A Quantum Interior Point Method for LPs and SDPs

We present a quantum interior point method with worst case running time $\widetilde{O}(\frac{n^{2.5}}{\xi^{2}} \mu \kappa^3 \log (1/\epsilon))$ for SDPs and $\widetilde{O}(\frac{n^{1.5}}{\xi^{2}} \mu \kappa^3 \log (1/\epsilon))$ for LPs, where the output of our algorithm is a pair of matrices $(S,Y)$ that are $\epsilon$-optimal $\xi$-approximate SDP solutions. The factor $\mu$ is at most $\sqrt{2}n$ for SDPs and $\sqrt{2n}$ for LP's, and $\kappa$ is an upper bound on the condition number of the intermediate solution matrices. For the case where the intermediate matrices for the interior point method are well conditioned, our method provides a polynomial speedup over the best known classical SDP solvers and interior point based LP solvers, which have a worst case running time of $O(n^{6})$ and $O(n^{3.5})$ respectively. Our results build upon recently developed techniques for quantum linear algebra and pave the way for the development of quantum algorithms for a variety of applications in optimization and machine learning.Comment: 32 page

Quantum Recommendation Systems

A recommendation system uses the past purchases or ratings of $n$ products by a group of $m$ users, in order to provide personalized recommendations to individual users. The information is modeled as an $m \times n$ preference matrix which is assumed to have a good rank-$k$ approximation, for a small constant $k$. In this work, we present a quantum algorithm for recommendation systems that has running time $O(\text{poly}(k)\text{polylog}(mn))$. All known classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for a real world application.Comment: 22 page

On the sum-of-squares degree of symmetric quadratic functions

We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $\ell_{\infty}$-norm as well as in $\ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on $\ell_1$-approximation of $f_k$; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.Comment: 33 pages. Second version fixes some typos and adds reference

Neutral Higgs boson pair production at the LC in the Noncommutative Standard Model

We study the Higgs boson pair production through $e^+e^-$ collision in the noncommutative(NC) extension of the standard model using the Seiberg-Witten maps of this to the first order of the noncommutative parameter $\Theta_{\mu \nu}$. This process is forbidden in the standard model with background space-time being commutative. We find that the cross section of the pair production of Higgs boson (of intermediate and heavy mass) at the future Linear Collider(LC) can be quite significant for the NC scale $\Lambda$ lying in the range $0.5 - 1.0$ TeV. Finally, using the direct experimental(LEP II, Tevatron and global electro-weak fit) bound on Higgs mass, we obtain bounds on the NC scale as 665 GeV $\le \Lambda \le 998$ GeV.Comment: 14 pages, 8 figure