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 [FH14] 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.
Classical approximation schemes for the ground-state energy of quantum and classical ising spin hamiltonians on planar graphs
A classical approximation algorithm for evaluating the ground state energy of the classical Ising Hamiltonian with linear terms on an arbitrary planar graph, which contrasts the well known fact that exact computation of the groundstate energy for the two-dimensional Ising spin glass model is NP-hard.