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Discretizing quantum field theories for quantum simulation

Terry Farrelly, Julien Streich

arXiv:2002.02643v2quant-phhep-lathep-th

TL;DR

Quantum algorithms for QFT have generally used circuits to approximate continuous-time Hamiltonian lattice dynamics, leaving open how large the timestep can be and whether other formulations can work. This paper constructs scalar and gauge-theory circuits, including one exactly equivalent to the discrete-time path integral. It finds that δt = a suffices for the stated continuum behavior and that path-integral lattice QFT can have a direct circuit interpretation.

  • Problem

    Quantum QFT algorithms had used circuits to approximate continuous-time Hamiltonian lattice dynamics, while the suitable timestep and alternatives to that recipe matter for experiments with a lower bound on δt/a.

    Source: §I · p. 1

  • Method

    The paper constructs scalar-field circuits, including a Shift circuit analyzed with circuit Feynman rules, and a finite-depth local circuit for non-abelian gauge theory.

    Source: §I · p. 2, §III · p. 2, §V · p. 4

  • Results

    The authors report that δt = a suffices for the correct continuum limit, with the scalar path-integral circuit exactly represented locally and the gauge path-integral amplitude equal to a finite-depth circuit.

    Source: §I · p. 2, §III · p. 2, §VI · p. 5

  • Takeaways & Limitations

    The results support using discrete-time Lagrangian lattice QFT in quantum simulation and indicate circuit possibilities beyond those derived from lattice QFT.

    Source: §VI · p. 5

    The work focuses on dynamics; initial-state preparation, measurement, and discretization convergence rates remain additional simulation concerns.

    Source: §VI · p. 5

Abstract

from arXiv · show

To date, all proposed quantum algorithms for simulating quantum field theory (QFT) simulate (continuous-time) Hamiltonian lattice QFT as a stepping stone. Two overlooked issues are how large we can take the timestep in these simulations while getting the right physics and whether we can go beyond the standard recipe that relies on Hamiltonian lattice QFT. The first issue is crucial in practice for, e.g., trapped-ion experiments which actually have a lower bound on the possible ratio of timestep to lattice spacing. To this end, we show that a timestep equal to or going to zero faster than the spatial lattice spacing is necessary for quantum simulations of QFT, but far more importantly a timestep equal to the lattice spacing is actually sufficient. To do this, first for $\phi^4$ theory, we give a quantum circuit exactly equivalent to the real-time path integral from the discrete-time Lagrangian formulation of lattice QFT. Next we give another circuit with no lattice QFT analogue, but, by using Feynman rules applied to the circuit, we see that it also reproduces the correct continuum behaviour. Finally, we look at non-abelian gauge fields, showing that the discrete-time lattice QFT path-integral is exactly equivalent to a finite-depth local circuit. All of these circuits have an analogue of a lightcone on the lattice and therefore are examples of quantum cellular automata. Aside from the potential practical benefit of these circuits, this all suggests that the path-integral approach to lattice QFT need not be overlooked in quantum simulations of physics and has a simple quantum information interpretation.

I. INTRODUCTION

The paper asks whether QFT circuits need very small timesteps and whether they must approximate continuous-time Hamiltonian lattice dynamics. It argues that δt = a suffices and presents circuit formulations that extend beyond that standard recipe.

I. INTRODUCTION

  • Quantum algorithms had approximated continuous-time Hamiltonian lattice dynamics, while the timestep-to-spacing ratio matters for trapped-ion experiments and fixed-range circuits.The paper relates this constraint to avoiding unwanted spin-motion coupling and preserving a sufficiently wide circuit lightcone.Source: §I · pp. 1–2
  • As a → 0, δt/a ≤ 1 can reproduce the correct physics, whereas an excessively large ratio prevents high-velocity physics even under spatial refinement.For a nearest-neighbour circuit with δt = 10a, the maximum speed remains v = 3/5 in units with c = 1.Source: Figure 1 · p. 1, §I · pp. 1–2
  • The paper presents scalar-field circuits, including a Shift circuit without a lattice-QFT analogue, and a finite-depth circuit equivalent to the discrete-time path integral for non-abelian gauge theory.The circuits have bounded information propagation and are quantum cellular automata; earlier proposals mostly concerned free or one-dimensional integrable QFTs.Source: §I · p. 2

II. SCALAR FIELDS ON A LATTICE

This section sets up scalar fields on a spatial lattice and contrasts continuous-time Hamiltonian dynamics with the discrete-time Lagrangian formulation used for quantum circuits.

II. SCALAR FIELDS ON A LATTICE

  • The paper considers continuous-time Hamiltonian lattice QFT for φ4 theory, while motivating the discrete-time Lagrangian formalism as a natural alternative for quantum simulation.All formalisms agree in the continuum limit.Source: §II · p. 2
  • Space is discretized at spacing a, with integer-vector site coordinates n and field operators φ_n and π_n obeying lattice commutation relations.The factor 1/a^d in the commutator ensures recovery of continuum commutation relations as a → 0.Source: §II · p. 2
  • Finite lattices with periodic boundary conditions are included, although the setup initially takes an infinite spatial lattice.On qubit hardware, continuous on-site fields must be discretized and truncated; the field representation is left to the architecture.Source: §II · p. 2

III. STRANG-SPLIT CIRCUIT FOR SCALAR FIELD THEORY

The Strang-split circuit gives a local-circuit realization of the discrete-time scalar-field path integral. With δt = a, it retains the correct continuum limit while using only one additional gate layer relative to the first-order Trotter circuit.

III. STRANG-SPLIT CIRCUIT FOR SCALAR FIELD THEORY

  • The circuit uses Strang splitting of the lattice Hamiltonian evolution, with each kinetic or potential layer composed of commuting local unitaries.This makes the circuit finite-depth and local.Source: §III · p. 2
  • Its path-integral action uses a symmetrized potential containing spatial derivatives, the mass term, and the interaction.The symmetrized potential averages left- and right-Riemann-sum approximations to the continuum action.Source: §III · pp. 2–3
  • At δt = a, the real-time transfer matrix is exactly a local quantum circuit and yields the correct continuum limit.The circuit is exactly equivalent to the discrete-time Lagrangian path-integral amplitude, rather than merely approximating continuous-time lattice Hamiltonian dynamics.Source: §III · p. 3
  • The Strang-split circuit is expected to converge faster than first-order Trotter and requires only one additional gate layer: 2τ + 1 rather than 2τ.The first-order circuit alternates 2τ layers; the Strang-split circuit adds one layer.Source: §III · p. 3

IV. SHIFT CIRCUIT FOR SCALAR FIELD THEORY

The Shift circuit is a scalar-field circuit not obtained from a Suzuki–Trotter decomposition or lattice path integral. Its free theory has the correct continuum propagator, while a modified interaction corrects a high-momentum contribution to the mass correction.

IV. SHIFT CIRCUIT FOR SCALAR FIELD THEORY

  • The Shift circuit uses U = W_X W_P W_X with δt = a and is a natural discretization of the massless, noninteracting theory in one spatial dimension.In that case, the continuum theory is solvable in left- and right-moving fields, a description that also works for this circuit.Source: §IV · p. 3
  • With λ = 0, the circuit’s Feynman propagator approaches the usual Feynman propagator as a → 0.The momentum integral is over the lattice Brillouin zone (−π/a, π/a]^D.Source: §IV · p. 3
  • The unmodified Shift circuit gives an order-λ mass correction twice as large as other regulators because high-momentum modes act like a second particle.The equal-time propagator has a symmetry under p_i ↔ π/a − p_i.Source: §IV · p. 3
  • A smeared interaction changes the Feynman rules at high energy and restores the correct first-order mass correction for p_in ≪ 1/a.The modification suppresses high-momentum particles.Source: §IV · pp. 3–4

V. LOCAL CIRCUIT FOR NON-ABELIAN GAUGE THEORY

For non-abelian gauge theory, the real-time transfer operator is an exact finite-depth local circuit at δt = a, equivalent to the discrete-time path integral with the Wilson action.

V. LOCAL CIRCUIT FOR NON-ABELIAN GAUGE THEORY

  • The real-time transfer operator is exactly a finite-depth local circuit at δt = a and is equivalent to the discrete-time path integral with the Wilson action.The path-integral amplitude is for initial and final gauge configurations obeying Gauss’ law.Source: §V · p. 4
  • Link Hilbert spaces use group-valued basis states, and physical states must be invariant under spatial gauge transformations.The gauge group is assumed to be U(1) or SU(n), though non-compact groups can also be considered.Source: §V · p. 4
  • Commuting plaquette operators make Wmag finite-depth; its depth is two in d = 2 and at most d! in general.Wel has depth one because its gates act on separate links.Source: §V · p. 4
  • The precise gates depend on the representation of link Hilbert spaces, and finding a suitable local truncation scheme is difficult.Group-theoretic methods and quantum link models are suggested as truncation options.Source: §V · p. 4

VI. DISCUSSION

The discussion argues that δt = a suffices for bosonic QFT simulations and that discrete-time path integrals can have a direct circuit interpretation. It also identifies scope and practical issues that remain.

VI. DISCUSSION

  • For bosonic QFTs, δt = a suffices, and real-time transfer matrices for scalar and non-abelian gauge fields are exactly finite-depth local circuits.The analogous factorization does not work as easily for Wilson fermions.Source: §VI · p. 5
  • The Shift circuit suggests that quantum computers may simulate QFTs using circuits beyond those arising from lattice QFT.This extends the paper’s circuit possibilities beyond its path-integral-derived constructions.Source: §VI · p. 5
  • Dynamics is only one part of simulation: initial states must also have the right continuum limit, and convergence rates affect computational complexity.The discussion also identifies state preparation and measurement as other key aspects.Source: §VI · p. 5
  • The paper suggests gradient descent for determining bare parameters from simulated physical processes, while noting that its computational complexity has no obvious guide.The procedure may be computationally expensive, but the parameter search is a one-time cost for each lattice spacing.Source: §VI · p. 5

0. Then we vary the bare parameters to minimize

The proposal tunes bare parameters by gradient descent, using simulated physical processes to compare simulated and target physical parameters.

0. Then we vary the bare parameters to minimize

  • Bare parameters are updated by gradient descent using simulated physical parameters obtained from processes that define the target values.The update uses slightly shifted bare parameters to estimate how the simulated values change.Source: §0 · p. 5
  • The learning rate η is chosen empirically, and gradient descent carries the usual risk of local minima.The authors note that the procedure’s computational complexity has no obvious guide, though tuning is a one-time cost per lattice spacing.Source: §0 · p. 5

Note added

The supplied passages note a related discrete-time scattering preprint, outline a field-configuration basis for the path-integral derivation, and give parameters for the Shift circuit.

Note added

  • A preprint [45] posted near completion studied discrete-time scattering in the Thirring model, unlike this paper.Source: Note added · p. 5

Appendix A: Relation of the Strang-split circuit to the path integral

  • Field-configuration states are eigenstates of the field operators; the expressions are mathematically well defined on finite spatial lattices but formal on infinite ones.Source: Appendix A · p. 7

Appendix B: The Shift circuit

  • For the Shift circuit, D = d + 1, M = 1 − m^2a^2/2, and N is the set of lattice vectors with components ±1.Here m is the mass.Source: Appendix B · p. 8

1. Continuum limit in the free case

In the free case, the circuit recovers Klein–Gordon dynamics for low momenta. In one dimension, its massless form discretizes the propagation of left- and right-moving fields.

1. Continuum limit in the free case

  • At low momenta, the continuum limit gives the Klein–Gordon equations and usual scalar-field dynamics on scales large compared with the lattice spacing.The small-momentum expansion c(p) = 1 − (p² + m²)a²/2 + O(a⁴) supports this limit.Source: §1 · p. 9

2. Motivation

  • The Shift circuit is a natural discretization of the massless, noninteracting continuum model in one dimension.The continuum free dynamics can be expressed using left- and right-moving fields.Source: §2 · p. 9
  • In one dimension with m = 0, the Shift circuit moves the lattice left- and right-moving fields in opposite directions each timestep.These lattice fields approach the continuum movers with the correct commutation relations.Source: §2 · pp. 9–10

3. Solution in the free case

The free Shift circuit is solved through its momentum-space evolution and annihilation operators. Its low-momentum dynamics and vacuum correlations recover continuum behavior, while high-momentum modes expose a doubling-like problem.

3. Solution in the free case

  • The momentum-space evolution is solved as an eigenvalue problem for the annihilation operator; real circuit energies require m > 0.The other eigenvalue corresponds to the creation operator.Source: §3 · p. 10
  • At small momenta, the circuit’s equal-time propagator reproduces the continuum propagator, and its vacuum approximates the continuum vacuum at length scales large compared with a.This follows from ω(p) approaching E(p) for small pa.Source: §3 · p. 11
  • The Shift circuit’s phase θ(p) closely approximates continuum E(p), but ω(p) becomes a poor approximation at high momenta.The field operators use θ(p) in their phase and ω(p) = sin[θ(p)a]/a in the denominator.Source: §3 · p. 11
  • The equal-time propagator’s denominator is small at both low and high momenta, making high-momentum modes analogous to fermion doubling.The passage proposes smearing the interaction over several sites or adding a Wilson-like artificial mass to mitigate the issue.Source: §3 · p. 11

Appendix C: Feynman propagator

This appendix develops the circuits’ discrete-time propagator, interaction-picture perturbation theory, and Feynman rules, including a limitation of the scattering framework. It also defines amputated diagrams and relates them to S-matrix elements through LSZ reduction.

Appendix C: Feynman propagator

  • As the spatial lattice spacing a tends to zero, the momentum-space propagator recovers the usual Feynman propagator; contour integration derives the discrete-time expression.The derivation allows general δt, whether or not it equals a.Source: Appendix C · pp. 11–12

Appendix D: Interacting fields

  • Interactions can be added to free circuit dynamics with a unitary, enabling discrete-time interaction-picture evolution and perturbation theory for the Shift circuit and other analogous discretizations.The approach also applies to the Strang-split circuit and other interaction terms when a solvable free evolution and a nontrivial interaction unitary are available.Source: Appendix D · pp. 12–13

Appendix E: Feynman rules

  • For the Strang-split circuit, the Feynman rules match discrete-time Lagrangian lattice QFT, while the Shift circuit has no obvious lattice-QFT analogue and yields a new propagator.The rules use the circuit propagator and discrete energy-momentum integration ranges; their symmetry factors match conventional φ4 perturbation theory.Source: Appendix E · p. 14

Appendix F: Deriving the Feynman rules

  • The derivation gives the same diagram symmetry factors as QFT and motivates connected diagrams by cancellation of vacuum bubbles, but its justification of the asymptotic vacuum assumption is explicitly nonrigorous.The cancellation follows from dividing by ⟨0|S|0⟩; the authors defer a rigorous argument to future work.Source: Appendix F · pp. 15–16
  • Amputated diagrams exclude external-leg loops, which reflect modified in- and out-going particle propagators in the interacting theory [39].The LSZ reduction theorem motivates relating S-matrix elements to correlation functions; a thorough investigation is deferred to future work.Source: Appendix F · p. 16

Appendix G: One-loop calculation

The one-loop calculation finds logarithmic cutoff growth, with the Shift circuit’s unmodified interaction giving a prefactor twice that of the momentum-cutoff regulator. Averaging the interaction over nearby sites restores agreement at small incoming momenta, while the non-abelian path integral is specified by spacetime plaquettes and link integrations.

Appendix G: One-loop calculation

  • The continuum one-loop mass correction has logarithmic divergence with the cutoff.Source: Appendix G · p. 16
  • The Shift circuit’s one-loop prefactor is off by a factor of two relative to the momentum-cutoff regulator because of high-momentum modes.Source: Appendix G · p. 17
  • Averaging the interaction over nearby sites modifies the vertex factor, with weights formed from v(−1) = v(1) = 1/4 and v(0) = 1/2.Source: Appendix G · p. 17
  • For small incoming momenta, the modified interaction agrees with the continuum mass correction, and the plotted corrections grow logarithmically with the same prefactor.The numerical comparison uses a momentum cutoff Λ = π/a; the plotted curves are normalized to agree at a = 2.Source: Appendix G · p. 17, Figure 8 · p. 18

Appendix H: Non-abelian circuit and path integral

  • For δt = a, the non-abelian path-integral action sums spacetime plaquettes except spatial plaquettes at the final time, with link integrations over specified time ranges.The formalism is mathematically well defined on finite lattices for compact gauge groups.Source: Appendix H · pp. 18–19
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