Quantum Approximate Optimization algorithm (QAOA) is one of the candidates to achieve a near-term quantum advantage. To search for such a quantum advantage in solving any problem, it is crucial to first understand the difference between problem instances' empirical hardness for QAOA and classical algorithms. We identify a computational phase transition of QAOA when solving hard problems such as 3-SAT -- the performance is worst at the well-known SAT-UNSAT phase transition, where the hardest instances lie. We connect the transition to the controllability and the complexity of QAOA circuits. Such a transition is absent for 2-SAT and QAOA achieves close to perfect performance at the problem size we studied. Then, we show that the high problem density region, which limits QAOA's performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA can be identified. The computational phase transition generalizes to other Hamiltonian-based algorithms, such as the quantum adiabatic algorithm.
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A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear representations, such as the Koopman representation and Koopman von Neumann mechanics, have regained attention from the dynamical-systems research community. Here, we aim to present a unified theoretical framework, currently missing in the literature, with which one can compare and relate existing methods, their conceptual basis, and their representations. We also aim to show that, despite the fact that quantum simulation of nonlinear classical systems may be possible with such linear representations, a necessary projection into a feasible finite-dimensional space will in practice eventually induce numerical artifacts which can be hard to eliminate or even control. As a result a practical, reliable and accurate way to use quantum computation for solving general nonlinear dynamical systems is still an open problem.
The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the state of the art in the exact solution of both problems -- by using mathematical programming techniques on digital computers. The main focus lies on sparse problem instances, although also dense ones can be solved. We enhance several algorithmic components such as reduction techniques and cutting-plane separation algorithms, and combine them in an exact branch-and-cut solver. Furthermore, we provide a parallel implementation. The new solver is shown to significantly outperform existing state-of-the-art software for sparse MaxCut and QUBO instances. Furthermore, we improve the best known bounds for several instances from the 7th DIMACS Challenge and the QPLIB, and solve some of them (for the first time) to optimality.
Inertial Navigation Systems (INS) are a key technology for autonomous vehicles applications. Recent advances in estimation and filter design for the INS problem have exploited geometry and symmetry to overcome limitations of the classical Extended Kalman Filter (EKF) approach that formed the mainstay of INS systems since the mid-twentieth century. The industry standard INS filter, the Multiplicative Extended Kalman Filter (MEKF), uses a geometric construction for attitude estimation coupled with classical Euclidean construction for position, velocity and bias estimation. The recent Invariant Extended Kalman Filter (IEKF) provides a geometric framework for the full navigation states, integrating attitude, position and velocity, but still uses the classical Euclidean construction to model the bias states. In this paper, we use the recently proposed Equivariant Filter (EqF) framework to derive a novel observer for biased inertial-based navigation in a fully geometric framework. The introduction of virtual velocity inputs with associated virtual bias leads to a full equivariant symmetry on the augmented system. The resulting filter performance is evaluated with both simulated and real-world data, and demonstrates increased robustness to a wide range of erroneous initial conditions, and improved accuracy when compared with the industry standard Multiplicative EKF (MEKF) approach.
We show that Gottesman's semantics (GROUP22, 1998) for Clifford circuits based on the Heisenberg representation can be treated as a type system that can efficiently characterize a common subset of quantum programs. Our applications include (i) certifying whether auxiliary qubits can be safely disposed of, (ii) determining if a system is separable across a given bi-partition, (iii) checking the transversality of a gate with respect to a given stabilizer code, and (iv) typing post-measurement states for computational basis measurements. Further, this type system is extended to accommodate universal quantum computing by deriving types for the $T$-gate, multiply-controlled unitaries such as the Toffoli gate, and some gate injection circuits that use associated magic states. These types allow us to prove a lower bound on the number of $T$ gates necessary to perform a multiply-controlled $Z$ gate.
Quantum Annealing (QA) is a computational framework where a quantum system's continuous evolution is used to find the global minimum of an objective function over an unstructured search space. It can be seen as a general metaheuristic for optimization problems, including NP-hard ones if we allow an exponentially large running time. While QA is widely studied from a heuristic point of view, little is known about theoretical guarantees on the quality of the solutions obtained in polynomial time. In this paper we use a technique borrowed from theoretical physics, the Lieb-Robinson (LR) bound, and develop new tools proving that short, constant time quantum annealing guarantees constant factor approximations ratios for some optimization problems when restricted to bounded degree graphs. Informally, on bounded degree graphs the LR bound allows us to retrieve a (relaxed) locality argument, through which the approximation ratio can be deduced by studying subgraphs of bounded radius. We illustrate our tools on problems MaxCut and Maximum Independent Set for cubic graphs, providing explicit approximation ratios and the runtimes needed to obtain them. Our results are of similar flavor to the well-known ones obtained in the different but related QAOA (quantum optimization algorithms) framework. Eventually, we discuss theoretical and experimental arguments for further improvements.
We study expected runtimes for quantum programs. Inspired by recent work on probabilistic programs, we first define expected runtime as a generalisation of quantum weakest precondition. Then, we show that the expected runtime of a quantum program can be represented as the expectation of an observable (in physics). A method for computing the expected runtimes of quantum programs in finite-dimensional state spaces is developed. Several examples are provided as applications of this method, including computing the expected runtime of quantum Bernoulli Factory -- a quantum algorithm for generating random numbers. In particular, using our new method, an open problem of computing the expected runtime of quantum random walks introduced by Ambainis et al. (STOC 2001) is solved.
We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyperparameter estimation in Gauss--Markov regression, which tends to be considerably easier. The method's relation and benefits in comparison to classical numerical integration and gradient matching approaches is elucidated. In particular, the method can, in contrast to gradient matching, handle partial observations, and has certain routes for escaping local optima not available to classical numerical integration. Experimental results demonstrate that the method is on par or moderately better than competing approaches.
Determining the proper level of details to develop and solve physical models is usually difficult when one encounters new engineering problems. Such difficulty comes from how to balance the time (simulation cost) and accuracy for the physical model simulation afterwards. We propose a framework for automatic development of a family of surrogate models of physical systems that provide flexible cost-accuracy tradeoffs to assist making such determinations. We present both a model-based and a data-driven strategy to generate surrogate models. The former starts from a high-fidelity model generated from first principles and applies a bottom-up model order reduction (MOR) that preserves stability and convergence while providing a priori error bounds, although the resulting reduced-order model may lose its interpretability. The latter generates interpretable surrogate models by fitting artificial constitutive relations to a presupposed topological structure using experimental or simulation data. For the latter, we use Tonti diagrams to systematically produce differential equations from the assumed topological structure using algebraic topological semantics that are common to various lumped-parameter models (LPM). The parameter for the constitutive relations are estimated using standard system identification algorithms. Our framework is compatible with various spatial discretization schemes for distributed parameter models (DPM), and can supports solving engineering problems in different domains of physics.
We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). In Berry et al., (2017), a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices. The algorithm here can also be exponentially faster for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in Liu et al., (2021)). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by Xue et al., (2021), but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse, invertible matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas Liu et al., (2021) and Xue et al., (2021) additionally require normality.
We present an end-to-end framework for solving the Vehicle Routing Problem (VRP) using reinforcement learning. In this approach, we train a single model that finds near-optimal solutions for problem instances sampled from a given distribution, only by observing the reward signals and following feasibility rules. Our model represents a parameterized stochastic policy, and by applying a policy gradient algorithm to optimize its parameters, the trained model produces the solution as a sequence of consecutive actions in real time, without the need to re-train for every new problem instance. On capacitated VRP, our approach outperforms classical heuristics and Google's OR-Tools on medium-sized instances in solution quality with comparable computation time (after training). We demonstrate how our approach can handle problems with split delivery and explore the effect of such deliveries on the solution quality. Our proposed framework can be applied to other variants of the VRP such as the stochastic VRP, and has the potential to be applied more generally to combinatorial optimization problems.
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