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arxiv: 2606.29848 · v1 · pith:B4DGY46Znew · submitted 2026-06-29 · 🪐 quant-ph · cs.NA· math-ph· math.MP· math.NA

Quantum Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation: A Weyl Calculus Approach

Hongkang Ni , Lexing Ying This is my paper

Pith reviewed 2026-06-30 06:09 UTC · model grok-4.3

classification 🪐 quant-ph cs.NAmath-phmath.MPmath.NA
keywords quantum eigenvalue transformationlinear combination of Hamiltonian simulationWeyl calculusmatrix functionsnon-normal operatorsquantum algorithmsFourier approximationtime-dependent simulation
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The pith

Weyl calculus reduces non-normal matrix function approximation to scalar Fourier problems, enabling optimal O(log 1/ε) quantum eigenvalue transformation via LCHS.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops formulas based on linear combination of Hamiltonian simulation to evaluate analytic functions f(A) on quantum computers, even when A is non-normal. Weyl calculus converts the non-commuting operator task into ordinary scalar Fourier approximation. This produces a quantum eigenvalue transformation algorithm whose query cost scales as O(log 1/ε). The same framework supplies a convex optimization procedure that generates discrete LCHS formulas without ansatzes and improves simulation of time-dependent dissipative differential equations by a factor of 2.1.

Core claim

Weyl calculus reduces the task of constructing LCHS formulas for f(A), with f analytic on the numerical range of A, to scalar Fourier approximation problems. The resulting quantum algorithm for eigenvalue transformation achieves O(log 1/ε) query complexity. The approach also yields an ansatz-free convex optimization method for discrete LCHS formulas and extends directly to time-dependent dissipative ODE simulation.

What carries the argument

Weyl calculus, which reduces construction of LCHS formulas for non-commuting operators to scalar Fourier approximation problems.

If this is right

  • Quantum eigenvalue transformation achieves optimal O(log 1/ε) query scaling.
  • An ansatz-free convex optimization framework produces discrete LCHS formulas optimized for coherent quantum implementation.
  • Simulation of time-dependent dissipative ODEs achieves a 2.1 times cost reduction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique could be tested on other quantum tasks that involve functions of non-normal operators.
  • The convex optimization procedure might be specialized to produce formulas for functions analytic only in smaller regions.
  • Numerical experiments on low-dimensional non-normal matrices would directly check whether the claimed complexity holds in practice.

Load-bearing premise

The function f is analytic on the numerical range of A and the Weyl calculus reduction incurs no hidden costs that change the stated query complexity.

What would settle it

A concrete counter-example computation for an analytic f and a non-normal A in which the number of queries needed to reach error ε exceeds O(log 1/ε).

Figures

Figures reproduced from arXiv: 2606.29848 by Hongkang Ni, Lexing Ying.

Figure 1
Figure 1. Figure 1: Integration contour C in the lower half-plane, with small indentations around the branch points ±R and a single enclosed pole at −i. Inside C, the analytically continued integrand ˆgR,C(z)e −izx has a single simple pole at z = −i, coming from the factor (1 + z 2 ) −1 . Its residue gives −2πi Resz=−i  1 √ 2π gˆR,C(z)e −izx = e −x . The contour C is traversed clockwise. By Cauchy’s residue theorem, the int… view at source ↗
Figure 2
Figure 2. Figure 2: Numerical illustration of the optimized LCHS construction for [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cost comparison for f(z) = p (1 + ρ) + z for several values of ρ. The dashed lines are linear fits of Λ versus log(1/ϵ). on the whole square. In this case, we instead optimize the residual directly on D1. We still approximate f(x + iy) on D1 by S(x, y) = X (j,k)∈GR wj,ke −i π 2 (jx+ky) , (81) where wj,k ∈ C. We use the Neumann Fourier–Bessel basis on the disc to measure the residual. Write x = r cos θ and … view at source ↗
Figure 4
Figure 4. Figure 4: Required truncation radius R for f(z) = z 4 with fixed normalization factor λ = 12. Since λ is fixed, R is proportional to the LCHS cost Λ = λR. depends on the LCHS parameters. The material is standard in LCHS-based methods; see [ALL23, ACL26, ACLY26, LS25] for more details. There are two common ways of implementing LCHS formulas. The first is a coherent im￾plementation, which combines LCU [CW12] with Hami… view at source ↗
read the original abstract

Linear combination of Hamiltonian simulation (LCHS) provides an efficient method for implementing matrix exponentials $e^{-tA}$ on quantum computers. In this paper, we develop LCHS formulas for computing general matrix functions $f(A)$ when $f$ is analytic on the numerical range of $A$, with $A$ possibly non-normal. The essential technical tool is Weyl calculus, which reduces the construction of LCHS formulas for noncommuting operators to scalar Fourier approximation problems. Our construction yields a quantum eigenvalue transformation algorithm with optimal $\mathcal{O}(\log\frac{1}{\epsilon})$ query complexity scaling. Furthermore, our Weyl-calculus-based theory gives rise to an ansatz-free convex optimization framework that directly produces discrete LCHS formulas. This circumvents the inefficiencies of traditional quadrature rules and yields formulas highly optimized for coherent implementation on quantum computers. In addition, both our theory and optimization framework apply to the simulation of time-dependent dissipative ODE $\frac{\mathrm{d}}{\mathrm{d} t} \psi(t) = -A(t)\psi(t)$, for which we achieve a $2.1\times$ cost reduction over prior art.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript develops LCHS formulas for analytic matrix functions f(A) on the numerical range of possibly non-normal A by reducing the problem via Weyl calculus to scalar Fourier approximation. This yields a quantum eigenvalue transformation algorithm with claimed optimal O(log 1/ε) query complexity, an ansatz-free convex optimization framework for discrete LCHS formulas, and a 2.1× cost reduction for simulating time-dependent dissipative ODEs dψ/dt = −A(t)ψ(t).

Significance. If the Weyl reduction achieves the stated complexity with no hidden overheads, the work would deliver an optimal quantum eigenvalue transformation algorithm together with a reproducible convex-optimization route to coherent LCHS formulas; the concrete 2.1× improvement on the time-dependent ODE benchmark is a measurable practical contribution.

major comments (1)
  1. [Abstract (complexity statement) and the Weyl-calculus reduction section] The optimality claim O(log 1/ε) query complexity rests on the assertion that every term in the Weyl expansion maps to an independent LCHS call whose aggregate cost remains strictly O(log 1/ε) with no extra logarithmic multiplier arising from symbol ordering, numerical-range enlargement, or discretization. The manuscript must supply an explicit bound on the number of LCHS invocations and demonstrate that this bound is independent of the condition number of the numerical range.
minor comments (1)
  1. The analyticity assumption on the numerical range should be stated with an explicit radius or strip width to make the Fourier-approximation error bounds fully quantitative.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that strengthens the complexity analysis. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract (complexity statement) and the Weyl-calculus reduction section] The optimality claim O(log 1/ε) query complexity rests on the assertion that every term in the Weyl expansion maps to an independent LCHS call whose aggregate cost remains strictly O(log 1/ε) with no extra logarithmic multiplier arising from symbol ordering, numerical-range enlargement, or discretization. The manuscript must supply an explicit bound on the number of LCHS invocations and demonstrate that this bound is independent of the condition number of the numerical range.

    Authors: We agree that the manuscript does not yet contain an explicit, self-contained bound isolating the total number of LCHS invocations from the condition number of the numerical range. The Weyl-calculus reduction reduces the problem to scalar Fourier approximation, which in principle yields a term count depending only on the analyticity radius of f, the diameter of the numerical range, and ε. However, this independence is asserted rather than proved in a dedicated lemma. In the revised version we will insert a new lemma (in the Weyl-calculus reduction section) that supplies the required bound: the number of invocations is at most C(log(1/ε) + 1), where C depends only on the analyticity parameters and the numerical-range diameter and is independent of its condition number. Symbol-ordering overhead is absorbed into the Weyl calculus without extra logarithmic factors. We will also add a short remark in the abstract clarifying that the O(log 1/ε) claim is now supported by this explicit bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents Weyl calculus as an external technical tool that reduces non-commuting operator functions to scalar Fourier approximation problems, from which LCHS formulas and the O(log 1/ε) complexity are derived as outputs of the construction. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. The central claims (optimal query scaling, ansatz-free optimization framework, 2.1× cost reduction) are positioned as consequences of the new method rather than presupposed by definition or prior self-citation chains. The derivation chain remains independent of its target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; the central claims rest on the analyticity assumption for f and the applicability of Weyl calculus to non-normal operators. No free parameters, invented entities, or additional axioms are stated.

axioms (2)
  • domain assumption f is analytic on the numerical range of A
    Stated in abstract as prerequisite for the LCHS formulas.
  • domain assumption Weyl calculus reduces non-commuting operator problems to scalar Fourier approximation
    Described as the essential technical tool.

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