Quantum spectral method for gradient and Hessian estimation

Changpeng Shao; Yuxin Zhang

arxiv: 2407.03833 · v2 · submitted 2024-07-04 · 🪐 quant-ph · cs.DS

Quantum spectral method for gradient and Hessian estimation

Yuxin Zhang , Changpeng Shao This is my paper

Pith reviewed 2026-05-23 23:15 UTC · model grok-4.3

classification 🪐 quant-ph cs.DS

keywords quantum gradient estimationanalytic functionsphase oraclesHessian estimationquantum optimizationspectral methodquery complexity

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The pith

Given phase oracles for the real and imaginary parts of an analytic function over the complex field, a quantum algorithm approximates its gradient to accuracy ε using Õ(1/ε) queries.

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

The paper develops a quantum algorithm for estimating the gradient of analytic functions defined over the complex numbers. Using separate phase oracles for the real and imaginary parts of the function, it achieves an ε-approximation with query complexity Õ(1/ε), independent of dimension. This improves on earlier quantum gradient methods whose cost scaled with the square root of dimension. The work also gives two quantum algorithms for Hessian estimation and proves a lower bound of Õ(d) queries in the general case.

Core claim

The authors present a quantum spectral method that, given phase oracles to the real and imaginary parts of an analytic function f over the complex field, returns an ε-approximation to its gradient using Õ(1/ε) queries. They extend this to two algorithms for Hessian estimation with query complexities Õ(d/ε) and Õ(d^{1.5}/ε) under different assumptions, and improved bounds Õ(s/ε) and Õ(sd/ε) when the Hessian is s-sparse. A lower bound of Õ(d) is proven for general Hessian estimation.

What carries the argument

The quantum spectral method that uses separate phase oracles for the real and imaginary parts of an analytic function to estimate its gradient and Hessian.

Load-bearing premise

The function must be analytic and well-defined over the complex field, and the algorithm must be given access to phase oracles that separately encode its real and imaginary parts.

What would settle it

A numerical simulation or implementation on a simple analytic function such as f(x) = exp(sum x_i) that requires more than O(1/ε) queries to reach ε-accurate gradient estimates, or that shows query cost growing with dimension d.

read the original abstract

Gradient descent is one of the most basic algorithms for solving continuous optimization problems. In [Jordan, PRL, 95(5):050501, 2005], Jordan proposed the first quantum algorithm for estimating gradients of functions close to linear, with exponential speedup in the black-box model. This quantum algorithm was greatly enhanced and developed by [Gily\'en, Arunachalam, and Wiebe, SODA, pp. 1425-1444, 2019], providing a quantum algorithm with optimal query complexity $\widetilde{\Theta}(\sqrt{d}/\varepsilon)$ for a class of smooth functions of $d$ variables, where $\varepsilon$ is the accuracy. This is quadratically faster than classical algorithms for the same problem. In this work, we continue this research by proposing a new quantum algorithm for another class of functions, namely, analytic functions $f(\boldsymbol{x})$ which are well-defined over the complex field. Given phase oracles to query the real and imaginary parts of $f(\boldsymbol{x})$ respectively, we propose a quantum algorithm that returns an $\varepsilon$-approximation of its gradient with query complexity $\widetilde{O}(1/\varepsilon)$. As an extension, we also propose two quantum algorithms for Hessian estimation, aiming to improve quantum analogs of Newton's method. The two algorithms have query complexity $\widetilde{O}(d/\varepsilon)$ and $\widetilde{O}(d^{1.5}/\varepsilon)$, respectively, under different assumptions. Moreover, if the Hessian is promised to be $s$-sparse, we then have two new quantum algorithms with query complexity $\widetilde{O}(s/\varepsilon)$ and $\widetilde{O}(sd/\varepsilon)$, respectively. We also prove a lower bound of $\widetilde{\Omega}(d)$ for Hessian estimation in the general case.

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.

Desk Editor's Note private letter to a colleague

The paper's core advance is a dimension-independent Õ(1/ε) quantum gradient estimator for analytic functions under separate phase oracles for real and imaginary parts.

read the letter

The main thing here is the claimed Õ(1/ε) query bound for gradient estimation of analytic f over the complex field, with no dependence on dimension d. This sits on top of the Jordan and Gilyen lines of work but exploits analyticity plus the split phase oracles to remove the sqrt(d) factor that appeared in the smooth-function case. The two Hessian algorithms (Õ(d/ε) and Õ(d^{1.5}/ε)) plus the sparse variants and the Ω(d) lower bound are the other concrete pieces. The lower bound in particular gives a clear target for any future Newton-style quantum method. The paper lays out these query complexities cleanly and positions them as direct extensions of the earlier results. The analyticity assumption is what makes the dimension-free scaling possible, and the authors are explicit about the oracle model they use. That said, the separate Re and Im phase oracles are a non-trivial modeling choice whose implementation cost is left for later work. The analyticity requirement itself is restrictive enough that it will only apply in limited settings. Without the full derivations it is hard to judge whether the error bounds hide extra logarithmic factors or rely on post-selection. The lower bound looks standard for the query model but would benefit from a short discussion of how tight it is under the same oracle assumptions. Overall this is aimed at people who already work on quantum query algorithms for continuous optimization. Anyone tracking gradient and Hessian estimation in the black-box setting will want to see the details. It is worth sending to referees because the claimed scalings are specific, the lower bound is new, and the extension to analytic functions is a natural next step even if the practical reach is narrow.

Referee Report

0 major / 3 minor

Summary. The paper proposes a quantum spectral method for gradient estimation of analytic functions f(x) defined over the complex field. Given separate phase oracles for Re(f) and Im(f), it claims an ε-approximate gradient with query complexity Õ(1/ε), independent of dimension d. It extends this to two quantum algorithms for Hessian estimation with complexities Õ(d/ε) and Õ(d^{1.5}/ε) under different assumptions, plus sparse variants Õ(s/ε) and Õ(sd/ε), and proves a lower bound of Õ(d) for general Hessian estimation.

Significance. If the claimed complexities hold under the stated analyticity and oracle model, the dimension-independent gradient result would be a notable advance over prior quantum algorithms (e.g., Gilyén et al. with √d dependence), as analyticity correlates derivatives and enables bypassing dimension factors. The Hessian algorithms and lower bound would support quantum second-order methods. The separate Re/Im phase oracles are a precise access model that makes the result possible.

minor comments (3)

[Abstract] Abstract, lines 8-10: the phase oracle model (separate oracles for Re(f) and Im(f)) should be formalized with a precise definition of the unitary in the introduction or §2, including how the analytic continuation is handled.
[Abstract] The lower bound of Õ(d) for Hessian estimation is stated without reference to the precise oracle model or function class; a short proof sketch or citation to the relevant section would clarify its scope.
Notation for the Õ notation and the precise dependence on d in the Hessian results should be consistent between abstract and main text; any hidden log(1/ε) or log d factors should be stated explicitly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The referee's description of the contributions is accurate. Since no specific major comments are listed in the report, we have no points requiring point-by-point rebuttal at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a quantum algorithm for ε-approximate gradient estimation of analytic functions over the complex field, given separate phase oracles for Re(f) and Im(f), achieving Õ(1/ε) query complexity independent of dimension. It extends prior independent results from Jordan (2005) and Gilyén et al. (2019) without load-bearing self-citations, fitted parameters renamed as predictions, or self-definitional reductions. Hessian extensions and lower bounds are presented as new contributions under stated assumptions. The derivation chain relies on external prior work and analyticity constraints rather than internal tautologies, making the central claims self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The central claims rest on the unelaborated assumption that suitable phase oracles exist and that analyticity over C supplies the necessary analytic continuation properties.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Given phase oracles to query the real and imaginary parts of f(x) respectively, we propose a quantum algorithm that returns an ε-approximation of its gradient with query complexity Õ(1/ε).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

our algorithm is based on the spectral method [For81, Tre00]

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