2.4 Matrix Product States
2.4.9 Variational Handling of Matrix Product States
Now we want to introduce a more efficient method to add many-body states in MPS representation and to apply operators to many-body states in MPS representation.
We have already seen direct methods to perform these operations in Section2.4.6and Section2.4.8, however, those increase the bond dimension of the resulting many-body state in MPS representation. In this section we want to describe a procedure that variationally optimizes a many-body state in MPS representation to MPS addition or MPO application with the constraint of a previously chosen maximum bond dimen-sion. This means we are performing the operation (addition or operator application),
while simultaneously limiting the bond dimension of the MPS representation of the resulting many-body state.
Let us start with the example of addition of two many-body states in MPS representa-tion|AiMPSand|BiMPS. Further, we want to obtain the resulting many-body state represented as an MPS|CiMPSwith a previously fixed bond dimensionDC. Instead of performing the direct addition followed by truncation (see Section2.4.6and Section 2.4.5), we vary the tensors in|CiMPSto find the global minimum of the functional
L[|CiMPS] =|| |CiMPS −(|AiMPS+|BiMPS)||2. (2.90) We find the tensorC[i]in|CiMPS that minimize Equation2.90by taking the first derivative of the functional with respect to the tensorC[i]∗inhC|MPS and set it to zero
0 = ∂
∂C[i]∗L[C] = ∂
∂C[i]∗( MPShC|CiMPS− MPShC|AiMPS− MPShC|BiMPS) . (2.91) Here, we already canceled terms not depending on C[i]∗. Solving Equation 2.91 minimizes the value of the functional in Equation 2.90with respect to C[i] and therefore brings the MPS|CiMPSclose to the sum of the two many-body states in MPS representation|AiMPSand|BiMPS.
At this point, the graphical representation of tensor networks is particularly handy.
The derivative of Equation2.91reads in the graphical representation
0 = C[h]
C[h]
· · ·
· · · C[i] C[j]
C[j]
· · ·
· · ·
− A[h]
· · · C[h]
· · · A[i] A[j]
C[j]
· · ·
· · ·
− B[h]
· · · C[h]
· · · B[i] B[j]
C[j]
· · ·
· · ·
, (2.92)
where the first part depictshC|Ciin Equation2.91, the second part depictshC|Ai in Equation2.91and the third part depictshC|Biin Equation2.91. The tensorC[i]∗ simply vanishes when taking the derivative of Equation2.91, since Equation2.91is linear inC[i]∗.
To solve Equation2.92, let us remember the (left-) right-orthonormalization property of many-body states in MPS representation. In Section2.4.4we discussed the feature of many-body state in MPS representation to be in a special form, where parts of tensor networks such as given in Equation2.92form identity operators (see Equation 2.66and Equation2.68). At this point of the MPS addition, we can take advantage of
this property. Assume the MPS|CiMPShappens to have its orthonormal center at orbitali, which means, all tensor to the left of orbitaliare left-orthogonalized and all tensors to the right of orbitaliare right-orthogonalized. Using the properties of left-and right-orthonormalization, we can simplify Equation2.92to
0 = C[i] −
A[h]
· · · C[h]
· · · A[i] A[j]
C[j]
· · ·
· · ·
− B[h]
C[h]
· · ·
· · · B[i] B[j]
C[j]
· · ·
· · ·
,
(2.93) which we can rearrange to
C[i] =
A[h]
· · · C[h]
· · · A[i] A[j]
C[j]
· · ·
· · ·
+ B[h]
C[h]
· · ·
· · · B[i] B[j]
C[j]
· · ·
· · ·
.
(2.94) This is a linear equation that we can solve directly. The (left-) right-orthonormal-ization property of the many-body state in MPS representation enabled us to turn Equation2.91into a linear equation (Equation2.94). By solving Equation2.94we find a minimum of the functional in Equation2.90for the particular tensorC[i].
However, the minimum we found from this procedure is not necessary the global minimum of Equation2.90. The solution of Equation2.91for orbitalistrongly depends on all tensorsC[j]∀j 6=i. To approach the global minimum, we also need to vary the other tensors in the MPS|CiMPS. We do this in the form ofsweeps[42,43], where we repeat the procedure we performed for orbitalifor the next tensori →i±1in the MPS|CiMPS, and all following. Thereby, we optimize the MPS|CiMPS to represent the sum in Equation2.90variationally, until we reached convergence to the desired accuracy. Most importantly, the bond dimension of the MPS|CiMPSremains fixed within this procedure. It is chosen for the initial guess of the MPS|CiMPS, and does not change in this procedure. The optimization happens in the fixed manifold of the many-body Hilbert space the MPS state|CiMPScovers. This manifold is not changed within this optimization procedure, which improves efficiency compared to the direct approach outlined above in Section2.4.6.
This procedure is also able to perform many other operations beyond the addition of two MPS. For example, we can easily use it to add more than two MPS at once.
Then the Equation2.94is just appended by more addends on the right hand side.
Further allows this procedure to truncate many-body states in MPS representation variationally [178]. Such an operation starts from the functional
L[|CiMPS] =|| |CiMPS− |AiMPS||2, (2.95)
and all following steps of the variation procedure remain similar to those for MPS addition. Most importantly we can also use the variational method to apply body operators in MPO representation, i.e. to perform level 2 operations on many-body states in MPS representation. Here, the functional to minimize reads
L[|CiMPS] =|| |CiMPS−ˆO|AiMPS||2, (2.96) but the concept of the procedure is again similar to what has been explained for MPS addition. For application of many-body operators in MPO representation the linear equation to solve reads
C[i] =
A[h]
O[h]
C[h]
· · ·
· · ·
· · · A[i]
O[i]
A[j]
O[j]
C[j]
· · ·
· · ·
· · ·
, (2.97)
which shows the additional summations that increase complexity of operator appli-cation. However, the resulting method is still more efficient than the direct approach to apply MPO to MPS (see Section2.4.8).
Another advantage of the variational method is that we can combine various opera-tions into one. For example, we can combine operator applicaopera-tions with an arbitrary large sum of other MPS. Then the functional to minimize reads
L[|CiMPS] =|| |CiMPS− ˆH|AiMPS+X
p
|piMPS
!
||2 (2.98) This feature will become in particular handy in the time evolution methods that we will outline in Section2.5.
The computational cost of a variational optimization is small compared to the direct approaches explained in Section2.4.6and Section2.4.8. The complexity of the varia-tional method for the addition of two MPS with bond dimensionDinto an MPS with bond dimensionDis per sweepO(LD3)and therefore only grows linearly with the system size. For application of MPO in quantum chemistry to MPS, the complexity is O(L4D2+L3D3), which is one order smaller inLcompared to the direct approach.
In this procedure to add many-body states in MPS representation, we took the deriva-tive of a single siteC[i]∗, which has the advantage that the MPS dimension does not change within the optimization. However, this comes at the cost that the MPS representation of the many-body state remains in a fixed manifold of the many-body Hilbert space. It is not free to adapt to a new manifold of the many-body Hilbert
space (see Section2.4.2) by changing the virtual basis. It can adjust the coefficient in the tensorsC[i]only, however, without changing the virtual basis. It cannot leave the corner of the Hilbert space the initial MPS|CiMPSwas constructed to represent. For this reason, we will use a dynamic extension, that is able to adjust the corner of the Hilbert space the MPS represents. Instead of taking the derivative of a single tensor C∗[i], we work with neighboring pairs of tensors
∂
∂C[i]∗ → ∂
∂C[i]∗C[j]∗, (2.99)
C[i] → C[i] C[i] = S[i,j] , (2.100)
which allows for adaption of the virtual basis and therefore to adapt the manifold it represents in the many-body Hilbert space.S[i,j]is known as thetwo-site object [51,106] and is equivalent to the two-site DMRG algorithm [42,43]. After the linear equation has been solved using the two-site object, it needs to be decomposed into two matrices to fit back in the resulting MPS representation of the many-body state. This can be done using the SVD introduced in Section2.4.5. It is known that the two-site procedure of doing variational operations improves the overall MPS performance. There are existing extensions to enhance the single-site procedure as well [179], however, we will use the two-site procedure throughout this thesis.