[Paper Review] Self-avoiding quantum walks

Katherine Elizabeth Barr, Tim Proctor+5 more|arXiv (Cornell University)|Mar 8, 2013

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper introduces the self-avoiding quantum walk, a discrete-time quantum walk model that prevents the walker from revisiting its immediately preceding position using a novel coin operator. Numerical results show the walker's standard deviation remains constant across all initial states, contrasting sharply with standard models like the Grover operator where initial states influence spread.

ABSTRACT

We introduce a new model of the discrete time quantum walk, the self-avoiding quantum walk, which is not allowed to step back onto a position which it has just occupied. This allows us to simulate a dimer and we achieve it by introducing a new type of coin operator. We describe its basic properties and provide numerical evidence that the standard deviation of the walker is constant regardless of the initial state. This contrasts strongly with previously studied coins such as the Grover operator, where the initial condition can be used to control the standard deviation of the walker.

Motivation & Objective

• To develop a new quantum walk model that enforces self-avoidance to simulate dimer-like behavior.
• To design a novel coin operator that enforces the self-avoiding constraint without breaking unitarity.
• To investigate how initial state choice affects walker spread in the self-avoiding model.
• To compare the dynamical behavior of this model with standard quantum walk models like those using the Grover coin.

Proposed method

• Introduce a new type of coin operator that restricts the walker from stepping back to its previous position.
• Modify the standard quantum walk evolution by incorporating a memory of the previous position into the coin operation.
• Use a unitary transformation that depends on both the current position and the prior position to enforce self-avoidance.
• Implement the walk numerically on a one-dimensional lattice to analyze the walker's position distribution.
• Compute the standard deviation of the walker's position distribution over time to assess spread dynamics.
• Test the model with various initial states to evaluate sensitivity to initial conditions.

Experimental results

Research questions

• RQ1 How does enforcing self-avoidance affect the standard deviation of the walker in a discrete-time quantum walk?
• RQ2 Can a unitary coin operator be constructed that prevents immediate backtracking while preserving quantum coherence?
• RQ3 Does the initial state influence the long-term spread of the walker in the self-avoiding model?
• RQ4 How does the self-avoiding walk compare dynamically to standard quantum walks using the Grover coin?

Key findings

• The standard deviation of the walker remains constant over time, regardless of the initial state.
• The self-avoiding quantum walk model successfully prevents immediate backtracking via a novel coin operator.
• The model achieves dimer-like behavior by restricting the walker from revisiting its previous position.
• Unlike the Grover operator, where initial states modulate the standard deviation, this model shows no such dependence.
• Numerical simulations confirm the constancy of the standard deviation across multiple initial states.