[Paper Review] A quantum walk with both a continuous-time and a continuous-spacetime limit

Di Molfetta Giuseppe, Pablo Arrighi|arXiv (Cornell University)|Jun 11, 2019

Quantum Computing Algorithms and Architecture33 references4 citations

TL;DR

This paper introduces a novel quantum walk (the 'Plastic QW') that unifies continuous-time and continuous-spacetime limits: it converges to lattice fermions under a continuous-time discrete-space limit and to the Dirac equation under a continuous-spacetime limit. The key contribution is a single discrete model that supports both non-relativistic and relativistic continuum limits via a scaling parameter α, enabling a unified framework for quantum simulation.

ABSTRACT

Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a Quantum Walks or Quantum Cellular Automata-based) enjoying a relativistic continuous spacetime limit. We provide a first example of a quantum simulation scheme that unifies both approaches. The proposed scheme supports both a continuous-time discrete-space limit, leading to lattice fermions, and a continuous-spacetime limit, leading to the Dirac equation. The transition between the two can be thought of as a general relativistic change of coordinates, pushed to an extreme. As an emergent by-product of this procedure, we obtain a Hamiltonian for lattice-fermions in curved spacetime with synchronous coordinates.

Motivation & Objective

• To unify Hamiltonian-based (continuous-time, discrete-space) and quantum walk-based (discrete-spacetime) quantum simulation schemes.
• To resolve the tension between non-relativistic lattice fermions and relativistic Dirac field theory in discrete models.
• To construct a single quantum walk model that supports both a continuous-time limit (lattice fermions) and a continuous-spacetime limit (Dirac equation).
• To provide a discrete-spacetime quantum cellular automaton (QCA) that can serve as a foundation for simulating interacting quantum field theories.
• To explore the emergence of a curved spacetime lattice fermion Hamiltonian in synchronous coordinates as a by-product.

Proposed method

• The model is a one-particle quantum walk on a (1+1)-dimensional spacetime grid with parameters m (mass), c (speed of light), ε (discretization scale), and α (scaling parameter).
• The spacetime discretization is governed by Δt = ε and Δx = ε^(1−α), allowing control over the limit taken via α.
• The walk is defined by a unitary evolution operator G that acts on a two-component spinor at each spacetime point.
• The continuous-time limit (α = 1) is derived by taking ε → 0 while keeping Δx finite, yielding the Kogut-Susskind Hamiltonian for lattice fermions.
• The continuous-spacetime limit (α = 0) is obtained by taking ε → 0 with Δx = Δt, recovering the Dirac equation.
• The model is extended to many non-interacting particles, yielding a QCA that simulates free Dirac quantum field theory in the continuum limit.

Experimental results

Research questions

• RQ1 Can a single discrete quantum walk model support both a continuous-time discrete-space limit and a continuous-spacetime limit?
• RQ2 Does the proposed quantum walk model yield the Kogut-Susskind lattice fermion Hamiltonian in the continuous-time limit?
• RQ3 Does the same model recover the Dirac equation in the continuous-spacetime limit?
• RQ4 Can the model be generalized to curved spacetime with a spacetime-dependent speed of light c(x,t)?
• RQ5 Does the model avoid the fermion doubling problem inherent in standard lattice fermion formulations?

Key findings

• The Plastic QW model successfully supports a continuous-time discrete-space limit (α = 1), yielding the Kogut-Susskind Hamiltonian for lattice fermions.
• The model also supports a continuous-spacetime limit (α = 0), recovering the Dirac equation in (1+1) dimensions.
• For intermediate values of α (0 < α < 1), the model interpolates between the two limits, demonstrating a unified scaling behavior.
• The model generalizes to curved spacetime with c(x,t), yielding a lattice fermion Hamiltonian in synchronous coordinates.
• The QCA formulation of the model correctly reproduces free Dirac quantum field theory in the many-particle continuum limit.
• An unexpected by-product is a new discretization of the Dirac equation in curved spacetime using synchronous coordinates, providing a novel lattice fermion Hamiltonian.