d ge 1) and 4 (d ge 2), respectively.

Motivation & Objective

• To determine the minimum number of lattice sites supporting stationary measures for the Grover walk on $\mathbb{Z}^d$.
• To resolve discrepancies and improve upon prior results on support size using eigenvalue analysis.
• To establish that no finite support exists for the 1D Grover walk with moving shift.
• To unify and confirm results from spectral mapping and Fourier analysis through direct solution of the eigenvalue problem.

Proposed method

• Solve the eigenvalue problem $U_G\Psi = \lambda\Psi$ for $|\lambda|=1$ to identify stationary amplitudes.
• Use the Grover coin matrix $G$ to define the time evolution operator $U_G$ on $\mathbb{Z}^d$.
• Apply projection operators $P_{2i-1}, P_{2i}$ to decompose the unitary evolution into directional shifts.
• Employ the spectral mapping theorem and algebraic constraints to derive support size bounds.
• Construct explicit examples of stationary states with minimal support for $d=2$ and $d \geq 3$.
• Verify that the support size is invariant under the walk's time evolution by checking invariance of the amplitude configuration.

Experimental results

Research questions

• RQ1 What is the minimal number of lattice sites that can support a stationary measure for the Grover walk on $\mathbb{Z}^d$ with moving shift?
• RQ2 What is the minimal number of lattice sites supporting a stationary measure for the Grover walk on $\mathbb{Z}^d$ with flip-flop shift?
• RQ3 Does a finite support exist for the stationary measure in the 1D Grover walk with moving shift?
• RQ4 How do the results compare with prior Fourier-analytic and spectral mapping approaches?

Key findings

• For the moving shift model on $\mathbb{Z}^d$ with $d \geq 1$, the minimum support size is $2^d$.
• For the flip-flop shift model on $\mathbb{Z}^d$ with $d \geq 2$, the minimum support size is 4.
• No finite support exists for the 1D Grover walk with moving shift, as shown by contradiction in the eigenvalue system.
• Explicit stationary states with exactly 4 non-zero entries are constructed for $d \geq 2$ under flip-flop shift.
• The results for $d=2$ and $d \geq 3$ are consistent with and generalize prior results from Stefanak et al. and Higuchi et al.
• The construction of $\Psi^{(\lambda)}_\star$ with $\#(S(\Psi^{(\lambda)}_\star)) = 4$ confirms the tightness of the bound for flip-flop shifts.