arXiv subjects

Brandon Augustino

Publications and source records attributed to Brandon Augustino.

18 recordsLinked to original sources

arXiv2026

An analysis of iterative refinement for quantum linear system solvers

We present and analyze an iterative refinement (IR) framework for improving the precision dependence of algorithms that combine a quantum linear system algorithm (QLSA) with quantum state tomography. Existing QLSAs achieve polylogarithmic dependence on the inverse error tolerance to prepare a quantum state encoding the solution, but extracting a classical description of such a state via tomography typically introduces a polynomial dependence on the target precision. To retain polylogarithmic dependence in the inverse error tolerance throughout the entire process, we develop an IR scheme that solves a sequence of related linear systems to progressively refine the solution while requiring only fixed-precision quantum subroutines, finally obtaining a classical description of a high-precision solution. We analyze the proposed framework under three input models: quantum-read/classical-write RAM (QRAM), linear combinations of unitaries (LCU), and sparse-access oracles. We evaluate the proposed scheme through numerical experiments on both quantum simulators and real quantum hardware. The results demonstrate that iterative refinement efficiently improves the precision of the solution and exhibits robustness to hardware noise.

quant-ph↗

arXiv2026

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

We establish a near-linear quantum query lower bound for high-accuracy convex optimization over an explicit family of $n$-dimensional ellipsoids. We focus on linear optimization with an explicitly given objective, where the feasible set is accessed through a membership oracle. We show that any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error $\Theta(n^{-2})$ requires $\Omega\!\left(\frac{n}{\log n\,\log\log n}\right)$ membership queries. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within $\Theta(n^{-2})$ distance from the feasible set. This resolves, up to logarithmic factors, an open question posed by Chakrabarti, Childs, Li, and Wu~(\textit{Quantum}, 2020) and by van Apeldoorn, Gily\'en, Gribling, and de Wolf~(\textit{Quantum}, 2020). Coupled with the upper bounds in these papers, the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The proof is built around a lower bound for determinant computation that is derived via a novel polynomial method based on Fourier-rank. In the continuous matrix phase-query model, computing the determinant of a real $n\times n$ matrix requires at least $n/2$ matrix-vector product queries. The construction also yields an $\Omega(n)$ phase-query lower bound for estimating the minimum eigenvalue of a real symmetric $n\times n$ matrix to additive accuracy $\Theta(n^{-2})$. These results extend the determinant and minimum-eigenvalue lower bounds of Childs, Hung, and Li~(ICALP 2021) from finite fields to the real-valued setting. Based on the same constructions, we also prove a near-optimal gradient-query lower bound for constant-accuracy optimization of smooth and strongly convex functions.

quant-ph↗

arXiv2026

Global Multi-Maturity SPX-VIX Calibration Beyond Markovian Stitching

We develop a global framework for joint S&P 500 (SPX)-VIX smile calibration across multiple maturities without the conditional-independence restriction induced by Markovian stitching. Exact local and global feasibility are equivalent: every globally feasible law has a block-preserving SPX-Markovization that leaves each monthly $(S_i,V_i,S_{i+1})$ law unchanged. Nevertheless, stitched laws can form a strict subset of globally feasible path laws because Markovization discards dependence on earlier history beyond the current SPX level. Adjacent smiles therefore cannot identify this dependence, and laws with identical monthly calibrations can price multi-period claims differently. Under the standard Markov reference, relative entropy selects the stitched minimum-information completion; non-Markov dependence requires cross-period information, an appropriate objective, or a history-dependent prior. For finite discretizations, we introduce an augmented-Bregman mirror-descent scheme. It preserves the fit to observable quote moments while controlling martingale and dispersion residuals. In a controlled infeasible affine system, this split keeps prescribed marginals about $25$ times tighter than cyclic row projection by exposing the discrepancy in the conditional rows. An exact finite-state example verifies block preservation and exhibits material cross-period price changes after Markovization. On smoothed SPX and VIX surfaces, numerical calculations illustrate a finite-budget penalty path: the worst fitted-smile error remains below $0.70$ volatility points across the reported sweep while the bulk conditional diagnostics improve substantially.

q-fin.CP↗

arXiv2026

Faster Algorithms for Multimarginal Optimal Transport

We study algorithms for approximating the multimarginal optimal transport (MOT) distance, a generalization of the classic optimal transport distance, between $m$ discrete probability distributions each supported on at most $n$ points. We give a classical algorithm that computes a coupling between these marginals whose expected transportation cost is within an additive $\varepsilon > 0$ of the MOT distance in time $O(m^2 n^m \varepsilon^{-1}\mathrm{polylog}(m,n,\varepsilon^{-1}))$. This is, to our knowledge, the first bound for general MOT problems with simultaneous linear dependence on the dimension $n^m$ and on the accuracy parameter $\varepsilon^{-1}$, improving the prior state of the art. On the quantum side, we give two algorithms that achieve speedups in dimension, though with worse accuracy dependence than classical approaches. First, we construct a quantum projected subgradient method for estimating the MOT distance within an additive $\varepsilon >0$ with runtime $O( m^3 n^{\frac{m}{2}+1} \varepsilon^{-2} \mathrm{polylog}(m,n,\varepsilon^{-1}))$. This algorithm works with the linear programming dual of the MOT problem, and does not return a coupling. We also give a quantum multimarginal Sinkhorn algorithm for entropy-regularized MOT. This algorithm returns an implicit description of an approximately optimal coupling with runtime $O(m^8n^{\frac{m+1}{2}} \varepsilon^{-5} \mathrm{polylog}(m,n,\varepsilon^{-1})))$ after the usual reduction from entropic MOT to unregularized MOT. We also record query lower bounds: for any precision $\varepsilon<1/2$, randomized classical algorithms require $\Omega(n^m/(1+\varepsilon n))$ queries and quantum algorithms require $\Omega(\sqrt{n^m/(1+\varepsilon n)})$ queries.

quant-ph↗

arXiv2026

Regularized Warm-Started Quantum Approximate Optimization and Conditions for Surpassing Classical Solvers on the Max-Cut Problem

Demonstrating quantum heuristics that outperform strong classical solvers on large-scale optimization remains an open challenge. Here we introduce Regularized Warm-Started QAOA (RWS-QAOA), which initializes qubits by minimizing expected energy with a regularizer that penalizes near-bitstring states, preventing QAOA from stalling. We further propose a protocol that yields fixed, instance-independent parameters, enabling RWS-QAOA to operate as a non-variational algorithm in which the quantum circuit parameters are fixed and only a classical warm starting step is instance-dependent. We evaluate RWS-QAOA on the Max-Cut problem for random regular graphs, where this protocol yields a constant-depth quantum circuit, across three complementary settings. First, on Quantinuum's trapped-ion processor, RWS-QAOA outperforms the classical algorithms with the best provable guarantees for Max-Cut on $3$-regular graphs, namely Goemans-Williamson and Halperin-Livnat-Zwick, on $96$-node instances. Second, tensor-network simulations on graphs with up to $N{=}10{,}000$ nodes show that depth-$6$ RWS-QAOA, achieving an average cut fraction of $0.9167$, surpasses the best classical heuristics under matched restrictions (no local-search post-processing and no iterative refinement). Third, we remove these restrictions and benchmark against the strongest unrestricted classical heuristics, including an optimized parallel Burer-Monteiro solver that improves upon the MQLib implementation. Even against this stronger baseline, we project that surface-code RWS-QAOA reaches a quantum-classical runtime crossover below $0.2$ seconds on $3{,}000$-node graphs with fewer than $1.3$ million physical qubits. Our results show that constant-depth quantum circuits combined with a classical warm start have a credible potential to surpass classical solvers on the Max-Cut problem when executed on future quantum computers.

quant-ph↗

arXiv2026

Quantum Speedups for Group Relaxations of Integer Linear Programs

Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to obtain because classical algorithms for many-constraint ILPs are global and exhaustive, whereas quantum frameworks that offer super-quadratic speedup exploit local structure of the objective and feasible set. We address this via quantum algorithms for Gomory's group relaxation. The group relaxation of an ILP is obtained by dropping nonnegativity on variables that are positive in the optimal solution of the linear programming (LP) relaxation, while retaining integrality of the decision variables. We present a competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup. When the group relaxation satisfies a nondegeneracy condition analogous to, but stronger than, LP non-degeneracy, our approach yields the optimal solution to the original ILP. Otherwise, the group relaxation tightens bounds on the optimal objective value of the ILP, and can improve downstream branch-and-cut by reducing the integrality gap; we numerically observe this on several practically relevant ILPs. To achieve these results, we derive efficiently constructible constraint-preserving mixers for the group relaxation with favorable spectral properties, which are of independent interest.

quant-ph↗

arXiv2025

On Speedups for Convex Optimization via Quantum Dynamics

We explore the potential for quantum speedups in convex optimization using discrete simulations of the Quantum Hamiltonian Descent (QHD) framework, as proposed by Leng et al., and establish the first rigorous query complexity bounds. We develop enhanced analyses for quantum simulation of Schr\"odinger operators with black-box potential via the pseudo-spectral method, providing explicit resource estimates independent of wavefunction assumptions. These bounds are applied to assess the complexity of optimization through QHD. Our findings pertain to unconstrained convex optimization in $d$ dimensions. In continuous time, we demonstrate that QHD, with suitable parameters, can achieve arbitrarily fast convergence rates. The optimization speed limit arises solely from the discretization of the dynamics, mirroring a property of the classical dynamics underlying QHD. Considering this cost, we show that a $G$-Lipschitz convex function can be optimized to an error of $\epsilon$ with $\widetilde{\mathcal{O}}(d^{1.5}G^2 R^2/\epsilon^2)$ queries. Moreover, under reasonable assumptions on the complexity of Hamiltonian simulation, $\widetilde{\Omega}(d/\epsilon^2)$ queries are necessary. Thus, QHD does not offer a speedup over classical zeroth order methods with exact oracles. However, we demonstrate that the QHD algorithm tolerates $\widetilde{\mathcal{O}}(\epsilon^3/d^{1.5}G^2 R^2)$ noise in function evaluation. We show that QHD offers a super-quadratic query advantage over all known classical algorithms tolerating this level of evaluation noise in the high-dimension regime. Additionally, we design a quantum algorithm for stochastic convex optimization that provides a super-quadratic speedup over all known classical algorithms in the high-dimension regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization achieved through a dynamical algorithm.

quant-ph↗

arXiv2025

Fast Convex Optimization with Quantum Gradient Methods

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.

quant-ph↗

arXiv2024

Generalized Short Path Algorithms: Towards Super-Quadratic Speedup over Markov Chain Search for Combinatorial Optimization

We analyze generalizations of quantum algorithms based on the short path framework first proposed by Hastings~[\textit{Quantum} 2, 78 (2018)], which has been extended and shown by Dalzell~et~al.~[STOC~'23] to achieve super-Grover speedups for certain binary optimization problems. We demonstrate that, under some commonly satisfied technical conditions, an appropriate generalization can achieve super-quadratic speedups not only over unstructured search but also over a classical optimization algorithm that searches for the optimum by drawing samples from the stationary distribution of a Markov chain. We employ this framework to obtain algorithms for problems including variants of Max Bisection, Max Independent Set, and finding the ground states of the Antiferromagnetic Ising Model and the Sherrington-Kirkpatrick Model, whose runtimes are asymptotically faster than those obtainable with previous short path techniques. In certain cases, our algorithms achieve super-quadratic speedups compared to the best known classical algorithms with rigorously established runtimes.

quant-ph↗

arXiv2024

Strategies for running the QAOA at hundreds of qubits

We explore strategies aimed at reducing the amount of computation, both quantum and classical, required to run the Quantum Approximate Optimization Algorithm (QAOA). First, following Wurtz et al. [Phys.Rev A 104:052419], we consider the standard QAOA with instance-independent "tree" parameters chosen in advance. These tree parameters are chosen to optimize the MaxCut expectation for large girth graphs. We provide extensive numerical evidence supporting the performance guarantee for tree parameters conjectured in [Phys.Rev A 103:042612] and see that the approximation ratios obtained with tree parameters are typically well beyond the conjectured lower bounds, often comparable to performing a full optimization. This suggests that in practice, the QAOA can achieve near-optimal performance without the need for parameter optimization. Next, we modify the warm-start QAOA of Tate et al. [Quantum 7:1121]. The starting state for the QAOA is now an optimized product state associated with a solution of the Goemans-Williamson (GW) algorithm. Surprisingly, the tree parameters continue to perform well for the warm-start QAOA. We find that for random 3-regular graphs with hundreds of vertices, the expected cut obtained by the warm-start QAOA at depth $p \gtrsim 3$ is comparable to that of the standard GW algorithm. Our numerics on random instances do not provide general performance guarantees but do provide substantial evidence that there exists a regime of instance sizes in which the QAOA finds good solutions at low depth without the need for parameter optimization. For each instance studied, we classically compute the expected size of the QAOA distribution of cuts; producing the actual cuts requires running on a quantum computer.

quant-ph↗

arXiv2023

Quantum Computing Inspired Iterative Refinement for Semidefinite Optimization

Iterative Refinement (IR) is a classical computing technique for obtaining highly precise solutions to linear systems of equations, as well as linear optimization problems. In this paper, motivated by the limited precision of quantum solvers, we develop the first IR scheme for solving semidefinite optimization (SDO) problems and explore two major impacts of the proposed IR scheme. First, we prove that the proposed IR scheme exhibits quadratic convergence toward an optimal solution without any assumption on problem characteristics. We also show that using IR with Quantum Interior Point Methods (QIPMs) leads to exponential improvements in the worst-case overall running time of QIPMs, compared to previous best-performing QIPMs. We also discuss how the proposed IR scheme can be used with classical inexact SDO solvers, such as classical inexact IPMs with conjugate gradient methods.

math.OC↗

arXiv2023

Challenges and Opportunities in Quantum Optimization

Recent advances in quantum computers are demonstrating the ability to solve problems at a scale beyond brute force classical simulation. As such, a widespread interest in quantum algorithms has developed in many areas, with optimization being one of the most pronounced domains. Across computer science and physics, there are a number of different approaches for major classes of optimization problems, such as combinatorial optimization, convex optimization, non-convex optimization, and stochastic extensions. This work draws on multiple approaches to study quantum optimization. Provably exact versus heuristic settings are first explained using computational complexity theory - highlighting where quantum advantage is possible in each context. Then, the core building blocks for quantum optimization algorithms are outlined to subsequently define prominent problem classes and identify key open questions that, if answered, will advance the field. The effects of scaling relevant problems on noisy quantum devices are also outlined in detail, alongside meaningful benchmarking problems. We underscore the importance of benchmarking by proposing clear metrics to conduct appropriate comparisons with classical optimization techniques. Lastly, we highlight two domains - finance and sustainability - as rich sources of optimization problems that could be used to benchmark, and eventually validate, the potential real-world impact of quantum optimization.

quant-ph↗

arXiv2023

A quantum central path algorithm for linear optimization

We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a single simulation working directly with the nonlinear complementarity equations. This approach yields an algorithm for solving linear optimization problems involving $m$ constraints and $n$ variables to $\varepsilon$-optimality using $\mathcal{O} \left( \sqrt{m + n} \frac{R_{1}}{\varepsilon}\right)$ queries to an oracle that evaluates a potential function, where $R_{1}$ is an $\ell_{1}$-norm upper bound on the size of the optimal solution. In the standard gate model (i.e., without access to quantum RAM) our algorithm can obtain highly-precise solutions to LO problems using at most $$\mathcal{O} \left( \sqrt{m + n} \textsf{nnz} (A) \frac{R_1}{\varepsilon}\right)$$ elementary gates, where $\textsf{nnz} (A)$ is the total number of non-zero elements found in the constraint matrix.

quant-ph↗

arXiv2023

Improvements to Quantum Interior Point Method for Linear Optimization

Quantum linear system algorithms (QLSA) have the potential to speed up Interior Point Methods (IPM). However, a major challenge is that QLSAs are inexact and sensitive to the condition number of the coefficient matrices of linear systems. This sensitivity is exacerbated when the Newton systems arising in IPMs converge to a singular matrix. Recently, an Inexact Feasible Quantum IPM (IF-QIPM) has been developed that addresses the inexactness of QLSAs and, in part, the influence of the condition number using iterative refinement. However, this method requires a large number of gates and qubits to be implemented. Here, we propose a new IF-QIPM using the normal equation system, which is more adaptable to near-term quantum devices. To mitigate the sensitivity to the condition number, we use preconditioning coupled with iterative refinement to obtain better gate complexity. Finally, we demonstrate the effectiveness of our approach on IBM Qiskit simulators

math.OC↗

arXiv2023

Generating Linear, Semidefinite, and Second-order Cone Optimization Problems for Numerical Experiments

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems.

math.OC↗

arXiv2023

An Inexact Feasible Quantum Interior Point Method for Linearly Constrained Quadratic Optimization

Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems. Interior Point Methods (IPMs) yield a fundamental family of polynomial-time algorithms for solving optimization problems. IPMs solve a Newton linear system at each iteration to find the search direction, and thus QLSAs can potentially speed up IPMs. Due to the noise in contemporary quantum computers, such quantum-assisted IPM (QIPM) only allows an inexact solution for the Newton linear system. Typically, an inexact search direction leads to an infeasible solution. In our work, we propose an Inexact-Feasible QIPM (IF-QIPM) and show its advantage in solving linearly constrained quadratic optimization problems. We also apply the algorithm to $\ell_1$-norm soft margin support vector machine (SVM) problems and obtain the best complexity regarding dependence on dimension. This complexity bound is better than any existing classical or quantum algorithm that produces a classical solution.

math.OC↗

arXiv2023

Solving the semidefinite relaxation of QUBOs in matrix multiplication time, and faster with a quantum computer

Recent works on quantum algorithms for solving semidefinite optimization (SDO) problems have leveraged a quantum-mechanical interpretation of positive semidefinite matrices to develop methods that obtain quantum speedups with respect to the dimension $n$ and number of constraints $m$. While their dependence on other parameters suggests no overall speedup over classical methodologies, some quantum SDO solvers provide speedups in the low-precision regime. We exploit this fact to our advantage, and present an iterative refinement scheme for the Hamiltonian Updates algorithm of Brand\~ao et al. (Quantum 6, 625 (2022)) to exponentially improve the dependence of their algorithm on precision. As a result, we obtain a classical algorithm to solve the semidefinite relaxation of Quadratic Unconstrained Binary Optimization problems (QUBOs) in matrix multiplication time. Provided access to a quantum read/classical write random access memory (QRAM), a quantum implementation of our algorithm exhibits a worst case running time of $\mathcal{O} \left(ns + n^{1.5} \cdot \text{polylog} \left(n, \| C \|_F, \frac{1}{\epsilon} \right) \right)$.

quant-ph↗

arXiv2021

Quantum Interior Point Methods for Semidefinite Optimization

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem $n$, but has worse dependence on other numerical parameters.

quant-ph↗