[Paper Review] Periodicity for the Hadamard walk on cycles

Norio Konno, Yuki Shimizu+1 more|arXiv (Cornell University)|Apr 24, 2015

Quantum Computing Algorithms and Architecture15 references3 citations

TL;DR

This paper proves that the Hadamard walk on a cycle $C_N$ is periodic only for $N = 2, 4, 8$, with periods $T_2 = 2$, $T_4 = 8$, and $T_8 = 24$, respectively. Using path counting and cyclotomic polynomial analysis, the authors show that for all other $N$, the walk lacks periodicity ($T_N = ar{ ext{infty}}$), contrasting with Dukes' eigenvalue-based method by introducing a novel algebraic approach grounded in polynomial factorization.

ABSTRACT

The present paper treats the period T_N of the Hadamard walk on a cycle C_N with N vertices. Dukes (2014) considered the periodicity of more general quantum walks on C_N and showed T_2 =2, T_4=8, T_8=24 for the Hadamard walk case. We prove that the Hadamard walk does not have any period except for his case, i.e., N=2, 4, 8. Our method is based on a path counting and cyclotomic polynomials which is different from his approach based on the property of eigenvalues for unitary matrix that determines the evolution of the walk.

Motivation & Objective

• To determine the exact set of cycle sizes $N$ for which the Hadamard walk on $C_N$ exhibits periodic behavior.
• To resolve the open question of whether periodicity occurs beyond the known cases $N=2,4,8$.
• To develop and apply a new method based on path counting and cyclotomic polynomial factorization to analyze periodicity.
• To provide a proof that $T_N = ar{ ext{infty}}$ for all $N \notin \{2,4,8\}$, confirming non-periodicity in all other cases.

Proposed method

• The authors analyze the characteristic polynomial $\det(\lambda I_{2N} - U_N^{(s)})$ of the evolution operator $U_N^{(s)}$ for the Hadamard walk on $C_N$.
• They decompose the characteristic polynomial into a product of cyclotomic polynomials $F_r(\lambda)$ and an additional polynomial $G(\lambda)$.
• The presence of a non-cyclotomic factor $G(\lambda)$ in the factorization implies that no finite $n$ satisfies $(U_N^{(s)})^n = I_{2N}$, hence $T_N = \infty$.
• For odd $N$, they prove the existence of a non-cyclotomic factor $G(\lambda)$, ruling out periodicity.
• For $N = 2^n \times M$ with $M$ odd and $n \geq 1$, they reduce the problem to the odd $M$ case and extend the non-periodicity result.
• They explicitly compute the characteristic polynomial for $N=16$ and show it contains non-cyclotomic factors, confirming $T_{16} = \infty$, and generalize this to $N = 2^n$ for $n \geq 5$.

Experimental results

Research questions

• RQ1 For which cycle sizes $N$ is the Hadamard walk on $C_N$ periodic?
• RQ2 Does periodicity persist beyond the known cases $N=2,4,8$?
• RQ3 Can periodicity be determined using path counting and cyclotomic polynomial factorization rather than eigenvalue analysis?
• RQ4 What is the algebraic structure of the characteristic polynomial of the evolution operator $U_N^{(s)}$ for general $N$?
• RQ5 Why does the walk fail to be periodic for $N \notin \{2,4,8\}$, and what algebraic invariant signals this failure?

Key findings

• The Hadamard walk on $C_N$ is periodic only for $N = 2, 4, 8$, with $T_2 = 2$, $T_4 = 8$, and $T_8 = 24$.
• For $N = 3$, the characteristic polynomial contains a non-cyclotomic factor $G(\lambda)$, so $T_3 = \infty$.
• For all odd $N$, the characteristic polynomial includes a non-cyclotomic factor, implying $T_N = \infty$.
• For $N = 16 = 2^4$, the characteristic polynomial contains non-cyclotomic factors $\lambda^4 - \frac{1}{2}\lambda^2 + 1$ and $\lambda^4 - \frac{3}{2}\lambda^2 + 1$, so $T_{16} = \infty$.
• For $N = 2^n$ with $n \geq 5$, the characteristic polynomial contains non-cyclotomic factors, so $T_N = \infty$.
• The method based on path counting and cyclotomic polynomial factorization provides a new, algebraically distinct approach to periodicity analysis, differing from eigenvalue-based methods.