A staggered gauge-invariant quantum cellular automaton for both the Kogut-Susskind Schwinger model and the Dirac equation

Abstract

We build a quantum cellular automaton (QCA) which coincides with 1+1 QED on its known continuum limits. It consists in a circuit of unitary gates driving the evolution of particles on a one dimensional lattice, and having them interact with the gauge field on the links. The particles are massive fermions, and the evolution is exactly U(1) gauge-invariant. We show that, in the continuous-time discrete-space limit, the QCA converges to the Kogut-Susskind staggered version of 1+1 QED. We also show that, in the continuous spacetime limit and in the free one particle sector, it converges to the Dirac equation—a strong indication that the model remains accurate in the relativistic regime. Introduction Quantum physical phenomena can always be modelled classically by means of matrices and vectors. But, as far as we know, the dimension of these vectors grows exponentially with the number of particles, making these model intractable for classical computers. To simulate quantum physical phenomena efficiently, it seems we have no choice but to harness the laws of quantum mechanics themselves, as Feynman first suggested1. Quantum simulation could be applied to better understand condensed matter problems2, simulate molecules, find ground states of an Hamiltonian, or even simulate the dynamics of quantum field theories (QFT)3–5. It is the latter application that motivates this paper. Amongst QFT, gauge theories are of fundamental importance to Physics, as they capture the fundamental interactions. Some of them have been recast in discrete space. Lattice QCD6 is the most famous example as it is extensively used to obtain theoretical numerical values, to be compared against experimental values coming out of particle accelerators: this procedure is partly how physicists are searching for new physics. Simulation has therefore taken a central role in the scientific method of particle physics. But these techniques are computationally heavy: finding a way to simulate lattice gauge theories efficiently and accurately with a quantum simulation device would be a game changer. Lattice gauge theories are also key for condensed matter through their application in spin liquids, and for quantum error correction e.g. via Kitaev’s toric code7, 8. The 1+ 1 QED, aka the Schwinger model9, is a good candidate for a first step towards the quantum simulation of the dynamics of a gauge theory. Indeed, it is based on the U(1) gauge group just like 3+1 QED. It captures many non trivial physical properties such as a mass gap, fermion confinement and chiral symmetry breaking10. It is exactly solvable in the massless limit9. These features explain why it is often used as a testbed for new techniques and ideas. The standard ways to quantum simulate QFT are fundamentally non-relativistic, as they all begin by expressing the theory in continuous-time discrete-space Hamiltonian form, using Kogut-Susskind methods11, 12. They then map the matter (fermions) and the gauge field (bosons) onto quantum systems on a lattice, whose interactions will mimic those of the target Hamiltonian3. Sometimes these interactions are implemented as discrete-time products of quantum gates, but even then these are obtained by approximating the target Hamiltonian via the Trotter formula, an approximation which remains valid only in the non-relativistic regime ∆t ∆x. This approach was recently realized experimentally on an ion trap architecture13. Numerical techniques exist that come to complement the standard approach, based on tensors networks. Those use compact, approximate descriptions of quantum states14 such as the Density Matrix Renormalization Group (DMRG)15, 16, discarding unwanted information about the states as they evolve, so that their description remain of manageable size—whilst attempting to keep track of the interesting physical ingredients. 1 ar X iv :2 10 3. 13 15 0v 1 [ qu an tph ] 2 4 M ar 2 02 1 In order to achieve quantum simulation in the relativistic regime ∆x ≈ ∆t on must keep space and time on an equal footing, discretizing both at the same time. This is sometimes referred to as digital quantum simulation. Digital simulation has indeed been very successful at describing relativistic particles in different fields17, but so far it has not been able to produce simulation scheme for interacting QFT in the > 2 particles sector. One exception is18. This work mimics the actual construction of QED in a natively discrete setting. It starts from a quantum cellular automaton (QCA) simulating the relativistic Dirac equation, imposes U(1) gauge-invariance, and obtains a QCA, which is argued to converge to the Schwinger model in the continuous space-time limit. Whilst the construction is very convincing, the argument for the convergence towards the Schwinger model, or any non-trivially interacting QFT for that matter, was bound to be a little weak, mainly because it is not even clear that interacting QFT themselves have such a continuum limit. In many ways, the classical Lagrangian that serves as departure point of a QFT is but a partial prescription for a numerical scheme (e.g. a regularized Feynman path integral), whose convergence is often challenging (renormalization). Continuous-spacetime does not seem to be the friendly place where QCA and QFT should meet. In19, 20 a quantum walk (i.e. the one particle sector of a QCA) was proposed which unifies non-relativistic analog quantum simulation with relativistic digital quantum simulation. Just by imposing ∆x = ∆1−α t and tuning the α , the operator is found to have well defined limits lattice fermions both in continuous-time discrete-space, and the relativistic Dirac equation in the continuous spacetime limit—a property referred to as plasticity. The QCA presented in this paper is closely related to these last two models. It is again based upon a QCA that recovers the relativistic Dirac equation, extended to become natively discrete gauge-invariant as in18. But this time, the QCA is plastic, allowing us to prove its continuum limit towards 1+1 QED, in the regime where 1+1 QED does have a limit, i.e. the nonrelativistic regime. In other words, we recover the Hamiltonian of the Kogut-Susskind Schwinger model in the continuous-time discrete-space limit. In the continuous spacetime limit we show that the QCA yields the Dirac equation in the free one particle sector, allowing to make the bridge between the non-relativistic and the relativistic regimes. Altogether, the QCA coincides with 1+1 QED on its mathematical continuum limits, whenever these are defined. The natively discrete digital circuit for staggered Schwinger model we propose is not seen in the literature. The QCA is staggered which is not usual in the QCA formalism21. One may wonder whether the approach, beyond the quantum simulation application, could be used to reframe QFT. Indeed, the fact that the QCA is gauge-invariant by construction, contains explicit relativistic and non-relativistic limits, is expressible by means of path integrals22, 23, suggests that Quantum Computing point of view upon QFT may bring both rigour and pedagogy to the table—reviving the line of thought initiated by Feynman with his checkerboard propagator for 1+1 Dirac equation24. The paper is organized as follows. We first define the QCA model, that is the spacetime structure and the gates. Second we show the continuous-time and discrete-space limit towards the Kogut-Susskind version of the Schwinger model, by means of the Jordan-Wigner transformation from qubits to fermions. Third we show that in a continuous spacetime limit, we recover the Dirac equation for the free one particle sector. Finally, we prove that the model is gauge-invariant and conclude by giving some perspectives. Model The Kogut-Susskind staggered version of the Schwinger model The Schwinger model9 is a (1+ 1)D model invariant under the U(1) gauge group. It models spinless electrons and their antiparticles, positrons, propagating on a 1D lattice and interacting with a U(1) gauge field. We briefly summarize it by giving its Hamiltonian, which can be written using a temporal gauge (A0(x) = 0, and A(x) = A1(x)) as : H = ∫ dx ( ψ†(x) [(i∂x + igA(x))σz +mσx]ψ(x)+ 1 2 E2(x) ) , (1) where E(x) is the electric field observable at x, and A(x) is its conjugate momentum, meaning [A(x),E(y)] = iδ (x− y). (2) Here ψ(x) = (ψ1(x),ψ2(x)) is a two component fermion field satisfying {ψα(x),ψ β (y)}= δαβ δ (x− y). (3) We now describe the staggered Kogut-Susskind version of the Schwinger model. This Kogut-Susskind procedure consists in putting fermion fields on the nodes of an infinite 1D lattice and bosonic gauge fields on the links between them, as depicted in Fig.1. We can interpret occupied odd sites as electrons and unoccupied even sites as positrons, therefore particles and