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Digitization of Scalar Fields for Quantum Computing

Natalie Klco, Martin J. Savage

arXiv:1808.10378v2quant-phhep-lathep-phnucl-th

TL;DR

The paper examines how to digitize lattice λφ^4 scalar field theories on quantum computers while controlling qubit, operator, and gate resources for NISQ devices. It uses Nyquist-Shannon sampling to compare field-space bases and shows that QFT-based Hamiltonian improvement can make digitization errors negligible relative to propagator and hardware errors. For localized and delocalized wavefunctions, roughly 4 and 6-or-more qubits per site, respectively, are sufficient in the reported estimates.

  • Problem

    The paper addresses how accurately scalar field theories can be represented with modest qubit registers, given digitization, propagator-approximation, and hardware-noise constraints.

  • Method

    The study estimates qubit, operator, and gate resources, applies Nyquist-Shannon sampling, compares field-operator and harmonic-oscillator bases, and improves the Hamiltonian using QFT phases.

  • Results

    When the Nyquist-Shannon bound is satisfied, low-lying-state digitization precision scales as |log|log|ε|||∼n_Q, and the reported requirements are n_Q∼4 for localized and n_Q>∼6 for delocalized wavefunctions.

  • Takeaways & Limitations

    Digitization need not limit near-term scalar-field simulations; propagator approximations and hardware gate errors are expected to dominate the simulation error budget.

Abstract

from arXiv · show

Qubit, operator and gate resources required for the digitization of lattice $λφ^4$ scalar field theories onto quantum computers are considered, building upon the foundational work by Jordan, Lee and Preskill, with a focus towards noisy intermediate-scale quantum (NISQ) devices. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin, Spentzouris, Amundson and Harnik building on the work of Somma, provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan, Lee and Preskill, and eigenstates of a harmonic oscillator, to describe $0+1$- and $d+1$-dimensional scalar field theory. We show how techniques associated with improved actions, which are heavily utilized in Lattice QCD calculations to systematically reduce lattice-spacing artifacts, can be used to reduce the impact of the field digitization in $λφ^4$, but are found to be inferior to a complete digitization-improvement of the Hamiltonian using a Quantum Fourier Transform. When the Nyquist-Shannon sampling theorem is satisfied, digitization errors scale as $|\log|\log |ε_{\rm dig}|||\sim n_Q$ (number of qubits describing the field at a given spatial site) for the low-lying states, leaving the familiar power-law lattice-spacing and finite-volume effects that scale as $|\log |ε_{\rm latt}||\sim N_Q$ (total number of qubits in the simulation). For localized(delocalized) field-space wavefunctions, it is found that $n_Q\sim4(7)$ qubits per spatial lattice site are sufficient to reduce theoretical digitization errors below error contributions associated with approximation of the time-evolution operator and noisy implementation on near-term quantum devices.

I. INTRODUCTION

The paper studies how representing scalar fields with qubits limits accuracy on NISQ devices, separating digitization from simulation and hardware errors. It evaluates resource requirements, sampling constraints, field-space bases, and Hamiltonian improvements for scalar field theories.

  • I. INTRODUCTION: The paper targets near-term quantum calculations even though NISQ devices are not expected to outperform classical methods for the scalar-field evolution considered.The framework is intended to prepare for future simulations requiring substantial quantum resources.
  • I. INTRODUCTION: The Nyquist-Shannon theorem guides the required field-space range and sampling interval, which determine the qubits needed for accurate scalar-field digitization.The paper examines both field-operator eigenstates and harmonic-oscillator eigenstates as field-space bases.
  • I. INTRODUCTION: NISQ simulations contain separate digitization, propagator-approximation, and hardware-noise errors, with the paper’s main text treating only digitization errors.The first two error sources are hardware-independent; simulation and noise effects are addressed separately.
  • I. INTRODUCTION: The study focuses on digitization errors when mapping 0+1- and 1+1-dimensional λφ^4 scalar fields onto qubits, while estimating qubit, operator, and gate requirements.The analysis also extrapolates these estimates to d+1 dimensions and considers localized and delocalized field-space wavefunctions.
  • I. INTRODUCTION: Quantum Fourier Transform phases can remove polynomial digitization errors and make the remaining effects exponentially small once Nyquist-Shannon conditions are satisfied.This provides a complete digitization improvement of the Hamiltonian rather than only a partial improved-action correction.

II. LATTICE SCALAR FIELD THEORY WITH QUBITS

The lattice formulation discretizes space and represents each scalar-field value with a finite qubit register, producing a dimensionless Hamiltonian with tunable bare parameters. Lattice artifacts are power-law at low energy, motivating improved actions analogous to those used in lattice QCD.

  • II. LATTICE SCALAR FIELD THEORY WITH QUBITS: The discretized Hamiltonian contains kinetic, nearest-neighbor gradient, mass, and self-interaction terms, with bare parameters tuned to reproduce physical observables.The finite-difference gradient uses neighboring sites separated by the lattice spacing a.
  • II. LATTICE SCALAR FIELD THEORY WITH QUBITS: A spatial lattice with spacing a and extent L represents the scalar field at L/a sites per direction, while n_Q qubits per site give N_Q=n_Q(L/a)^d total qubits.The field values are restricted to a finite range and digitization interval.
  • II. LATTICE SCALAR FIELD THEORY WITH QUBITS: Low-energy lattice observables reproduce continuum quantities with corrections polynomial in aE and exponentially suppressed finite-volume effects for spatially localized states.The lattice theory therefore functions as a low-energy effective field theory with ultraviolet cutoff set by 1/a.
  • II. LATTICE SCALAR FIELD THEORY WITH QUBITS: Improved actions add symmetry-consistent operators whose coefficients are tuned to suppress lattice-spacing artifacts at higher orders.The paper applies this lattice-QCD-inspired strategy to scalar-field Hamiltonians, while noting that sufficiently elaborate improvements can become impractical.

III. IMPLICATIONS OF THE NYQUIST-SHANNON SAMPLING THEOREM

The Nyquist-Shannon theorem links accurate reconstruction to the field-space support and sampling interval, thereby constraining qubit requirements. Basis choice affects both qubit counts and operator or gate complexity, while localized lattice observables can retain exponentially accurate sampling.

  • III. IMPLICATIONS OF THE NYQUIST-SHANNON SAMPLING THEOREM: Sampling a function over an interval larger than its position-space support with spacing below the Nyquist bound enables reconstruction up to exponentially small corrections.The bound is expressed through the maximum momentum-space support k_max and sampling interval δx.
  • III. IMPLICATIONS OF THE NYQUIST-SHANNON SAMPLING THEOREM: Plane-wave bases are efficient for smooth functions with exponential or Gaussian decay, whereas other functions may favor different bases for covering position- and momentum-space support.The theorem itself does not specify the best basis for a given function.
  • III. IMPLICATIONS OF THE NYQUIST-SHANNON SAMPLING THEOREM: For quantum field computations, the theorem determines the qubits required for a target accuracy, while the basis also changes the number and complexity of operators and gates.Optimal basis selection therefore requires considering both precision and implementation resources.
  • III. IMPLICATIONS OF THE NYQUIST-SHANNON SAMPLING THEOREM: For localized lattice-QCD quantities, satisfying the sampling bounds makes eigenvalues and eigenstates exponentially close to those of the lattice Hamiltonian, leaving power-law lattice-spacing deviations.This separates sampling accuracy from discretization artifacts controlled by the lattice spacing.

IV. 0+1 DIMENSIONAL SCALAR FIELD THEORY

The paper first uses the 0+1-dimensional harmonic oscillator as a non-interacting test case for field digitization. Its continuum Hamiltonian has oscillator eigenstates and energies, while conjugate momentum acts as a field-space derivative.

  • III. 0+1 DIMENSIONAL SCALAR FIELD THEORY: The 0+1-dimensional non-interacting scalar theory reduces to a harmonic oscillator after field and Hamiltonian redefinitions.The digitized variables are identified with the oscillator’s dimensionless field and momentum operators.
  • III. 0+1 DIMENSIONAL SCALAR FIELD THEORY: The conjugate momentum is represented as the derivative operator -i d/dφ̄, satisfying the canonical equal-time commutation relation.This identifies how momentum is implemented in the continuous field-space description.

A. Jordan-Lee-Preskill Basis

The Jordan-Lee-Preskill basis digitizes the field on a finite, regularly sampled field-space grid and evaluates the field and momentum terms in complementary representations. Choosing the field range near Nyquist-Shannon saturation can yield high precision for low-energy harmonic-oscillator states, while complete momentum-operator digitization improves convergence.

  • JLP field digitization: The field is sampled uniformly over a bounded range, with n_Q qubits encoding the resulting field-space basis states.The field values span from -φ̄max to φ̄max at intervals set by the number of basis states.
  • JLP field digitization: A 4-qubit field representation with φ̄max = 4.7 achieves better than 10^-3% precision for the lowest five harmonic-oscillator energies on an ideal quantum computer.The reported optimum values are φ̄max = 3.1, 4.7, and 6.9 for 3, 4, and 5 qubits, respectively.
  • Nyquist-Shannon sampling: Increasing the field range until the Nyquist-Shannon bound is satisfied yields exponentially small further gains beyond the saturation point.The sampling range must cover the relevant field- and momentum-space support of the target harmonic-oscillator states.
  • JLP field digitization: The Hamiltonian evaluates the field-squared operator directly and uses a quantum Fourier transform to diagonalize the conjugate-momentum-squared operator.This alternates between field-space and momentum-space representations for efficient evaluation of the two terms.
  • Momentum representation: The exact conjugate-momentum implementation makes low-lying eigenvalues and eigenvectors exponentially close to their undigitized harmonic-oscillator values once sampling is adequate.This contrasts with the polynomial dependence on the field-space spacing associated with the finite-difference momentum operator.

1. Perturbatively Improved Hamiltonian

Perturbative improvement adds terms that cancel finite-difference errors in the conjugate-momentum operator. For the harmonic oscillator, this reduces the leading digitization error and improves ground-state energy accuracy by roughly one to two orders of magnitude.

  • Perturbative improvement: Finite-difference momentum errors can be systematically canceled by adding correction terms derived in conjugate-momentum space and transformed into field space.The resulting Hamiltonian contains additional operators representing the perturbative improvement.
  • Perturbative improvement: Adding the improvement term produces one to two orders of magnitude greater ground-state energy accuracy than the unimproved calculation.The improved residual dependence on the field-space spacing is reduced to a higher-order form.
  • Perturbative improvement: At sufficiently large φ̄max, digitization errors with polynomial dependence on δφ̄ can also be removed by fitting calculations across multiple digitization scales.This provides an extrapolation-based alternative when the finite-difference error structure is known.

2. The Impact of Noise

Noise limits the practical value of exponentially small digitization errors, so qubit precision should be matched to hardware noise while retaining the exact conjugate-momentum operator when possible.

  • Exact conjugate-momentum implementation gives exponential precision for low-lying states, but imperfect gates and decoherence can reduce this improvement to practical irrelevance.The paper contrasts the exact QuFoTr-based operator with perturbative approximations that exhibit only polynomial precision.
  • Below a noise-dependent digitization scale, decreasing the field spacing further no longer improves precision, allowing digitization errors to fall beneath other error sources with few qubits.For each gate-noise level, the study finds a field-spacing threshold below which finer digitization does not improve the calculation.
  • Once Nyquist-Shannon saturation is reached, precision increases exponentially with the number of field states and double-exponentially with qubits for localized low-energy wavefunctions.The fitted scaling is ϵ ∼(1.8(2)×10^3) 2^-2.234(4)ns, with ns = 2^nQ.
  • At fixed noise, increasing the field range beyond the hardware-resolvable precision provides no benefit; for the illustrated case, precision does not improve beyond ¯φmax ≈ 3.5.The broader conclusion is that qubit and gate resources should be matched to the precision available from the device.
  • Post-measurement noise mitigation can justify mapping the system at higher theoretical precision, because the relevant target becomes the extrapolated hardware precision.Without such mitigation, exceeding hardware-resolvable precision wastes qubits and gates.

B. Harmonic Oscillator Basis

A harmonic-oscillator basis can greatly improve precision when tuned near the system frequency, but detuning increases resource demands and makes field digitization more robust for unknown systems.

  • B. Harmonic Oscillator Basis: The basis truncation controls coverage of field and conjugate-momentum space through the number of states and ωφ, with Hamiltonian matrix diagonalization yielding the approximate eigenstates and eigenvalues.This first-quantized mapping places harmonic-oscillator basis states directly on the quantum register.
  • B. Harmonic Oscillator Basis: Near ωφ = 1, the harmonic-oscillator basis outperforms field-space digitization, while poor frequency choices produce inferior precision.At exactly ωφ = 1, the lowest basis state is an eigenstate and the ground-state error vanishes.
  • B. Harmonic Oscillator Basis: When the harmonic-oscillator basis is tuned exactly, time evolution uses commuting single-qubit phases without a Trotter decomposition.Detuning determines the required Trotter-step size and introduces interactions in the basis representation.
  • B. Harmonic Oscillator Basis: NISQ gate limitations make systems with nQ ≤ 4 the practically accessible regime in the resource estimates discussed.
  • B. Harmonic Oscillator Basis: A tuned harmonic-oscillator basis requires fewer operations for a free oscillator, whereas detuning produces exponentially many multi-qubit operations and can exceed field-digitization costs.The tuned basis works efficiently because its states are exact eigenstates; detuned bases share features of self-interacting systems.
  • B. Harmonic Oscillator Basis: For an arbitrary unknown system, the JLP field-digitization basis is more robust because it remains limited to two-body operators, unlike a generic detuned harmonic-oscillator basis.The comparison concerns quantum computational resources and may also favor field digitization in qubit requirements.

C. λφ4 Scalar Field Theory: Comparing Bases

The study compares JLP field eigenstates with harmonic-oscillator bases for digitizing interacting scalar fields, balancing sampling precision, tuning robustness, and circuit resources. Both bases can achieve accurate results when field- and momentum-space support satisfy Nyquist-Shannon sampling, but their resource trade-offs differ.

  • Interacting-field sampling: For λ = 32 and φ̄_max = 2.5, the highest attainable precision differs from λ = 0 by approximately 5 orders of magnitude, while saturation moves from 6 to 18 states.The interaction shrinks field-space support but increases momentum-space support, requiring finer field sampling.
  • Precision and tuning: For n_Q ≥3, tuning either φ̄_max in JLP or ω_φ with the HO state count can make digitization errors significantly smaller.Both bases require balanced sampling in field and conjugate-momentum space; undersampling either domain reduces precision.
  • Circuit resources: With NISQ-era gate-count constraints, only systems with n_Q ≤4 may be practical for the analyzed one-site time-evolution circuits.The comparison treats one first-order-Trotterized step and standard multi-Pauli CNOT decompositions.
  • Precision and tuning: At fixed qubit count, the HO basis can provide higher precision and a broader acceptable tuning window than tuned JLP digitization in the localized-wavefunction system.The broader window reduces sensitivity to the exact angles used in Z-axis rotation gates.
  • Circuit resources: JLP limits interaction operators to at most the Hamiltonian’s highest field power, whereas the HO basis requires tensor-product operators acting on up to all 2n_Q qubits.The additional QFT CNOTs in JLP are quickly outweighed by the HO basis’s higher-body operator costs.

1. Delocalized Wavefunctions: m2 < 0

Delocalized low-lying wavefunctions make ground- and first-excited-state resolution demanding because their energy separation becomes small. In this regime, tuned JLP digitization can outperform the HO basis at equal qubit count, while noise can strongly affect ground-state identification.

  • Wavefunction structure: For m² < 0, increasing μ produces wavefunctions with support near two separated potential minima, enlarging the field-space coverage required by JLP.The corresponding JLP requirement is larger φ̄_max, while the HO basis uses smaller ω_φ values when tuned.
  • State resolution: For delocalized states, low-precision calculations may return arbitrary combinations of the nearly degenerate ground and first-excited states.Resolving these states requires higher precision and therefore more qubits.
  • Basis comparison: At equal qubit count, tuned JLP digitization can achieve higher ground-state-energy precision than the HO basis for non-Gaussian delocalized wavefunctions.The HO basis loses its expected coverage advantage when the wavefunction is far from Gaussian.
  • NISQ implications: Noise from propagator approximation or intrinsic gate implementation is expected to significantly limit correct ground-state identification in systems with multiple degenerate extrema.The limitation is especially relevant for delocalized low-lying states with near-degenerate energies.

V. 1+1 DIMENSIONAL λφ4 SCALAR FIELD THEORY

The two-site theory extends the digitization analysis to spatially coupled λφ4 fields, where local finite-difference interactions preserve manageable circuit structure. Digitization converges rapidly with site-register qubits, while residual lattice-spacing errors and global Fourier-transform costs constrain improvements.

  • Scaling to higher dimensions: A d+1-dimensional position-space evolution can be assembled from site-by-site applications involving at most d neighboring two-site interactions per site.The two-site 1+1-dimensional theory therefore supplies the operation and gate-count inventory for larger spatial lattices.
  • Digitization and lattice effects: The two-site λφ4 theory shows double-exponential convergence of digitization errors with n_Q toward the un-digitized value.Finite-difference spatial operators introduce separate polynomial lattice-spacing deviations that are not included in the displayed digitization convergence.
  • Digitization and lattice effects: The continuum approach retains lattice-spacing errors scaling as ε ∼ 2^-N_Q unless Fourier-space phases are implemented through a Quantum Fourier Transform.The QuFoTr can remove polynomial lattice-spacing artifacts, but requires nonlocal operations across the lattice.
  • Spatial implementation: Position-space time evolution requires only local interactions between neighboring spatial sites, making finite-difference gradients preferable to a global lattice-wide Fourier transform on NISQ devices.Global operations are expected to be prohibitive because of gate fidelity and coherence-time limitations.
  • Spatial implementation: The JLP basis uses two-body intersite qubit interactions, whereas the HO basis requires tensor-product Pauli operators acting on up to all 2n_Q qubits.This operator-structure difference makes the physical representation of the field important for resource estimates.

VI. SUMMARY AND OUTLOOK

The paper compares field-operator and harmonic-oscillator digitizations for scalar-field simulation, emphasizing resource-aware choices for NISQ hardware. It concludes that digitization can be made subdominant to propagator-approximation and hardware-gate errors when sampling and tuning are appropriate.

  • The JLP basis is more robust for delocalized or nonsmooth wavefunctions, whereas the harmonic-oscillator basis can be advantageous when tuning flexibility matters.
  • nQ ∼4 qubits per site suffice for m2 > 0, while nQ ≳6 are needed for m2 < 0 to reduce digitization and discretization errors below near-term hardware noise.
  • |log|log|ε||| ∼ nQ when the Nyquist-Shannon bound is saturated, yielding comparable accuracy in field and conjugate-momentum space.
  • Exact momentum-space phases implemented with a quantum Fourier transform produce the double-exponential convergence, but noise imposes a precision barrier.
  • Higher-body operators make the harmonic-oscillator basis more burdensome, while truncation artifacts can affect gate decompositions and symmetries.
  • Propagator approximation and hardware gate errors, currently above 10^-4 in the cited model, are expected to dominate once digitization is sufficiently controlled.
  • Hardware-specific resource maps support choosing bases and digitization parameters according to whether qubits, gates, precision, or noise are most constrained.

Appendix A: Jordan-Lee-Preskill Basis Example: Three Qubits

The three-qubit JLP example decomposes the digitized Hamiltonian into Pauli operators and constructs Trotterized evolution by alternating field-space operations with quantum Fourier transforms. The exact momentum-space implementation preserves a simple operator structure while requiring nonlocal field-space action.

  • A three-qubit register represents eight states, and the Hamiltonian is decomposed using tensor products of Pauli operators and the identity.
  • The three-qubit JLP Hamiltonian contains only two-qubit nontrivial operators, with no higher-body terms in its free-field decomposition.
  • Exact momentum operators remain simple in Fourier space, whereas finite-difference operators introduce higher-body Pauli terms and substantially greater resource requirements.
  • The Trotterized evolution applies a field-space phase, a symmetric quantum Fourier transform, a momentum-space phase, and the inverse transform.
  • The three-qubit phase circuit uses six CNOT gates and three single-qubit phase operations for one application of the basic operator.

Appendix B: Jordan-Lee-Preskill Circuit Compilation

The appendices describe circuit compilation and operator representations for JLP and harmonic-oscillator digitizations, including CNOT cancellation, Fourier-transform choices, and increasingly nonlocal momentum operators. These constructions expose the trade-off between simpler operator structure and the resources needed for improved digitization.

  • Neighboring CNOT pairs can be systematically cancelled in JLP-basis circuits because operators between Fourier transforms are diagonal.
  • CNOT cancellation reduces redundant gates but does not change the expected fourth-power scaling with qubits per site for λφ4 operators.
  • The harmonic-oscillator basis has a more cumbersome compilation structure, including three-body operators and many-body terms absent from the simplest JLP decomposition.
  • The symmetric quantum Fourier transform removes odd-body operators from Fourier-space phase decompositions, providing roughly a factor-of-two reduction in their number.
  • Finite-difference, improved, and exact momentum operators become progressively more nonlocal in field space, while exact Fourier-space implementation directly realizes the desired phases.

Appendix H: Noisy Simulations

For a three-qubit, one-site free scalar field, noisy Trotterized evolution produces errors that can exceed theoretical digitization errors, especially as gate noise accumulates over many small steps.

  • Simulation errors from first-order Trotterization and representative near-term quantum noise exceed the theoretical systematic errors of digitization.The authors identify simulation errors in steps (2) and (3) as the dominant uncertainty source.
  • The Schatten 1-norm distance from the exactly digitized propagator scales linearly with Trotter step size for idealized Trotter evolution.The study uses a one-site free scalar field digitized onto three qubits with φ̄max = 3.0.
  • At small δt, accumulated noise from the increasing number of Trotter steps causes the noisy propagator to depart from ideal Trotter scaling and can reach O(1) operator-norm errors.For σCNOT = 10^-3, saturation appears near δt ≈ 3 × 10^-3 in the studied evolution.
  • Reducing gate-error rates extends the usable circuit before noisy-propagator saturation, but higher-order Trotterization may increase gate counts and therefore noise exposure.The hardware- or software-coherence boundary remains dependent on the implementation and noise model.
  • Ground-state persistence decreases at both large δt, from Trotterization errors, and small δt, from accumulated gate noise.Noiseless evolution becomes exact as δt approaches zero, whereas noisy evolution eventually loses persistence because many noisy steps accumulate.
  • For the field expectation value, step-(3) noise alone produces O(1) deviations at σCNOT = 10^-3, while σCNOT = 10^-4 and σθ = 10^-5 yield approximately 10^-2 precision.The observable is initialized from a ground state rotated by one site in φ-space.
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