Digitization of Scalar Fields for Quantum Computing

• Nyquist-Shannon sampling gives digitization errors scaling as |log|log|ε_dig||| ∼ n_Q, while spatial lattice and finite-volume errors scale as |log|ε_latt|| ∼ N_Q.The required field-space range and sampling interval depend on wavefunction localization and curvature.Source: §I · p. 4
• The paper isolates field-digitization error from time-evolution approximation and hardware noise, finding that digitization can be controlled while the latter sources dominate realistic NISQ error budgets.The main results assume an ideal device with exact time evolution; first-order Trotterization and unitary gate noise are treated in Appendix H.Source: §I · pp. 4–5, Figure 1 · p. 5
• The analysis examines φ-operator and harmonic-oscillator eigenstate bases for localized and delocalized scalar-field wavefunctions.It studies 0+1- and 1+1-dimensional λφ^4 theory and extrapolates resource estimates to d+1 dimensions.Source: §I · pp. 5–6
• The work provides operator structures for n_Q = 3, 4, 5 qubits per spatial site and quantum circuits for n_Q = 3.It estimates qubit, operator, and gate requirements for the digitized simulations.Source: §I · pp. 5–6
• The study concludes that Quantum Fourier Transform improvement can make digitization effects exponentially small once Nyquist-Shannon sampling conditions are satisfied, outperforming polynomial Hamiltonian improvements.The QuFoTr acts on the n_Q qubits at each spatial site and removes all polynomial dependence on δφ(x).Source: §I · p. 6

Lattice discretization

• A cubic lattice with spacing a and extent L discretizes space into L/a sites per direction for numerical evaluation of scalar-field observables.The discretization uses canonical field and momentum operators with the standard equal-time commutation relation.Source: §II · pp. 6–7
• The discretized Hamiltonian uses finite-difference gradients and bare parameters m_0 and λ_0 tuned to recover quantities such as the scalar mass and 2φ scattering amplitude.Fields, momentum, Hamiltonian, and mass can be rescaled into dimensionless variables for the lattice formulation.Source: §II · p. 7

Lattice artifacts

• At energies well below the ultraviolet cutoff π/a, lattice observables receive polynomial spacing corrections ∼(aE)^n and, for localized states, exponential finite-volume corrections ∼e^-MφL.The lattice theory is therefore treated as a low-energy effective field theory with ultraviolet cutoff set by the inverse lattice spacing.Source: §II · p. 7

Improved actions

• Symanzik-inspired improvements add lattice operators suppressed by powers of a, with coefficients fixed by matching and tuning to reduce discretization errors.For example, the Sheikholeslami-Wohlert term improves Wilson fermion spatial errors from O(a) to O(a^2), while higher-order improvements can become impractical.Source: §II · pp. 7–8

• MSAH emphasized the theorem’s relevance to quantum computations, building on its implicit use by Somma and its broader role in signal processing and LQCD analysis.Source: §III · p. 8
• Full support coverage in position and momentum space yields exponentially small discretization errors, constraining lattice spacing and volume for accurate low-lying eigenstates and eigenvalues.This underpins Lüscher’s finite-volume analysis of QCD observables.Source: §III · p. 8
• The theorem does not select a basis: plane waves suit smoothly decaying functions, while tuned harmonic-oscillator eigenstates can efficiently represent localized Gaussian-like functions.Source: §III · p. 8
• Basis choice determines the qubit count and operator complexity required for a target precision, so optimal quantum implementations must balance qubit and gate resources.Source: §III · pp. 8–9
• In LQCD, power-law deviations scaling as ~(aE)^n arise from lattice-Hamiltonian errors rather than Nyquist-Shannon undersampling when sampling bounds are satisfied.The authors note that applying the theorem explicitly in classical quantum Monte Carlo warrants further exploration.Source: §III · p. 9

Setup

• The 0+1-dimensional non-interacting scalar field theory is simply a harmonic oscillator used to demonstrate the preceding construction.The formulation applies field and Hamiltonian redefinitions, including φ̂ = 1/√m̄₀ φ̄̂, Π̂ = √m̄₀ Π̄̂, and Ĥ = m̄₀ H̄.Source: §IV · p. 9

Digitized formulation

• Digitization identifies the rescaled field and conjugate momentum with X and P, following studies by Somma and MSAH.The digitized Hamiltonian is denoted H_h in the supplied formulation.Source: §IV · p. 9

Continuum limit

• Without field digitization, the system is a harmonic oscillator with energy eigenstates |ψ_n⟩ and energy eigenvalues E_n = n + 1/2.The conjugate momentum is represented as the field-space derivative Π̄ = −i d/dφ̄, satisfying the equal-time commutation relation.Source: §IV · p. 9

A. Jordan-Lee-Pressel Basis

• The digitized field uses ns = 2^nQ regularly spaced values between finite field cutoffs, replacing the conjugate momentum by a finite-difference operator in field space.The field operator is diagonal in the field basis, while the conjugate momentum is handled in its complementary representation.Source: A. Jordan-Lee-Preskill Basis · pp. 9–10
• A Quantum Fourier Transform makes the conjugate-momentum operator diagonal, and replacing the finite-difference momentum with the exact momentum operator removes the residual even-power δφ^2 discretization differences from the undigitized oscillator.The resulting lowest-state quantities are exponentially close to the continuum harmonic-oscillator values when the sampling condition is met.Source: A. Jordan-Lee-Preskill Basis · p. 11

A. Jordan-Lee-Preskill Basis

• After the Nyquist-Shannon bound is satisfied, eigenvalue and eigenvector errors become exponentially small; before saturation, deviations decrease polynomially with δφ^2.The effective operating point is near sampling-bound saturation, where increasing qubits changes the error-scaling regime.Source: A. Jordan-Lee-Preskill Basis · p. 11
• 4 qubits per site suffice to eliminate field-digitization errors as an important NISQ-era error source, while a 4-qubit grid with φmax = 4.7 achieves better than 10^-3% energy precision for the lowest five harmonic-oscillator states.The ideal simulations optimize φmax at 4.7 for four qubits; the broader study compares 3-, 4-, and 5-qubit digitizations.Source: A. Jordan-Lee-Preskill Basis · pp. 10, 12, Figure 2 · p. 11

1. Perturbatively Improved Hamiltonian

• Adding powers of δ^2 to the finite-difference conjugate-momentum operator systematically cancels deviations from the true operator and produces a parametrically improved Hamiltonian.The improvement term is derived in conjugate-momentum space and transformed into ˜φ space.Source: §1 · pp. 12–13
• One to two orders of magnitude improvement in harmonic-oscillator ground-state accuracy is obtained with the quadratic Hamiltonian improvement versus the unimproved calculation.Figure 3 shows that the residual dependence on δ˜φ becomes O(...).Source: §1 · p. 13
• Polynomial digitization errors can be removed by fitting their leading δ˜φ dependence across digitization scales and adding cancellation terms to the Hamiltonian.The procedure follows Symanzik improvement, with terms informed perturbatively or by available higher-dimension operators.Source: §1 · p. 13

• Exact conjugate-momentum operators retain exponential low-lying-state precision, but imperfect gates and decoherence can reduce the achievable precision to polynomial behavior.The perturbatively improved Hamiltonians illustrate this degradation, making exponentially small digitization improvements potentially irrelevant below quantum-noise levels.Source: §2 · p. 13
• ϵ ∼(1.8(2)×10^3) 2^-2.234(4)n_s at the Nyquist-Shannon saturation point, giving exponential precision in states and double-exponential precision in qubits for localized low-energy fields.The scaling follows n_s = 2^n_Q and provides a general estimate for qubit requirements.Source: Figure 4 · p. 14, §2 · pp. 14–15
• For each gate-noise level, reducing the field digitization below a threshold no longer improves precision, while digitization errors remain below other errors for arbitrarily many low-lying eigenstates with few qubits.With φ̄_max = 5.5, ideal calculations can reach ∼10^-12 below a critical δ̃φ.Source: §2 · p. 14
• Noise below ∼10^-13 has little effect for the explored parameters, and increasing φ̄_max beyond ∼3.5 does not improve precision at the fixed noise level.Precision should therefore be matched in both directions: excess qubits or gates are wasteful when hardware cannot resolve the represented accuracy.Source: §2 · p. 15
• Post-measurement noise mitigation can justify exceeding raw hardware precision, but the extrapolated hardware precision must then be balanced against qubit-mapping precision.The cited variational-eigensolver implementations use extrapolation in a parameter that increases with two-qubit interactions.Source: §2 · p. 15
• The primitive Gaussian phase-noise model does not significantly alter results above the effective noise level, but more accurate architecture-specific noise and decoherence models remain necessary.The model represents Gaussian over- or under-rotations, while ongoing error-correction research is expected to improve hardware-specific modeling [72].Source: §2 · p. 15

Convergence

• When the basis spans the low-lying states’ field and momentum spaces, Nyquist-Shannon sampling ensures exponential convergence of eigenstates and eigenvalues.For ωφ = 1, the evolution matrix is diagonal and N basis states suffice to converge the lowest N eigenstates.Source: B. Harmonic Oscillator Basis · p. 16

Encoding distinction

• HO eigenstates are mapped directly onto quantum-register states in a first-quantized encoding, unlike prior work that digitized continuous HO wavefunctions in a field basis.This approach also differs from the earlier deuteron calculation using a second-quantized occupancy encoding.Source: B. Harmonic Oscillator Basis · p. 16

Digitization method

• HO digitization is set directly by the maximum basis-state count and ωφ, so Hamiltonian circuits can be constructed entirely in field space without a Quantum Fourier Transform.The basis simultaneously determines coverage in field and conjugate-momentum spaces.Source: B. Harmonic Oscillator Basis · p. 16

Precision

• Near ωφ = 1, the HO basis outperforms field-space digitization, while poor frequency choices produce inferior precision.At ωφ = 1, the lowest basis state is an exact eigenstate and the error vanishes.Source: B. Harmonic Oscillator Basis · p. 17, Figure 5 · p. 17

Resource comparison

• For an arbitrary unknown system, field digitization is more robust than a detuned HO basis because detuning causes exponentially many multi-qubit operations.The field-digitization basis remains limited to 2-body operators, whereas detuned HO evolution can exceed its resource requirements substantially.Source: B. Harmonic Oscillator Basis · p. 18
• A tuned HO basis requires significantly fewer evolution operations than field digitization for a free HO because its basis states are exact system eigenstates.The tuned basis requires 0 two-qubit gates, while detuning introduces interactions among all qubits and increases Trotter demands.Source: B. Harmonic Oscillator Basis · p. 18

JLP digitization

• 18 states are required for λ = 32 at φ̄_max = 2.5 to reach Nyquist-Shannon saturation, versus 6 states for λ = 0.The interaction expands the momentum-space domain to approximately Π̄ = ±10 while shrinking field-space support, creating a trade-off that preserves similar precision for comparable resources.Source: C. λφ4 Scalar Field Theory: Comparing… · p. 19, Figure 6 · p. 20

HO basis tuning

• Increasing ω_φ localizes HO basis functions in field space and extends momentum-space coverage, whereas increasing the state count expands both momentum- and field-space support.These correlated effects make ω_φ and the number of states jointly analogous to JLP’s independently tuned field-space and sampling parameters.Source: C. λφ4 Scalar Field Theory: Comparing… · pp. 20–21

Basis comparison

• For n_Q ≥ 3, tuning either basis can suppress digitization errors below expected NISQ hardware errors, while HO can outperform φ̄_max-tuned JLP at fixed qubit count.For n_Q < 3, field digitization is expected to limit NISQ accuracy; HO also provides a larger acceptable tuning window and reduced rotation-angle sensitivity.Source: C. λφ4 Scalar Field Theory: Comparing… · pp. 21–22

NISQ implementation

• Only systems with n_Q ≤ 4 may be practical on NISQ devices because of time-evolution gate-count constraints.Manual JLP compilation removes adjacent CNOT pairs, whereas systematic cancellation is more difficult for HO circuits because of frequent Pauli-basis changes.Source: C. λφ4 Scalar Field Theory: Comparing… · pp. 22–23

Circuit resources

• JLP operators involve at most as many qubits as the highest field-interaction power in the Hamiltonian, whereas HO operators have k limited by site-register qubits.For JLP field digitization, additional 2-qubit CNOT gates for the QuFoTr are quickly outnumbered by CNOTs implementing HO k-body operators; JLP CNOT scaling is limited to n4.Source: C. λφ4 Scalar Field Theory: Comparing… · p. 23

1. Delocalized Wavefunctions: m2 < 0

• As μ increases, the symmetric ground state and antisymmetric first excited state become exponentially close in energy, so insufficient precision can produce arbitrary mixtures.Resolving these low-lying states requires more qubits, because their splitting is controlled by barrier penetration between the two minima.Source: §1 · pp. 23–25
• For μ = 2 and 5 with λ = 1, the field-space wavefunction has two localized support regions, demanding larger JLP φ̄max or smaller HO ωφ when parameters are tuned.Figure 8 explores sensitivity to φ̄max for JLP and to ωφ and the number of states per site for the HO basis.Source: §1 · p. 23, Figure 8 · p. 24
• With φmax = 9 and nQ = 7, JLP represents the ground and first-excited wavefunctions for λ = 1 and μ = 1, 2 in field and momentum space.The momentum-space plots mark where the Nyquist-Shannon bound is saturated, indicating when increasing φ̄max is preferable to increasing the number of quantum states.Source: Figure 9 · p. 24
• The JLP basis can achieve higher ground-state-energy precision than the HO basis at equal qubit counts for tuned delocalized states.This reflects the wavefunctions’ strongly non-Gaussian, two-region support; JLP remains effective without assuming their detailed structure.Source: Figure 8 · p. 24, §1 · p. 25
• Software and hardware noise can bury the small energy splittings of systems with multiple degenerate extrema, limiting reliable ground-state identification.The relevant noise arises from propagator approximation and intrinsic gate implementation.Source: §1 · p. 25

Position-space implementation

• The 2-site theory inventories the operations and gate counts needed for d+1-dimensional λφ4 evolution, using position-space interactions among neighboring sites.Position-space evolution requires at most d neighboring two-site interactions per site and can be parallelized to constant depth as lattice size increases.Source: §V · pp. 26–27

Digitization convergence

• Figure 11 shows double-exponential convergence in n_Q toward the un-digitized ground-state energy for the 2-site λφ4 theory.The plotted precision excludes deviations from the infinite-spatial-site continuum limit.Source: Figure 11 · p. 27, §V · p. 27

Continuum-limit errors

• Finite-difference implementation of φ∇2φ introduces polynomial lattice-spacing deviations, giving continuum errors that scale as ε ∼ 2^N_Q.These lattice-spacing artifacts are distinct from the digitization convergence shown in Figure 11.Source: §V · p. 27

NISQ constraints

• Although a lattice-wide QuFoTr could remove polynomial gradient artifacts through exact Fourier-space phases, global entangling operations are expected to be prohibitive on NISQ hardware.The finite-difference gradient therefore appears optimal because it requires only local interactions between neighboring qubit registers.Source: §V · pp. 27–28

Resource comparison

• For the intersite φ∇2φ operator, the JLP basis requires fewer resources than the HO basis because JLP uses two-body interregister operators, whereas HO requires operators acting on up to all 2n_Q qubits.This comparison emphasizes that the physical representation mapped onto qubits strongly affects quantum-resource requirements.Source: §V · pp. 28–29

Basis selection

• The JLP basis is more robust for delocalized or non-smooth wavefunctions, while the HO basis can be more noise-tolerant but requires all k-body operators through k = nQ.Those HO operators increase CNOT counts, and two-qubit entangling gates often dominate near-term hardware errors.Source: §VI · pp. 29–30

Resource requirements

• nQ ∼4 qubits per site for m2 > 0 and nQ>∼6 for m2 < 0 reduce digitization and discretization errors below near-term hardware noise.These requirements extend prior estimates to delocalized field-space wavefunctions and apply across JLP, momentum-mode, and harmonic-oscillator digitizations.Source: §VI · p. 30

Nyquist-Shannon scaling

• | log | log |ϵ||| ∼nQ when the Nyquist-Shannon bound is saturated, because field and conjugate-momentum spaces then achieve comparable accuracy.For a free 0+1-dimensional theory, ϵ(%) = (1.8(2) × 103) 2−2.234(4)ns.Source: §VI · p. 30

Quantum Fourier transform

• Exact momentum-space phases implemented with a quantum Fourier transform produce the required convergence through a highly non-local field-space operator.For spatial dimensions greater than zero, the QFT register grows with digitization precision rather than spatial-lattice size, making global entanglement a plausible NISQ goal.Source: §VI · p. 30

Noise matching

• Noise limits double-exponential convergence at a precision barrier set by phase-noise magnitude, so digitization precision should be matched to each device’s noise level.Exact phases remain preferred because they minimize the required state count, cost no more than local momentum operators, and symmetrize field- and momentum-space gate structures.Source: §VI · p. 30

Outlook

• An ideal ∼60-qubit device could represent up to ∼20 spatial sites at the 10−6 level, while propagator approximation and hardware gate errors remain dominant.The study also identifies truncation and boundary-condition choices as important for avoiding edge effects and preserving Fourier-space symmetry.Source: §VI · pp. 30–31

Operator decomposition

• The digitized harmonic-oscillator Hamiltonian has a Pauli-operator decomposition split into an overall identity and non-identity terms, with only nontrivial operators acting on two qubits and extension to larger systems.Source: Appendix A · p. 32
• Finite-difference conjugate momentum introduces higher-qubit operators, increasing resource costs and producing power-law deviations compared with the exact Hamiltonian’s exponentially close ideal results.For n_Q = 4, a σz ⊗ σz ⊗ σz ⊗ σz term appears; for n_Q = 6, two-, four-, and six-qubit σz combinations occur.Source: Appendix A · p. 33

Quantum circuits

• A symmetric QuFoTr circuit implements one Trotterized evolution step by alternating field- and momentum-space operator actions for the n_Q = 3 system.The sequence applies Φ3, a symmetric QuFoTr, a second Φ3, and the inverse symmetric QuFoTr; equivalent controlled-rotation circuits were presented by MSAH.Source: Appendix A · pp. 33–34, Figure 12 · p. 34
• One n_Q = 3 Trotterized time-evolution step requires 24 CNOT gates, 6 Hadamard gates, and 24 single-qubit phase rotations.The count includes the symmetric QuFoTr circuits and implements the unimproved Trotterized evolution operator.Source: Appendix A · p. 34

Appendix B: Jordan-Lee-Preskill Circuit Compilation

• CNOT cancellations in the JLP basis reduce redundant gates but do not change the expected fourth-power scaling with qubits per lattice site for λφ4 operators.Cancellations arise systematically for neighboring diagonal operators and are expected primarily in position-space implementations of interacting fields; HO-basis compilation is more cumbersome but likewise lacks modified scaling.Source: Appendix B · pp. 34–36, Figure 13 · p. 35

Appendix C: Harmonic Oscillator Basis Example: Three Qubits

• A tuned three-qubit HO basis requires only three nontrivial commuting single-qubit operators, whereas detuning introduces three three-qubit, four two-qubit, and four single-qubit operators.Interactions are substantially more expensive: the mapped field-coordinate interaction contains seven three-qubit, four two-qubit, and one single-qubit operators, and tuned free-HO evolution is therefore not representative of interacting resource costs.Source: Appendix C · pp. 37–38, Figure 14 · p. 37

Appendix D: Symmetric Quantum Fourier Transform

• The symmetric QuFoTr makes position- and momentum-space gate decompositions identical for a free theory and removes odd-body phase operators, yielding roughly a factor-of-two reduction when all k-body terms are needed.A single layer of single-qubit phase gates symmetrizes the momentum-space register, while qubit-order reversal can be handled by classical readout rather than swap networks.Source: Appendix D · pp. 38–39

Appendix E: Field Conjugate-Momentum Operators

• Finite-difference, improved, and exact conjugate-momentum operators become increasingly nonlocal in field space, while direct Fourier-space implementation provides a diagonal qubit formulation.This progression is shown for both periodic and twisted boundary conditions.Source: Appendix E · p. 40, Figure 15 · p. 41, Figure 16 · p. 42

Appendix F: Basic Circuit Construction

• A k-body Pauli exponential is implemented using 2(k −1) CNOTs, with parity accumulation and basis changes converting available z-axis rotations into arbitrary Pauli rotations.Hadamard and phase-gate transformations provide x- and y-axis rotations, while CNOT strings compute and uncompute the register parity.Source: Appendix F · pp. 42–43

Appendix G: Lowest-Lying Energy Eigenvalues

• The lowest-lying ground-state and first-excited-state energies for the one-site and two-site systems are reported in Table IV, including the exactly known one-site harmonic-oscillator case.The remaining ground-state values are given with sufficient digits for the tuning and precision plots in the main text.Source: Table IV · p. 43, Appendix G · p. 43

• Simulation errors from Trotterization and noisy gate implementation exceed theoretical digitization errors, making them the dominant uncertainty for near-term scalar-field simulations.The appendix concludes that digitization can be made subdominant, whereas errors from steps (2) and (3) remain significant barriers.Source: Appendix H · pp. 43, 47
• For n_Q = 3 and φ̄_max = 3.0, the noiseless Schatten 1-norm error scales linearly with δt, whereas accumulated noise causes low-δt saturation at O(1).The saturation reflects a software coherence-time limit from imperfect gates, even without decoherence or amplitude damping; lowering gate errors extends the usable circuit length.Source: Appendix H · p. 44, Figure 17 · p. 44
• The simulations use local unitary gate errors, with σ_CNOT one order of magnitude larger than σ_θ to represent anticipated two-qubit-error dominance.The model applies sampled SU(2)^⊗m errors before and after each gate, while omitting non-unitary, non-Markovian, and non-local fluctuations.Source: Appendix H · p. 45
• With σ_CNOT = 10^-3, ground-state persistence can remain at the 1% level for selected δt, but σ_CNOT = 10^-2 prevents this for every tested δt.Noise reduces persistence both at large δt through Trotter errors and at small δt through accumulated gate errors, despite structural suppression of half the Hilbert-space mixing.Source: Figure 18 · p. 45, Appendix H · p. 46
• At σ_CNOT = 10^-3, noise produces O(1) deviations in the field expectation value, while ∼10^-2 precision requires σ_CNOT = 10^-4 and σ_θ = 10^-5.The observable is initialized from a rotated ground state and is especially sensitive to low-energy-state mixing; the required precision is presently unavailable on NISQ hardware.Source: Appendix H · pp. 46–47
• Extrapolating collections of biased or lower-precision simulations toward the zero-error limit is identified as a leading route to recovering precision beyond available hardware noise.Higher-order Trotterization reduces temporal errors but increases gate counts and therefore exposure to implementation noise, so mitigation and algorithmic design must be considered together.Source: Appendix H · p. 47