# Improved approximation algorithms for bounded-degree local Hamiltonians

@article{Anshu2021ImprovedAA, title={Improved approximation algorithms for bounded-degree local Hamiltonians}, author={Anurag Anshu and David Gosset and Karen J. Morenz Korol and Mehdi Soleimanifar}, journal={ArXiv}, year={2021}, volume={abs/2105.01193} }

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

- PhysicsArXiv
- 2022

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

- Computer ScienceICALP
- 2021

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

- Physics
- 2022

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

- Computer ScienceArXiv
- 2022

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.

## References

SHOWING 1-10 OF 27 REFERENCES

Almost Optimal Classical Approximation Algorithms for a Quantum Generalization of Max-Cut

- Computer ScienceAPPROX-RANDOM
- 2019

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

- Computer Science, MathematicsESA
- 2021

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

- Computer ScienceJournal of Mathematical Physics
- 2019

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

- Computer Science, Mathematics2011 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

- Computer ScienceAPPROX-RANDOM
- 2020

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

- Computer ScienceQuantum Inf. Comput.
- 2019

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

- Computer ScienceITCS
- 2022

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

- Physics, Computer ScienceNature Physics
- 2021

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

- Computer ScienceTQC
- 2020

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

- MathematicsQuantum Inf. Comput.
- 2014

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.