Metriplectic Integrators for the Landau Collision Operator

We present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a Galerkin discretization for the velocity space, we transform the infinite-dimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the production of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the monotonic behavior of entropy are guaranteed for finite element discretizations in general, independently of the mesh configuration.Comment: 24 pages. Comments welcom

Quantum Algorithms for Matrix Problems and Machine Learning

This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.Science and Engineering Research Board (Department of Science and Technology), Government of Indi

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 Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving

Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, "spectral sparsification" reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. In this work we demonstrate a polynomial quantum speedup for spectral sparsification and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with $n$ nodes and $m$ edges, outputs a classical description of an $\epsilon$-spectral sparsifier in sublinear time $\tilde{O}(\sqrt{mn}/\epsilon)$. This contrasts with the optimal classical complexity $\tilde{O}(m)$. We also prove that our quantum algorithm is optimal up to polylog-factors. The algorithm builds on a string of existing results on sparsification, graph spanners, quantum algorithms for shortest paths, and efficient constructions for $k$-wise independent random strings. Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut.Comment: v2: several small improvements to the text. An extended abstract will appear in FOCS'20; v3: corrected a minor mistake in Appendix

Translationally Invariant Universal Quantum Hamiltonians in 1D

. Recent work has characterized rigorously what it means for one quantum system to simulate another and demonstrated the existence of universal Hamiltonians—simple spin lattice Hamiltonians that can replicate the entire physics of any other quantum many-body system. Previous universality results have required proofs involving complicated ‘chains’ of perturbative ‘gadgets.’ In this paper, we derive a significantly simpler and more powerful method of proving universality of Hamiltonians, directly leveraging the ability to encode quantum computation into ground states. This provides new insight into the origins of universal models and suggests a deep connection between universality and complexity. We apply this new approach to show that there are universal models even in translationally invariant spin chains in 1D. This gives as a corollary a new Hamiltonian complexity result that the local Hamiltonian problem for translationally invariant spin chains in one dimension with an exponentially small promise gap is PSPACE-complete. Finally, we use these new universal models to construct the first known toy model of 2D–1D holographic duality between local Hamiltonians

Classical simulations of quantum systems using stabilizer decompositions

One of the state of the art techniques for classically simulating quantum circuits relies on approximating the output state of the circuit by a superposition of stabilizer states. If the number of non-Clifford gates in the circuit is small, such simulations can be very effective. This thesis provides various improvements in this framework. First, we describe an improved method of computing approximate stabilizer decompositions, which reduces the time cost of computing a single term in the decomposition from $O(\ell n^2)$ to $O(m n^2)$, where $\ell$ is the total number of gates in the circuit, and $m$ is the number of non-Clifford gates. Since this subroutine has to be repeated exponentially many times, this improvement can be significant in practice whenever $\ell \gg m$. Our method uses a certain re-writing of the circuit, which in some cases allows for a significant amelioration of the exponential scaling of the required classical resources. Furthermore, we describe a method of constructing exact, low-rank stabilizer decompositions of $\ket{\psi}^{\otimes m}$, where $\ket{\psi}$ is either a magic state or an equatorial state. For any single qubit magic state $\ket{\psi}$, we find stabilizer decompositions of $\ket{\psi}^{\otimes m}$ with $2^{m\log_2(3)/4}$ terms. This improves on the best known bound of $2^{m \log_2(7)/6}$. Similarly, for any single qubit equatorial state $\ket{\psi}$, we give a stabilizer decomposition of $\ket{\psi}^{\otimes m}$ with $2^{m/2}$ terms. To our knowledge no such decompositions were previously known. These results translate to milder exponential scaling of the classical resources required for estimating probabilities of quantum circuits up to a polynomially small multiplicative error, as well as allowing more types of circuits to be simulated in this way. We also consider certain obstructions to classical simulations. It has been argued in various contexts that contextuality and non-locality hamper classical simulations of quantum circuits. Linear constraint systems (LCSs) are a generalization of the well-known Peres-Mermin magic square, which has been recently used to prove a separation between the power of constant depth classical and quantum circuits. While binary LCSs have been studied in detail, $d$-ary LCSs are less well-understood. In this thesis we consider linear constraint systems modulo $d > 2$. We give a simple proof, of the previously known fact, that any linear constraint system which admits a quantum solution consisting of generalized Pauli observables in odd dimension must be classically satisfiable. We further prove that, for odd $d$, if a Pauli-like commutation relation between two variables in the LCS arises, then it has no quantum solutions in any dimensions, in stark contrast to the even $d$ case. We apply this result to various examples, for instance showing that many generalizations of the Peres-Mermin magic square do not give rise to a quantum vs. classical satisfiability gap