Quantum Simulation on Riemannian Manifolds

1 Introduction

• Spatially discretized field theories and constrained optimization motivate Riemannian dynamics because their kinetic operators or feasible-set geometry are naturally manifold-based.Lattice sigma models and gauge theories use sums of manifold Laplacians, while barrier-induced metrics can support quantum-dynamical methods on constrained domains.Source: §1 · pp. 4–5

1.1.1 Global spectral methods

• Explicit global spectral algorithms cover tori, spheres, simplices, truncated positive orthants and affine images, and ℓp balls.For orthants and ℓp domains, coordinate changes reduce the chosen metrics to Euclidean ones; torus, sphere, and simplex algorithms use structured spectral transforms and quadrature.Source: §1.1 · p. 6, §1.1.1 · pp. 8–9
• Slow spectral growth controls truncation, while uniform potential approximation and oversampling quadrature bound approximation and aliasing errors.With a degree-K potential approximation, each multiplication raises spectral degree by at most K; quadrature exact through degree 2N+K reproduces its matrix on the retained space without aliasing.Source: §1.1.1 · p. 8

1.1.2 Coherent-chart methods

• Coherent charts use local Fourier expansions, coherent amplitude recombination, and an IMS correction that restores the exact global Hamiltonian before truncation.The correction accounts for derivatives acting on the chart cutoffs; the corrected local operators together represent the global Riemannian Hamiltonian.Source: §1.1.2 · p. 9
• For compact manifolds, polynomial local Fourier cutoffs suffice at fixed accuracy when wavefunction and geometric data satisfy specified derivative bounds.The result applies when the number of controlled derivatives is proportional to dimension d and the corresponding bounds grow no faster than poly(d)^d.Source: §1.1.2 · p. 10
• For products of constant-dimensional manifolds, finite-dimensional coherent-chart simulation also admits polynomial local resolution under smoothness, approximation, and differentiability assumptions.A global source space connects the different chart grids, with evaluations and local Fourier transforms applied factor by factor.Source: §1.1.2 · pp. 10–11

1.1.3 Applications

• The applications generally do not establish complete end-to-end quantum speedups, which may also require state preparation, measurement, spectral-gap or convergence analyses, resource optimization, and classical comparisons.The paper identifies torus simulation as a notable exception in its analysis of polynomially scaling resources, but says that even natural end-to-end algorithms do not immediately imply a quantum speedup.Source: §1.1.3 · p. 11
• Spatially discretized field theories have Laplace–Beltrami kinetic terms, and the sphere algorithm gives an end-to-end digital simulation of an O(N) nonlinear sigma model with explicit error and resource bounds.The example uses nearest-neighbor chordal interactions; the broader applications include curved-space quantum mechanics, nonlinear sigma models, CPN−1 models, and lattice gauge theories.Source: §1.1.3 · p. 11
• Theorem 3.11 digitally simulates the controlled Riemannian evolution required by Gribling, Apers, Nieuwboer, and Walter’s quantum path-following method [Gri+25] for the logarithmic barrier on a positive orthant truncated to a compact box.This implements a central primitive assumed by their method, rather than establishing a complete end-to-end optimization algorithm.Source: §1.1.3 · p. 11
• Theorem 5.6 extends Quantum Hamiltonian Descent [Len+23] to geodesically convex potentials on Riemannian manifolds, and the sphere and simplex specializations yield end-to-end optimization algorithms.The extension generalizes the Euclidean convergence analysis based on a Lyapunov function.Source: §1.1.3 · pp. 11–12
• The Riemannian Real-Space Adiabatic Algorithm extends the approach of Herman et al. [Her+25], with an adiabatic guarantee and a digital implementation on a periodic Duffy ℓp domain.The Duffy map makes the adiabatic path unitarily equivalent to the path on a Euclidean box, carrying over the mechanisms identified by Herman et al.Source: §1.1.3 · p. 12

1.2 Related work

• The flat-torus global spectral method removes aliasing exactly for a finite-mode comparison potential, avoiding a high-order regularity bound on the evolution and using O(d log Q) qubits for Q = poly(d, 1/ϵ).This contrasts with prior grid-based analyses that bound discretization error through regularity of the evolving wavefunction.Source: §1.2 · p. 12
• The global spectral methods provide end-to-end discretization error bounds for black-box-potential Schrödinger dynamics on non-flat spaces, while the coherent-chart method gives continuum convergence under explicit assumptions.The paper characterizes these as, to its knowledge, the first such global-spectral bounds and specifies geometric and regularity assumptions for the coherent-chart analysis.Source: §1.2 · p. 13

2 Preliminaries and Notation

• A Riemannian metric is a smooth, positive-definite symmetric tensor that defines local inner products and, through its determinant, the coordinate-invariant volume measure.The volume measure defines the Hilbert space of wavefunctions on the manifold.Source: §2 · pp. 13–14
• The Laplace–Beltrami operator generalizes the Euclidean Laplacian as the divergence of the gradient; on compact manifolds its spectrum is discrete and its eigenfunctions form a smooth orthogonal basis.Its eigenspaces are finite dimensional.Source: §2 · p. 14
• The Riemannian logarithm maps a point to the tangent vector of its unique minimizing geodesic from a base point, and is defined under the unique-geodesic assumption.It generalizes the Euclidean displacement vector.Source: §2 · p. 14
• Geodesic convexity requires a unique minimizing geodesic between each pair of points in the set, with that geodesic remaining inside the set.A function on such a set is geodesically convex when its restriction to every unit-velocity geodesic arc is convex.Source: §2 · p. 15
• Duhamel’s formula provides an identity for bounding the difference between two Hamiltonian evolutions that start from the same initial state.The stated setting assumes self-adjoint operator families with unitary propagators.Source: §2 · p. 15

3 Global Spectral Methods

• On a compact manifold, the global spectral method retains finitely many Laplace–Beltrami eigenfunctions, applies kinetic evolution diagonally, and evaluates the potential through weighted quadrature values and a transform back.This adapts the Fourier pseudospectral approach, where the kinetic operator is diagonal in the Fourier basis and the potential is evaluated in position space.Source: §3 · p. 15
• A formal eigenfunction-basis replacement does not ensure an efficient algorithm: analysis must control both leakage from the retained spectral space and aliasing from finite quadrature.The cited Euclidean analysis can require grid resolution exponential in dimension, so it does not yield polynomial implementations of the spherical and simplex constructions described here.Source: §3 · p. 16
• For the considered manifolds, a uniformly close, exactly band-limited proxy for the potential supports finite-bandwidth analysis, after which sufficient Gauss-quadrature oversampling exactly evaluates Hamiltonian matrix elements.The proxy is used in the analysis to connect the continuous evolution to a finite-bandwidth truncated evolution.Source: §3 · p. 16
• An efficient implementation requires case-specific constructions of the weighted evaluation transform, its adjoint, and the quadrature rule; these are not generally efficient in arbitrary settings.The paper proceeds to analyze examples where these transformations are tractable and yield polynomial-time algorithms.Source: §3 · p. 17

3.1 General framework

• The framework uses slow spectral growth, a uniformly accurate band-limited smoothing of the potential, and an efficiently implementable transform over an exact quadrature grid.Together, these conditions are intended to simplify global-spectral error analysis and support resource-efficient discretizations.Source: §3.1 · pp. 19–20
• The quadrature-transform condition permits gate complexity polynomial in the local resolution RQ while keeping qubit requirements logarithmic in the total node count.For the torus, sphere, and simplex constructions, RQ = Θ(Q).Source: §3.1 · p. 20
• The transform condition is not automatic for generic polynomial systems on nonrectangular domains; the sphere and simplex meet it through recursive factorizations into controlled one-dimensional polynomial transforms.The framework requires eigenfunction loading on a Q-exact quadrature grid so oversampling can make discrete sums equal integrals.Source: §3.1 · p. 20
• For band-limited potentials, the Galerkin error analysis uses a Dyson-series comparison; a stated cutoff condition is N ≥ K0 + (q − 1)K for q ≥ 2eB1(t)∥fK∥∞.The result concerns initial data in VK0 and a band-limited potential fK.Source: §3.1 · p. 20
• If the filtration has slow spectral growth and the potential proxy lies in VK, a Q-exact quadrature rule with Q ≥ N + K exactly implements the Galerkin projector.The framework then uses this exactness to connect the continuum Galerkin estimate to quadrature evolution.Source: §3.1 · p. 23

3.2 Flat Torus

• The torus algorithm uses Fourier modes to diagonalize the Laplace–Beltrami operator and yields improved discretization spacing and a quadratic reduction in qubits in d.Through coordinate changes, the approach also gives algorithms for positive orthants and full-rank polyhedral constraints with log-barrier Hessian metrics, and for ℓp balls.Source: §3.2 · p. 24
• For sufficiently large Fourier degree m, any Lipschitz function on T^d can be made O(ϵ)-close to a Fourier polynomial, with approximation error falling as O(m^-1).The smoothed function has Fourier support in [−2(m − 1), 2(m − 1)]^d.Source: §3.2 · pp. 24, 27

The final grid state satisfies

• The discretized evolution uses O(d log M) qubits, and its sampled-observable error is bounded by 4∥h∥∞ϵ.The grid has M points per coordinate; the observable bound follows from the state error and uniform approximation error.Source: §3.2 · p. 28, The final grid state satisfies · p. 30
• Jackson band limitation and oversampling make the grid resolution polynomial in dimension under the theorem’s assumptions, avoiding the prior analysis’s generally exponential-in-dimension resolution.The potential proxy has an exact quadrature match to the Galerkin potential, so state error depends on the uniform proxy error and Dyson-series tail rather than evolved-wavefunction Sobolev norms.Source: The final grid state satisfies · p. 30
• Under the stated assumptions, the digital interaction-picture circuit produces a normalized coefficient state with success probability at least 1 − O(ϵ).The theorem assumes piecewise C1 schedules, finitely many sign intervals for b, and specified state-preparation, potential-query, and circuit routines.Source: The final grid state satisfies · pp. 30–31

• The rescaled evolution is simulated with operator error O(ϵ) and failure probability O(ϵ), using the time-controlled query count in (77).Each query uses one sampled-potential block, two binary-oracle calls, forward and inverse padded QFTs, and two controlled kinetic phases; returning to the Schrödinger picture adds one final kinetic phase.Source: The final grid state satisfies · p. 32
• Synthesis and simulation errors sum to O(ϵ), while initial-state preparation adds at most ϵprep to the final error.The Hamiltonian perturbation contributes B1ϵg + O(τgδ) by Duhamel’s formula, and the hybrid argument contributes O(ϵ).Source: The final grid state satisfies · p. 32
• The final-grid-state and joint-readout estimates, together with the coefficient estimate, establish bounds (75) and (76).The readout transform W adds at most δ in state norm.Source: The final grid state satisfies · pp. 32–33
• The compiled implementation has cost and workspace bounds (78) and (79), with each padded QFT costing d polylog(dM/δ).The assumed schedule and kinetic-phase costs are polylogarithmic at the same tolerance; total cost also includes state preparation and final readout.Source: The final grid state satisfies · p. 33

3.3 Positive orthant (and its invertible affine images) with log-barrier metrics

• The log-coordinate change turns the log-barrier metric into the Euclidean metric and its Laplace–Beltrami operator into the ordinary Laplacian.After rescaling the logarithmic box, the evolution can be treated using the torus simulation, assuming periodic extensions of the pulled-back potential and observable.Source: §3.3 · pp. 33–34, The final grid state satisfies · p. 35
• The full orthant has continuous spectrum and an infinite domain, so the construction truncates it to a compact box with opposite faces identified in logarithmic affine coordinates.The potential and observable are required to be face-compatible for this identification.Source: §3.3 · pp. 33–34
• An invertible affine change of coordinates maps the log-barrier metric on the polyhedron back to the orthant case.The truncated polyhedron is specified by bounds on Ax − c, and its coordinates are transformed by z = Ax − c.Source: §3.3 · pp. 33–34
• For bounded, face-compatible Lipschitz potentials and observables, the discretization result gives a finite-grid evolution using O(d log M) qubits.The grid has M = 2Q + 1 nodes along each transformed coordinate.Source: §3.3 · pp. 34–35

The final grid state satisfies

• The final-grid theorem produces a normalized coefficient state with success probability at least 1 − O(ϵ), with coefficient labels given by ordinary torus Fourier labels.The coordinate transformation preserves the stated state and observable estimates.Source: The final grid state satisfies · pp. 35–37

3.4 ℓp balls with a Duffy metric

• The Duffy metric makes simulation on compact subsets of ℓp balls and upper hemispheres reducible to evolution on a flat torus.This choice addresses the lack of a comparably tractable eigenbasis for the ambient Euclidean metric when p is not 1 or 2.Source: §3.4 · p. 37
• The construction assumes p ≥ 2 for the required map regularity and restricts coordinates to a compact box because the Duffy-map derivatives are unbounded near the boundary.The coordinate truncation need not produce a smaller concentric ℓp ball.Source: §3.4 · pp. 37–38
• The coordinate maps define the metric by isometry, rather than using the metric induced from the ambient Euclidean space.The resulting volume measures are the Riemannian volume measures of the defined metrics.Source: §3.4 · p. 38
• The periodic-boundary discretization applies when the pulled-back potential and observable match on opposite faces of the coordinate box.Under this condition, the result covers both the truncated ball and upper-hemisphere images.Source: §3.4 · p. 39, The final grid state satisfies · p. 40

The final grid state satisfies

• The discretization uses a finite grid with M = 2Q + 1 nodes per coordinate and supports an O(d log M)-qubit evolution.The final-grid theorem gives a quantum circuit producing a normalized coefficient state with success probability at least 1 − O(ϵ).Source: §3.4 · p. 40, The final grid state satisfies · p. 41

3.5 Sphere with the standard metric

• Spherical harmonics satisfy the paper’s slow spectral-growth condition, by the Fischer decomposition and recursive containment of homogeneous polynomials.The supplied passage identifies this as condition (27), without stating its formula.Source: §3.5 · pp. 42–43
• The heat-smoothing analysis bounds the spherical squared-distance Laplacian by 2d, including the limiting argument needed at the antipode.The direct squared-distance function is not differentiable at the antipode, so the proof smooths it before applying Itō’s lemma and then takes a limit.Source: §3.5 · pp. 44–45
• The spherical heat-kernel smoother approximates bounded geodesically G-Lipschitz functions arbitrarily closely while keeping them band-limited.The corollary states this for 0 < ϵ < 1; its displayed quantitative conditions are incomplete in the supplied passages.Source: §3.5 · p. 46
• A weighted evaluation map built from spherical quadrature is an isometry on V_N, and its inverse implements the spherical harmonic transform.The construction uses recursively structured spherical coordinates and a centered azimuthal QFT.Source: §3.5 · pp. 47, 49, 50
• Algorithm 1 is not quantumly efficient in the usual sense because its dependence on the radial grid size R is not logarithmic, although R is polynomial in d and 1/ϵ when the other parameters are fixed.This is a qualification on the transform's efficiency, not a claim that the discretization requires exponentially many radial points.Source: §3.5 · p. 52
• Theorem 3.17 constructs the spherical-transform circuit using O(d(N + 1)R^3) classical arithmetic operations and O(d(N + 1)R^2) classical memory.The transform maps coefficient labels to node registers; its weighted sample state has amplitudes given by the spherical harmonic expansion evaluated on the quadrature nodes.Source: §3.5 · pp. 51–52

Let the exact finite-dimensional coefficient state solve

• For sphere evolution with bounded geodesically Lipschitz potential and observable, the discretization theorem gives polynomially bounded spectral and quadrature degrees N and Q, using O(d log(Q + 1)) data qubits.The proof uses heat-kernel smoothing to choose the cutoffs.Source: §3.5 · p. 54, Let the exact finite-dimensional… · pp. 54–55
• The digital interaction-picture algorithm produces a normalized coefficient state with success probability at least 1 - O(ϵ), using truncated-Dyson queries [LW18; Ber+20].Each query uses a sampled-potential block, forward and inverse spherical transforms, and controlled kinetic phases.Source: Let the exact finite-dimensional… · pp. 55–57

3.6 Simplex with Wright–Fisher metric

• The simplex Laplace–Beltrami eigenfunctions are multivariate Jacobi polynomials, and their recursive factorization yields a quantum simplex transform.The transform construction uses O(d(N + 1)R^3) classical arithmetic operations and O(d(N + 1)R^2) classical storage.Source: §3.6 · pp. 58, 60, 61
• Multinomial averaging, or Bernstein smoothing, maps a bounded Euclidean-Lipschitz function on the simplex to a polynomial in V_K without increasing its supremum norm.The smoothing is used to support the polynomial spectral cutoff.Source: §3.6 · pp. 58–60

Let the exact finite-dimensional coefficient state solve

• Under the stated boundedness and Lipschitz assumptions, the simplex discretization theorem supplies spectral and quadrature degrees and bounds the coefficient-state and observable approximations.Its construction sets N = L0 + (q − 1)K and Q = N + K.Source: §3.6 · p. 63, Let the exact finite-dimensional… · p. 64
• The simplex coefficient and quadrature representations require O(d log(Q + 1)) data qubits.The grid and coefficient registers use the same asymptotic qubit count.Source: Let the exact finite-dimensional… · pp. 63–64
• The discretization uses the simplex Jacobi polynomials’ slow spectral growth property to choose a cutoff and construct a continuum quadrature evolution.The construction chooses K, then N = L0 + (q − 1)K and Q = N + K.Source: Let the exact finite-dimensional… · pp. 63–64

Let R and C denote the quadrature and coefficient registers and define

• The digital interaction-picture method simulates the finite-dimensional simplex evolution with O(ϵ) error and failure probability, including oracle, kinetic-phase, interaction-picture, and Jacobi-transform synthesis errors.Each time-controlled query uses two potential-oracle calls, forward and inverse Jacobi transforms, and two controlled Wright–Fisher kinetic phases.Source: Let the exact finite-dimensional… · p. 64, Let R and C denote the quadrature and… · pp. 65–66

Coherent-chart method

• The method simulates Riemannian Schrödinger dynamics by local Euclidean spectral methods on a finite atlas and coherently recombines the solutions, requiring extended access to localize the global wavefunction.The approach is presented as a more generic algorithm for Riemannian Schrödinger equations.Source: §4 · p. 66

Atlas and localization

• A box-adapted atlas and subordinate partition of unity localize functions inside chart interiors, so their zero extensions have no boundary jump and permit standard Fourier or Chebyshev discretizations.A suitable finite atlas exists for compact manifolds, and smooth bump functions can be normalized to form the partition of unity.Source: §4 · p. 67

Localization map

• The localization map is an isometry, and its adjoint recombines chart sections; J†J = I while JJ† projects onto the physically consistent subspace.The direct-sum chart representation is redundant, and not every tuple of chart functions represents a global function.Source: §4 · p. 67

Correct dynamics

• Overlapping chart amplitudes interfere when recombined, but directly conjugating the Hamiltonian by J does not yield the correct dynamics because cutoff derivatives alter the kinetic term.The potential commutes with each cutoff, whereas the kinetic part requires correction.Source: §4 · p. 68

IMS correction

• The Riemannian IMS identity supplies an exact compressed Hamiltonian representation, with the negative full IMS potential canceling the localization error from differentiating the cutoffs.The IMS identity is used here for exact representation of the Riemannian Hamiltonian.Source: §4 · pp. 68–69

J UBNUJ (238)

• The coherent-chart method combines chart-local Fourier–Galerkin blocks through partition-of-unity localization while keeping the evolving state in the manifold’s physical L2 space.The localization map is an isometry, and the synthesis map recombines chart data; the projection onto consistent chart tuples is JJ†.Source: J UBNUJ (238) · pp. 71–72

4.1 Analysis of the coherent chart algorithm in continuous space

• Theorem 4.9 bounds Galerkin error using manifold properties and the evolved state’s Sobolev norm, assuming s ≥ 3, regular coefficient extensions, and self-adjoint unitary evolutions.The theorem requires the exact solution to satisfy the stated Sobolev regularity condition.Source: §4.1 · p. 74

Define the ordinary-box chart distortion constant

• Under the stated bounds scaling as O(poly(d)^d) with s = Θ(d), the corollary guarantees N = poly(d) for fixed target accuracy.The sufficient conditions include bounds on chart, coefficient, geometry, and evolution quantities.Source: Define the ordinary-box chart… · p. 78
• On general high-dimensional manifolds, overlapping charts may have nonintersecting local grids, and the global source interpolant may be inefficient; the factorized construction is conditional on constant-dimensional product structure and implementation assumptions.Source: Define the ordinary-box chart… · p. 78

4.2 Tensor products of manifolds and a discretized algorithm

• Polynomial cutoff does not guarantee polynomial runtime: that additionally requires polynomial source-space, block-encoding, oracle, and Hamiltonian-normalization costs, while kinetic simulation grows with maxν Nν.Global interpolation may be inefficient beyond product manifolds, and the number of interpolation functions and grid points can grow with dimension.Source: §4.2 · p. 79, §4.2.1 · pp. 82–83
• A global source interpolant resolves disjoint Fourier grids on overlapping charts and, for product manifolds, can be assembled coherently from constant-dimensional factor interpolants.The construction does not prescribe a particular global interpolation basis; its cutoff bound depends on the stated source-space approximation and implementation properties.Source: §4.2 · p. 79, §4.2.1 · p. 80

4.2.1 Digital Coherent-Charts Method

• Under the stated raw-block, potential-oracle, and factor-isometry encoding assumptions, the sandwiched operators admit block encodings; when Npad ≤ 7N, αK = O(mN^2) and αf = O(Λf).The near-isometry bound controls normalization but does not itself construct a circuit; controlled implementations and their costs remain assumptions.Source: §4.2.1 · pp. 82–83
• With explicit block-encoding and oracle-access assumptions, the algorithm avoids materializing an Nd × Nd resampling matrix or iterating over all product-grid nodes, applying factorwise primitives coherently instead.The formal product grid still contains N^Θ(d) nodes.Source: §4.2.1 · p. 83

4.2.2 Discretization error analysis for products of constant dimensional manifolds

• Under uniform factor bounds and the theorem’s approximation assumptions, each of four discretization errors is at most (C d^β/N)^d, so polynomial resolution suffices for fixed accuracy.The analysis compares exact and discrete evolutions through source projection, coefficient approximation, local-grid approximation, and final-operator comparison, combining the errors with Duhamel’s formula.Source: §4.2.2 · pp. 83–84
• The iterated Jackson smoother approximates sufficiently regular periodic functions with coordinatewise Fourier bandwidth at most κJ, preserves real-valuedness, and applies entrywise to matrix fields.The high-order approximation bounds are stated for functions in W^{q+ℓ,∞}(S).Source: §4.2.2 · p. 87
• Continuum and source-space lemmas bound the kinetic-operator error from replacing raw coefficients with Jackson-smoothed coefficients, under the stated smoother condition and approximate-isometry assumption.The continuum bound uses factorwise geometric coefficient norms; the source-space result assumes Aν is a δν-approximate isometry and the condition on nJ from Lemma 4.15.Source: §4.2.2 · pp. 90, 92
• Sufficient padding makes coefficient-multiplication aliasing zero; when nJ = N, the condition is Npad ≥ N + 6(N −1).With minimal padding, Npad = N + 6(N −1) ≤ 7N; the two differentiation matrices contribute O(N^2), converting an N^−(d+2) coefficient approximation rate into an N^−d padded-matrix rate.Source: §4.2.2 · p. 95
• The Fourier consistency comparison decomposes the Galerkin defect into output and input Fourier tails, with the second-order operator acting on the input tail.The retained-space Fourier projection is the best approximation in each Hr norm.Source: §4.2.2 · pp. 96–97
• For products of constant-dimensional manifolds, combining the component estimates and applying (353) and (402) proves the discretization bound (395).The combined estimate first has a 3ηsrc term.Source: §4.2.2 · p. 98

Local Fourier Error – Potential

• Product-grid interpolation bounds chart-function error for periodic H^{d+2}(T^d) extensions, and the potential discretization bound follows by applying this interpolation estimate to regular chart functions.The potential proposition assumes the stated Sobolev regularity for each chart and factor-coordinate derivatives.Source: Local Fourier Error – Potential · pp. 98, 100, 101
• The dynamical error is bounded by the sum of source, continuum Jackson, kinetic Fourier, potential, and discrete Jackson error budgets.This bound compares the exact evolution with the discretized evolution under the assumptions stated for Proposition 4.22.Source: Local Fourier Error – Potential · p. 103
• Theorem 4.23 gives a polynomial cutoff sufficient for error at most ε when factor dimensions and chart data are uniformly bounded and its source and error-envelope assumptions hold.The theorem assumes products of at most d factors of constant dimension and imposes source-resolution and comparison-evolution bounds.Source: Local Fourier Error – Potential · p. 103
• The source, continuum Jackson, Fourier, potential, and discrete Jackson contributions each admit bounds of the form (C d^β/N)^d under the theorem’s assumptions.The proof combines their bounds by absorbing factor sums, coefficient envelopes, and fixed constants into a polynomial base.Source: Local Fourier Error – Potential · p. 104

5.1 Simulations of Physics

• Spatial discretization turns a field theory into a configuration space that is a product of copies of a target manifold, with kinetic and potential terms determined by its metric and interactions.The product metric makes the kinetic operator a sum of factor Laplace–Beltrami operators; spatial-gradient, on-site, and plaquette terms contribute to the potential.Source: §5.1 · p. 105

5.1.1 Quantum Mechanics on Curved Spaces

• Constrained mechanical systems have curved configuration spaces and Hamiltonians with Laplace–Beltrami kinetic terms; curved-spacetime quantum field theory is outside this treatment.The field-theory models considered here are on flat Minkowski spacetime, not curved spacetime.Source: §5.1.1 · pp. 105–106

5.1.2 Nonlinear Sigma Models

• Discretizing nonlinear sigma models on a spatial lattice gives a product configuration space of target-manifold copies, with a product metric and a quantum kinetic operator equal to the sum of sitewise Laplace–Beltrami operators.The lattice interaction uses geodesic distance on the target manifold as a chart-independent discretization of the spatial-gradient term.Source: §5.1.2 · pp. 107–108

5.1.3 CPN−1 Models

• The CP^{N−1} model fits the nonlinear sigma model construction by taking complex projective space with the Fubini–Study metric as its target manifold.With the usual choice V ≡ 0, its lattice Hamiltonian is again of the form −∆G + V on a product of target-manifold copies.Source: §5.1.3 · pp. 108–110

5.1.4 Lattice Gauge Theories

• On a fixed lattice, the Kogut–Susskind Hamiltonian is already −∆G + V on a product of copies of the compact gauge group, with link rather than site degrees of freedom.The discussion is restricted to the pure-gauge sector; a theory such as lattice QCD adds matter fields while retaining this Hamiltonian as a summand.Source: §5.1.4 · pp. 110–111

5.1.5 Concrete simulation of the spatially discretized O(N) nonlinear sigma model

• For the spatially discretized O(N) model, the sphere-valued field has O(N)-invariant nearest-neighbor chordal interactions that raise spherical-harmonic degree by at most one at each endpoint.On the unit sphere, 1 − σ · σ′ equals half the squared chordal distance; the paper applies its sphere-simulation algorithm to this lattice model.Source: §5.1.5 · p. 112
• The initial coefficient-state preparation has state error O(ϵ), and reversible potential evaluation is assumed to satisfy Tϵch = O(ϵ); for Q = J + 1, discretization error is bounded by 2^(2−q) ≤ ϵ.The tensor product of the single-site quadrature rules is Q-exact for the product filtration.Source: §5.1.5 · p. 113
• For fixed D, N, a, T, and J0, the algorithm uses polynomial resources in nΛ and 1/ϵ and produces the simulated coefficient state with success probability at least 1 − O(ϵ).The theorem also gives bounds on potential-evaluation calls and total qubits.Source: §5.1.5 · p. 113

• The barrier Hessian metric determines the Newton direction, Dikin ellipsoid, and proximity measure; self-concordance controls its variation in classical path-following analysis.Source: §5.2 · p. 114
• The prior Riemannian quantum path-following framework has a spectral-gap estimate independent of Euclidean Hessian condition numbers and ground states concentrated near the central path for sufficiently large γ.Its use assumes, among other things, sufficient initial ground-state overlap, ground-energy estimates, and controlled Hamiltonian evolution.Source: §5.2 · p. 114
• For the standard logarithmic barrier, the paper’s simulation result covers fixed-η evolution on a box with Dirichlet boundaries and a bounded Lipschitz potential, subject to the theorem’s hypotheses.This gives a digital simulation primitive for a compactly truncated barrier geometry, not the complete path-following algorithm.Source: §5.2 · p. 115
• An end-to-end discrete quantum interior-point method remains open, requiring uniform truncation and boundary-effect control plus accounting for simulation, energy estimation, state preparation, and readout.Source: §5.2 · p. 115

5.3 Riemannian Quantum Hamiltonian Descent

• The extension complements operator- and path-integral formulations [AN26] with rigorous convergence guarantees, including boundary terms, and explicit digital simulation algorithms on the sphere and Fisher–Rao simplex.The section also situates this work alongside Quantum Hamiltonian Descent [Len+23] and notes that the prior oracle-level analysis assumed phase-oracle access to the objective and sparse access to a Laplace–Beltrami discretization.Source: §5.3 · p. 115

5.3.1 Additional Preliminaries

• The Riemannian commutator identity includes Hessian and Ricci-curvature terms, distinguishing it from the flat-space case where covariant derivatives commute.The interior identity combines Hessian contractions with gradient terms involving the Laplacian and Ricci curvature.Source: §5.3.1 · p. 116, Also define the total boundary… · p. 119

Also define the total boundary functional

• The boundary functional identity extends from a smooth operator core to the full operator domain when the stated graph-continuity and embedding assumptions hold.For Dirichlet boundary conditions, the text notes that boundary terms simplify and the symmetric operator extends to the domain.Source: Also define the total boundary… · pp. 121–122
• Theorem 5.6 gives QHD convergence for geodesically convex objectives on a compact, geodesically convex smooth domain under its stated Dirichlet and favorable-boundary-form conditions.The theorem assumes the domain lies in a normal neighborhood of the minimizer and imposes a Hessian bound on ϕ; the associated quadratic form is shown to be nonnegative.Source: Also define the total boundary… · pp. 123–124
• The optimal schedule saturates the frequency-growth constraint, minimizing the potential-oracle count; the remaining parameter λ can also be optimized for the leading small-ϵ cost.The optimal operational path has a constant-speed representative. The stated query counts exclude kinetic implementation, discretization, state preparation, and measurement costs.Source: Also define the total boundary… · pp. 126–128

5.3.2 Riemannian QHD over the Sphere

• Theorem 5.8 treats spherical QHD convergence for x⋆∈Ω⊂S^d, where Ω is smooth and geodesically convex inside B(x⋆,R), R<π, with explicit geometric additive error O_R(d^2/m^2).Source: §5.3.2 · p. 130
• For spherical QHD, the optimal-schedule corollary gives a constant-speed schedule and minimizes the leading small-ϵ query count over λ when its stated positivity conditions hold.The dimension estimates count only potential-oracle queries and omit kinetic, discretization, state-preparation, and measurement costs.Source: §5.3.2 · pp. 131–132

5.3.3 Riemannian QHD over the Simplex

• On a compact smooth geodesically convex domain in the open spherical orthant, QHD converges for real W1,∞ potentials that are Fisher–Rao geodesically convex and minimized at x⋆, under the theorem’s stated condition.The coordinate-based implementation may depend on the domain’s face margin; kinetic implementation, discretization, state preparation, and measurement costs are additional.Source: §5.3.3 · pp. 133, 135
• For simplex QHD, the optimal-schedule corollary gives an operational path and minimizes the leading small-ϵ potential-oracle count over λ when its stated positivity conditions hold.The potential-oracle model uses the Wright–Fisher Hamiltonian and the L∞ norm of f.Source: §5.3.3 · pp. 134–135

5.4 Real-Space Adiabatic Algorithm on ℓp Balls

• The Riemannian RSAA replaces the Euclidean Dirichlet Laplacian with the Laplace–Beltrami operator for the manifold's metric.The section describes this as an extension of the ground-state preparation algorithm introduced by Herman et al. [Her+25].Source: §5.4 · p. 135
• For optimization on a compact connected manifold, the algorithm evolves a Hamiltonian along a schedule to prepare a ground state concentrated near the potential's optimum.The initial ground-state preparation is trivial, and the final state is designed to be sufficiently concentrated around the optimum.Source: §5.4 · pp. 135–136
• The digital ℓp-ball implementation uses a sampled grid, performs finite-dimensional evolution, and measures the grid register; Dirichlet evolution can be obtained by reflection and restriction to an odd sector.The presentation assumes periodic boundary conditions for simplicity.Source: §5.4 · p. 137
• Under Theorem 5.13's assumptions, Algorithm 3 returns a point in S_epsilon with probability at least 1-eta using O(d log(2Q+1)) qubits.The proof bounds the probability of returning outside S_epsilon by eta.Source: §5.4 · pp. 137–138
• For rotated block-separable objectives, the lifted Duffy Hamiltonian has polynomial adiabatic runtime, while known black-box algorithms have exponential runtime for the same class.The gap is the same along both paths; the polynomial-gap result for the box objective relies on the assumptions in [Her+25, Theorem 3.6].Source: §5.4 · p. 139

AI Use Disclosure

• Large language model tools were used interactively throughout preparation, from exploratory research to exposition and writing.The disclosure says most provable statements began as ideas from the author.Source: AI Use Disclosure · p. 139
• The authors substantially revised the final manuscript and remain solely responsible for its mathematical claims, proofs, references, and conclusions.Source: AI Use Disclosure · p. 139
• GPT-5.6 Sol made substantial contributions to the analysis and suggested using IMS Localization in the coherent-charts method.Source: AI Use Disclosure · p. 139

Disclaimer

• The paper is prepared for informational purposes by JPMorgan Chase's Global Technology Applied Research center.Source: Disclaimer · p. 139
• JPMorgan Chase and its affiliates disclaim representations, warranties, and liability concerning the information and its potential legal, compliance, tax, or accounting effects.Source: Disclaimer · p. 139
• The document is not investment research or advice, and is not a recommendation, offer, or solicitation involving financial products or transactions.Source: Disclaimer · pp. 139–140