We provide algorithms that compute $\epsilon$-estimates of the $\ell_p$-Lewis weights of a matrix $A \in \mathbb{R}^{m \times n}$ for $p \geq 4$ using $O(p^2 \log(m/\epsilon))$ rounds of leverage score computation, where $\ell_p$-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of $O(p^3 \log(m/\epsilon))$ due to Fazel, Lee, Padmanabha, and Sidford (2022). We obtain our results by carefully applying a local variant of relatively smooth gradient descent to primal and dual forms of the $\ell_p$-Lewis weight optimization problem and providing tools to convert between different notions of approximate $\ell_p$-Lewis weights. This work subsumes the note “On computing approximate Lewis weights” by Apers, Gribling, and Sidford.

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.

Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the query complexity of volume estimation to a membership oracle has been a longstanding open question. Most of the previous work focuses on the high-dimensional limit. In this work, we tightly characterize the deterministic, randomized and quantum query complexity of this problem in the high-precision limit, i.e., when the dimension is constant.

Hamiltonian Monte Carlo (HMC) is a Markov chain algorithm for sampling from a high-dimensional distribution with density $e^{-f(x)}$, given access to the gradient of $f$. A particular case of interest is that of a $d$-dimensional Gaussian distribution with covariance matrix $\Sigma$, in which case $f(x) = x^\top \Sigma^{-1} x$. We show that Metropolis-adjusted HMC can sample from a distribution that is $\varepsilon$-close to a Gaussian in total variation distance using $\widetilde{O}(\sqrt{\kappa} d^{1/4} \log(1/\varepsilon))$ gradient queries, where $\varepsilon>0$ and $\kappa$ is the condition number of $\Sigma$. Our algorithm uses long and random integration times for the Hamiltonian dynamics, and it creates a warm start by first running HMC without a Metropolis adjustment. This contrasts with (and was motivated by) recent results that give an $\widetilde\Omega(\kappa d^{1/2})$ query lower bound for HMC with a fixed integration times or from a cold start, even for the Gaussian case.

Solving systems of linear equations is one of the most important primitives in many different areas, including in optimization, simulation, and machine learning. Quantum algorithms for solving linear systems have the potential to provide a quantum advantage for these problems. In this work, we recall the Chebyshev iterative method and the corresponding optimal polynomial approximation of the inverse. We show that the Chebyshev iteration polynomial can be efficiently evaluated both using quantum singular value transformation (QSVT) as well as linear combination of unitaries (LCU). We achieve this by bounding the 1-norm of the coefficients of the polynomial expressed in the Chebyshev basis. This leads to a considerable constant-factor improvement in the runtime of quantum linear system solvers that are based on LCU or QSVT (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth).

The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O’Donnell [16] and later Raghavendra & Weitz [20] show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz [20]. In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible.

The Pythagoras number of a sum of squares is the shortest length among its sums of squares representations. In many algebras, for example real polynomial algebras in two or more variables, there exists no upper bound on the Pythagoras number for all sums of squares. In this paper, we study how Pythagoras numbers in ⁎-algebras over C behave with respect to small perturbations of elements. More precisely, the approximate Pythagoras number of an element is the smallest Pythagoras number among all elements in its ε -ball. We show that these approximate Pythagoras numbers are often significantly smaller than their exact versions, and allow for (almost) dimension-independent upper bounds. Our results use low-rank approximations for Gram matrices of sums of squares and estimates for the operator norm of the Gram map.

Matrix scaling is a simple to state, yet widely applicable linear-algebraic problem: the goal is to scale the rows and columns of a given non-negative matrix such that the rescaled matrix has prescribed row and column sums. Motivated by recent results on first-order quantum algorithms for matrix scaling, we investigate the possibilities for quantum speedups for classical second-order algorithms, which comprise the state-of-the-art in the classical setting. We first show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime: any quantum algorithm that solves the matrix scaling problem for $n \times n$ matrices with at most $m$ non-zero entries and with $\ell_2$-error $\varepsilon=\widetildeΘ(1/m)$ must make $\widetildeΩ(m)$ queries to the matrix, even when the success probability is exponentially small in $n$. Additionally, we show that for $\varepsilon\in[1/n,1/2]$, any quantum algorithm capable of producing $\frac{\varepsilon}{100}$-$\ell_1$-approximations of the row-sum vector of a (dense) normalized matrix uses $Ω(n/\varepsilon)$ queries, and that there exists a constant $\varepsilon_0>0$ for which this problem takes $Ω(n^{1.5})$ queries. To complement these results we give improved quantum algorithms in the low-precision regime: with quantum graph sparsification and amplitude estimation, a box-constrained Newton method can be sped up in the large-$\varepsilon$ regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an $\varepsilon$-$\ell_1$-scaling in time $\widetilde O(n^{1.5}/\varepsilon^2)$, whereas the best previously known bounds were $\widetilde O(n^2\mathrm{polylog}(1/\varepsilon))$ (classical) and $\widetilde O(n^{1.5}/\varepsilon^3)$ (quantum).

We extend upper bounds on the quantum independence number and the quantum Shannon capacity of graphs to their counterparts in the commuting operator model. We introduce a von Neumann algebraic generalization of the fractional Haemers bound (over$\mathbb {C}$) and prove that the generalization upper bounds the commuting quantum independence number. We call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product. In particular, this makes it an upper bound on the Shannon capacity. The tracial Haemers bound is incomparable with the Lovász theta function, another well-known upper bound on the Shannon capacity. We show that separating the tracial and fractional Haemers bounds would refute Connes’ embedding conjecture. Along the way, we prove that the tracial rank and tracial Haemers bound are elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam, Combinatorica, 2019). We also show that the inertia bound (an upper bound on the quantum independence number) upper bounds the commuting quantum independence number.

Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn’s algorithm for matrix scaling and Osborne’s algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√{mn}/ε⁴) for scaling or balancing an n × n matrix (given by an oracle) with m non-zero entries to within 𝓁₁-error ε. Their classical analogs use time Õ(m/ε²), and every classical algorithm for scaling or balancing with small constant ε requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of n, at the expense of a worse polynomial dependence on the obtained 𝓁₁-error ε. Even for constant ε these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn’s and Osborne’s algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn’s algorithm for entrywise-positive matrices to the 𝓁₁-setting, obtaining an Õ(n^{1.5}/ε³)-time quantum algorithm for ε-𝓁₁-scaling. We also prove a lower bound, showing our quantum algorithm for matrix scaling is essentially optimal for constant ε: every quantum algorithm for matrix scaling that achieves a constant 𝓁₁-error w.r.t. uniform marginals needs Ω(√{mn}) queries.

Brandao and Svore recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m is approximately n, which is the same as classical.