## Learn to Use ITensor

# Algorithms for MPS and MPO (also IQMPS and IQMPO)

## Summing MPS

`sum(MPS psi1, MPS psi2, Args args = Args::global()) -> MPS`

`sum(IQMPS psi1, IQMPS psi2, Args args = Args::global()) -> IQMPS`

Return the sum of the MPS psi1 and ps2. The returned MPS will have an orthogonality center on site 1. Before being returned, the MPS representing the sum will be compressed using truncation parameters provided in the named arguments

`args`

.Show Exampleauto psi3 = sum(psi1,psi2,{"Maxm",500,"Cutoff",1E-8});

`sum(vector<MPS> terms, Args args = Args::global()) -> MPS`

`sum(vector<IQMPS> terms, Args args = Args::global()) -> IQMPS`

Returns the sum of all the MPS provided in the vector

`terms`

as a single MPS, using the truncation accuracy parameters (such as "Cutoff" or "Maxm") provided in the named arguments`args`

to control the accuracy of the sum.This function uses a hierarchical, tree-like algorithm which first sums pairs of MPS, then pairs of pairs, etc. so that the largest bond dimensions are only reached toward the end of the process for maximum efficiency. Therefore using this algorithm can be much faster than calling the above two-argument

`sum`

function to sum the terms one at a time.Show Exampleauto terms = vector<MPS>(4); terms.at(0) = psi0; terms.at(1) = psi1; terms.at(2) = psi2; terms.at(3) = psi3; auto res = sum(terms,{"Cutoff",1E-8});

## Overlaps, Matrix Elements, and Expectation Values

`overlap(MPS psi1, MPS psi2) -> Real`

`overlap(IQMPS psi1, IQMPS psi2) -> Real`

`overlapC(MPS psi1, MPS psi2) -> Cplx`

`overlapC(IQMPS psi1, IQMPS psi2) -> Cplx`

Compute the exact overlap @@\langle \psi_1|\psi_2 \rangle@@ of two MPS or IQMPS. If the overlap value is expected to be a complex number use

`overlapC`

.The algorithm used scales as @@m^3 d@@ where @@m@@ is typical bond dimension of the MPS and @@d@@ is the site dimension.

(In ITensor version 1.x this function was called

`psiphi`

. This name is still supported for backwards compatibility.)`overlap(MPS psi1, MPO W, MPS psi2) -> Real`

`overlap(IQMPS psi1, IQMPO W, IQMPS psi2) -> Real`

`overlapC(MPS psi1, MPO W, MPS psi2) -> Cplx`

`overlapC(IQMPS psi1, IQMPO W, IQMPS psi2) -> Cplx`

Compute the exact overlap (or matrix element) @@\langle \psi_1|W|\psi_2 \rangle@@ of two MPS psi1 and psi2 with respect to an MPO W.

The algorithm used scales as @@m^3\, k\,d + m^2\, k^2\, d^2@@ where @@m@@ is typical bond dimension of the MPS, @@k@@ is the typical MPO dimension, and @@d@@ is the site dimension.

(In ITensor version 1.x this function was called

`psiHphi`

. This name is still supported for backwards compatibility.)`overlap(MPS psi1, MPO W1, MPO W2, MPS psi2) -> Real`

`overlap(IQMPS psi1, IQMPO W1, IQMPO W2, IQMPS psi2) -> Real`

`overlapC(MPS psi1, MPO W1, MPO W2, MPS psi2) -> Cplx`

`overlapC(IQMPS psi1, IQMPO W1, IQMPO W2, IQMPS psi2) -> Cplx`

Compute the exact overlap (or matrix element) @@\langle \psi_1|W_1 W_2 |\psi_2 \rangle@@ of two MPS psi1 and psi2 with respect to two MPOs W1 and W2.

The algorithm used scales as @@m^3\, k^2\,d + m^2\, k^3\, d^2@@ where @@m@@ is typical bond dimension of the MPS, @@k@@ is the typical MPO dimension, and @@d@@ is the site dimension.

(In ITensor version 1.x this function was called

`psiHKphi`

. This name is still supported for backwards compatibility.)

## Multiplying MPOs

`nmultMPO(MPO A, MPO B, MPO & C, Args args = Args::global())`

`nmultMPO(IQMPO A, IQMPO B, IQMPO & C, Args args = Args::global())`

Multiply MPOs A and B. On return, the result is stored in C. MPO tensors are multiplied one at a time from left to right and the resulting tensors are compressed using the truncation parameters (such as "Cutoff" and "Maxm") provided through the named arguments

`args`

.Show ExampleMPO C; nmultMPO(A,B,C,{"Maxm",500,"Cutoff",1E-8});

## Applying MPO to MPS

`applyMPO(MPO K, MPS psi, Args args = Args::global()) -> MPS`

`applyMPO(IQMPO K, IQMPS psi, Args args = Args::global()) -> IQMPS`

Apply an MPO K to an MPS psi, resulting in an approximation to the MPS phi:

@@|\phi\rangle = K |\psi\rangle@@ .

The resulting MPS is returned. The algorithm used is chosen with the parameter "Method" in the named arguments`args`

.The default algorithm used is the "density matrix" algorithm, chosen by setting the parameter "Method" to "DensityMatrix". If the input MPS has a typical bond dimension of @@m@@ and the MPO has typical bond dimension @@k@@ , this algorithm scales as @@m^3 k^2 + m^2 k^3@@ .

No approximation is made when applying the MPO, but after applying it the resulting MPS is compressed using the truncation parameters provided in the named arguments

`args`

.An alternative algorithm can be chosen by setting the parameter "Method" to "Fit". This is a sweeping algorithm that iteratively optimizes the resulting MPS @@|\phi\rangle@@ (analogous to DMRG). This algorithm has better scaling in the MPO bond dimension @@k@@ compared to the "DensityMatrix" method, but is not guaranteed to converge (depending on the input MPO and MPS). The number of sweeps can be chosen with the parameter "Nsweep".

It is recommended to try the default "DensityMatrix" first because it is more reliable. Then, the "Fit" method can be tried if higher performance is required.

Named arguments recognized:

`"Method"`

— (default: "DensityMatrix") algorithm used for applying the MPO to the MPS. Currently available options are- "DensityMatrix"
- "Fit"

`"Cutoff"`

— (default: 1E-13) truncation error cutoff for compressing resulting MPS`"Maxm"`

— maximum bond dimension of resulting compressed MPS`"Verbose"`

— (default: false) if true, prints extra output`"Normalize"`

— (default: false) choose whether or not to normalize the output wavefunction`"Nsweep"`

— (default: 1) sets the number of sweeps of the "Fit" algorithm

Show Example//Use the method "DensityMatrix" auto phi = applyMPO(K,psi,{"Method=","DensityMatrix","Maxm=",100,"Cutoff=",1E-8}); //Use the method "Fit" with 5 sweeps auto phi2 = applyMPO(K,psi,{"Method=","Fit","Maxm=",100,"Cutoff=",1E-8,"Nsweep=",5});

`applyMPO(MPO K, MPS psi, MPS phi, Args args = Args::global()) -> MPS`

`applyMPO(IQMPO K, IQMPS psi, IQMPS phi, Args args = Args::global()) -> IQMPS`

Similar to

`applyMPO`

above, but accepts a guess for the output wavefunction (the guess wavefunction`phi`

is not overwritten).Currently, this version of

`applyMPO`

only accepts "Fit" for the parameter "Method". Choosing a good guess state`phi`

can improve the convergence of the "Fit" method.Show Example//Use the method "Fit" with 5 sweeps and a guess state phi auto Kpsi = applyMPO(K,psi,phi,{"Method=","Fit","Maxm=",100,"Cutoff=",1E-8,"Nsweep=",2});

`checkMPOProd(MPS psi2, MPO K, MPS psi1) -> Real`

`checkMPOProd(IQMPS psi2, IQMPO K, IQMPS psi1) -> Real`

Computes, without approximation, the difference @@||\, |\psi_2\rangle - K |\psi_1\rangle ||^2@@ , where K is an arbitrary MPO. This is especially useful for testing methods for applying an MPO to an MPS.

Show Example//Approximate K*psi auto phi = applyMPO(K,psi,{"Maxm=",200,"Cutoff=",1E-12}); //Check Print(checkMPOProd(phi,K,psi)); //should be close to zero

*This page current as of version 2.0.7*

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