Quantum Computation of Scattering in Scalar Quantum Field Theories

Stephen P. Jordan; Keith S. M. Lee; John Preskill

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Quantum Computation of Scattering in Scalar Quantum Field Theories

Stephen P. Jordan, Keith S. M. Lee, John Preskill

arXiv:1112.4833v2Dec 20, 2011hep-thquant-ph

TL;DR

Quantum field theory has important physical applications, but scattering calculations are difficult at strong coupling and high precision. This paper develops a quantum algorithm for massive φ4 scattering using adiabatic preparation and lattice simulation. The algorithm has polynomial runtime in particle number, energy, and precision and offers exponential speedups over known classical methods within its stated scope.

Problem
Scattering calculations in quantum field theory can require great computational complexity, while perturbative methods fail at strong coupling and beyond certain precision.
Method
The paper develops a quantum algorithm using adiabatic turn-on to prepare interacting wavepacket states and spatial discretization with continuum-limit error analysis.
Results
The algorithm applies at weak and strong coupling, runs polynomially in precision, particle number, and energy, and provides exponential speedups over the fastest known classical algorithms.
Takeaways & Limitations
The results establish an efficient quantum approach for determining scattering amplitudes in massive scalar φ4 theory in four or fewer spacetime dimensions.
Takeaways & Limitations
The work focuses on massive scalar φ4 theory in four or fewer spacetime dimensions; extensions to fermions, gauge symmetries, and massless particles remain future work.

Abstract

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Quantum field theory provides the framework for the most fundamental physical theories to be confirmed experimentally and has enabled predictions of unprecedented precision. However, calculations of physical observables often require great computational complexity and can generally be performed only when the interaction strength is weak. A full understanding of the foundations and rich consequences of quantum field theory remains an outstanding challenge. We develop a quantum algorithm to compute relativistic scattering amplitudes in massive phi-fourth theory in spacetime of four and fewer dimensions. The algorithm runs in a time that is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. Thus, it offers exponential speedup over existing classical methods at high precision or strong coupling.

1 Introduction

The paper motivates quantum simulation of quantum field theories by their physical importance, computational richness, and unresolved simulation complexity. It develops a digital quantum algorithm for scattering in φ4 theory, addressing weak and strong coupling, precision, state preparation, and continuum-limit errors.

Motivation: Quantum field theory underlies the Standard Model and supports exceptionally precise experimental predictions.
Motivation: Scattering calculations become difficult because Feynman-diagram counts grow factorially with perturbative order and particle number.
Problem: Strong-coupling lattice methods can compute static quantities but cannot predict dynamical observables such as scattering amplitudes.
Contribution: The algorithm calculates φ4 scattering amplitudes with complexity polynomial in time, volume, particle number, energy scale, and desired precision.
Problem: Quantum circuits had efficiently simulated several quantum systems, but efficient simulation of interacting quantum field theories remained open because of nonlinear interactions and infinitely many local degrees of freedom.
Contribution: The work combines adiabatic preparation of interacting wavepackets, digital time evolution, observable construction, and explicit discretization and continuum-limit error analysis.

2 Background

The background introduces scalar φ4 theory, its lattice formulation, particle interpretation, and continuum-limit requirements. It emphasizes renormalization, finite-size and discretization convergence, and efficient local-Hamiltonian simulation.

2 Background: The paper restricts its discussion to scalar φ4 theory and presents the exposition as self-contained for a broad audience.
2 Background: Continuum φ4 theory is relevant to Higgs self-interactions and to critical phenomena and universality in statistical-mechanical systems.
Lattice formulation: The lattice theory uses spatial sites with field operators φ(x), conjugate momenta π(x), and a Hamiltonian whose ground state is the vacuum.
Particle states: Free-theory creation and annihilation operators provide momentum eigenstates, while adiabatically turning on the interaction produces corresponding interacting single-particle states.
Continuum limit: A meaningful continuum limit requires tuning lattice parameters as the spacing decreases; this procedure is called renormalization.
Algorithmic analysis: The analysis studies convergence rates to the continuum and infinite-volume limits, while quantum circuits simulate local Hamiltonian evolution with a novel volume-scaling analysis.

3 Quantum Algorithm

The algorithm prepares interacting wavepackets, evolves them through scattering, and measures outgoing particles using field-based quantum simulation. Its resource analysis gives polynomial scaling and identifies adiabatic preparation as a dominant cost in relevant dimensions, while diffuse wavepackets and incoming bound states constrain the procedure.

Algorithm overview: The circuit prepares the free vacuum, excites wavepackets, adiabatically turns on interactions, evolves through scattering, and measures outgoing states.Measurements either return to the free theory and measure momentum-mode number operators or use localized detectors with phase estimation.
Representation: Field representation is used because the quartic interaction is nonlocal in momentum space, making Suzuki-Trotter simulation less efficient there.The field representation stores field values in qubit registers at lattice points.
State preparation: Kitaev–Webb Gaussian-state preparation constructs the free vacuum by preparing a diagonal-covariance Gaussian and reversibly changing basis to the desired covariance matrix.The dominant classical subroutine for large volume is the LDLT decomposition of the inverse covariance matrix, with a stated ˜O(V^2.376) cost.
Wavepacket preparation: Quasilocal wavepacket operators can be truncated to fully local operators with exponentially small error, allowing gate complexity to depend on wavepacket support rather than total volume.The truncation distance is chosen as c1/m0 beyond the support, where c1 ≫ 1.
State-preparation constraints: Adiabatic preparation can broaden wavepackets and reduce efficiency because diffuse packets often pass through one another without significant scattering.The resulting low event rate can require many simulation repetitions before interesting scattering events are observed.

4 Analysis of Algorithm

The algorithm analyzes cutoff, discretization, state-preparation, evolution, measurement, and adiabatic errors for simulating φ4 scattering on quantum computers. The analysis establishes logarithmic qubit requirements and identifies dominant costs across coupling regimes and dimensions.

Error sources: Cutoffs discretize space and the field, while quantum-computing primitives introduce additional approximation errors that must be bounded.The field is truncated to [−φmax, φmax] and discretized in increments of δφ; errors also arise from adiabatic evolution and time simulation.
Adiabatic preparation: Adiabatic wavepacket preparation is controlled by propagation and diabatic errors, including vacuum particle creation and one-particle-to-three-particle splitting.The phase induced during preparation scales with τ/J^2, and the resulting criteria determine suitable J, τ, and gate complexity.
Representation by qubits: The field representation uses nb = ⌈log2(1 + 2φmax/δφ)⌉ qubits per lattice site.The finite register follows from imposing a field-magnitude cutoff and resolution at each lattice point.
Representation by qubits: The non-perturbative analysis achieves fidelity 1 − εtrunc with nb logarithmic in 1/a, 1/εtrunc, and V for weakly and strongly coupled φ4 processes.The result applies to simulations at energy scale E while retaining the stated inner-product fidelity with the exact state.
Adiabatic preparation: Adiabatic paths must avoid the quantum phase transition, with weak- and strong-coupling preparations following distinct paths in parameter space.The weakly coupled continuum-like theory is prepared along a path that does not cross the transition; the strongly coupled path approaches it closely.
Weak coupling: At weak coupling, many-particle scattering is exponentially rare and perturbative series cannot provide arbitrarily high precision, motivating the algorithm’s high-precision analysis.The suppression follows from the number of vertices in connected diagrams and the asymptotic, nonconvergent character of perturbation theory.

4.3 Suzuki-Trotter Formulae for Large Lattices

The section analyzes Suzuki-Trotter simulation of local lattice Hamiltonians and derives linear scaling with the number of lattice sites for fixed formula order. It also examines momentum-dependent costs in strongly coupled scattering.

Large-lattice scaling: A kth-order Suzuki-Trotter formula gives linear scaling in the number of lattice sites when the Hamiltonian is local.The derivation uses locality and commutator structure to show that the leading correction has norm O(V) for fixed k.
Large-lattice scaling: The same locality argument applies when non-neighboring terms commute and to Suzuki-Trotter decompositions for time-dependent Hamiltonians.This extends the scaling result beyond the specific lattice evolution setting analyzed first.
Strong-coupling momentum scaling: In strong coupling, high-energy incoming particles enlarge local Hamiltonian terms and increase the Suzuki-Trotter order required for accurate evolution.Using E as an estimate of local-term magnitude yields bounds scaling with powers of p and the lattice volume.

4.4 Measurement of Occupation Numbers

The paper develops measurement strategies for particle occupation, momentum, position, energy, and bound-state detection using phase estimation and localized detectors. Locality makes detector simulation depend on detector volume rather than the full simulated volume, while Gaussian envelopes control measurement noise.

Momentum-mode occupation: Phase estimation measures occupation numbers of momentum modes by simulating the associated operators with Suzuki-Trotter formulas.The required evolution can be implemented by diagonalizing Π_p and Φ_p, computing induced phases, and applying phase kickback.
Localized detectors: Momentum-mode occupation measurements initially cannot distinguish particles with momentum p from those with momentum −p and scale quadratically with V.Localized detectors correct both issues by restricting measurements to spatial regions.
Localized detectors: Localized detectors measure particle occupation within spatial regions and provide partial momentum and position resolution.Their momentum lattice is more coarsegrained than the full lattice, consistent with the uncertainty principle.
Localized detectors: Surrounding the collision region with a small number of localized detectors can resolve the p versus −p ambiguity after scattering.A proposed arrangement uses 2^d detectors corresponding to the faces of a d-dimensional cube.
Bound-state detection: For bound-state detection, regional energy and momentum measurements identify particles when detector regions are small compared with particle separations.The detection location and measured energy estimate velocity and momentum, enabling bound-state identification.
Detector complexity: Localized detector simulation has complexity scaling with detector volume rather than the total volume V.This follows from the quasilocality of the creation and annihilation operators used to construct the detectors.

5 Some Field-Theoretical Aspects

This section develops the field-theoretical framework for the spatially discretized massive φ4 theory, including renormalization, effective-field-theory errors, and finite-volume effects.

Finite volume modifies Wilson coefficients and asymptotic wavepackets, while short-range interactions imply these corrections should be small in the perturbative regime.
The physical mass is related to bare parameters through renormalization, with hybrid perturbation theory used to obtain the mass to second order in the coupling.
The lattice formulation leaves out operators whose Wilson coefficients determine discretization errors, including φ2n terms and Lorentz-violating operators.
In D = 4, the coefficients of φ6 and higher φ2n operators scale as c ∼a2 and c′′ ∼a4, making φ6 the more significant omitted operator.
The φ6 coefficient is obtained by matching the six-point function of the full and effective theories at leading order in the lattice spacing.

6 Conclusions

The paper establishes an efficient quantum algorithm for scattering amplitudes in scalar φ4 theory and analyzes discretization and continuum-limit issues. It reports exponential speedups for weak and strong interactions, while restricting the present study to massive scalar theories in four or fewer dimensions.

The algorithm uses adiabatic turn-on to prepare interacting wavepacket states and discretizes space while analyzing discretization errors and the continuum limit.
The algorithm provides exponential speedups over the fastest known classical algorithms for scattering-amplitude calculations.
Its runtime is polynomial in desired precision, particle number, and particle energy, and it applies to both weakly and strongly interacting theories.
The analysis focuses on massive scalar φ4 theory in spacetime of four and fewer dimensions; extensions to fermions, gauge symmetries, and massless particles remain future work.

A. Notation

This appendix provides a notation table for the paper.

Table 5 collects the notation used throughout the paper.

B. Loop Integrals for Mass Renormalization

This appendix evaluates loop integrals used in mass renormalization, including one-loop and two-loop contributions under the chosen renormalization condition.

The mass-renormalization calculation uses one-loop 1PI insertions into the propagator and introduces a Feynman-parameter integral.
The one-loop renormalization condition sets iδm equal to the value of the one-loop diagram.
The renormalization condition makes the first two first-order diagrams cancel, leaving a two-loop contribution to evaluate.
The renormalization condition M(p = (m, 0)) = 0 fixes the evaluation point for the self-energy calculation.
As a →0, the momentum integral converges in D = 4 but becomes singular in D = 2 and D = 3, requiring extraction of the singular part.

C. Loop Integrals for Matching

This section develops loop-integral matching calculations for effective operators, including the φ6 coefficient and infrared behavior.

The φ6 coefficient is obtained by evaluating the relevant six-point Feynman diagram at zero external momentum.This choice reflects that φ6 is a non-derivative operator.
Matching the six-point function equates full-theory diagrams with effective-theory diagrams, including permutations of loop diagrams.
The propagator and the Lorentz-violating operator series are treated together in the matching construction.
The effective theory must reproduce the full theory’s infrared divergences, which requires including the relevant operators.Omitting them produces an incorrect infrared behavior.

D. Loop Sums for Matching

This section evaluates loop sums using Riemann-sum convergence and Euler–Maclaurin corrections, then examines finite-volume effects on Wilson coefficients.

The loop sums converge to their corresponding integrals as the lattice extent ˆL becomes large.
Euler–Maclaurin summation quantifies the difference between each Riemann sum and its corresponding integral.
The one-dimensional and two-dimensional sums are evaluated by applying the summation formula once and twice, respectively.
The remaining terms are evaluated exactly where possible and expanded around ma = 0, including the D = 4 result.
Finite ˆL increases the magnitude of the Wilson coefficient in all dimensions.

E. Integrals for Effective Potential

This section relates one-loop scattering amplitudes to Mandelstam variables and derives the effective potential through channel decomposition and contour methods.

The 2-body scattering amplitude is expressed using the Mandelstam variables s, t, and u.
A renormalization condition defines λ as the magnitude of the amplitude at zero momentum in D = 4 dimensions.
In the nonrelativistic center-of-momentum frame, the s-channel contribution vanishes while the t- and u-channels produce the two potential terms.
The asymptotic form of one integral is obtained with Laplace’s method.
The q integral is evaluated by contour completion in the upper half-plane, accounting for a branch cut from q = 2im to q = i∞.
The resulting expression for the amplitude is obtained by substituting the evaluated integral into the preceding formula.

F. Minimal Qubit Requirement

The paper estimates the resources needed for a minimal 2 →4 scattering demonstration and analyzes how qubit requirements depend on precision, lattice separation, and coupling.

F. Minimal Qubit Requirement: Approximately 1,000–10,000 noiseless qubits should suffice for a 2 →4 scattering simulation in 1 + 1 dimensions.The range depends on the desired precision; noisy physical qubits can replace perfect logical qubits through error correction.
F. Minimal Qubit Requirement: The simulation energy is chosen as E = 5m so that 2 →4 scattering has available phase space.
F. Minimal Qubit Requirement: Discretization errors are of order (pa)2, and setting this quantity to ǫ determines the lattice spacing required for the target precision.
F. Minimal Qubit Requirement: Sufficient lattice volume is needed to separate incoming and outgoing particles and approximate the asymptotic states defining the S-matrix.
F. Minimal Qubit Requirement: The interaction range and wavepacket width constrain the separation needed for substantial scattering without unwanted scattering in the in and out states.
F. Minimal Qubit Requirement: The number of qubits per site is not very sensitive to λ; replacing one parameter changes nb from 20 to 19 at ǫ = 0.01 and leaves it at 13 for ǫ = 0.1.
F. Minimal Qubit Requirement: The qubit estimate is N = 6 × (r/a) × nb, with the factor six allowing space for four outgoing particles and uneven spacing.
F. Minimal Qubit Requirement: The asymptotic qubit scaling is intended to hold at both strong and weak coupling, unlike the perturbative estimate shown in Figure 2.