In this section, we present the most commonly used operations performed on MPSs. This includes the more technical truncation of an MPS, which is nevertheless essential to the use of MPSs, and also arithmetic operations such as the scalar product.
Normalization of MPSs
5.3.1
Here we explain how to obtain a right-canonical MPS. There are infinite ways to transform MPSs by inserting arbitrary unitary transformationsP
m0jUj;m
j,m0jUj;m0
j,mj =1j;mj,mj between the site tensors. In particular, we can choose them in a way that the new site tensors fulfill normalization constraints.
|ψi= M M · · · M M M
σ1 σ2 σL−2 σL−1 σL
= M M · · · M M B
σ1 σ2 σL−2 σL−1 σL
QR(ML) ML0−1
= M M · · · M B B
σ1 σ2 σL−2 σL−1 σL
QR(ML−1) ML0−2
...
= B B · · · B B B
σ1 σ2 σL−2 σL−1 σL
Figure 5.4: Schematic of the transformation of a non-canonical MPS into a right-canonical MPS.
Starting from an MPS with non-normalized tensors Eq. (5.2), we reshape the rightmost tensor, ML;m0
L−1,(σLmL) =ML;mσL
L−1,mL. (5.11)
As before,m0=m0=mL=mL = 1 are dummy indices. Then,M0Lis decomposed via the QR decomposition,
ML;m0
L−1,(σLmL) = X
m0L
RL−1;m
L−1,m0L−1QL;m0
L−1,(σLmL) , (5.12) in which BL;mσL
L−1,mL=Q
L;mL−1,(σLmL) fulfills the condition Eq. (5.4) for right-normalized ten-sors due to the orthonormal rows ofQL. In the next step, the residual part of the decomposition RL is incorporated into the site tensorML−10 on the left,
ML0−1;m
L−2,σL−1mL−1 = X
mL−1
ML−1;mσL−1
L−2,mL−1Rm
L−1,mL−1 . (5.13)
36 Chapter 5. Matrix-Product States
This is repeated with the new site tensor ML0−1 until the first site is reached. The complete procedure is illustrated in Fig. 5.4. Once the system is normalized, i.e., the first site is reached, the residual of the decomposition of the first site will be a scalar number. The square of the absolute value of this number is the norm of the state and can be discarded if a state with norm hψ|ψi= 1 is desired.
The procedure to obtain a left-canonical MPS is analogous and also similar to the construction of an MPS from an arbitrary state (see Sec. 5.2.1).
In order to obtain a mixed-canonical state (see Sec. 5.1 and Fig. 5.5) with active site j it is necessary to normalize the state from the left and from the right up to sitej. The residualsRj−1 and Rj are both incorporated into the site tensor Mj.
A · · · A M B · · · B
σ1 σj−1 σj σj+1 σL
Figure 5.5: MPS with active sitej and consequently left (right) normalized site tensor left (right) of site j as defined in Eq. (5.4).
Truncation
5.3.2
One of the main benefits of MPSs is the possibility to restrict the bond dimension m, but to still obtain a very good approximation for certain classes of many-body quantum states. This is particularly important as m grows with, e.g., the application of a matrix-product operator (MPO) (see Sec. 6.1). The problem we want to tackle is therefore to approximate|ψiwith bond dimensionm by|ψ˜i with bond dimensionm < m˜ so that
k|ψi − |ψ˜ik22 (5.14)
is minimized. Note that if Eq. (5.14) was zero, it would be equivalent to hψ|ψ˜i = 1 (assuming both states are normalized), which would also be equivalent to |ψi = |ψ˜i. In this section, we present two procedures to obtain such a truncated MPS.
The first procedure is based on the SVD and gives direct access to the error induced by the trun-cation; the second is a variational approach that results in an optimally truncated representation of the input state.
Direct Truncation via SVD
The truncation via the SVD is a well established method and described in the literature several times [Sch13, Man17, Hub17]. Nevertheless, we explain it here in full detail in order to deliver a clean and complete description.
Consider a general decomposition of a pure state |ψi ∈ H=HA⊗ HB,
|ψi=X
a,b
Ψa,b|aiA⊗ |biB , (5.15)
with orthonormal bases {|aiA} ∈ HA, {|biB} ∈ HB, and the coefficient matrix Ψa,b with rank r = min(dim(HA),dim(HB)). The goal is to reduce the number of used basis states of the
Section 5.3. Operations on MPSs 37
Hilbert spaces HA and HB while still representing the initial state as accurately as possible.
In the following, we assume, without loss of generality, that the parts denoted with A and B represent the left and the right side of an one-dimensional (1D) system, which is cut at bondj. Because the bases are orthonormal, the 2-norm of the state k|ψik22 and the Frobenius norm of the coefficient matrixkΨa,bk2F are equal:
k|ψik22 =hψ|ψi=X
Hence, it is sufficient to find an approximationΨ˜a,bfor Ψa,bwith a lower rank in order to obtain a representation of|ψi with a reduced number of used basis states. In Sec. 4.1, we have shown that the SVD provides exactly this approximation. Applied on Eq. (5.15), we obtain
|ψi=X
aVs,b|biB again being orthonormal basis elements. Note that the right side of Eq. (5.17) is the Schmidt decomposition, which states that every pure state can be decomposed into two sets of orthonormal basis elements, one for each side, while sharing the same eigenvalues, which are called Schmidt values.
At this point, we can limits and therefore restrict the bond dimension between the subsystem A and the subsystem B. The error ε introduced by this truncation is approximately given by the square root of the discarded weight
= Xm s= ˜m+1
Σ2s. (5.18)
If the Schmidt values decay fast enough, the discarded weight is small and we obtained a good approximation in terms of Eq. (5.14).
It is surprisingly simple to incorporate these considerations into an algorithm that sequentially truncates every bond of an MPS. As a pre-requirement, we have to ensure that the state is in the mixed-canonical form (see Fig. 5.5). The truncation sweeps in practice usually start from one of the edges, but in the following we nevertheless keep the assumption that j is the active site.
Depending on the direction in which we want to sweep through the system, we need to adjust the operations of the procedure. In the following, we concentrate on the case of a right sweep, whereas Figs. 5.6 and 5.7 cover both cases. The procedure starts by reshaping the site tensor,
Mj;(σjmj−1),mj =Mj;mσj j
−1,mj . (5.19)
Next, an SVD is performed and only the largestmmax singular values are kept,2 Mj;(σjmj−1),mj = Up to this point, we have only obtained a representation ofM with a reduced rank.
2Note that here we usemmaxinstead ofm˜ in order to indicate that this parameter is not specific to the matrix but a parameter of the MPS.
38 Chapter 5. Matrix-Product States
Figure 5.6: Pictorial representation of the singular-value decomposition of a rank-three tensorM into U·S·V† within a truncation sweep to the right (top) or to the left (bottom).
In order to reduce the bond dimension m between site j and j + 1, the parts of the matrix decomposition are assigned to the different site tensors (see Fig. 5.7),
Aσj;mj
Figure 5.7: Pictorial representation of the assignment of the result of the singular-value decomposition from Fig. 5.6 into the new left (right) normalized rank-three tensorAj (Bj) and the new active siteMj+1 (Mj−1) at the top (bottom). Note that the indexsj becomes the newmj and the indexs0j becomes the newmj−1.
Now, the active site has been moved one site to the right and the procedure can restart with j=j+ 1until the right edge is reached.
Due to this sweeping through the system, the truncation of siteL−1becomes dependent on the truncation of site1 but not vice versa. In the case of small truncations, the error resulting from this asymmetry is small and can be ignored, but for a better approximation further steps, e.g., a variational truncation, are necessary.
Variational Truncation
The variational truncation is an iterative method that starts from an initial guess state|ψ˜i with a chosen bond dimensionmmaxand variationally minimizes the distance to the untruncated state
|ψi. A good choice for the initial state is the result of the previous paragraph.
The distance is given by
k|ψi − |ψ˜ik22=hψ|ψi − hψ|ψ˜i − hψ˜|ψi+hψ˜|ψ˜i . (5.23) In order to minimize this global distance, we sweep through the system and minimize only locally with respect to a single site tensor M˜j. Because this tensor only occurs in the second half of
Section 5.3. Operations on MPSs 39
Eq. (5.23), the new (optimized)M˜j can be obtained via
∂
∂M˜σj;mj j
−1,mj
hψ˜|ψ˜i − hψ˜|ψi
= 0. (5.24)
Let us now consider that the truncated state is in a mixed canonical form with the active site
hψ˜| · · · ·
Figure 5.8: Summary of the iterative truncation considering the truncated state to be in a mixed canonical form. The left-hand side can then be reduced to the active sitej that we want to obtain. The right-hand side, which needs to be considered completely, can nevertheless be computed iteratively via the bond tensorsLandR.
at positionj. Then, the new tensor is given by
M˜j;mσj
in which the left (right) part of the tensor network (Fig. 5.8, right hand side) are contracted into L˜j−1 (R˜j+1), Based on this optimization, we start with a canonical state and sweep back and forth through the system until the distance is smaller than a threshold. Note that except at the edges, it is never necessary to calculate the complete contraction of the boundary tensors Eqs. (5.26) and (5.27), because the next tensor in sweep direction has already been calculated in the sweep before, and the other tensor is obtained by enlarging the one from the previous sweep step. Additionally to this single-site variational truncation, there is also a two-site variant, which is more stable against local minima but less performant. Nevertheless, the two-site variant is often used, because it offers the possibility to dynamically change the bond dimension.
40 Chapter 5. Matrix-Product States
|ψi hψ˜|
Figure 5.9: The scalar product hψ˜|ψi of two (possibly different) states represented as MPS. The optimal contraction order is sideways, e.g., from left to right, as indicated by the shading and Eq. (5.30).
Scalar Product
5.3.3
In order to calculate the overlap of two states, which is needed, e.g., to calculate the norm of a state, we need to evaluate the scalar product between two MPSs that is defined as
hψ˜|ψi= X
σ1···σL, σ10···σL0
hσ01· · ·σ0L| X
m0,...,mL
M˜σL;m0L L
−1,mL· · ·M˜σ1;m01 0,m1M1;mσ1 0,m1· · ·ML,mσL L
−1,mL|σ1· · ·σLi . (5.28) Because the result of every tensor contraction in Eq. (5.28) is a number andhσ10 · · ·σL0|σ1· · ·σLi= δσ0
1···σ0L,σ1···σL, the sum shrinks to hψ˜|ψi= X
σ1···σL, m0,...,mL
M˜σL;mL
L−1,mL· · ·M˜σ1;m1 0,m1M1;mσ1 0,m1· · ·ML,mσL
L−1,mL . (5.29) The computation of this expression is costly, because it consists ofdL·(2L−1)tensor contractions.
Hence, the complexity increases exponentially with the system size. It is therefore beneficial to reformulate the expression into
hψ˜|ψi= X
σL,mL−1, mL
M˜σ1;mLL
−1,mL · · · X
σ1,m0
M˜σ1;m1 0,m1M1;mσ1 0,m1
!
· · ·
! ML;mσL L
−1,mL, (5.30) as depicted in Fig. 5.9. For the evaluation of Eq. (5.30), only dLadditions and 2dL−dtensor contractions are necessary, i.e., the evaluation scales polynomially with the system size. Note that the sum over the dummy index mL is distinct, because it does not follow the rule for the intermediate sums.
Addition and Scaling of MPSs
5.3.4
In order to add two MPSs|ψiwith site tensorsMj and|ψ˜iwith site tensorsM˜j, as it is necessary in the Lanczos algorithm (Sec. 4.2), the direct sum of all site tensors needs to be performed. That
Section 5.3. Operations on MPSs 41
Note that the dimension of the new state|ψ0iis given by the sum of the dimensions of the initial states. The indicies of the new state are therefore given bym0j =mj+ ˜mj.
If the goal is only to scale the MPS, this approach is far from optimal, because the dimension is doubled while it could stay constant by rescaling the norm of the MPS. Such a rescaling can for example be performed at the end of a truncation sweep.
Reduced Density Operator and Entanglement Entropy
5.3.5
A last quantity that can be accessed directly from an MPS is the reduced density operator for a specific partition and the closely related von Neumann entanglement entropy or short entanglement entropy. The density operator of a pure state|ψi is given by
ˆ
ρ=|ψi hψ| . (5.34)
In order to obtain the reduced density operator of a specific part, e.g., partA that includes all sites left of sitej, the other part (B with all sites right of site j−1) need to be traced out
ˆ
ρA=TrB|ψi hψ| . (5.35)
This quantity becomes particularly accessible if we express the state in its Schmidt decomposition (Eq. (5.17)),
And analogously, we obtain the reduced density operator for the partB, ˆ
ρB =TrA|ψi hψ|=X
s
Σ2s|siB Bhs| . (5.39)
42 Chapter 5. Matrix-Product States
Note that the reduced density operators share the same spectrum, but of course, act on different parts of the system. Furthermore, this spectrum is given by the square of the singular values of the SVD of an active sitej, if a truncation sweep to the left is performed.3
At this point, we can directly read off the von-Neumann entanglement entropy SA|B(|ψi) =−TrρAlog2ρA=−X
s
Σ2slog2Σ2s, (5.40) which is a quantity that describes how much|ψi differs from a product state, i.e., how entangled the parts Aand B of the system are with each other.