[Paper Review] Continuum Limits of the 1D Discrete Time Quantum Walk

Michael Manighalam, Mark Kon|arXiv (Cornell University)|Sep 17, 2019

Quantum Computing Algorithms and Architecture2 references4 citations

TL;DR

This paper develops a formal framework for taking continuum limits of the 1D discrete time quantum walk (DTQW), proving that only specific coin operations yield nontrivial limits. It shows that simultaneous space-time continuum limits yield massless Dirac equations, and that the continuous time limit of the DTQW recovers the canonical continuous time quantum walk (CTQW) for any compatible coin, establishing a rigorous bridge between discrete and continuous quantum dynamics.

ABSTRACT

The discrete time quantum walk (DTQW) is a universal quantum computational model. Significant relationships between discrete and corresponding continuous quantum systems have been studied since the work of Pauli and Feynman. This work continues the study of relationships between discrete quantum models and their ostensive continuum counterparts by developing a formal transition between discrete and continuous quantum systems through a formal framework for continuum limits of the DTQW. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coins") admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk (CTQW) when the coin is allowed to transition through the continuous limit process.

Motivation & Objective

• To establish a general formal framework for taking continuum limits of the 1D discrete time quantum walk (DTQW).
• To determine which internal coin operations admit nontrivial continuum limits in the DTQW model.
• To investigate the dynamics resulting from sequential and simultaneous space and time continuum limits of the DTQW.
• To clarify the relationship between the DTQW and the canonical continuous time quantum walk (CTQW) in the continuous time limit.
• To rigorously connect discrete quantum walk dynamics to the Dirac equation in the continuum limit.

Proposed method

• Formalizing the DTQW evolution as a unitary operator sequence: $\Psi(x,t+\Delta t) = S C \Psi(x,t)$, with $S$ as shift and $C$ as coin operation.
• Using discrete Fourier transforms to analyze the time evolution in momentum space and derive the dispersion relation.
• Applying asymptotic expansions in $\Delta t$ and $\Delta x$ to derive partial differential equations (PDEs) in the continuum limit.
• Employing eigenvalue decomposition of effective Hamiltonians in the Fourier domain to identify limiting dynamics.
• Introducing a time evolution equation in the continuous time limit and relating it to the CTQW via a phase transformation.
• Deriving the continuous space-time limit by taking both $\Delta t, \Delta x \to 0$ with a fixed coin, leading to a Dirac-type equation.

Experimental results

Research questions

• RQ1 Which internal coin operations in the DTQW admit nontrivial continuum limits?
• RQ2 What dynamical equations emerge when both space and time are continuized in the DTQW?
• RQ3 Can the continuous time limit of the DTQW be identified with the canonical CTQW for arbitrary coins?
• RQ4 What is the nature of the Dirac-like equation obtained in the simultaneous space-time continuum limit?
• RQ5 How do the solutions of the DTQW's continuous time limit relate to those of the CTQW?

Key findings

• Only specific coin operations—those satisfying certain symmetry and parameter constraints—admit nontrivial continuum limits in the DTQW.
• The simultaneous continuum limit of space and time in the DTQW yields a massless Dirac equation, regardless of the initial coin, provided step size scaling is consistent.
• The continuous space limit of the continuous time limit of the DTQW results in a massless Dirac equation, confirming consistency across sequential limits.
• The continuous time limit of the DTQW can be mapped exactly to the canonical CTQW for any coin that allows a smooth transition to continuous time, establishing a direct dynamical equivalence.
• The limiting dynamics are governed by a Dirac-type PDE: $i\partial_t \Psi = (i\hat{A}\partial_x + \hat{B})\Psi$, where $\hat{A}$ and $\hat{B}$ are derived from the coin and shift operators.
• The eigenmodes of the limiting Hamiltonian are shown to satisfy a time evolution equation that reduces to the CTQW form under a phase transformation, proving the equivalence.