Title: Fast Zeroth-Order Convex Optimization with Quantum Gradient Methods
Venue: NeurIPS 2025
Accepted: yes
Keywords: quantum computing

convex optimization

quantum gradient methods

semidefinite programming

linear programming

zero-sum games
Abstract: We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies quantum algorithms for zeroth-order convex optimization using quantum gradient estimation with noisy function evaluation oracles and shows that by relying solely on noisy function value queries, the quantum algorithms match the first-order query complexities of classical gradient descent methods. The framework are then generalized to non-Euclidean settings via quantum mirror descent and its variants. Leveraging connections between SDP and eigenvalue optimization, the framework is applied to solve white-box problems, including SDPs, LPs, and zero-sum games, and achieves a better time complexity on the dependences of $r_d, r_p$ and $\\varepsilon$.  For smooth objectives with Lipschitz gradients, quantum gradient descent and mirror prox achieve convergence rates matching classical first-order methods. A fast query complexity is also proven under the PL condition assumption.\n\nSTRENGTHS AND WEAKNESSES:\nIn previous work, quantum gradient estimation methods have demonstrated the capability for zeroth-order black-box approaches to match the performance of classical first-order black-box methods. However, when applied to concrete optimization problems, quantum advantages may fail to materialize—particularly in the presence of erroneous black-box settings. Therefore, rigorous analysis and ensuring correct parameter choices are both challenging and significant.\n\nThis work provides a rigorous proof that quantum gradient estimation with erroneous black-boxes, when employed as a gradient estimator subroutine in projected gradient descent and mirror descent, maintains the original convergence rate. This establishes a solid theoretical foundation for broader applications. Furthermore, the authors demonstrate quantum advantages in applications to SDPs, LPs, and zero-sum games, achieving superior complexity compared to prior approaches relying on the HHL algorithmic family. Consequently, this work holds substantial theoretical value and lays an important groundwork for future investigation to the application of quantum gradient estimation.\n\nQUESTIONS:\nSuggestions:\n1. In the white-box section, the authors' algorithms achieve improvements in terms of $r_d$, $r_p$, and precision $\\varepsilon$, but exhibit slightly worse dimensionality dependence compared to other algorithms. Though such phenomena are common in recent works of quantum optimization, the authors are suggested to briefly explain the source of the additional dimensionality dependence.\n\n2. In Theorem D.3 and Corollary D.1, the authors are advised to explicitly specify the form of the error rather than solely referring to it as \"additive error\".\n\n3. The caption of the result table is suggested to include the meaning of symbols and explicitly highlight which factors demonstrate quantum advantage.\n\n4. The statement about dimension independence may not be entirely rigorous. Under the error function value black-box setting, the quantum gradient estimation method still exhibits a logarithmic dependence on the dimension in its query complexity. Based on my knowledge, the quantum community only proved that in the error-free black-box setting, the bounded error quantum gradient estimation can achieve a query complexity of complete dimension independence.\n\n5. If I understand right, the \"domain\" in Table 1 refers to a normed space, not the feasible domain. The authors are suggested to add a brief clarification to avoid potential confusion.\n\nQuestions:\n1. Regarding SDP problems, the quantum algorithms in this work output a dual solution under additive error, is it possible to derive error bounds for primal solutions? \n\n2. Selecting an appropriate mirror map is one of the crucial steps in implementing quantum mirror descent. Note that the authors provided several examples (e.g., for feasible domains like the probability simplex and Schatten-1 norm). For other feasible domains, are there general methodologies for choosing the mirror map and the Bregman divergence?\n\n3. Regarding the result in Table 1, is there any classical or quantum lower bound?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper shows that quantum algorithms can solve zeroth-order convex optimization problems, using only noisy function evaluations, with query complexity matching that of classical first-order methods. By combining quantum gradient estimation with standard optimization routines like projected subgradient and mirror descent, the authors achieve exponential improvements in dimension over classical zeroth-order methods. They extend the framework to smooth functions, non-Euclidean settings, and structured problems like SDPs, LPs, and zero-sum games, demonstrating improved complexities in several regimes.\n\nSTRENGTHS AND WEAKNESSES:\n* Strengths:\n    * The key positive result is that zeroth-order quantum methods match the rates of classical first-order methods, exhibiting an exponential separation from classical zeroth-order optimization.\n    * The paper offers a unified and complete treatment across various settings—nonsmooth and smooth objectives, Euclidean and non-Euclidean geometries—using tools like mirror descent, dual averaging, and mirror prox.\n    * The framework is broadly applicable and extends to important structured problems such as semidefinite programming, linear programming, and zero-sum games.\n    * The paper is well written. It is mathematically precise, and the results are well contextualized within the broader literature on quantum optimization.\n* Weakness\n    * The core technical contribution seems limited to plugging known quantum gradient (or subgradient) estimators into standard first-order convex optimization routines. Once an oracle with sufficiently controlled bias and variance is assumed (or constructed), the convergence analyses for mirror descent, projected subgradient, or gradient descent follow in a largely black-box, modular fashion. The analysis mirrors classical proofs, accounting for estimation error.\n\nQUESTIONS:\nIn view of the \"Weaknesses\" section:\n\n* Are there genuinely new technical challenges in adapting classical convergence guarantees to the quantum zeroth-order setting, beyond tracking bias and variance from oracle errors?\n* Does any part of the analysis fundamentally rely on quantum-specific insights, or could the results be derived by combining known quantum gradient estimation techniques with standard optimization proofs?\n\nIt would be important for the authors to clarify this, as it directly impacts the novelty and technical contribution of the work..\n\nLIMITATIONS:\nNo limitations", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents new quantum algorithms for zeroth-order convex optimization—that is, optimization using only noisy function value queries, without access to gradients. The main contributions are:\n\nExponential Quantum Speedup: The authors show that quantum algorithms can match the first-order query complexities of classical gradient-based methods, even though they only use zeroth-order (function value) oracles. This results in an exponential speedup in the dimension d compared to classical zeroth-order methods, whose complexity is typically linear in d\n\nDimension-Independent Query Complexity: The proposed quantum algorithms achieve dimension-independent query complexities (up to polylogarithmic factors) for both smooth and nonsmooth convex optimization problems, a significant improvement over classical zeroth-order approaches.\n\nGeneralization to Non-Euclidean Settings: By leveraging quantum (sub)gradient estimation, the authors extend their methods to non-Euclidean spaces using quantum mirror descent and its variants (including dual averaging and mirror prox), broadening the applicability of their techniques.\n\nApplications to White-box Problems: The framework is applied to important structured optimization problems such as semidefinite programming (SDP), linear programming (LP), and zero-sum games. For these, the quantum algorithms achieve better or comparable complexities to the best known classical and quantum methods in certain parameter regimes.\n\nTechnical Advances: The paper develops robust quantum subgradient and gradient estimation procedures that work with noisy oracles, and provides convergence guarantees for quantum versions of projected subgradient, mirror descent, and mirror prox methods\n\nApplications and Speedups:\n\nSDP and LP: The quantum mirror descent framework yields SDP and LP solvers with gate complexities that provide quadratic speedups in certain regimes over classical matrix multiplicative weights methods and some previous quantum approaches.\n\nZero-sum Games: The algorithm achieves improved complexity for finding Nash equilibria in zero-sum games, outperforming prior classical and quantum algorithms in some parameter ranges\n\nSTRENGTHS AND WEAKNESSES:\nOverall, i think this is a nice paper  making a good contribution to Quantum optimization. \n\nThe results are fairly theoretical (which i think is OK) and assume access to sufficiently precise quantum oracles; practical implementation awaits advances in quantum hardware. I guess it'd be nice if the authors discuss what happens if the oracles are a bit noisy.\n\nThe quantum algorithms match the rates of standard (non-accelerated) classical first-order methods, but do not (yet) achieve the accelerated rates of Nesterov-type methods. Again, it'd be good to have a discussion on this.\n\nThe precision requirements for the oracles are stringent; relaxing these is an open direction for future research\n\nQUESTIONS:\nStated above.\n\nLIMITATIONS:\nNone.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a unified algorithmic framework for zeroth-order convex optimization with certain quantum oracles to achieve dimension-independent optimization errors. By assuming the errors in the function values of the quantum oracle to be vanishingly small, the proposed achieved rate is comparable to the errors achievable under the classical first-order-oracle setting under stochastic unit-variance errors. The key technical idea in the proposed approach is to leverage a gradient estimator analyzed in [74] (presented as in Lemma 2.1) to adapt first order algorithms to the quantum zeroth order setting. This approach is extended beyond zeroth order convex and smooth optimization, and achievability analysis was provided form SDP, LP, and zero-sum games.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:  Zeroth-order optimization is important in settings where gradients are unavailable or expensive to compute, and understanding potential speedup of quantum algorithms is an important topic that could have significant impact in Neurips society. Any significant result in this direction on improving the best known results and approaches could provide valuable theoretical insights. In terms of clarity, the technical sessions of this paper are well-written and accessible. The assumptions are clearly stated and proved, and the analysis is carefully structured. \n\n Weaknesses: A main issue of the submitted manuscript is the mismatch between the noise amplitude between the proposed quantum framework and the classical oracles, which results in an unfair advantage of the proposed scheme on achieving an optimization error that could have been achieved and outperformed by the classical oracles under the same noise condition, even under zeroth order access. \n\nFor instance, definition 1 assumes that the quantum oracle has a zeroth order measurement error of theta which is conventionally presumed to be constant. But through a careful investigation, the achievability results presented in this work in fact requires the measurement error theta to be polynomially smaller than the targeting estimation error epsilon (e.g., see the 2.1). Knowing that under such vanishing error condition, the classical zeroth order oracle can in fact achieve rates that are orderwise better than the rates in e.g. Table 1 that was used in the comparison. Particularly, the bound on the gradient estimation error this manuscript relies on (Lemma 2.1) is weaker than the guarantees of the classical zeroth order two point gradient estimator under the same regime of theta used in the main theorems. Therefore, if we apply the same analysis under the same error conditions, the rates achievable for classical algorithms would outperform the rates of quantum algorithms presented in this work, which is the opposite of what is claimed in the abstract. \n\nTwo other important but relatively minor issues: 1. the abstract claims an exponential separation between quantum and classical optimization but this is not clearly elaborated in the main text. 2. the quantum oracle considered in this work assumes a pure state access which is well known to be algorithmically easier than its classical counterpart. Therefore, it is not a generalization of the classical access. However, this distinction is not made clear in the abstract, which could potentially mislead the readers into thinking the proposed result is a quantum speedup.\n\nQUESTIONS:\nWhile the topic investigated in the manuscript is clearly of significance and there might be technical merits of the proposed results. As mentioned in the earlier discussion on the weakness of the submitted draft, we recommend the authors to provide following discussions for the manuscript to be considered publishable. \n\n1. The achieved rates of the quantum algorithm must be compared to the ones achieved by the best known classical algorithms under the same vanishing error condition, i.e., the noise variance decays with the same polynomial dependency of epsilon and d. \n\n2. The comparison between classical and quantum algorithms needs to be based on the above fair comparison, and any claims in the abstract needs to be justified in the manuscript. \n\n3. We recommend the authors to clearly mention in the abstract that the quantum oracle considered in this work under the pure state assumption is known to be strictly easier than its classical counterpart. There has been abundant works in recent years that demonstrated this distinction, which should also be mentioned.\n\n4. Section 2 mentioned that one main benefit of first-order methods vs newton's approach is the convergence of global optimum irrespective of the starting point (line 103-107 on page 3), however we would point out that this global convergence is achievable with a modified newton's method with second-order smoothness condition.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-04"}