Complexity of simulating quantum systems

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Complexity of simulating quantum systems on
classical computers
Barbara Terhal IBM Research
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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/approxim
  ately 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.

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Physically-Relevant Quantum States
local interactions are between O(1) degrees of
freedom (e.g. qubits)
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Efficient Classical Descriptions
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Matrix Product States
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1st Generalization Tree Tensor Product States
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2nd Generalization Tensor Product States or PEPS
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Properties of MPS and Tree-TPS
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Properties of tensor product states
PEPS and TPS perhaps too general for classical
simulation purposes
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Quantum Circuit Point of View
Past Light Cone





Max width
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Quantum Circuit Point of View
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Quantum Circuit Point of View
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Area Law
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Classical Simulations of Dynamics
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Lieb-Robinson Bounds
Lieb-Robinson Bound Commutator of operator A
with backwards propagated B decays
exponentially with distance between A and B,
when A is outside Bs effective past light-cone.
Bulk Past Light Cone
A
B
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Stoquastic Hamiltonians
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Examples of Stoquastic Hamiltonians
Particles in a potential Hamiltonian is a sum
of a diagonal potential term in position xgt 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-Waves 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.
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Stoquastic Hamiltonians
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Frustration-Free Stoquastic Hamiltonians
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Conclusion