Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank

arXiv:2609.09035 · quant-ph, cs.DS, math.OC · Submitted 2026-09-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank".

Kai: Near-optimal quantum query lower bounds for high-accuracy convex optimization are established by leveraging Fourier rank to derive near-linear query complexity results.

Mira: First, who's behind it and why it matters.

Title and authors: Kai: So we’ve seen some really interesting work on this paper called "Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank," and it seems like they’ve established a new speed limit for how hard high-accuracy convex optimization is when you use quantum computers.

Mira: Exactly, Kai, the paper dives into the theoretical limits of solving linear optimization problems defined over explicit families of n-dimensional ellipsoids, which is pretty concrete in terms of what we're trying to optimize.

Lev: From my side, I'm thinking about how this relates to actual hardware; if we were to implement this on current noisy devices, the required number of queries would be substantial.

Kai: It shows that any algorithm aiming for a solution with an additive objective error of (n-two) needs at least (n n n) membership queries in the worst case <ref:2609.09035#pg0,n \log n \log \log n)$ membership queries>.

Mira: That result is significant because it directly addresses an open question about whether quantum algorithms can achieve faster query complexities in this specific regime, resolving what Chakrabarti, Childs, Li, and Wu asked back in their two thousand twenty work <ref:2609.09035#pg0,Chakrabarti, Childs, Li, and Wu>.

Lev: For real hardware considerations, that (n n n) complexity means we're looking at a very demanding resource allocation if we want to tackle these problems directly with current error mitigation techniques.

Kai: The paper uses a novel polynomial method based on Fourier rank as its foundation, which is essentially a continuous version of the polynomial method for quantum query complexity.

Mira: That's where the theoretical meat of it is; they track the matrix rank of Fourier frequencies instead of just the degree, and show that after T queries, every amplitude has a Fourier rank at most T.

Lev: If we translate that to error correction, it suggests that achieving good bounds might require a lot of entanglement or very high-fidelity operations to keep those ranks bounded across the computation.

Kai: The core idea is an obstruction: if you only use fewer than n/two queries, the Haar-average success probability for predicting the determinant of the hidden matrix Q is exactly one/two which suggests a stronger limitation than standard bounded-error bounds <ref:2609.09035#pg0>.

Mira: It shows that any function whose Fourier rank is below n has zero Haar correlation with the determinant, which really proves that you can't get away from these query limits for solving optimization problems over these ellipsoid families.

Title and authors: Lev: That connection to the determinant computation is interesting; if we think about running this on a quantum computer, we’re essentially hitting a wall related to how hard it is to compute that matrix determinant in the first place.

Kai: They achieve this by reducing the optimization problem to computing the determinant of a real matrix Q, which requires at least n/two queries in the continuous matrix phase-query model <ref:2609.09035#pg0>.

Mira: They do this by showing that a low-rank frequency cannot correlate with that determinant because a suitable reflection preserves the frequency and reverses the determinant, leading to Theorem three point five (Continuous matrix phase determinant lower bound).

Lev: That reduction strategy is powerful; it means the difficulty isn't just in checking membership, but in fundamentally dealing with the algebraic structure of linear algebra problems under these quantum query constraints.

Kai: The next step involves combining that determinant lower bound with a membership query lower bound for ellipsoids, showing that an accurate optimizer can predict the determinant through approximate inverse iteration.

Mira: Because the membership predicate for these centered ellipsoids has a Fourier rank at most one, it implies that M membership queries can only produce output probabilities of Fourier rank at most 2M <ref:2609.09035#pg0>.

Lev: So, to get the final near-linear bound, they're essentially combining a query count from solving linear systems with another query count related to how well you can probe the set itself.

Kai: And for smooth quadratics, they prove a near-optimal gradient-query lower bound of (q (two + q) (three + q)), where q is related to the condition number kappa.

Mira: That result establishes that the quantum optimality of the principal classical parameter dependences in these nonsmooth, smooth convex, and constant-accuracy strongly convex regimes is characterized by these specific functional dependencies.

Lev: If we're dealing with a function whose condition number kappa is very large—which happens often in real-world systems—the required queries scale up significantly because of that

sqrt kappa, n: factor.

Kai: Looking at the final scaling, for exactly feasible optimization with error epsilon = (n-two), the query complexity is T at least Clb n n n <ref:2609.09035#pg0,n \log n \log \log n>.

Title and authors: Mira: And for approximately feasible optimization where the tolerance scales as delta n = (n-two), the complexity remains T at least Clb n n n with a scaled ellipsoid family Kapp n <ref:2609.09035#pg0,n \log n \log \log n>.

Lev: That final characterization is tight up to logarithmic factors, which gives us a clear metric for what to expect when deploying these optimization routines in practical quantum applications.

Kai: Before we wrap up this discussion on "Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank," I want to quickly mention that the authors did sketch a way to remove the n factor by using a certified proposal mechanism based on residual estimation.

Mira: That approach involves running the optimizer multiple times at every step, which leads to an overall complexity of W = O(d d) for the total number of calls, though they noted this isn't essential for the main asymptotic result.

Lev: That certified proposal mechanism sounds like it’s exactly what we need to see if we can build a more practical quantum iteration scheme that doesn't rely on just theoretical worst-case bounds.

Kai: So, to summarize, this paper provides a near-linear query complexity bound for high-accuracy convex optimization over explicit ellipsoid families by using Fourier rank arguments derived from determinant computation hardness.

Mira: It also gives us tighter bounds for gradient queries on smooth quadratics and distinguishes between exactly and approximately feasible solutions based on their required error scaling.

Lev: For running this on real hardware, the main hurdle is that the complexity scales with n, so we need to be really careful about how large n we can realistically handle before the query count becomes prohibitive.

Kai: The implication for us is that we now have a very precise speed limit for solving these types of optimization problems on quantum hardware, which helps us design better algorithms.

Mira: It also gives us a clearer picture of when and where quantum advantage might be achievable versus where classical methods are simply more efficient given the problem structure.

Lev: I think this work gives us concrete numbers to benchmark our error correction overhead against, which is something we need as we move toward fault-tolerant systems.

Kai: So that's a wrap on "Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank," and I think it’s a really solid piece of theoretical groundwork for the community to build upon.

The paper's summary: Kai: So, we just got through that deep dive into the paper's summary of "Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank," which essentially boils down to establishing a new speed limit for solving high-accuracy convex optimization over those specific ellipsoid families.

Mira: Exactly, Kai, what I find most compelling is how they use this concept of Fourier rank as a continuous tool to prove that any algorithm trying to solve these problems has to make at least (n n n) queries in the worst case.

Lev: And from my standpoint in error correction, that lower bound is pretty telling because it gives us a hard target for the complexity we have to contend with when trying to run these routines on actual hardware, especially considering how sensitive those Fourier ranks are to noise.

Kai: Right, and what really stands out is how they connect this optimization problem directly to the classic difficulty of computing determinants and eigenvalues of real matrices Q, which sets up a really solid foundation for the whole argument.

Mira: That reduction step is crucial because it shows that you can’t just focus on the membership queries; you have to deal with the underlying algebraic structure of those matrices, and that's where Fourier rank comes in as the measuring stick.

Lev: If we think about running this on a real quantum processor, that means any implementation would need to be extremely careful with how it handles those matrix operations because they are inherently tied to these lower bounds.

Kai: It’s also interesting how the authors distinguish between exactly feasible optimization and approximately feasible ones, showing that the scaling of the required queries changes depending on whether you need a perfect answer or just one that's within a certain tolerance level.

Mira: That distinction is important because it shows us how robust an algorithm needs to be when dealing with real-world noise, not just theoretical perfect solutions, and it dictates how we design error mitigation strategies.

Lev: So, if we look at the implications for practical applications, this means that designing a quantum optimization routine isn't just about finding a clever circuit; it’s fundamentally about understanding the algebraic bottlenecks dictated by Fourier rank.

Kai: It really puts things into perspective for us in the experimental side because it tells us exactly where to focus our efforts when we start building these systems and measuring the query counts.

Mira: And this entire framework suggests that for problems with a very large dimension, like those in machine learning or materials science, we might be hitting fundamental limits that classical methods might actually bypass more easily in certain regimes.

Lev: That’s why understanding these bounds is vital; it helps us predict the scaling of our required resources and whether the quantum approach is even going to be feasible for high-dimensional problems.

The paper's improvements: Tom: So, we're moving on to what the paper suggests to do next regarding its findings on Fourier rank and optimization limits, which basically focuses on ways to clean up those theoretical bounds.

Kai: It seems like the authors sketched out a method for removing that pesky n factor by using a "certified proposal" mechanism based on residual estimation, which is essentially running the optimizer multiple times at every step to ensure reliability.

Mira: That approach sounds promising because it directly addresses one of the scaling issues we saw earlier, and if that mechanism works as they claim, it would give us a much tighter complexity bound without sacrificing accuracy.

Lev: From an error correction standpoint, a certified proposal mechanism is interesting because it implies we have a way to ensure that our iterative process stays within the bounds of what's computationally feasible for a given fidelity level.

Kai: Right, and they note that while this isn't strictly necessary for the main asymptotic result, it’s useful if you want to see how practical the complexity looks when you start talking about actual execution.

Mira: It also suggests that we can get a more concrete idea of how many actual calls we'd need in practice, which is something I always appreciate because theory needs to connect to what's actually happening in the physical system.

Lev: If we can prove those certified proposal bounds hold rigorously, it gives us a better blueprint for designing hybrid quantum-classical systems where the classical part manages the budget of quantum calls effectively.

Kai: That leads into something really important: they also have an appendix discussing continuous query registers in L two(Y, nu) C squared, which is a more formal way to prove that exact membership queries can be mapped to rank growth rules.

Mira: That continuous setting confirms the mechanism for the Fourier rank growth rule, showing how one exact membership query effectively maps a certain rank structure to another one that's at most one higher, which solidifies their core technique.

Lev: If we look at how this relates to our work on quantum regular language states or maybe even tensor separability criteria, this continuous model provides a more flexible mathematical framework for analyzing the underlying quantum information flow.

Kai: It shows that the mathematical structure is robust enough to handle these continuous settings, which is a nice piece of consistency when you’re building things that have to scale up.

Mira: Ultimately, these improvements suggest a clearer path toward optimizing the complexity bounds themselves without having to rely on those potentially overly conservative n factors in our final numbers.

Lev: So, the implication here is that we might be able to design more efficient quantum protocols by focusing our efforts on these certified iterative methods rather than just chasing the absolute worst-case theoretical limits.

Conclusion: Kai: So, to wrap things up, this paper on "Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank" establishes a precise query complexity limit for solving high-accuracy optimization problems over ellipsoid families using Fourier rank arguments.

Mira: That result really hammers home how the algebraic structure of those matrices dictates the computational difficulty in the quantum setting, tying it back to those determinant lower bounds we discussed.

Lev: It’s a lot to take in, but for me, the real impact is seeing exactly what kind of resource scaling we need to anticipate if we ever try to actually implement these optimization routines on physical hardware.

Kai: I think the implication for us is that it gives us a very clear benchmark for when we can expect quantum speedups versus when classical methods are just more efficient for certain problem structures.

Mira: Exactly, and this framework helps us design algorithms that are inherently aware of the constraints imposed by Fourier rank, which could lead to much more efficient resource allocation in future optimization solvers.

Lev: I agree; if we can use these bounds to guide our error correction choices, it could make building fault-tolerant systems for these kinds of problems a lot more targeted and less wasteful.

Kai: It’s really exciting because it gives us a concrete theoretical target to aim for when we start designing the actual quantum circuits and setting up the measurements.

Mira: This work really solidifies the connection between continuous Fourier analysis and discrete quantum query complexity, which is a neat piece of theoretical machinery we can use elsewhere.

Lev: So, as we look ahead, I think this paper sets a strong foundation for how we should approach designing hybrid solvers that need to manage those complex algebraic dependencies efficiently.

Global Technology Applied Research Center of JPMorgan Chase & Co. · CNRS, Université Paris Cité

quant-ph, cs.DS, math.OC

Submitted: 2026-09-08

Updated: 2026-10-05

Comments: Updates from Version 1: Updated proof removing logarithmic factors from lower bound, detailed comparison to independent concurrent work, minor clarifications

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 89/100

The gist: Near-optimal quantum query lower bounds for high-accuracy convex optimization are established by leveraging Fourier rank to derive near-linear query complexity results.

Key concepts

Fourier Rank: This concept is a measure used to track how complex the input data's frequencies are. In this context, it relates to the matrix rank of a Fourier frequency. The paper uses it to show that after $T$ queries, every amplitude in the system has a Fourier rank at most $T$, which acts as an obstruction for low-query algorithms.
Continuous Matrix Phase Oracle: This is a model used in continuous quantum query complexity where the algorithm interacts with a matrix via phase queries. The paper proves that computing the determinant of such a matrix requires at least $n/2$ queries using this oracle, providing a strong lower bound for related problems.
Membership Queries: These are specific queries an algorithm makes to determine if a point lies inside or outside an ellipsoid. The paper derives near-linear query complexity bounds for these membership queries by reducing the optimization problem to the more fundamental task of computing matrix determinants.
Optimization Lower Bounds for Smooth Quadratics: This result focuses on finding lower bounds for optimizing smooth, strongly convex functions. It shows that achieving an additive objective error of $\epsilon$ requires $\Omega(q \log(2 + q) \log \log(3 + q))$ gradient-query queries, where $q$ is related to the condition number and dimension.

Terminology

Accuracy — 51× in this paper · explained in 1 episode
Query Complexity — 24× in this paper · explained in 2 episodes
Trace — 5× in this paper · explained in 2 episodes
Core — 4× in this paper · explained in 2 episodes
Hilbert Space — 3× in this paper · explained in 2 episodes
Ambient Dimension — 2× in this paper · explained in 1 episode
Benchmark — 2× in this paper · explained in 2 episodes
Fidelity — 2× in this paper · explained in 2 episodes
Adam — 1× in this paper · explained in 2 episodes
Adaptive Algorithm — 1× in this paper · explained in 1 episode
Combinatorial Optimization — 1× in this paper · explained in 2 episodes
Consistency — 1× in this paper · explained in 3 episodes
Entanglement — 1× in this paper · explained in 3 episodes
Generalization — 1× in this paper · explained in 13 episodes
Quantum Circuits — 1× in this paper · explained in 2 episodes
Spectral Gap — 1× in this paper · explained in 5 episodes
Spectral Geometry — 1× in this paper · explained in 2 episodes

Summary

Near-optimal quantum query lower bounds for high-accuracy convex optimization are established by leveraging Fourier rank to derive near-linear query complexity results. This work characterizes the computational limits of solving linear optimization problems over explicit families of ellipsoids, showing that any algorithm must make a number of membership queries proportional to the dimension, resolving open questions regarding quantum speedups in this regime.

The gist: Any algorithm that solves high-accuracy convex optimization over an explicit family of n-dimensional ellipsoids requires at least omega(n log n log log n) membership queries in the worst case.

Lower Bounds via Fourier Rank Polynomial Method

The proof is built around a novel polynomial method based on Fourier rank, which serves as a continuous analogue to the polynomial method for quantum query complexity. Instead of tracking the degree of input variables, this method tracks the matrix rank of a Fourier frequency, denoted as Fourier rank. A key result is that after T queries, every amplitude has a Fourier rank at most T. The argument shows that if an algorithm outputs a prediction in two states (like +1 or-1), its Haar-average success probability for predicting the determinant of the hidden matrix Q is exactly 1/2 if fewer than n/2 queries are used. This obstruction is stronger than standard bounded-error lower bounds, as it shows that every function of Fourier rank below n has zero Haar correlation with the determinant.

Reduction from Determinant Computation

The core strategy involves reducing the problem to computing the determinant of a real matrix Q. The continuous matrix phase-query model requires at least n/2 queries to compute the determinant. This is achieved by showing that a low-rank frequency cannot correlate with the determinant because a suitable reflection preserves the frequency and reverses the determinant, leading to "Theorem 3.5 (Continuous matrix phase determinant lower bound): Any algorithm which predicts det Q for Q ∼ µn with average success probability strictly greater than 1/2, using the matrix phase oracle (9), makes at least n/2 queries."

Reduction from Optimization to Determinant Prediction

The hard family of convex sets are centered ellipsoids, defined by a positive-definite matrix AQ. The membership predicate for these ellipsoids is shown to have a Fourier rank at most one, which implies that M membership queries can produce output probabilities of Fourier rank at most 2M. By combining the determinant lower bound (requiring d/2 queries) with the membership query lower bound (requiring M queries), and using an explicit reduction where an accurate optimizer can predict the determinant through approximate inverse iteration, a near-linear membership query lower bound is derived.

Optimization Lower Bounds for Smooth Quadratics

The paper also proves a near-optimal gradient-query lower bound for constant-accuracy optimization of smooth and strongly convex functions. This result is achieved by studying quadratic functions where the condition number is at most κ, leading to a dimension d related to min√κ, n. The analysis shows that any quantum algorithm requiring additive objective error at most εquad requires omega(q log(2 + q) log log(3 + q)) queries to the gradient phase oracle, where q = min√κ, n. This establishes quantum optimality of the principal classical parameter dependences in the respective nonsmooth, smooth convex, and constant-accuracy strongly convex regimes.

Scaling and Final Bounds

The final query complexity is determined by scaling factors related to accuracy. For exactly feasible optimization with error ε = Θ(n − 2), the required number of membership queries is T ≥ Clb n log n log log n. For approximately feasible optimization, where the tolerance scales as δn = Θ(n − 2), the required query complexity is T ≥ Clb n log n log log n, with a scaled ellipsoid family Kapp n. This establishes that the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The results extend from finite fields to the real-valued setting for determinant and minimum-eigenvalue lower bounds.

Appendix on Removing Log Log n Factor

A sketch is provided for removing the log log n factor by treating each optimizer output as a proposal and using a certified proposal mechanism based on residual estimation. This involves running the optimizer multiple times at every step to ensure reliability, leading to an overall complexity of W = O(log d log log d) for the total number of calls. This demonstrates how the lower bound can be refined, although it is not essential for the main asymptotic result.

Appendix on Continuous Query Registers

The continuous exact-membership determinant lower bound is proven using a continuous query register in L2(Y, ν) ⊗ C2. The key insight here is that one exact membership query and its inverse map rank r amplitudes to rank at most r + 1 by Lemma A.1, allowing the application of the Fourier-rank growth rule to the continuous setting, which confirms that "Theorem 3.

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed the provided scientific paper on Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank. This work establishes fundamental theoretical limits on quantum query complexity for high-accuracy convex optimization and relates these limits to the hardness of computing determinants and eigenvalues of real matrices.

Here are the specific improvements that can be made to AI systems, derived from this research:

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Improved Theoretical Benchmarking for Optimization Complexity:

A quantum lower bound of approximately omega(n log n log log n) for high-accuracy convex optimization provides a rigorous speed limit benchmark. This allows researchers to definitively categorize the computational difficulty of solving specific classes of problems (like those defined by centered ellipsoids) in both classical and quantum regimes.

Tighter Bounds on Quantum Speedup:

The paper demonstrates that for strongly convex quadratics, the optimal quantum query complexity is omega(min[√κ, n] log(2 + min[√κ, n]) log log(3 + min[√κ, n])). This provides a precise roadmap for identifying where quantum advantages are achievable and where they are fundamentally limited by the problem structure (condition number κ).

Enhanced Robustness Against Approximation Errors:

The paper provides distinct lower bounds for exactly feasible (Theorem 5.1) versus approximately feasible optimization (Theorem 5.5), showing that the required query complexity scales differently with respect to the tolerance parameter ε (e.g., scaling by s=n vs s=n2). This allows AI systems to be designed and optimized not just for idealized perfect solutions, but for real-world scenarios where output accuracy is inherently limited, leading to more robust algorithms that are less susceptible to small input perturbations.

Optimized Quantum Gradient Estimation:

The derivation links the optimization lower bound directly to lower bounds on minimum eigenvalue estimation of symmetric matrices (omega(n) queries). This suggests that improving quantum gradient estimation techniques—the primary source of speedup in high-dimensional regimes—will yield direct, quantifiable improvements in solving complex optimization tasks.

Algorithmic Design Guided by Matrix Hardness:

The connection between optimization and determinant/eigenvalue computation (via the Fourier-rank polynomial method) informs how to construct hard instances for training or testing AI models. By understanding which matrix structures (like Haar-random matrices or specific orthogonal families) are computationally intractable, researchers can design adversarial test cases that probe the limits of a model's ability to solve related linear algebraic problems efficiently.

Guidance for Hybrid Quantum-Classical Architectures:

The paper details how quantum optimization algorithms simulate inverse iteration and approximate matrix inversion using a sequence of optimizer calls and majority-ball amplification (Proposition 4.11). This provides a blueprint for designing hybrid quantum-classical systems where the quantum processor performs the core search/iteration steps, while classical post-processing handles the necessary error amplification and decision logic.

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The improved AI system can achieve the following:

Identify and classify optimization problems based on their inherent complexity (e.g., determining if a specific machine learning loss function optimization falls into the omega(n log n log log n) regime).
Develop quantum gradient estimation subroutines that are provably optimal for high-dimensional, strongly convex functions, ensuring the resulting AI models benefit from the maximum possible quantum speedup allowed by their condition number.
Design and deploy robust optimization algorithms capable of handling inherent output inaccuracies (e.g., in noisy sensor data or limited model precision), guaranteeing a solution that remains within a specified tolerance (e.g., 0(n−2)) regardless of the difficulty of the problem instance, by leveraging the exact trade-offs between approximation error and query complexity derived from Theorem 5.5.
Build specialized quantum kernels for AI training that exploit the hardness of related linear algebraic problems (determinant/eigenvalue computation) to create rigorous, high-dimensional test suites for neural network optimization methods.
Implement hybrid quantum-classical solvers where the classical component intelligently manages a budget of quantum optimizer calls, using certified success probabilities (like those derived from Lemma 4.10) to ensure that the final result is separated with a probability strictly greater than 1/2, even when facing worst-case matrix inputs.