Faster quantum and classical SDP approximations for quadratic binary optimization

We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. The class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into constant factor approximations of the original quadratic optimization problem

Faster quantum and classical SDP approximations for quadratic binary optimization

We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. The class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into constant factor approximations of the original quadratic optimization problem

Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints

Semidefinite programs are optimization methods with a wide array of applications, such as approximating difficult combinatorial problems. One such semidefinite program is the Goemans-Williamson algorithm, a popular integer relaxation technique. We introduce a variational quantum algorithm for the Goemans-Williamson algorithm that uses only $n{+}1$ qubits, a constant number of circuit preparations, and $\text{poly}(n)$ expectation values in order to approximately solve semidefinite programs with up to $N=2^n$ variables and $M \sim O(N)$ constraints. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit, a technique known as the Hadamard Test. The Hadamard Test enables us to optimize the objective function by estimating only a single expectation value of the ancilla qubit, rather than separately estimating exponentially many expectation values. Similarly, we illustrate that the semidefinite programming constraints can be effectively enforced by implementing a second Hadamard Test, as well as imposing a polynomial number of Pauli string amplitude constraints. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems, including MaxCut. Our method exceeds the performance of analogous classical methods on a diverse subset of well-studied MaxCut problems from the GSet library.Comment: 21 pages, 6 figures. Updated files to the version of manuscript accepted by Quantu

A quantum view on convex optimization

In this dissertation we consider quantum algorithms for convex optimization. We start by considering a black-box setting of convex optimization. In this setting we show that quantum computers require exponentially fewer queries to a membership oracle for a convex set in order to implement a separation oracle for that set. We do so by proving that Jordan's quantum gradient algorithm can also be applied to find sub-gradients of convex Lipschitz functions, even though these functions might not even be differentiable. As a corollary we get a quadraticly faster algorithm for convex optimization using membership queries. As a second set of results we give sub-linear time quantum algorithms for semidefinite optimization by speeding up the iterations of the Arora-Kale algorithm. For the problem of finding approximate Nash equilibria for zero-sum games we then give specific algorithms that improve the error-dependence and only depend on the sparsity of the game, not it's size. These last results yield improved algorithms for linear programming as a corollary. We also show several lower bounds in these settings, matching the upper bounds in most or all parameters

Quantum SDP-Solvers: Better upper and lower bounds

Brand\~ao and Svore very 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 $m\approx n$, which is the same as classical.Comment: v4: 69 pages, small corrections and clarifications. This version will appear in Quantu