We provide a generic toolkit for sparsifying the constraint set of linear programs (LPs). To this end, we reduce solving a linear program with n constraints and d variables (n≫ d), to solving a sequence of LPs defined over only a small subset of the constraints, obtained by adaptively sub-sampling the original set. We provide results for both the low and high precision regimes. To achieve the former result, we streamline and generalize the techniques from [Assadi '25] for approximately computing maximum matchings in the semi-streaming setting to the case of general LPs. For the latter, we robustify the methods of [Clarkson '95], which were originally designed for exact LP solvers. As a consequence we obtain fast approximate LP solvers which reduce the dependence on width and error from quadratic to linear, compared to vanilla multiplicative-weights based approaches. Additionally, we leverage our findings to obtain fast LP solvers in the quantum query access model, where the running time scales with √n. This completely decouples the component responsible for quantum speed-ups, solely represented by a generalization of Grover’s search, from its classical algorithmic counterpart.
We provide a generic toolkit for sparsifying the constraint set of linear programs (LPs). To this end, we reduce solving a linear program with n constraints and d variables (n≫ d), to solving a sequence of LPs defined over only a small subset of the constraints, obtained by adaptively sub-sampling the original set. We provide results for both the low and high precision regimes. To achieve the former result, we streamline and generalize the techniques from [Assadi '25] for approximately computing maximum matchings in the semi-streaming setting to the case of general LPs. For the latter, we robustify the methods of [Clarkson '95], which were originally designed for exact LP solvers. As a consequence we obtain fast approximate LP solvers which reduce the dependence on width and error from quadratic to linear, compared to vanilla multiplicative-weights based approaches. Additionally, we leverage our findings to obtain fast LP solvers in the quantum query access model, where the running time scales with √n. This completely decouples the component responsible for quantum speed-ups, solely represented by a generalization of Grover’s search, from its classical algorithmic counterpart.
Abstract. We describe a quantum algorithm based on an interior point method for solving a linear program with [Formula: see text] inequality constraints on [Formula: see text] variables. The algorithm explicitly returns a feasible solution that is [Formula: see text]-close to optimal and runs in time [Formula: see text], which is sublinear for tall linear programs (i.e., [Formula: see text]). Our algorithm speeds up the Newton step in the state-of-the-art interior point method of Lee and Sidford [ Solving Linear Programs with Sqrt(rank) Linear System Solves, 2019]. This requires us to efficiently approximate the Hessian and gradient of the barrier function, and these are our main contributions. To approximate the Hessian, we describe a quantum algorithm for the spectral approximation of [Formula: see text] for a tall matrix [Formula: see text]. The algorithm uses leverage score sampling in combination with Grover search and returns a [Formula: see text]-approximation by making [Formula: see text] row queries to [Formula: see text]. This generalizes an earlier quantum speedup for graph sparsification by Apers and de Wolf [ SIAM J. Comput., 51 (2022), pp. 1703–1742]. To approximate the gradient, we use a recent quantum algorithm for multivariate mean estimation by Cornelissen, Hamoudi, and Jerbi [ Near-optimal quantum algorithms for multivariate mean estimation, 2022]. While a naive implementation introduces a dependence on the condition number of the Hessian, we avoid this by preconditioning our random variable using our quantum algorithm for spectral approximation.
Linear programming (LP) decoding approximates optimal maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of linear inequalities derived from the constraints represented by the rows of a parity-check matrix of the code. Adaptive linear programming (ALP) decoding significantly reduces the complexity of LP decoding by iteratively and adaptively adding necessary constraints in a sequence of smaller LP problems. Adaptive introduction of constraints derived from certain additional redundant parity check (RPC) constraints can further improve ALP performance. In this paper, we propose a new and effective algorithm to identify RPCs that produce linear constraints, referred to as “cuts,” that can eliminate non-ML solutions generated by the ALP decoder, often significantly improving the decoder error-rate performance. The cut-finding algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. Simulation results for several low-density parity-check codes demonstrate that the modified ALP decoding algorithm significantly narrows the performance gap between LP decoding and ML decoding.
While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. By modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, we first prove that LP decoding can be performed by solving a number of LP problems that each contains at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate solving such systems iteratively. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.
We give the first sublinear-time classical solvers for sparse semidefinite programs in the bounded-radius regime, without low-rank assumptions or Frobenius norm dependence on the constraint matrices. For constant precision and bounded primal and dual radii, prior quantum algorithms of Brandão et al. (2019) and van Apeldoorn and Gilyén (2019) achieved O(n+m) dependence on matrix dimension n and constraint number m. Compared with the O(mn) runtime of existing classical methods, this suggests a quartic quantum speedup when m≈n. Beyond a usual Grover speedup, this separation relies on the Quantum OR lemma, whose sample-reuse mechanism decouples the cost of Gibbs-state preparation from constraint search. We show that this reuse mechanism is classically realizable for sparse SDPs.
Our main technical contribution is a classical procedure for simultaneously estimating many expectation values with respect to a sparse Hamiltonian's Gibbs state. This combines randomized Lánczos filtering with an efficient sampling-based estimator. We also introduce a stochastic online-learning framework for SDP solving, substantially improving accuracy-dependence over standard oracle-based MMWU approaches. Let s denote the the input matrix sparsity and γ:=Rr/ε capture dependence on the primal (R) and dual (r) radii as well as target accuracy (ε). When γ2≤min{m,n/s}, our solver runs in time O(nsγ4.5+msγ2). For γ=O(1), this is O((n+m)s) and sublinear in the O(mns) input size. Similar to the quantum algorithms, this matches known lower bounds with respect to m and n, up to logarithmic factors. This implies that, with respect to dimensions m and n, there is no super-quadratic quantum advantage for generic sparse SDP solving.
While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem, and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. We first prove, by modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, that LP decoding can be performed by solving a number of LP problems that contain at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate iterative solution of such systems. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.