Complexity of simulating quantum systems on classical computers

Barbara TerhalIBM Research

Computational Quantum Physics

Computational quantum physicists (in condensed-matter physics, quantum chemistry etc.) have been in the business of showing how to simulate and understand properties of many-body quantum systems using a classical computer.

Heuristic and ad-hoc methods dominate, but the claim has been that these methods often work well in practice.

Quantum information science has and will contribute to computational quantum physics in several ways:

• Come up with better simulation algorithms

• Make rigorous what is done heuristically/approximately in computational physics.

• Delineate the boundary between what is possible and what is not. That is: show that certain problems are hard for classical (or even quantum) computers in a complexity sense.

Physically-Relevant Quantum States

local interactions are between O(1) degrees of freedom (e.g. qubits)

Efficient Classical Descriptions

Matrix Product States

1st Generalization: Tree Tensor Product States

2nd Generalization: Tensor Product States or PEPS

Properties of MPS and Tree-TPS

Properties of tensor product states

PEPS and TPS perhaps too general for classical simulation purposes

Quantum Circuit Point of View

Past Light ConeMax width

Quantum Circuit Point of View

Quantum Circuit Point of View

Area Law

Classical Simulations of Dynamics

Lieb-Robinson Bounds

Bulk Past Light Cone B

ALieb-Robinson Bound: Commutator of operator A with backwards propagated B decays exponentially with distance betweenA and B, when A is outside B’s effective past light-cone.

Stoquastic Hamiltonians

Examples of Stoquastic Hamiltonians

Particles in a potential; Hamiltonian is a sum of a diagonal potential term in position |x> and off-diagonal negative kinetic terms (-d2/dx2).All of classical and quantum mechanics.Quantum transverse Ising model Ferromagnetic Heisenberg models (modeling interacting spins on lattices)Jaynes-Cummings Hamiltonian (describing atom-laser interaction), spin-boson model, bosonic Hubbard models, Bose-Einstein condensates etc. D-Wave’s Orion quantum computer…Non-stoquastic are typically fermionic systems, charged particles in a magnetic field.

Stoquastic Hamiltonians are ubiquitous in nature.

Note that we only consider ground-state properties of these Hamiltonians.

Stoquastic Hamiltonians

Frustration-Free Stoquastic Hamiltonians

Conclusion