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Fixed-Point Quantum Circuits for Quantum Field Theories

Natalie Klco, Martin J. Savage

arXiv:2002.02018v1quant-phhep-thnucl-th

TL;DR

The paper addresses the challenge of preparing scalar-field QFT ground states on digital quantum devices. It derives localized fixed-point unitaries from small non-interacting systems, with exponentially localized correlations and double-exponentially suppressed digitization artifacts supporting larger-register preparation and anticipated interacting-theory applications.

  • Problem

    Ground-state preparation for scalar QFTs requires many entangling gates because spatial-gradient correlations entangle sites, while arbitrary real wavefunctions are exponentially costly to prepare.

    Source: §I · p. 1

  • Method

    The paper uses ground-state correlations, circuit symmetries, and effective-operator ideas to derive spatially localized fixed-point circuit elements from small systems.

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

  • Results

    Digitization artifacts are double-exponentially suppressed with the number of field-digitization qubits, and fixed-point angles agree sufficiently with digitized circuits for small near-term instances.

    Source: §IV · p. 7

  • Takeaways & Limitations

    Classically computed ground states in small volumes can determine preparation operators for larger lattices than can be stored classically, with anticipated applicability to interacting scalar theories.

    Source: §V · p. 8

Abstract

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Renormalization group ideas and effective operators are used to efficiently determine localized unitaries for preparing the ground states of non-interacting scalar field theories on digital quantum devices. With these methods, classically computed ground states in a small spatial volume can be used to determine operators for preparing the ground state in a beyond-classical quantum register, even for interacting scalar field theories. Due to the exponential decay of correlation functions and the double exponential suppression of digitization artifacts, the derived quantum circuits are expected to be relevant already for near-term quantum devices.

I. INTRODUCTION

Preparing QFT ground states digitally is difficult because spatial correlations entangle field values across sites, while arbitrary wavefunction preparation generally requires exponentially many entangling gates. The paper addresses this using localized effective operators whose fixed-point parameters can be inferred from small systems.

I. INTRODUCTION

  • Ground-state preparation is a leading resource challenge for digital simulation of scalar QFTs.Digitization and quantum Fourier transforms can protect against some systematic errors, but state preparation remains difficult.Source: §I · p. 1
  • Correlations induced by the spatial gradient operator entangle sites and increase the number of gates needed to prepare the scalar-field ground state.For massive QFTs, exponential correlation localization makes the required number of unitary operators scale linearly with spatial volume.Source: §I · pp. 1–2
  • The proposed method derives spatially localized effective operators whose parameters rapidly approach fixed points as distant lattice sites are integrated out.The construction uses properties and circuit symmetries of non-interacting scalar-field ground states.Source: §I · p. 2
  • Small systems sized by the correlation length can determine operators for preparing larger ground states beyond classical storage limits, including interacting scalar theories.The associated systematic errors are described as exponentially suppressed.Source: §I · p. 2

II. SCALAR FIELD GROUND STATE

The digitized scalar-field ground state is represented in the field basis on a finite lattice and prepared from a fiducial state through an amplitude-setting operator. Correlation structure and truncation determine how the corresponding preparation operators can be localized.

II. SCALAR FIELD GROUND STATE

  • The ground state can be prepared operationally from a fiducial state using a non-unitary operator Γ̂.Γ̂ can also be viewed as the density matrix of the digitized ground state, supporting the effective-operator construction.Source: §II · p. 2
  • The scalar-field wavefunction is sampled at spacing δφ between field cutoffs ±φmax on each of N lattice sites, with nQ qubits per site.The resulting register represents the digitized ground state in the field-operator eigenbasis.Source: §II · p. 2
  • The correlation matrix K is exponentially localized, while mutual information decays approximately as e^-2mr and negativity remains localized at the ultraviolet scale.These properties connect intrinsic field correlations to localized ground-state preparation circuits.Source: §II · p. 2
  • Truncating K to a band-diagonal matrix makes circuit elements depend only on neighboring sites within a maximum distance d.The discussion takes d = 1 unless otherwise stated, retaining the largest contributions in the correlation hierarchy.Source: §II · p. 3

III. QUANTUM CIRCUITRY

The circuit uses controlled y-rotations whose angles can be reorganized from qubit-local θ parameters into site-localized α parameters. Correlation decay and digitization structure make these localized operators converge toward fixed patterns as lattice size grows.

III. QUANTUM CIRCUITRY

  • Each θℓ,k specifies a y-axis rotation target qubit and the binary control value k, with operators of equal ℓ commuting.The circuit contains 2^ℓ controlled operators for each target-qubit level.Source: §III · p. 3
  • Figure 1 represents arbitrary real-wavefunction preparation using controlled operators for binary strings k and uniquely angled Ry rotations.The controls are organized by qubit level, with the most significant control bit at the top.Source: Figure 1 · p. 3
  • The θ angles are expanded site-wise into α angles, which isolate sensitivity to controls at shorter spatial distances.Figure 2 shows intermediate controlled operators and the grouping of rotations by target qubit.Source: Figure 2 · p. 3, §III · p. 4
  • α angles are linear combinations and averages of θ angles, and with d = 1 they match distance-one θ angles while vanishing at greater spatial extent.This construction is controlled by the exponentially decaying correlation matrix K.Source: §III · pp. 3–4
  • As nQ increases, last-qubit θ angles approach π/4, yielding a single-qubit Ry(π/4) limit for long-distance controlled operators.Early digitization qubits instead approach a fixed, non-constant distribution; the discussion focuses on fixed nQ ≤ 5.Source: §III · p. 4
  • The number of unique local operators becomes independent of lattice volume as fixed nQ circuits approach spatial fixed points.This supports preparing lattices containing many correlation lengths with a volume-independent set of local operators.Source: §III · pp. 4–5

IV. FIXED POINT CIRCUIT

The fixed-point circuit replaces volume-dependent digitized calculations with localized effective operators derived from marginalized ground-state information, while preserving exponentially accurate rotation angles. Continuum and digitized calculations converge rapidly, with ∼1% agreement at nQ = 2 and ∼0.001% differences at nQ = 3 in small volumes.

Circuit construction

  • The operator indices specify the site, inner-site qubit, and binary control value that determine each localized rotation.The inner-site index records how many lower qubits must be marginalized, while κφn labels field values at each site.Source: §IV · p. 5

Effective operators

  • The effective operator for a site is obtained by marginalizing the full operator over field values of sites beneath the rotation location.For the θ-circuit, the relevant angles are determined from ratios of squared wavefunction amplitudes or diagonal reduced-density-matrix elements.Source: §IV · p. 5

Fixed-point limits

  • Continuum fixed-point expressions are derived using infinite-volume, infinite-truncation, and continuous-register limits, with finite-size corrections treated separately.Analytic asymptotic correlation-matrix elements and determinant ratios are convenient but not required because they can be computed inexpensively for large volumes.Source: §IV · p. 6

Numerical convergence

  • ∼1% agreement is obtained between continuum and digitized angles with coarse nQ = 2 digitization on each site.This comparison appears in the left panel of Fig. 3 for the center-lattice angle.Source: §IV · p. 7, Figure 3 · p. 7
  • ∼0.001% angle differences are found for nQ = 3 in small volumes, supporting fixed-point angles for initializing digitized ground states.The resulting precision matches expected near-term hardware precision; increasing nQ improves fidelity faster than gate costs grow.Source: §IV · pp. 7–8

Localization

  • Increasing the correlation-distance truncation d makes the effective operator less local, while the correction remains confined to a lower sub-block controlled by retained off-diagonal correlations.This connects the spatial range of circuit communication to the nonzero off-diagonal structure of K.Source: §IV · p. 8

V. REFLECTIONS

Fixed-point circuits prepare non-interacting scalar-field ground states using resources set by exponentially localized correlations, while extending to interacting theories and broader massive systems. The approach omits the continuum limit, which increases the required circuit angles as lattice spacing decreases.

Fixed-point preparation

  • Fixed-point circuits prepare non-interacting scalar-field ground states, with classical resources scaling only with the spatial correlation distance and applicability extending to interacting theories.Massive-field correlations decay exponentially, allowing larger lattices than can be stored classically; interacting ground states may be treated perturbatively or non-perturbatively.Source: §V · p. 8

Limitations

  • The continuum limit is not considered, although it is required for predictions of physical observables with complete uncertainty quantification.Reducing lattice spacing increases the number of lattice sites within a correlation volume according to a power law.Source: §V · p. 8
  • The fixed-point analysis remains valid as lattice spacing decreases, but the circuit requires an increased number of α-angles.This growth reflects the increasing number of lattice sites within a correlation volume.Source: §V · p. 8

Interacting theories

  • Interacting ground states can be approached from non-interacting states or informed directly by finite-volume calculations, avoiding adiabatic depth scaling with unpredictable spectral gaps.Strong interactions still require further exploration for perturbative corrections, while infinite-volume circuit elements can be extrapolated from exponentially localized correlation lengths.Source: §V · p. 8

Broader applicability

  • Fixed-point circuits are expected to apply to locally interacting massive fields with exponentially decaying correlations or area-law entanglement, potentially including confining gauge theories.The authors suggest that classical calculations of the QCD vacuum could inform state preparation on beyond-classical quantum devices.Source: §V · p. 8
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