We present a quantum interior-point method (IPM) for second-order cone programming (SOCP) that runs in time O~(nrζκδ2log(1/ϵ)) where r is the rank and n the dimension of the SOCP, δ bounds the distance of intermediate solutions from the cone boundary, ζ is a parameter upper bounded by n , and κ is an upper bound on the condition number of matrices arising in the classical IPM for SOCP. The algorithm takes as its input a suitable quantum description of an arbitrary SOCP and outputs a classical description of a δ -approximate ϵ -optimal solution of the given problem.Furthermore, we perform numerical simulations to determine the values of the aforementioned parameters when solving the SOCP up to a fixed precision ϵ . We present experimental evidence that in this case our quantum algorithm exhibits a polynomial speedup over the best classical algorithms for solving general SOCPs that run in time O(nω+0.5) (here, ω is the matrix multiplication exponent, with a value of roughly 2.37 in theory, and up to 3 in practice). For the case of random SVM (support vector machine) instances of size O(n) , the quantum algorithm scales as O(nk) , where the exponent k is estimated to be 2.59 using a least-squares power law. On the same family random instances, the estimated scaling exponent for an external SOCP solver is 3.31 while that for a state-of-the-art SVM solver is 3.11 .
We present a quantum interior-point method (IPM) for second-order cone programming (SOCP) that runs in time O~(nrζκδ2log(1/ϵ)) where r is the rank and n the dimension of the SOCP, δ bounds the distance of intermediate solutions from the cone boundary, ζ is a parameter upper bounded by n , and κ is an upper bound on the condition number of matrices arising in the classical IPM for SOCP. The algorithm takes as its input a suitable quantum description of an arbitrary SOCP and outputs a classical description of a δ -approximate ϵ -optimal solution of the given problem.Furthermore, we perform numerical simulations to determine the values of the aforementioned parameters when solving the SOCP up to a fixed precision ϵ . We present experimental evidence that in this case our quantum algorithm exhibits a polynomial speedup over the best classical algorithms for solving general SOCPs that run in time O(nω+0.5) (here, ω is the matrix multiplication exponent, with a value of roughly 2.37 in theory, and up to 3 in practice). For the case of random SVM (support vector machine) instances of size O(n) , the quantum algorithm scales as O(nk) , where the exponent k is estimated to be 2.59 using a least-squares power law. On the same family random instances, the estimated scaling exponent for an external SOCP solver is 3.31 while that for a state-of-the-art SVM solver is 3.11 .
We present a quantum interior point method (IPM) for semi-definite programs that has a worst-case running time of Õ( n 2.5 / ξ 2 μ κ 3 log(1/ϵ)). The algorithm outputs a pair of matrices ( S,Y ) that have objective value within ϵ of the optimal and satisfy the constraints approximately to error xi. The parameter mu is at most √2 n while kappa is an upper bound on the condition number of the intermediate solution matrices arising in the classical IPM. For the case where κ ≪ n 5/6 , our method provides a significant polynomial speedup over the best-known classical semi-definite program solvers that have a worst-case running time of Õ( n 6 ). For linear programs, our algorithm has a running time of Õ( n 1.5 / ξ 2 μ κ 3 log (1/ϵ)) with the same guarantees and with parameter μ < √2 n . Our technical contributions include an efficient quantum procedure for solving the Newton linear systems arising in the classical IPMs, an efficient pure state tomography algorithm, and an analysis of the IPM where the linear systems are solved approximately. Our results pave the way for the development of quantum algorithms with significant polynomial speedups for applications in optimization and machine learning.
We give classical and quantum algorithms for approximately solving second-order cone programs (SOCPs) based on the multiplicative weights (MW) update method. Our approach follows the MW framework previously applied to semidefinite programs (SDPs), of which SOCP is a special case. We show that the additional structure of SOCPs can be exploited to give better runtime with SOCP-specific algorithms. For an SOCP with m linear constraints over n variables partitioned into r≤n second-order cones, our quantum algorithm requires O(rγ5+mγ4) (coherent) queries to the underlying data defining the instance, where γ is a scale-invariant parameter proportional to the inverse precision. This nearly matches the complexity of solving linear programs (LPs), which are a less expressive subset of SOCP. It also outperforms (especially if n≫r) the naive approach that applies existing SDP algorithms onto SOCPs, which has complexity O(γ4(n+γn+m)). Our classical algorithm for SOCP has complexity O(nγ4+mγ6) in the sample-and-query model.
We study quantum interior point methods (QIPMs) for second-order cone programming (SOCP), guided by the example use case of portfolio optimization (PO). We provide a complete quantum circuit-level description of the algorithm from problem input to problem output, making several improvements to the implementation of the QIPM. We report the number of logical qubits and the quantity/depth of non-Clifford T-gates needed to run the algorithm, including constant factors. The resource counts we find depend on instance-specific parameters, such as the condition number of certain linear systems within the problem. To determine the size of these parameters, we perform numerical simulations of small PO instances, which lead to concrete resource estimates for the PO use case. Our numerical results do not probe large enough instance sizes to make conclusive statements about the asymptotic scaling of the algorithm. However, already at small instance sizes, our analysis suggests that, due primarily to large constant pre-factors, poorly conditioned linear systems, and a fundamental reliance on costly quantum state tomography, fundamental improvements to the QIPM are required for it to lead to practical quantum advantage.
The emergence of huge-scale, data-intensive linear optimization (LO) problems in applications such as machine learning has driven the need for more computationally efficient interior point methods (IPMs). While conventional IPMs are polynomial-time algorithms with rapid convergence, their per-iteration cost can be prohibitively high for dense large-scale LO problems. Quantum linear system solvers have shown potential in accelerating the solution of linear systems arising in IPMs. In this work, we introduce a novel almost-exact quantum IPM, where the Newton system is constructed and solved on a quantum computer, while solution updates occur on a classical machine. Additionally, all matrix-vector products are performed on the quantum hardware. This hybrid quantum-classical framework achieves an optimal worst-case scaling of O(n2) for fully dense LO problems. To ensure high precision, despite the limited accuracy of quantum operations, we incorporate iterative refinement techniques both within and outside the proposed IPM iterations. The proposed algorithm has a quantum complexity of O(n1.5κAlog(ε1)) queries to QRAM and O(n2log(ε1)) classical arithmetic operations. Our method outperforms the worst-case complexity of prior classical and quantum IPMs, offering a significant improvement in scalability and computational efficiency.
The growing demand for solving large-scale, data-intensive linear and conic optimization problems, particularly in applications such as artificial intelligence and machine learning, has highlighted the limitations of classical interior point methods (IPMs). Despite their favorable polynomial-time convergence, conventional IPMs often suffer from high per-iteration computational costs, especially for dense problem instances. Recent advances in quantum computing, particularly quantum linear system solvers, offer promising avenues to accelerate the most computationally intensive steps of IPMs. However, practical challenges such as quantum error, hardware noise, and sensitivity to poorly conditioned systems remain significant obstacles. In response, a series of Quantum IPMs (QIPMs) has been developed to address these challenges, incorporating techniques such as feasibility maintenance, iterative refinement, and preconditioning. In this work, we review this line of research with a focus on our recent contributions, including an almost-exact QIPM framework. This hybrid quantum-classical approach constructs and solves the Newton system entirely on a quantum computer, while performing solution updates classically. Crucially, all matrix-vector operations are executed on quantum hardware, enabling the method to achieve an optimal worst-case scalability w.r.t dimension, surpassing the scalability of existing classical and quantum IPMs.
This paper studies a fundamental problem in convex optimization, which is to solve semidefinite programming (SDP) with high accuracy. This paper follows from the existing robust SDP-based interior point method analysis due to [Huang, Jiang, Song, Tao and Zhang, FOCS 2022]. While, the previous work only provides an efficient implementation in the classical setting. This work provides a novel quantum implementation. We give a quantum second-order algorithm with high-accuracy in both the optimality and the feasibility of its output, and its running time depending on log(1/ε) on well-conditioned instances. Due to the limitation of quantum itself or first-order method, all the existing quantum SDP solvers either have polynomial error dependence or low-accuracy in the feasibility.