Creating an MPS from a state vector#

The Schmidt decomposition can be used sequentially, to split a multilegged tensor representing a quantum state, such as \(\psi_{ijk}\), or \(\psi_{i_1i_2i_3\cdots i_N}\) into a collection of three-legged tensors (See matrix-product state objects).

The rationale starts by understanding that Schmidt decomposition of the bipartite system as the separation of a two-legged tensor tensor into two smaller tensors. For instance, we can write

\[\psi_{ij} = B_{ik} A_{kj1}\]

identifying the matrices from the SVD form

\[B_{ik} = U_{ik} s_k, \; A_{kj1} = V_{kj}\]

The k index that bonds both tensors is called the “virtual dimension”, as opposed to the original indices i and j, which we associated to physical subsystems.

This procedure can be extended to larger tensors wiht more legs. Let us take an N-dimensional quantum system, whose state is represented by an N-leg tensor and reinterpret it as a bipartite system with N-1 and 1 component, respectively.

\[\psi_{i_1i_2\ldots i_{N-1}i_N} = \Psi_{I i_N}\]

Here, the index \(I\) groups all the possible configurations from the previous (N-1) indices and has a potentially large dimension, \(d_1\cdots d_{N-1}\). But let us ignore this fact and use the Schmidt decomposition to write

\[\Psi_{I i_N} = B_{Ik} A^{(N)}_{ki_N1} = \Psi^{(N-1)}_{i_1\ldots i_{N-1}k} A_{k{i_N}1}\]

We can repeat this process iteratively, now separating

\[\Psi^{(N-1)}_{i_1\cdots i_{N-2}i_{N-1}k} = \Psi^{(N-2)}_{i_1\cdots i_{N-2}j} A^{(N-1)}_{ji_{N-1}k}\]

If repeated N times, the outcome is a decomposition of the whole tensor into a contraction of 3-legged tensors:

\[\psi_{i_1i_2\ldots i_{N-1}i_N} = A^{(1)}_{1i_1\alpha_2} A^{(2)}_{\alpha_2i_2\alpha_3}\cdots A^{(N)}_{\alpha_Ni_N1}\]

In SeeMPS, this iterative procedure is performed in very different parts of the code. For the user, this algorithm is offered through the class methods from_tensor(), which transforms an N-legged tensor into a matrix-product state with N components.

Take for instance the following example, which creates an unnormalized random state of 8 qubits, and then uses this decomposition to create a matrix-product state:

>>> from seemps import MPS
>>> import numpy as np
>>> state = np.random.randn(2,2,2,2,2,2,2,2)
>>> mps = MPS.from_vector(state)

Alternatively, the same effect can be obtained from the wavefunction of a composite quantum system represented as a vector. Naturally, because we are inputting a vector, this function takes as an argument a list with the dimensions of the quantum subsystems.

>>> from seemps import MPS
>>> import numpy as np
>>> state = np.random.randn(2**8)
>>> mps = MPS.from_vector(state, [2]*8)
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