TEBD Time evolution#

The second but better known algorithm is the time-evolving block decimation method to approximate the evolution of a quantum state. This algorithm solves the Schrödinger equation

\[i \partial_t |\psi\rangle = H|\psi\rangle\]

with a Hamiltonian that consists entirely of nearest-neighbor interactions

\[H = \sum_{i=1}^{N-1} h_{i,i+1}\]

The algorithm proceeds by applying local gates made of small time-steps \(\exp(-i \delta{t} h_{i,i+1})\), adapting the matrix-product state so that the bond dimension does not grow too much.

Trotter2ndOrder

Class implementing a 2nd order Trotter algorithm.

Trotter3rdOrder

Class implementing a 3rd order Trotter algorithm.

The following is an example evolving a matrix-product state with 20 qubits under a spin-1/2 Heisenberg Hamiltonian:

>>> import seemps
>>> mps = seemps.random_uniform_mps(2, 20)
>>> H = seemps.hamiltonians.HeisenbergHamiltonian(20)
>>> dt = 0.1
>>> U = seemps.evolution.Trotter2ndOrder(H, dt)
>>> strategy = seemps.state.DEFAULT_STRATEGY.replace(tolerance = 1e-8)
>>> t = 0.0
>>> for steps in range(0, 50):
...   mps = U.apply_inplace(mps)
...   t += dt
>>> mps
<seemps.state.CanonicalMPS object at 0x000002166AFFC1D0>

See also#

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