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Related Experiment Video

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Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Computational difficulty of finding matrix product ground states.

Norbert Schuch¹, Ignacio Cirac, Frank Verstraete

¹Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany.

Physical Review Letters

|July 23, 2008

Summary

Finding ground states for certain one-dimensional Hamiltonians is computationally difficult, as hard as factoring integers. This impacts variational methods using matrix product states (MPS).

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Area of Science:

Quantum mechanics
Computational complexity theory
Condensed matter physics

Background:

One-dimensional (1D) Hamiltonians are often described by matrix product states (MPS).
Determining the ground state of quantum systems is a fundamental problem with significant computational challenges.

Purpose of the Study:

To determine the computational difficulty of finding ground states for 1D Hamiltonians that are matrix product states.
To establish new bounds for convergence proofs of variational methods utilizing MPS.

Main Methods:

Construction of a class of 1D frustration-free Hamiltonians with unique MPS ground states.
Analysis of the computational hardness of finding these ground states, relating it to the factoring problem.

Main Results:

Finding the ground state for a specific class of 1D Hamiltonians is proven to be at least as computationally hard as factoring.
Without ground state uniqueness, the problem becomes NP-complete, preventing certification of the found ground state.

Conclusions:

The computational hardness of finding ground states for 1D MPS Hamiltonians is established.
New limitations are imposed on the convergence proofs for variational methods employing MPS.