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Quantum Algorithms for Quantum Field Theories
Stephen P. Jordan, Keith S. M. Lee, John Preskill
TL;DR
The paper addresses scattering calculations in regimes where classical methods can break down or become inefficient, including strong coupling, many external particles, and high precision. It develops a quantum algorithm based on adiabatic state preparation and Hamiltonian simulation, with analyses covering discretization, Trotter, and preparation costs. The reported results establish logarithmic qubit scaling with precision and polynomial output-particle scaling, with nonperturbative applicability to weakly and strongly coupled φ4 theory.
Problem
Classical methods for computing scattering amplitudes can break down or be inefficient at strong coupling, large numbers of external particles, and high precision.
Method
The algorithm prepares free and interacting states adiabatically, simulates Hamiltonian evolution, reverses the interaction, and measures momentum-mode occupations.
Results
The qubit representation achieves 1 − ǫ fidelity with nb logarithmic in 1/a, 1/ǫ, and V, and the analysis applies nonperturbatively to weakly and strongly coupled φ4 theory.
Takeaways & Limitations
The efficiency analysis gives polynomial scaling in the number of outgoing particles, with the dominant preparation cost depending on spacetime dimension.
Abstract
from arXiv · showhide
Quantum field theory reconciles quantum mechanics and special relativity, and plays a central role in many areas of physics. We develop a quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory with quartic self-interactions (phi-fourth theory) in spacetime of four and fewer dimensions. Its run time is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. In the strong-coupling and high-precision regimes, our quantum algorithm achieves exponential speedup over the fastest known classical algorithm.
A.1 Steps of Algorithm and Comments
The algorithm prepares the free vacuum and wavepacket states, simulates interacting Hamiltonian evolution, and measures outgoing momentum occupations. Its state-preparation components use Gaussian construction and wavepacket methods, with errors controlled by particle separation.
- The procedure prepares the free vacuum before subsequent interaction and scattering steps.
- Kitaev and Webb’s method constructs V-dimensional multivariate Gaussian superpositions for state preparation.
- The dominant classical cost of Gaussian preparation is computing an LDLT decomposition of the inverse covariance matrix, requiring ˜O(V^2.376) time.
- Wavepacket states are obtained by simulating a wavepacket Hamiltonian Hψ, whose evolution can be simulated similarly to the field Hamiltonian H.
- Wavepacket-separation errors scale as ǫ ∼e^−δ/m, while the packets retain a constant momentum spread rather than precisely defined momenta.
3. Adiabatically turn on the interaction. For 0 ≤s ≤1, let
The interaction is turned on adiabatically through a sequence of controlled evolutions, then Hamiltonian evolution and measurement produce scattering data before the interaction is reversed. In strong coupling, the physical mass is measured progressively to guide a safe preparation rate.
- Adiabatic preparation uses J steps, with Uj representing unitary evolution generated by the time-dependent Hamiltonian.
- The algorithm simulates Hamiltonian time evolution, turns off the interaction by reversing the turn-on process, and measures momentum-mode occupation numbers.
- In the strongly coupled case, the allowable adiabatic rate cannot be calculated perturbatively, leaving the rate-selection problem to be determined.
- The physical mass is obtained by subtracting measured vacuum energy from measured single-particle energy.
- Repeating mass estimation at successively higher λ0 values enables reaching strong coupling while choosing the adiabatic rate from the estimated mass.
A.2 Efficiency
The efficiency analysis identifies spatial discretization and imperfect adiabaticity as dominant precision-dependent errors. The asymptotic cost is controlled by state preparation, with different dominant components across dimensions and only polynomial dependence on output multiplicity and precision.
- For a massive theory, imperfect particle-separation errors decrease exponentially with distance, so V and qubits per site scale logarithmically with ǫ.
- Trotter errors scale as ǫ ∼n^−2k, giving complexity scaling as ǫ^−1/2k.
- Spatial discretization and imperfect adiabaticity are the dominant contributions to scaling with ǫ.
- In d = 1, adiabatic state preparation dominates the cost, whereas in d = 2, 3, free-vacuum preparation dominates.
- The free-vacuum and interacting-theory preparation costs scale as n_out^2.376(d+1) and n_out^2d+3+o(1), respectively.
- In three-dimensional spacetime, free-vacuum preparation dominates total n_out scaling, while adiabatic turn-on dominates in two-dimensional spacetime.
A.3 Mass Renormalization
The mass-renormalization discussion combines perturbative calculations at weak coupling with known near-transition behavior at strong coupling. Lattice discretization changes propagators and regulates spatial loop-momentum integrals, while the theory’s phase structure differs across dimensions.
- At weak coupling, the mass-renormalization form is obtained using perturbation theory, while strong coupling uses known behavior near the phase transition.
- The squared-mass shift receives one-loop contributions at first order and two-loop contributions at second order in the coupling.
- Lattice calculations use a different propagator, and spatial loop-momentum components are cut off by π/a, making lattice spacing an ultraviolet regulator.
- The φ4 theory is believed to share a universality class with the Ising model, with critical behavior associated with second-order phase transitions.
- In D = 4 dimensions, believed triviality implies no non-trivial renormalization-group fixed point and hence no phase transition as parameters vary.
A.4 Representation by Qubits
The representation encodes field amplitudes and conjugate momenta on a finite range and discretizes them using qubits. The required qubits per site scale logarithmically with lattice spacing, fidelity, and volume.
- Qubit complexity: The required number of qubits per site is logarithmic in the inverse lattice spacing, inverse error tolerance, and physical volume.Polynomial upper bounds on field and momentum ranges establish this logarithmic scaling.
- Scope of the representation: The bounds apply to the parameter combinations used for adiabatic preparation and scattering of both strongly and weakly coupled wavepackets.The analysis is explicitly nonperturbative and therefore covers both coupling regimes.
- Field and momentum discretization: The field configuration probability density is ρ(φ1, …, φV) = |ψ(φ1, …, φV)|2.The wavefunction amplitude in the field basis determines the probability distribution used for truncation bounds.
- Truncation errors: Truncating each field value to [−φmax, φmax] produces an overlap loss bounded using the probability that any site lies outside this range.The union bound relates the total truncation error to the sitewise out-of-range probabilities.
- Field and momentum discretization: The field basis is discretized in increments of δφ, while the conjugate-momentum basis is related by a Fourier transform.This links field discretization to momentum truncation through the canonical commutation relation.
A.5 Adiabatic Preparation of Interacting Wavepackets
Interacting wavepackets are prepared adiabatically by evolving from a massive free theory along a path that avoids the quantum phase transition. The procedure controls propagation and diabatic particle creation while preserving wavepacket localization.
- Wavepacket preparation: Single-particle wavepackets can be prepared at finite evolution time, and multiple-particle preparation works similarly when particles are separated by more than 1/m.The separation condition keeps the particles beyond the characteristic interaction length.
- Relativistic dynamics: The construction uses the relativistic dispersion relation Ep(s) = √(p2 + m2(s)) in the continuum-like regime.The lattice approximation is expected to be valid when particle momentum is suitably controlled.
- Wavepacket propagation: For a narrowly concentrated momentum wavepacket, the phase expansion separates translation from broadening.The first-order phase translates the packet by distance D, while the second-order term induces broadening.
- Adiabatic path: The preparation starts from a massive free theory and follows an adiabatic path toward weakly or strongly coupled continuum-like theories.The path must not cross the quantum phase transition to maintain adiabaticity.
- Error control: The evolution parameters J and τ are chosen to keep propagation length small and the probability of diabatic particle creation small.These criteria determine the discretized preparation schedule and its runtime requirements.
A.5.1 Weak Coupling
In weak coupling, the analysis bounds adiabatic preparation errors from particle creation and splitting while examining the regimes where classical scattering calculations become inefficient. Perturbation theory is limited at arbitrarily high precision because its series is asymptotic rather than convergent.
- Error channels: Weak-coupling adiabatic preparation has two diabatic error channels: particle creation from the vacuum and splitting of incoming particles.The transition matrix element is decomposed into contributions from these two processes.
- Mass control: At first order in λ0, the φ4 and φ2 contributions to two-particle transitions cancel, keeping the physical mass fixed along the preparation path.This cancellation relies on the chosen path and requires tuning of µ.
- Motivation: Classical scattering methods can break down or become inefficient at strong coupling, large numbers of external particles, and high precision.The weak-coupling section focuses on the high-precision frontier after noting the other regimes.
- High-precision regime: Perturbative methods cannot be extended to arbitrarily high precision because quantum-field-theory perturbation series are asymptotic but not convergent.This motivates analyzing gate complexity directly in the high-precision regime.
- Continuum scaling: For d = 1, 2, 3, the splitting-probability integral is convergent as a → 0.This supports controlled continuum extrapolation for the splitting channel in the considered dimensions.
A.5.2 Strong Coupling
Strong coupling is approached by tuning the bare coupling near the critical value and preparing the state along a path toward the phase transition. The resulting scaling analysis covers high-momentum scattering but relies on approximations whose validity becomes most stringent near criticality.
- Strong-coupling construction: Strongly coupled continuum-like theories are obtained in two and three spacetime dimensions by approaching the quantum phase transition.The bare coupling is varied toward a critical value while the bare mass is held constant.
- Scattering regime: For ultrarelativistic incoming particles near critical coupling, the process can produce a shower with nout ∼ p/m outgoing particles.Strong coupling makes perturbation theory inapplicable in this regime.
- Adiabatic scaling: The scaling τ = O(p^(d+1)) for d = 1, 2 suffices to satisfy the adiabaticity and diabatic-error conditions.This result controls the time required for adiabatic state preparation at high momentum.
- Limitations: The strong-coupling analysis cannot provide a detailed quantitative treatment of adiabaticity and instead applies the traditional criterion under condition 100.The relevant approximations become most stringent at s = 1, where derivatives of m2 with respect to s become large.
- Gate complexity: Using a kth-order Suzuki–Trotter formula, the total gate count is derived from the preparation-time scaling for d = 1, 2.The gate complexity depends on the order of the formula and the physical parameters of the simulated system.
A.6 Suzuki-Trotter Formulae for Large Lattices
The section uses Suzuki-Trotter formulae and Lie-algebra arguments to establish linear lattice-size scaling, then derives gate scaling for strongly coupled theories at large momentum. The resulting quantum-gate complexity is O(pd+1+o(1)(tV )1+o(1)).
- Motivation: Little attention had been given to how quantum simulation algorithms scale with the number of lattice sites V.The section addresses scaling with V in addition to the previously studied dependence on t.
- Suzuki-Trotter construction: Suzuki and elementary Lie algebra theory derive linear scaling in V for Hamiltonians with non-neighboring terms.The construction uses a standard Suzuki-Trotter theorem for alternating evolutions under A and B.
- Commutator bounds: For local Hamiltonians Hφ and Hπ, the nested-commutator coefficient satisfies ∥∆2k+1∥ = O(V) for fixed k.This improves on the generic bound based on max{∥A∥, ∥B∥} by exploiting the canonical commutation relations.
- Strong-coupling estimate: In the strongly coupled, high-momentum regime, the commutator action is bounded heuristically by O(Vp2k+1+d).The estimate uses an energy bound on local positive terms and the relation that a scales as a small multiple of 1/p.
- Gate complexity: O(pd+1+o(1)(tV )1+o(1)) quantum gates simulate the strongly coupled theory at large p.Each timestep requires O(V) = O(V pd) gates, while the number of timesteps scales as n = O(p1+(1+d)/2kt1+1/2k).