Improved approximation algorithms for bounded-degree local Hamiltonians

  title={Improved approximation algorithms for bounded-degree local Hamiltonians},
  author={Anurag Anshu and David Gosset and Karen J. Morenz Korol and Mehdi Soleimanifar},
Anurag Anshu, David Gosset, Karen J. Morenz Korol, and Mehdi Soleimanifar 1 Department of EECS & Challenge Institute for Quantum Computation, University of California, Berkeley, USA and Simons Institute for the Theory of Computing, Berkeley, California, USA. 2 Department of Combinatorics and Optimization and Institute for Quantum Computing, University of Waterloo, Canada 3 Department of Chemistry, University of Toronto, Canada and 4 Center for Theoretical Physics, Massachusetts Institute of… 
4 Citations
An Optimal Product-State Approximation for 2-Local Quantum Hamiltonians with Positive Terms
We resolve the approximability of the maximum energy of the Quantum Max Cut (QMC) problem using product states. A classical 0.498-approximation, using a basic semidefinite programming relaxation, is
Application of the Level-$2$ Quantum Lasserre Hierarchy in Quantum Approximation Algorithms
This work provides the first ever use of the level-2 hierarchy in an approximation algorithm for a particular QMA-complete problem, so-called Quantum Max Cut, and indicates that higher levels of the quantum Lasserre Hierarchy may be very useful tools in the design of approximation algorithms for Q MA-complete problems.
A construction of Combinatorial NLTS
The NLTS (No Low-Energy Trivial State) conjecture of Freedman and Hastings [14] posits that there exist families of Hamiltonians with all low energy states of high complexity (with complexity
The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut
The first application of Boolean Fourier analysis methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next.


Almost Optimal Classical Approximation Algorithms for a Quantum Generalization of Max-Cut
This work studies classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model, and shows how to classically and efficiently obtain approximation ratios 0.649 (anti-feromagnetic XY model) and 0.498 ( anti-ferromagnetic Heisenburg XYZ model).
Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems
The first approximation algorithm beating this bound is presented, a classical polynomial-time 0.764-approximation for strictly quadratic instances, and it is conjecture these are the hardest instances to approximate.
Approximation algorithms for quantum many-body problems
Gaussian states can vastly outperform Slater determinant states commonly used in the Hartree-Fock method and an efficient algorithm is given that outputs a fermionic Gaussian state whose energy is at least λmax/O(n log n).
Approximation Algorithms for QMA-Complete Problems
  • Sevag Gharibian, J. Kempe
  • Computer Science, Mathematics
    2011 IEEE 26th Annual Conference on Computational Complexity
  • 2011
A natural approximation version of the QMA-complete local Hamiltonian problem is defined and a non-trivial approximation ratio can be obtained in the class NP using product states and a polynomial time algorithm is given providing a similar approximation ratio for dense instances of the problem.
An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem
This work works in the product state space and extends the framework of Goemans and Williamson for approximating MAX-2-CSPs, and achieves an approximation ratio of 0.328, which is the first example of an approximation algorithm beating the random quantum assignment ratio of0.25 by a constant factor.
Classical and quantum bounded depth approximation algorithms
The QAOA is considered and strong evidence is provided that, for any fixed number of steps, its performance on MAX-3-LIN-2 on bounded degree graphs cannot achieve the same scaling as can be done by a class of "global" classical algorithms.
Circuit lower bounds for low-energy states of quantum code Hamiltonians
New techniques based on entropic and local indistinguishability arguments that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes are proved.
Classical algorithms for quantum mean values
It is shown that a classical approximation is possible when the quantum circuits are limited to constant depth, and sub-exponential time classical algorithms are developed for solving the quantum mean value problem for general classes of quantum observables and constant-depth quantum circuits.
Beyond Product State Approximations for a Quantum Analogue of Max Cut
This work considers a computational problem where the goal is to approximate the maximum eigenvalue of a two-local Hamiltonian that describes Heisenberg interactions between qubits located at the vertices of a graph, and provides an efficient classical algorithm which achieves an approximation ratio of at least 0.53 in the worst case.
Quantum systems on non-k-hyperfinite complexes: a generalization of classical statistical mechanics on expander graphs
This work constructs families of cell complexes that generalize expander graphs, generalizing the idea of a non-hyperfinite (NH) family of graphs and considers certain quantum systems on these complexes.