MPS update#

There are many types of algorithmic updates that one might want to implement using matrix-product-state forms. For instance, in orden of complexity:

  • Apply a local quantum gate, such as a \(\sigma_x\) rotation onto a qubit.

  • Apply an entanging gate onto a pair of neighboring qubits.

  • Evolve the quantum state for a brief period of time with a Hamiltonian.

The first type of update is inconsequential. It involves contracting the matrix that represents the local transformation with the tensor in the MPS that is associated with the particular quantum subsystem.

However, the other operations are more complex and involve creating new tensors that approximate in some way the outcome of the operation. We will focus now on the two-site updates, such as the entangling gate mentioned in the second item, because this is an operation that can be implemented accurately when an MPS is in canonical form. Furthermore, this transformation is the basis for more complex evolution algorithms.

Let us consider a quantum operation \(U\) acting on two quantum subsystems that are “neighboring sites” in an MPS state representation. Say the first subsystem is labeled \(i_n\) and the second one is \(i_{n+1}\), in

\[|\psi\rangle = \cdots A_{\alpha_ni_n\alpha_{n+1}} B_{\alpha_{n+1}i_{n+1}\alpha_{n+2}}\cdots |\ldots i_n i_{n+1}\ldots\rangle\]

The quantum transformation can be represented as a four-legged tensor:

\[U = U_{j_{n}j_{n+1}i_{n}i_{n+1}}|j_nj_{n+1}\rangle\langle{i_ni_{n+1}}|\]

When we contract this operator with the MPS above, the two affected sites are now associated to a combined tensor

\[U|\psi\rangle = \cdots C_{\alpha_nj_nj_{n+1}\alpha_{n+2}}\cdots |\ldots j_n j_{n+1}\ldots\rangle\]

where the larger, four-legged tensor is

\[C_{\alpha_nj_nj_{n+1}\alpha_{n+2}} = U_{j_{n}j_{n+1}i_{n}i_{n+1}} A_{\alpha_ni_n\alpha_{n+1}} B_{\alpha_{n+1}i_{n+1}\alpha_{n+2}}\]

and we assume the Einstein convention of summing over repeated indices, \(i_n, i_{n+1}, \alpha_{n+1}\).

Our job is now to split the four-legged tensor \(C\) into two new tensors, \(\tilde{A}\) and \(\tilde{B}\).

\[C_{\alpha_nj_nj_{n+1}\alpha_{n+2}} = \tilde{A}_{\alpha_nj_n\beta{n+1}} \tilde{B}_{\beta{n+1}j_{n+1}\alpha_{n+2}}\]

The tool for this, as usual, is the Schmidt decomposition. However, note that now we are creating a new index \(\beta_{n+2}\) which can be significantly larger—it may be as large as the product of the dimensions of \(\alpha_{n}\) and \(j_n\), which represents an exponential growth!

It is therefore critical to implement this decomposition using a proper truncation strategy, that drops the Schmidt values that have little or no significance, and which keeps track of the truncation error caused by this update algorithm.

In SeeMPS, two functions are responsible for this update. These are the following two methods of the CanonicalMPS class:

update_2site_right(AA, site, strategy)

Split a two-site tensor into two one-site tensors by right orthonormalization and insert the tensor in canonical form into the MPS at the given site and the site on its right.

update_2site_left(AA, site, strategy)

Split a two-site tensor into two one-site tensors by left orthonormalization and insert the tensor in canonical form into the MPS Ψ at the given site and the site on its right.

Both functions take as input the combined tensor \(C\) and the site n into which its decomposition is to be inserted. Both functions are also destructive, in the sense that they change the state on which they operate.

The difference between these two methods is that they differ in which tensor, \(A\) or \(B\) they associate to the \(U\) and \(V\) isometries. If the state is in canonical form with respect to site n and you apply update_2site_right, the updated state will be in canonical form with respect to site n+1. Conversely, if the original state was in canonical form with respect to site n+1, update_2site_left will leave its output in canonical form with respect to n.

The following excerpt from apply_inplace() shows how a list of two-site unitaries U is sequentially applied from “left” to “right” on a quantum state in canonical form. The argument to the update_2site_right function is an efficient contraction of the four-legged tensor U[i] associated to the i-th unitary, and the corresponding tensors of sites j and j+1. The result is an in-place update of the MPS state.

...
if center > 1:
    state.recenter(1)
for j in range(L - 1):
    ## C = np.einsum("ijk,klm,nrjl -> inrm", state[j], state[j + 1], U[j])
    state.update_2site_right(
        _contract_nrjl_ijk_klm(U[j], state[j], state[j + 1]), j, strategy
    )
...
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