(Sub)Exponential Quantum Speedup for Optimization

Jiaqi Leng; Kewen Wu; Xiaodi Wu; Yufan Zheng

REVIEW 2 major objections 5 minor 2 cited by

(Sub)Exponential Quantum Speedup for Optimization

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves that two plain quantum optimization algorithms — adiabatic evolution and quantum Hamiltonian descent — can have (sub)exponential query advantage over any classical algorithm.

desk verdict Discrete half is a genuine reduction-based advance; continuous half is a real but conditional result resting on an unproved claim imported from an overlapping preprint—worth refereeing with that flag. read the letter →

arxiv 2504.14841 v1 pith:ZBZOHVDB submitted 2025-04-21 quant-ph cs.CCmath.OC

Jiaqi Leng , Kewen Wu , Xiaodi Wu , Yufan Zheng This is my paper · ORCID

classification quant-phcs.CCmath.OC MSC 81P6868Q1290C26

keywords quantumadiabaticoptimizationoracleseparationexponentialspeedupHamiltoniandescenttransversefieldIsingSchrödingeroperatorencodingperturbativegadgetscontinuous

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that optimization, both discrete and continuous, is a setting where quantum computation can provably beat every classical algorithm by an exponential gap in the oracle-query model. It constructs families of objective functions whose global minimum is hard for any classical method to find, yet is reached in polynomial time by the plain adiabatic algorithm (discrete case) and by quantum Hamiltonian descent (continuous case). If the construction is right, it settles the open question of whether such separations exist in optimization, and it does so without exotic modifications of the algorithms: only the natural linear Hamiltonian path and the Schrödinger operator evolution are used.

What carries the argument

The argument is carried by a chain of perturbative reductions. Starting from a stoquastic sparse adiabatic path with inverse-polynomial spectral gap, edge-subdivision gadgets embed its interaction graph into a hypercube, and then a second-order reduction maps it to a transverse-field diagonal Hamiltonian of the form $-\sum_i a_i X_i + D(t)$; a toy TFI Hamiltonian acting as a unary clock linearizes the path, and a small tilt extends it so that it starts as a pure TFI Hamiltonian and ends diagonal. For the continuous setting, the key object is the double-well operator $\hat X(\lambda)=-\frac{d^2}{d\xi^2}+\lambda^2(\xi^2-1/4)^2$, whose first two eigenstates form symmetric/antisymmetric pairs $|b_0\rangle,|b_1\rangle$ concentrated on one side of the well; the new error analysis shows that a diagonal Hamiltonian $D$ can be represented as a continuous function $\hat D$ over the box with error $\mathrm{poly}(n)\Lambda^{-\Omega(1)}$ rather than the naive $\Theta(2^n)$ bound.

What would settle it

Compute the two lowest eigenvalues of $-\frac{d^2}{d\xi^2}+\lambda^2(\xi^2-1/4)^2$ on $[-1,1]$ with vanishing boundary conditions for large $\lambda$ and check that the gap is $2/\Lambda$ with $\Lambda=e^{\lambda/6(1\pm o(1))}$ and that $\langle b_0|b_0\rangle_{[-1,w]}\le O(\Lambda^{-2/3})$; failure of any of these at increasing $\lambda$ would invalidate the error estimate $\varepsilon\le O(\sqrt{n}\,\Lambda^{-1/6}\|D\|)$ in Lemma 10.5.

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Extended reading notes

Core claim

The central claim is Theorem 4.1: for every $n$ there is an $n$-qubit transverse-field Ising Hamiltonian $H_{\mathrm{TFI}}$ and a family of diagonal Hamiltonians $D$ such that the linear path $H(t)=(1-t)H_{\mathrm{TFI}}+tD$ has exponential quantum advantage. Any classical algorithm must use $\exp(n^{\Omega(1)})$ queries to find the diagonal entry $u$ minimizing $\langle u|D|u\rangle$, while the path keeps spectral gap $n^{\Omega(1)}$ and norm $\mathrm{poly}(n)$, and simulating the Schrödinger dynamics for $T=\mathrm{poly}(n)$ yields a state $n^{-\Omega(1)}$-close to $|u\rangle$ with $\mathrm{poly}(n)$ queries and gates. The continuous analogue (Theorem 10.2) transfers the same separation to functions $f$ on a bounded box, where the QHD-type dynamics $H(t)=-\Delta+t f(x)+\nu(t)g(x)$ with $\nu(t)=e^{-\Theta(\sqrt{t})}$ finds the minimizer in polynomial time while classical query algorithms require exponentially many queries.

Load-bearing premise

The continuous part of the construction imports a spectral-gap and concentration bound for the double-well operator from prior work; if that bound has hidden parameter dependencies or errors, the continuous separation does not follow.

Editorial extensions

If this is right

Any classical query algorithm for the constructed discrete family needs $\exp(n^{\Omega(1)})$ queries even to reach $1/\mathrm{poly}(n)$ accuracy, so no classical black-box method can match the adiabatic algorithm on these instances.
The plain linear-path adiabatic evolution suffices for the speedup; the construction needs only the standard mixer $D$ plus a fixed TFI starting Hamiltonian, not problem-specific or nonlocal mixers.
The continuous construction gives the first provable super-polynomial quantum-classical separation for continuous optimization, carried by quantum Hamiltonian descent dynamics.
The reductions preserve an inverse-polynomial spectral gap and polynomial norms throughout, so the quantum evolutions are implementable by standard digital simulation in polynomial time.
The modular chain — sparse to hypercube to TFD to linear to Schrödinger — is a reusable template for compiling other adiabatic oracle separations into optimization separations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the separation is oracle-based, converting it into an unconditional complexity-class separation would require additional structure; removing the oracle is a natural next test of the framework.
The continuous construction suggests that QHD's advantage in this setting is driven by the adiabatic gap and the concentration of the double-well encoding, not by non-adiabatic higher-energy exploration; this could be tested by replacing $\hat X$ with other spectrally similar operators and checking whether the speedup persists.
The linearization gadget is essentially a quantum tunneling chain through a clock register; this may be reusable to turn other walk-based quantum speedups into optimization speedups.
The continuous result inherits the cited double-well spectral bounds; an independent re-derivation of Claim 10.6 with explicit $\lambda$-dependencies would make the error analysis self-contained and easier to verify.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: the discrete half of this paper looks like a genuine advance. Leng, Wu, Wu, and Zheng take the GHV oracle separation and compile it, through a chain of explicit perturbative reductions, into a standalone optimization problem solved by the plain adiabatic algorithm. That was not in GHV, and the technical machinery here—edge-subdivision gadgets, linearization via a clock register, path tilt and extension—is new and mostly well argued. The spectral gap and hardness preservation steps in Sections 5–9 are coherent, and the discrete result is anchored to the external GHV benchmark rather than to the authors' own prior work.

The continuous half is also real, but it is conditional in a specific place. The reduction from TFD Hamiltonians to Schrödinger operators relies on Lemma 10.5, which in turn imports Claim 10.6 from ZLLW24, an overlapping-author preprint. Three quantitative facts are load-bearing: the 2/Λ spectral gap of the double-well operator, the exact reflection symmetry of |b0⟩ and |b1⟩, and the O(Λ^{-2/3}) concentration bound. If any of these has a hidden λ- or n-dependence, the ε ≤ O(√n Λ^{-1/6}‖D‖) estimate in Lemma 10.5 collapses and Theorem 10.2 does not follow. The paper does not re-prove these bounds, and the stress-test note identifies exactly this as the least secure step. That is the correct concern, and it is the reason a referee rather than a desk rejection is needed.

Everything else in the continuous part appears consistent once Claim 10.6 is granted. The bD construction and the new error analysis that avoids the trivial 2^n Λ^{-1/3} bound are a genuinely nice piece of work. The classical-hardness argument in Theorem 10.2(i) is compressed, but the query-to-query reduction is straightforward and I would not call that a soft spot.

One minor note: no code or machine-checked proofs, which is normal for a theory paper, and the self-citation of ZLLW24 is not itself a flaw—but it does mean the load-bearing claim lacks independent verification.

Who this is for: researchers in quantum algorithms and complexity, especially anyone working on adiabatic optimization, oracle separations, or QHD. It deserves a serious referee. My recommendation: send it to peer review, assign a referee who knows the ZLLW24 inner-workings, and ask that referee to verify Claim 10.6 and the bD error analysis before accepting.

I would bring this to a reading group and cite it in my own work. Not because I'm fully convinced of the continuous half, but because the discrete result alone is a notable step and the continuous framework is worth engaging with even if the imported lemma needs patching.

Referee Report

2 major / 5 minor

Summary. The paper constructs oracle-based families of optimization problems for which it claims provable exponential quantum-classical query separation. For discrete optimization, Theorem 4.1 reduces the GHV stoquastic adiabatic separation to a linear Hamiltonian path H(t)=(1-t)H_TFI+tD, where H_TFI is a transverse-field Ising Hamiltonian and D is diagonal, and shows that plain adiabatic evolution solves the problem in poly(n) time while any classical algorithm needs exp(n^{Ω(1)}) queries. The proof is a sequence of perturbative reductions: sparse stoquastic paths to hypercube paths (Lemma 4.3), hypercube paths to TFD paths (Lemma 4.4), piecewise-linear interpolation (Lemma 4.5), a toy TFI ground-state shifter (Lemma 4.6), linearization via a clock register (Lemma 4.7), and a tilt-and-extension step (Lemma 4.8). For continuous optimization, Theorem 10.2 embeds the discrete construction into Schrödinger operators H(t)=-Δ+ν(t)g(x)+t f(x), with ν(t)=e^{-Θ(√t)}, yielding a Quantum Hamiltonian Descent speedup. The continuous proof depends on Lemma 10.5, whose error analysis in turn depends on Claim 10.6, a set of spectral, symmetry, and concentration properties of the double-well operator bX(λ), imported from the overlapping preprint [ZLLW24].

Significance. If the results are correct, this is a substantial contribution: it provides the first provable exponential quantum-classical separation for optimization in the oracle model using the plain adiabatic algorithm and QHD, with hardness inherited from the GHV separation. The discrete part is a significant technical achievement: the perturbative reductions in Sections 5–9 are explicit, preserve an inverse-polynomial spectral gap, preserve classical hardness, and are stated in reusable form in Propositions B.1 and B.2. The paper also ships detailed proofs rather than relying on numerical experiment. The main caveat is that the continuous result is not self-contained at a load-bearing point: Claim 10.6 is taken from an overlapping-author preprint, and the current manuscript does not prove the quantitative bounds that feed Lemma 10.5 and Theorem 10.2. With that claim supplied or independently verified, the significance is high; as it stands, the continuous theorem is conditional on an external source.

major comments (2)

[§10.3, Claim 10.6 and Lemma 10.5] The continuous half of the paper rests on Claim 10.6, which is imported from [ZLLW24, Theorem 4.22] rather than proved in this manuscript. The three quantitative properties in the claim—the spectral gap 2/Λ with Λ satisfying (10.16), the reflection identity (10.44), and the concentration bound (10.45)—are exactly the inputs to Claims 10.7 and 10.8, and they control the error ε≤O(√nΛ^{-1/6}∥D∥) in Lemma 10.5 and hence the gap and fidelity statements in Theorem 10.2(ii). If any of these bounds carries a hidden λ- or n-dependence, the estimates δ(λ)≥1/poly(n) and the n^{-Ω(1)} closeness in Theorem 10.2(ii) may fail. Because [ZLLW24] is an overlapping-author preprint, this is not an independent verification. Please include a self-contained proof of Claim 10.6, or state it from a peer-reviewed source with explicit uniform-in-λ constants for λ≥λ0=Θ(log n). As written, Theorem 10.2 is valid only conditionally on that external claim.
[§10.4, proof of Claim 10.10(i)] The displayed estimate for ⟨bx|bDp|bx⟩ reads "≤⟨x|D|x⟩⟨ex|ex⟩−2∥˚x∥∥D∥"; this inequality is not implied by the preceding line, since the two cross terms can have either sign and the triangle inequality gives an upper bound with a plus sign, not a minus sign. The final O(√nΛ^{-1/3}∥D∥) bound is still recoverable by applying the triangle inequality to obtain both upper and lower bounds, but the proof as printed is mathematically incorrect. Please correct the inequality and explicitly state both bounds.

minor comments (5)

[§10.3, Eq. (10.16)] The notation d^kΛ/dλ^k = (e^{λ/6})^{1±o_k(1)} for k∈N is not a well-defined definition of Λ; please state precisely what is assumed for each fixed k, whether the o_k(1) terms are uniform in the regime λ≥λ0, and how these derivative bounds are used in the estimates for ν(t), ν̇(t), and ν̈(t).
[Appendix B] The first sentence contains the typo "impiclitly"; it should read "implicitly".
[References] Weyl's inequality is cited to a Wikipedia article [Wik25]; please use a standard textbook or published reference for this classical result.
[Theorem 1.2 vs Theorem 10.2] The informal Theorem 1.2 states T,t_f=poly(n), while the formal Theorem 10.2 states t_f=poly(n,1/η); please align the two statements.
[§2.6] The text calls the reflection symmetry in (10.44) a "newfound symmetry property," but Claim 10.6 attributes it to [ZLLW24, Theorem 4.22]; please clarify what is new in this paper versus what is imported.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: discrete reduction anchored to an external GHV separation; continuous extension uses an overlapping-author spectral theorem as an independent premise, not as a restatement of the conclusion.

full rationale

The paper's claimed derivation chain is a sequence of reduction theorems, not a fitted prediction. Theorem 4.1 is proven by starting from the external GHV oracle separation (Theorem 4.2, [GHV21]) and applying in-paper perturbative reductions (Lemmas 4.3 through 4.8). Each reduction explicitly preserves the spectral gap and translates classical queries back to queries of the GHV path, so the classical lower bound is inherited rather than introduced by construction. The continuous result, Theorem 10.2, relies on Claim 10.6, which the paper explicitly imports: "We present a key claim about bX based on [ZLLW24, Theorem 4.22] and its proof." That cited theorem concerns spectral and concentration properties of the double-well operator bX and is stated independently of the target quantum-classical separation; it supplies the parameter Λ, reflection symmetry, and concentration bounds used later in the new error analysis of Claims 10.7-10.11 and Lemma 10.5. While [ZLLW24] has overlapping authors and is a load-bearing premise, it is a prior mathematical theorem with its own stated assumptions, not a rename or a fit of the conclusion, and no equation in the present paper is defined in terms of its own output. The remaining steps, including the construction of bD, the oracle reductions, and the time rescaling, are explicit and self-contained once Claim 10.6 is granted. Thus no circular step can be exhibited; the paper's findings rest on external benchmarks rather than on their own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The discrete claim rests on GHV21, an external separation; the continuous claim additionally rests on the ZLLW24 correspondence, which is self-cited. No new physical entities are postulated; the auxiliary functions g and ν in (1.4) are part of the construction, not invented entities.

assumptions (4)

domain assumption GHV21 oracle separation (Theorem 4.2): there exist stoquastic sparse adiabatic paths with spectral gap Ω(√m), norm O(m), and classical query lower bound exp(n^{1/5-o(1)}).
Used as the hardness anchor; the entire reduction chain starts from this theorem and preserves both gap and hardness.
domain assumption ZLLW24 double-well operator properties (Claim 10.6, i.e., [ZLLW24, Theorem 4.22]): bX(λ) has spectral gap 2/Λ with Λ=e^{λ/6(1±o(1))}, plus concentration and reflection properties (10.44) and (10.45).
Imported from a preprint by overlapping authors; not fully proved in this paper, but used to establish Lemma 10.5.
standard math Perturbative simulation lemmas (Lemmas 3.8 to 3.10) from [BH17]: first- and second-order perturbative reductions preserve low-energy spectra.
Quoted as standard results for Hamiltonian simulation; used throughout Sections 5 to 9.
standard math Quantum adiabatic theorem (Theorem A.3) and real-space simulation algorithm [CLLLZ22, Theorem 3] (Lemma A.2).
Used to convert spectral gap and norm bounds into actual quantum simulation complexity; standard in the field.

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Pith. "Pith review of (Sub)Exponential Quantum Speedup for Optimization." pith.science (2026). https://pith.science/paper/ZBZOHVDB

@misc{pith2026250414841,
  author       = {Pith},
  title        = {Pith review of: (Sub)Exponential Quantum Speedup for Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZBZOHVDB}},
  note         = {Machine review of arXiv:2504.14841}
}

read the original abstract

We demonstrate provable (sub)exponential quantum speedups in both discrete and continuous optimization, achieved through simple and natural quantum optimization algorithms, namely the quantum adiabatic algorithm for discrete optimization and quantum Hamiltonian descent for continuous optimization. Our result builds on the Gily\'en--Hastings--Vazirani (sub)exponential oracle separation for adiabatic quantum computing. With a sequence of perturbative reductions, we compile their construction into two standalone objective functions, whose oracles can be directly leveraged by the plain adiabatic evolution and Schr\"odinger operator evolution for discrete and continuous optimization, respectively.

Figures

Figures reproduced from arXiv: 2504.14841 by the authors.

**Figure 2.** An illustration of how we reduce an edge of weight [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** An illustration of how we reduce an edge of weight [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Spectrum of A(t) for different λ values. A small but nonzero λ perturbs the spectrum, ensuring an Θ(λ) spectral gap throughout and making the instantaneous ground state transition of A(t) smooth. for a sufficiently large ∆. By first-order perturbation theory, the degen…

**Figure 5.** Figure 5: Visualizations of |χ0⟩, |χ1⟩ regarding Xb and their linear combination |b0⟩, |b1⟩ defined in (2.9). The parameter for Xb is chosen to be λ = 30. The potential field fdw (amplified 10 times) is plotted in a dotted line for reference. Building on the established mapping …

**Figure 6.** Figure 6: The Construction of Db from a diagonal Hamiltonian D with parameter w = 1/4 in Definition 10.3. It is efficient in the sense that a query to Db(·) can be implemented by one query to D. a path of Schr¨odinger operators: H(t) = −∆ + tf(x) + ν(t)g(x), ∀t ≥ 0, (2.11) where…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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