228 research outputs found

    Metriplectic Integrators for the Landau Collision Operator

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    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

    Faster quantum and classical SDP approximations for quadratic binary optimization

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    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

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    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 nn nodes and mm edges, outputs a classical description of an ϵ\epsilon-spectral sparsifier in sublinear time O~(mn/ϵ)\tilde{O}(\sqrt{mn}/\epsilon). This contrasts with the optimal classical complexity O~(m)\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 kk-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

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    . 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

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    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(n2)O(\ell n^2) to O(mn2)O(m n^2), where \ell is the total number of gates in the circuit, and mm 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 m\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 ψm\ket{\psi}^{\otimes m}, where ψ\ket{\psi} is either a magic state or an equatorial state. For any single qubit magic state ψ\ket{\psi}, we find stabilizer decompositions of ψm\ket{\psi}^{\otimes m} with 2mlog2(3)/42^{m\log_2(3)/4} terms. This improves on the best known bound of 2mlog2(7)/62^{m \log_2(7)/6}. Similarly, for any single qubit equatorial state ψ\ket{\psi}, we give a stabilizer decomposition of ψm\ket{\psi}^{\otimes m} with 2m/22^{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, dd-ary LCSs are less well-understood. In this thesis we consider linear constraint systems modulo d>2d > 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 dd, 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 dd 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
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