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Breaking the Curse of Dimensionality in Quantum PDE Solvers via Gevrey Regularity

Pooya Ronagh, Mariia Sobchuk, Xiaoran Li, Grecia Castelazo, Ala Shayeghi

arXiv:2608.07893v3quant-ph

TL;DR

High-precision quantum PDE solving faces a dimensionality cost, while classical improvements rely on stronger structural assumptions. This paper uses Gevrey regularity to set Fourier cutoffs and analyze quantum solvers for periodic and Dirichlet problems; its results include Gevrey-dependent truncation bounds and an approximate quantum solution state.

  • Problem

    Classical PDE methods can improve on naïve discretization under stronger structural assumptions, but dimension remains in the exponent and the curse of dimensionality persists.

    Source: §I · p. 2

  • Method

    The paper uses Gevrey regularity to select Fourier cutoffs and analyzes quantum PDE solvers, including periodic discretization and restricted Dirichlet stability.

    Source: §I · p. 2, B. Dirichlet discretization and… · p. 20

  • Results

    Gevrey smoothness gives exponential-accuracy Fourier approximation, and the periodic solver returns a state ε-close in ℓ2 norm to the normalized discrete solution with O(1) success probability.

    Source: B. Discretization error and cutoff… · p. 7, C. Quantum algorithm and resource bounds · p. 14

  • Takeaways & Limitations

    The framework is demonstrated on isotropic and anisotropic Poisson equations with periodic and Dirichlet boundary conditions.

    Source: §I · p. 2

    For variable-coefficient operators, the corresponding estimate on the residual space must be established separately.

    Source: §VI · p. 28

Abstract

from arXiv · show

We connect different degrees of smoothness of real-valued periodic functions to the number of qubits required for their high-precision Fourier-basis amplitude encodings. Our central observation is that the Gevrey hierarchy, which stratifies the space between smooth and analytic functions, provides a natural class for high-precision quantum algorithms. We use Gevrey regularity to analyze a quantum algorithm for solving general linear partial differential equations (PDEs) with periodic boundary conditions. We then extend the solver to PDEs with Dirichlet boundary conditions using thin Gevrey collars and a regularized inverse. Our resource bounds specify precision-dependent Fourier cutoffs and account for restricted discrete stability, elementary gate costs, and restriction to the Dirichlet domain. Finally, we apply our approach to isotropic and anisotropic Poisson equations with periodic and Dirichlet boundary conditions.

I. INTRODUCTION

The paper addresses the persistent dimensionality cost of high-precision PDE solving and the limited understanding of quantum advantage for continuous-domain problems. It uses Gevrey regularity to analyze quantum Fourier-based solvers for periodic and Dirichlet PDEs, demonstrating the framework on Poisson equations.

I. INTRODUCTION

  • Prior quantum PDE solvers use Fourier methods or spectral lattices, but continuous-domain quantum advantage has received less systematic study than discrete problems.Existing approaches also include imposing all-zero Dirichlet conditions on a hypercube.Source: §I · p. 2
  • Classical methods can reach O(log^d(1/ε))-type complexity under stronger structural assumptions, yet dimension d remains in the exponent, so the curse of dimensionality persists.Naïve quantum discretization also requires O((1/ε)^d) points for target ℓ2 accuracy ε.Source: §I · p. 2
  • The paper quantifies how regularity weaker than analyticity affects Fourier-basis amplitude-encoding costs and constructs block encodings for higher-order differential operators.These constructions support concrete gate and qubit bounds for general linear PDEs with periodic boundary conditions.Source: §I · p. 2
  • The framework is extended to Dirichlet problems using thin Gevrey collars and a regularized inverse, and is demonstrated on isotropic and anisotropic Poisson equations.The table contrasts published classical complexity rates and required structural assumptions with the paper’s periodic quantum rate; it also identifies a Dirichlet counterpart.Source: §I · p. 2, Table I · p. 3

A. Aliasing sums and Gevrey smoothness

The section connects spatial samples, discrete Fourier coefficients, and Fourier-basis amplitude encodings. It explains how regularity controls Fourier decay and therefore the cutoff and qubit resources needed for accurate representations.

A. Aliasing sums and Gevrey smoothness

  • Applying the d-fold quantum Fourier transform to the spatial-lattice amplitude encoding produces an amplitude encoding of the discrete Fourier coefficients.The discrete coefficients are grid averages, while continuous Fourier coefficients are averages over the torus.Source: A. Aliasing sums and Gevrey smoothness · pp. 4–5
  • Discrete Fourier coefficients combine continuous coefficients at frequencies spaced by 2N + 1, so fast high-frequency decay is needed for the zero-offset term to dominate.The continuous and discrete coefficients are not generally identical.Source: A. Aliasing sums and Gevrey smoothness · p. 5
  • Gevrey regularity gives exponentially decaying Fourier coefficients and tails, whereas p-smooth functions have polynomial tails and require cutoff O((1/ε)^(1/p)).Table II relates regularity classes to coefficient decay, tail bounds, and sufficient cutoffs for a target error.Source: Table II · p. 4, A. Aliasing sums and Gevrey smoothness · p. 6
  • Analytic functions are the s = 1 Gevrey case, while entire functions have super-exponential decay; the Gevrey hierarchy also includes smooth functions that are not Gevrey.For analytic functions, the derivative bound guarantees a holomorphic extension to a strip of width r.Source: A. Aliasing sums and Gevrey smoothness · p. 6

B. Discretization error and cutoff selection

This section bounds Fourier truncation and discretization errors, then gives Gevrey-dependent cutoffs for approximating functions and their derivatives. It distinguishes accuracy of a truncated Fourier series from accuracy of the discrete Fourier transform.

B. Discretization error and cutoff selection

  • Differentiation is diagonal in the Fourier domain, and the analysis bounds the error from applying a cutoff derivative operator to the discretization of the true derivative.The Fourier coefficients of Dαf are (iω)^α times those of f.Source: B. Discretization error and cutoff… · p. 6
  • An ε-accurate truncated Fourier expansion exists with the cutoff specified by the regularity-class bounds, but the same conclusion does not automatically hold for the truncated QFT or DFT.The tail bound used is the tabulated upper bound, not necessarily the actual tail.Source: B. Discretization error and cutoff… · pp. 6–7
  • Theorem 2 bounds discrepancies between continuous and discrete Fourier representations, with Gevrey and Hölder cases subject to their respective cutoff or smoothness assumptions.For finite bandwidth, both errors are zero once N ≥ 1/r.Source: B. Discretization error and cutoff… · p. 7
  • For an s-Gevrey function, the function-approximation cutoff scales as eΩ((ds)^s/r), while the derivative cutoff scales as eΩ((s(d + |α|))^s/r).These bounds are stated for s ≥ 1; the remark gives corresponding bounds for 0 < s < 1 with adjusted cutoffs.Source: B. Discretization error and cutoff… · p. 7

III. LINEAR PDES WITH PERIODIC BOUNDARY CONDITIONS

The paper formulates a general inhomogeneous linear PDE solver for periodic functions using coefficient and source-state oracles. It selects a regularity-dependent Fourier cutoff, solves the discretized system with a quantum linear-system algorithm, and returns a state approximating the normalized discrete solution.

III. LINEAR PDES WITH PERIODIC BOUNDARY CONDITIONS

  • The solver assumes oracle access to block encodings of discretized coefficient functions and state preparation for the discretized source term.Its target precision accounts for Fourier truncation, oracle, block-encoding, and inversion errors.Source: §III · pp. 8–9
  • For the inhomogeneous case, the algorithm applies a quantum linear-system solver to the discretized operator, with query complexity O(κ log(1/ε)) to both the operator block encoding and source-state oracle.κ is the effective condition number based on the smallest nonzero singular value.Source: §III · p. 8
  • A continuous spectral gap alone does not ensure that the discrete smallest positive singular value converges usefully; additional conditions such as ellipticity may be required.The stated resource results are framed in terms of the discrete smallest positive singular value.Source: §III · p. 9
  • The cutoff is chosen to meet the discretization-error budget, after which the algorithm prepares the block encoding and applies the QLSA to approximate the minimum-norm solution.The output is ε-close in ℓ2 norm to the normalized discrete solution with O(1) success probability.Source: §III · p. 9

A. Choosing the Fourier cutoff

For s-Gevrey solutions with s ≥ 1, the Fourier cutoff required for discretization accuracy grows with the regularity, dimension, and PDE order. The normalized-solution error guarantee also requires discrete target-orthogonality.

A. Choosing the Fourier cutoff

  • For s-Gevrey solutions with s ≥ 1, a sufficient cutoff scales as eΩ((s(d + ⌈J⌉))^s/r).The discretization guarantee is stated for a minimum-norm solution whose discretization is orthogonal to ker(LN).Source: A. Choosing the Fourier cutoff · pp. 10–11
  • Without discrete target-orthogonality, the normalized vector-error bound acquires an additional term.The corollary's stated ε-discretization guarantee assumes uN ⊥ ker(LN).Source: A. Choosing the Fourier cutoff · pp. 10–11
  • The Fourier and PDE error estimates also extend to complex-valued functions under the same regularity and orthogonality assumptions.The paper identifies Schrödinger equations and other quantum-mechanical problems with complex-valued wavefunctions as applications.Source: A. Choosing the Fourier cutoff · p. 11

B. Block encoding of differential operators

The operator block encoding combines coefficient-dependent diagonal terms through an LCU with a custom encoding for differential terms. A controlled derivative block uses a swap network to route coordinates through derivative stages.

B. Block encoding of differential operators

  • The LCU constructs the operator from coefficient-dependent diagonal terms, while the differential term receives a custom encoding.The LCU index register enumerates α ∈ J; its preparation weights use the normalization factors for each term.Source: B. Block encoding of differential… · p. 12
  • A conditional swap network routes each spatial coordinate through its required derivative stages and restores the data registers.The controlled block uses O((⌈J⌉ + 1) log² N + ⌈J⌉(d log(1 + ⌈J⌉) log N + RN)) elementary gates.Source: B. Block encoding of differential… · p. 13, Figure 2 · p. 14
  • The controlled derivative block cancels phases from idle stages and uses at most ⌈J⌉ + 1 QFT and inverse-QFT pairs.Its auxiliary workspace and controlled failure-counter updates are omitted from the figure.Source: B. Block encoding of differential… · p. 13, Figure 2 · p. 14

C. Quantum algorithm and resource bounds

Under a Gevrey regularity assumption and discrete target-orthogonality, Algorithm 1 returns a state close to the normalized discretized solution with constant success probability. Its resource bounds include cutoff, oracle-query, and elementary-gate costs.

C. Quantum algorithm and resource bounds

  • With discrete target-orthogonality, Algorithm 1 returns an ε-close normalized solution state with O(1) success probability.The guarantee is for a general linear PDE with periodic boundary conditions and an s-Gevrey minimum-norm solution.Source: C. Quantum algorithm and resource bounds · p. 14
  • After suppressing logarithmic factors, each query costs eO(d(|J| + ⌈J⌉)) elementary gates, including state preparation.The full query and gate counts also depend on the LCU normalization and the minimum positive singular value.Source: C. Quantum algorithm and resource bounds · pp. 14–15
  • For fixed regularity parameters and PDE order, the sufficient cutoff has w = O(ds).The theorem also gives gate and query bounds in terms of N, with the least sufficient cutoff chosen.Source: C. Quantum algorithm and resource bounds · pp. 14–15
  • Continuous minimum norm alone does not establish the discrete target-orthogonality required by the theorem.The condition must be imposed or justified separately; otherwise the vector-error bound includes an additional term.Source: C. Quantum algorithm and resource bounds · p. 16

IV. EXTENSION TO DIRICHLET BOUNDARY CONDITIONS

For non-periodic boundary conditions, the periodic hypercube supplies a trial space rather than a periodic PDE reduction. A controlled-regularity periodic extension and collocation of the original equation and boundary condition underpin the construction.

IV. EXTENSION TO DIRICHLET BOUNDARY CONDITIONS

  • The periodic hypercube is used as a trial space, not as a periodic reformulation of the original boundary-value problem.The construction does not invoke the periodic-boundary result as a stated theorem.Source: §IV · p. 16
  • A periodic extension with controlled Gevrey parameters allows a Fourier cutoff polylogarithmic in target precision to resolve the solution on Ω.The original equation and boundary condition are collocated on that trial space.Source: §IV · p. 16
  • The construction depends on controlling both the extension's regularity and the probability of obtaining the solution state on Ω.PDE evaluation and solution-state output use the finer grid ΓM restricted to Ω.Source: §IV · p. 16

A. Gevrey extensions and ramp construction

Repeated averaging constructs smooth ramp functions that taper extensions across collars; multiplying the extension by a multidimensional ramp yields a periodic Gevrey extension matching the Dirichlet solution. The collar width balances boundary-series control, containment in the periodic cell, and collar mass.

Ramp construction

  • The descending ramp is built by integrating an infinite convolution of normalized interval indicators, producing a smooth transition with all positive-order derivatives vanishing at its endpoints.Finite-factor convolutions approximate the infinite-convolution definitions shown in the figure.Source: A. Gevrey extensions and ramp… · pp. 16–17, Figure 4 · p. 17
  • Auxiliary ramps localize successive boundary-derivative terms, while the multidimensional ramp equals one on the Dirichlet box and vanishes outside its collar.The multidimensional construction uses widths summing to δ and multiplies the entire extension.Source: A. Gevrey extensions and ramp… · pp. 17–18

Gevrey extension

  • The resulting periodic extension equals u on the domain, vanishes outside the enlarged box, and has controlled Gevrey parameters.Multiplication by the ramp preserves the solution and its Dirichlet values while tapering the continuation.Source: Figure 3 · p. 17, A. Gevrey extensions and ramp… · pp. 18–19
  • The collar width is capped by r, 1, and 1/(9dν^2) to control the boundary Taylor series, remain inside the periodic cell, and bound collar mass.For fixed r and s, the resulting radius is of order 1/(dν^2).Source: A. Gevrey extensions and ramp… · p. 19
  • The parameter ν controls mixed normal jets on faces of every codimension and may grow exponentially with dimension, so polynomial dimension bounds retain it explicitly.For product solutions, ν is bounded by a corresponding one-dimensional trace parameter independently of d.Source: A. Gevrey extensions and ramp… · p. 19

B. Dirichlet discretization and resource bounds

The Dirichlet solver collocates the original PDE and boundary values using a Fourier trial space, while restricted stability and regularization address small singular values. Under the stated stability and kernel conditions, it prepares an ε-close normalized solution state with constant success probability.

Dirichlet discretization

  • The stacked discretization evaluates the PDE in the interior and boundary values on each face, with boundary weights chosen to match the interior block’s scale.Scaling the boundary block as a contraction would suppress it by a factor Θ(N^d).Source: B. Dirichlet discretization and… · pp. 19–20

Restricted stability

  • A polynomial concentrated away from the domain can be nearly invisible to all collocation rows, so the smallest nonzero singular value is not the method’s stability constant.Restricted stability corresponds instead to an a priori estimate involving the PDE residual and boundary traces.Source: B. Dirichlet discretization and… · p. 20

Quantum algorithm

  • The algorithm regularizes small singular values with a scale-t inverse approximation, then restricts the successful branch to the interior and amplifies it.The theorem requires a positive lower bound on restricted stability and the kernel condition at the chosen cutoff.Source: B. Dirichlet discretization and… · pp. 21–22

Resource bounds

  • With the stated oracle access and cutoff hypotheses, the algorithm returns a normalized state ε-close to the restricted solution grid with constant success probability.No exact spectral support or source-consistency assumption is needed.Source: B. Dirichlet discretization and… · pp. 21–23
  • A sufficient Fourier cutoff follows from uniform Fourier approximation and requires N ≥ d; the stability and kernel hypotheses are evaluated at that cutoff.The resource statement also specifies elementary-gate costs and qubit counts, excluding oracle workspace.Source: B. Dirichlet discretization and… · pp. 22–23

V. APPLICATION TO THE POISSON EQUATION

The section applies the framework to isotropic and anisotropic Poisson equations, deriving resource bounds for periodic and Dirichlet boundary conditions. Its bounds depend on Gevrey regularity, the operator, and the boundary setting.

Periodic Poisson equations

  • For periodic anisotropic Poisson with symmetric positive-definite Σ and an s-Gevrey solution (s ≥ 1), Algorithm 1 prepares an ε-close normalized approximation using O(d log N + log(κ_LN/ε)) qubits, excluding oracle qubits.The periodic discretized solution is handled through a minimum-norm solution; the stated qubit bound excludes coefficient and source oracle workspace.Source: §V · pp. 24–25
  • For diagonal Σ, implementing the discretized Laplacian costs O(d log d log N + log^2 N + R_N) elementary gates.The proof identifies a simplified approximately O(d log N) gate cost for the diagonal case.Source: §V · p. 25
  • The periodic Poisson solve projects out the constant mode, since the discrete Laplacian has a constant-vector kernel and the consistent source must have zero sum.The projected source gives the minimum-norm solution; the discarded mean is included in the error bound.Source: §V · pp. 24, 26

Dirichlet Poisson equations

  • For Dirichlet Poisson with s ≥ 2 Gevrey regularity, Corollary 5 prepares an ε-close state approximating the solution restricted to Ω with constant success probability.The resource statement counts queries to a combined source and boundary-data oracle and gives elementary-gate and qubit bounds, excluding oracle workspace.Source: §V · p. 26
  • In the isotropic case with fixed r and s, the stated Dirichlet bound is polynomial in d when ν and log(C/µ) grow at most polynomially.The stated parameters include ||Σ||_1,1 = d, Θ_Σ = 1, and r̄ = Θ(1/(dν^2)).Source: §V · p. 27

VI. DISCUSSION

The discussion distinguishes periodic and Dirichlet stability and clarifies the scope of the resource claims. It also identifies source-consistency handling and possible extensions of the framework.

Stability and scope

  • Ellipticity gives a positive smallest singular value away from the kernel, but the paper does not establish a useful rate as N grows, except for constant-coefficient periodic Poisson.For isotropic and anisotropic periodic Poisson, Corollary 4 provides an explicit N-independent bound.Source: §VI · pp. 27–28
  • For Dirichlet discretizations, exterior-concentrated trial functions can produce exponentially small singular values; the stated query bound scales as 1/ε^2 and is not claimed optimal.The collar estimate also retains explicit dimension dependence through the mixed face-jet parameter ν.Source: §VI · p. 28

Source and consistency

  • Range-projected sources recover the pseudoinverse solution without requiring exact discrete consistency, while a target component in the operator kernel must still vanish or be included in the error bound.The periodic Poisson kernel is the constant mode; for Dirichlet Poisson, full column rank makes the target-kernel condition automatic.Source: §VI · p. 28

Comparison with Monte Carlo

  • The paper’s polylogarithmic precision dependence concerns periodic solution-state preparation; the discussed Dirichlet bound instead has polynomial precision dependence.A downstream bounded scalar expectation requires additional amplitude-estimation calls and end-to-end costs for preparation, normalization, and solution norm.Source: §VI · p. 28

Further applications

  • Potential extensions include time-dependent PDEs and coupled vector systems, while the latter would require extending the scalar framework.The discussion also identifies periodic Gevrey extensions and applications involving Poisson–Boltzmann and Stokes equations.Source: §VI · pp. 28–29

Appendix A: The Fourier method

The appendix develops Fourier tools for relating function regularity to coefficient decay, truncation accuracy, and derivative approximation bounds. It presents the dual-grid Fourier transform and describes convolutional Fourier derivatives and a block-encoding circuit for the derivative multiplier.

1. Fourier transform

  • The discrete Fourier coefficients are obtained by applying a d-fold QFT to the sampled-function state, and continuous and discrete coefficients are related by an aliasing sum.The construction uses frequency and spatial grids with the same number of points, (2N + 1)^d.Source: §1 · p. 32, §2 · p. 32

2. Smoothness and the decay of Fourier coefficients

  • Gevrey regularity yields stretched-exponential Fourier-coefficient bounds for s > 0, with different rates for s ≥ 1 and 0 < s < 1, subject to frequency thresholds.The stated envelopes use b = 3/4 for s ≥ 1 and b = s/2 for 0 < s < 1; the Gevrey bounds apply when (r∥ω∥∞)^(1/s) ≥ 16.Source: §2 · pp. 33, 37
  • Continuity alone implies Fourier coefficients tend to zero, while differentiability, Hölder regularity, and Gevrey bounds establish progressively stronger decay rates.For Gevrey classes, the displayed coefficient bounds hold with derivative-bound parameters C and r whenever (r∥ω∥∞)^(1/s) ≥ 16.Source: §2 · pp. 33–34
  • Proposition A.2 bounds Fourier-series tails by regularity-dependent envelopes and gives sufficient cutoffs for an ε-accurate truncated expansion.For entire functions, the tail is o(e^(-rN)) for every r > 0; for finite-bandwidth functions, it is exactly zero once N ≥ 1/r.Source: §2 · pp. 37–38
  • For s-Gevrey periodic functions, precision-dependent Fourier cutoffs yield exponential error bounds; finite-order Hölder regularity gives algebraic convergence but not the same exponential accuracy.The stated Gevrey cutoff scale is N ∈ eΩ((ds)^s/r) for s ≥ 1.Source: §2 · pp. 40–41

3. The Fourier approximation of derivatives

  • Fourier-truncated derivatives approximate discretized true derivatives with Euclidean and grid-uniform error bounds under stated Gevrey cutoff conditions, or for Hölder rows when p + q ≥ 2d + 3.For s ≥ 1, the corollary gives a cutoff scale N ∈ eΩ((s(d + |α|))^s/r).Source: §3 · pp. 41, 43

Appendix B: Convolutional structure of Fourier derivatives

  • The truncated Fourier derivative acts as a periodic convolution on the grid, with separable kernels and closed forms yielding standard first- and second-derivative spectral stencils.The kernel formulas extend to higher derivatives through derivatives of the Dirichlet kernel.Source: Appendix B · pp. 44, 46

Appendix C: Block-encoding of the derivative multiplier via LCU

  • The derivative multiplier admits a (1, log N + 2, 0)-block-encoding using O(log N) elementary gates, with comparator implementations trading scratch space and circuit depth.The ripple-carry comparator uses 2 log N − 1 Toffoli gates and one scratch qubit; a depth-optimized option uses Θ(log N) Toffoli gates, Θ(log log N) depth, and no scratch qubits.Source: Appendix C · pp. 47, 49

Appendix D: Prior literature on solving the Poisson equation

Appendix D compares quantum approaches to Poisson equations and describes a higher-order extension of the [CPP+13] Dirichlet solver. Their resource bounds depend on differing regularity, boundary, and solution-to-source assumptions.

Hamiltonian simulation [CPP+13]

  • The original [CPP+13] solver directly simulates the isotropic discretized Laplacian for zero Dirichlet data, using its sine-transform diagonalization.The extension adds a higher-order polynomial approximation while retaining the tensor sine eigenbasis.Source: Appendix D · pp. 49–50

Table IV

  • Table IV compares elementary gates excluding supplied input-state oracles, with rows conditioned on method-specific regularity, boundary, and ξ assumptions.Its [CPP+13] row includes independent repetitions for constant success probability, while the [TAWL21] rows assume ξ = Θ(1).Source: Table IV · p. 49

Higher-order extension of [CPP+13]

  • For fixed r ≥3, the [CPP+13] extension uses N = Θ(ε^-1/(r-2)); corrected boundary residual and solution error are O(h^(r-2)).The correction uses source-dependent boundary data, and the analysis assumes preparation of the boundary-corrected source state.Source: Appendix D · p. 50

Finite-difference and spectral methods [CLO21]

  • For [CLO21] adaptive FDM, stencil order k remains implicit in ε without independent high-order derivative-growth bounds; Dirichlet reflection also requires smoothness across the faces.Zero boundary values alone do not guarantee that the odd periodic extension has the required smoothness.Source: Appendix D · p. 51
  • The [CLO21] spectral method covers periodic and Dirichlet data, including nonzero Dirichlet values, but its derivative-dependent truncation bound requires stronger-than-C∞ all-orders control.The stated uniform derivative bound is stronger than C∞ smoothness or finite bandwidth.Source: Appendix D · pp. 51–52

QSVT and fast inversion [TAWL21]

  • The [TAWL21] QSVT comparison is restricted to periodic PDEs and assumes ξ = Θ(1); in the worst case, ξ = 1/∥A∥ adds a κ factor to both query counts.The worked example uses an invertible shifted operator, −∆ + I.Source: Appendix D · p. 52
  • Under ξ = Θ(1), [TAWL21] standard QSVT has depth and operator-block query count O(dN^2 log(1/ε)), while fast inversion reduces both to O(1).These counts treat supplied block encodings and related operations as unit-cost.Source: Appendix D · p. 53

QSVT gate implementation

  • The gate accounting computes and retains Fourier eigenvalues through QSVT; fast inversion implements the reciprocal with reversible division rather than a unit-cost oracle.The stated fast-inversion gate cost is O(d log^2 N + log^2(dN/ε)).Source: Appendix D · p. 53
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