Here’s a simple example that can be used to check state -> mps conversion code.

Representation

Suppose we are guaranteed that throughout a quantum computational process, the entanglement of the states is low.

Consider a quantum circuit built out of LOCAL one and two qubit gates only. What is meant by local is that the qubits can be arranged in a line and the qubit gates can only act on neighboring qubits. (Why is it a reasonable assumption? One can always swap qubits to be local over polynomial time over separation.)

$$ |\phi\rangle=\sum_i\alpha_i|A_i\rangle|B_i\rangle $$

The number of \(\alpha_i\) is determined by the dimension of the smaller system. Notice Schmit decomposition is important, because a naive split up

$$|\psi\rangle=\sum_\sigma|\sigma_1\sigma_2\sigma_3\sigma_4\rangle=|\sigma_1\sigma_2\rangle|\sigma_3\sigma_4\rangle$$

Schmit is essentially an SVD, which decomposes M into UDV, where U,V are unitary. The diagonal elements of D are like $\alpha_i$ and the U and V matrices contain orthogonal bases. Low entanglement in this context means the matrix D has a bunch of zero on its diagonal, which allows us to make it smaller.

Remember the original assumption is that overall entanglement is small, and in particular MPS only capture “local” entanglement. This is the reason why MPS are only good for 1D systems (2D systems make nonlocal segments of a line local).

Evaluation

overlap

Naively there are \(2^N\) operations (the big sum), each one involving \(N\) matrix multiplications. However, we can reduce it to roughly \(N\) 3 matrix multiplications.

$$\langle \psi|\psi \rangle=\sum\limits_{\sigma_1\sigma_2\dots\sigma_n}(M^{\sigma_1}M^{\sigma_2}\dots M^{\sigma_n})(M^{\sigma_1}M^{\sigma_2}\dots M^{\sigma_n})^T $$
$$=\sum\limits_{\sigma_1\sigma_2\dots\sigma_n}M^{\sigma_1}M^{\sigma_2}\dots(\sum\limits_{\sigma_n} M^{\sigma_n} {M^{\sigma_n}}^T)\dots{M^{\sigma_2}}^T{M^{\sigma_1}}^T $$

Invariance

This representation has \(U(1)^N\) symmetry. That is, the matrices can be transformed as \(M\rightarrow UMU^\dagger\) without changing the MPS. Therefore we can put the MPS in left canonical form, right canonical form or mixed canonical form.