Complexity of simulating quantum systems on classical computers
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Transcript of Complexity of simulating quantum systems on classical computers
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