Open adscft-quibt opened 2 years ago
Hi @adscft-qubit!
You can start a TEBD simulation in any state. For this it is, however, necessary to enter the state in a canonical Vidal form in Liouville space. This means that for a N site chain you'll need to enter the state in an MPS form consisting of N Gamma tensors and N-1 diagonal lambda matrices as shown in this picture:
I've labeled the axis of the Gamma tensors with "L" for "left bond", "R" for "right bond", "P" for "physical bond in vectorised Liouville space". The axis order should be [L, P, R] as described in the documentation of oqupy.AugmentedMPS
here.
Let's assume we'd like to start a PT-TEBD simulation in the Bell State |Ψ-> = |u, d> - |d, u>. (read "up" and "down" for "u" and "d"). The density matrix for this state is:
The last expression is an MPS in canonical form with a bond dimension of 4 due to the 4 terms. We can write this in python using the single site Liouville basis {uu, ud, du, uu} where xy = |x><y|.
uu = np.array([[1,0],[0,0]]).flatten()
ud = np.array([[0,1],[0,0]]).flatten()
du = np.array([[0,0],[1,0]]).flatten()
dd = np.array([[0,0],[0,1]]).flatten()
gamma0 = np.array([uu,(-1)*ud, (-1)*du, dd])
lambda0 = np.array([0.5, 0.5, 0.5, 0.5])
gamma1 = np.array([dd, du, ud, uu])
gamma0.shape=(1,4,4)
gamma0 = np.swapaxes(gamma0, 1, 2)
gamma1.shape=(4,4,1)
initial_augmented_mps = oqupy.AugmentedMPS(
gammas=[gamma0, gamma1],
lambdas=[lambda0,])
@adscft-qubit: Is this what you needed?
Best, gefux
Hi TEMPO developers,
I have a question about the setting of the initial state. How can i set the initial state of the system as the maximal entanglement state in PT-TEBD algorithm,if the system is composed by two qubits?
Best regards.