[Paper Review] Limit theorems and absorption problems for quantum random walks in one dimension

Norio Konno|ArXiv.org|Oct 2, 2002

Quantum Computing Algorithms and Architecture29 references4 citations

TL;DR

This paper establishes limit theorems and absorption probabilities for two types of one-dimensional quantum random walks (QRWs)—G-type (Gudder) and A-type (Ambainis et al.)—using a novel PQRS combinatorial method. It reveals distinct behaviors in limit distributions, symmetry, and absorption probabilities, showing that the A-type QRW has a higher absorption probability to the origin than the G-type, with exact bounds derived via spectral analysis and generating functions for finite and infinite systems.

ABSTRACT

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by 2 times 2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.

Motivation & Objective

• To clarify the fundamental differences in behavior between G-type and A-type quantum random walks in one dimension.
• To derive limit theorems for both types of QRWs using a combinatorial PQRS method, avoiding Fourier analysis.
• To analyze absorption probabilities and first hitting time moments for both QRW types in finite and infinite systems.
• To provide exact expressions for absorption probabilities and conditional moments using generating functions and spectral methods.

Proposed method

• The PQRS method decomposes the time evolution into operators P and Q acting on left and right chirality states, with P and Q defined via unitary matrix U.
• The amplitude evolution is modeled as |Ψ_{j,k}(n+1)⟩ = P_j |Ψ_{j,k+1}(n)⟩ + Q_j |Ψ_{j,k-1}(n)⟩, with distinct P and Q matrices for A-type and G-type QRWs.
• Generating functions and boundary conditions are used to derive expressions for the probability generating functions p_k^{(N)}(z) and r_k^{(N)}(z) in finite systems.
• Spectral analysis and contour integration are applied to compute absorption probabilities via integrals over the unit circle: P^{(N)}_{j,k}(φ) = ∫ |⋯|² dθ / (2π).
• The method enables exact computation of absorption probabilities and moments by expressing them in terms of Bessel functions and roots of characteristic equations.
• For the Hadamard walk (U = H), explicit formulas are derived for N ≥ 2, including closed forms for r_1^{(N)}(z) involving Bessel-like sums.

Experimental results

Research questions

• RQ1 How do the limit distributions of A-type and G-type quantum random walks differ in one dimension?
• RQ2 What is the symmetry structure of the probability distribution for each QRW type?
• RQ3 What are the exact absorption probabilities to the origin for both QRW types in finite and infinite systems?
• RQ4 How do the moments of the first hitting time to the origin behave conditionally on eventual absorption?
• RQ5 What is the precise relationship between the PQRS method and the spectral properties of the QRW evolution?

Key findings

• For the infinite system, the absorption probability to the origin from k=1 satisfies (4−π)/π ≤ P_{j,1}^{(∞)}(φ) ≤ 1 for both A-type and G-type QRWs.
• The conditional first moment of the first hitting time to 0 from k=1 is E_{j,1}^{(∞)}(T_0|T_0<∞) = 1 / P_{j,1}^{(∞)}(φ), with higher moments diverging for m≥2.
• For finite N≥2, the absorption probability P^{(N)}_{A,1}(φ) is expressed as a quadratic form in α, β with coefficients involving ∫ |r_1^{(N)}(e^{iθ})|² dθ / (2π), showing explicit dependence on initial state.
• For the Hadamard walk with initial state |R⟩ at k=1, P^{(N)}_{A,1}(|R⟩) satisfies the conjecture in Eq. (4.6) for N=2,…,6.
• Explicit formulas are derived for r_1^{(N)}(z), including r_1^{(2)}(z)=0, r_1^{(3)}(z)=z³/(2−z²), and for N≥4, r_1^{(N)}(z) = −z² J_{N-3}(z) J_{N-4}(z) / [√2 (J_{N-3}(z))² − z J_{N-3}(z) J_{N-4}(z) − √2 (−1)^{N-3}].
• The G-type QRW absorption probability at k=1 is P^{(N)}_{G,1}(φ) = |α|² + (∫ |r_1^{(N)}(e^{iθ})|² dθ / 2π) |β|², showing a simpler dependence on the initial state than the A-type.