Observer System for DMRG

An observer is an object which can be passed to the ITensor DMRG algorithm, to allow measurements to be performed throughout the DMRG calculation and to set conditions for early stopping of DMRG.

The only requirement of an observer is that it is a subtype of AbstractObserver. But to do something interesting, it should also overload at least one the methods measure! or checkdone!.

A general purpose observer type called DMRGObserver is included with ITensors which already provides some quite useful features. It accepts a list of strings naming local operators to be measured at each step of DMRG, with the results saved for later analysis. It also accepts an optional energy precision, and stops a DMRG calculation early if the energy no longer changes to this precision. For more details about the DMRGObserver type, see the DMRGObserver documentation page.

Defining a Custom Observer

To define a custom observer, just make a struct with any name and internal fields you would like, and make this struct a subtype of AbstractObserver.

For example, let's make a type called DemoObserver as:

using ITensors, ITensorMPS

mutable struct DemoObserver <: AbstractObserver
   energy_tol::Float64
   last_energy::Float64

   DemoObserver(energy_tol=0.0) = new(energy_tol,1000.0)
end

In this minimal example, our DemoObserver contains a field energy_tol which we can use to set an early-stopping condition for DMRG, and an field last_energy which our observer will use internally to keep track of changes to the energy after each sweep.

Now to give our DemoObserver type a useful behavior we need to define overloads of the methods measure! and checkdone!.

Overloading the checkdone! method

Let's start with the checkdone! method. After each sweep of DMRG, the checkdone! method is passed the observer object, as well as a set of keyword arguments which currently include:

• energy: the current energy
• psi: the current wavefunction MPS
• sweep: the number of the sweep that just finished
• outputlevel: an integer stating the desired level of output

If the checkdone! function returns true, then the DMRG routine stops (recall that checkdone! is called only at the end of a sweep).

In our example, we will just compare the energy keyword argument to the last_energy variable held inside the DemoObserver:

function ITensorMPS.checkdone!(o::DemoObserver;kwargs...)
  sw = kwargs[:sweep]
  energy = kwargs[:energy]
  if abs(energy-o.last_energy)/abs(energy) < o.energy_tol
    println("Stopping DMRG after sweep $sw")
    return true
  end
  # Otherwise, update last_energy and keep going
  o.last_energy = energy
  return false
end

(Recall that in order to properly overload the default behavior, the checkdone! method has to be imported from the ITensors module or preceded with ITensors.)

Overloading the measure! method

The other method that an observer can overload is measure!. This method is called at every step of DMRG, so at every site and for every sweep. The measure! method is passed the current observer object and a set of keyword arguments which include:

• energy: the energy after the current step of DMRG
• psi: the current wavefunction MPS
• bond: the bond b that was just optimized, corresponding to sites (b,b+1) in the two-site DMRG algorithm
• sweep: the current sweep number
• sweep_is_done: true if at the end of the current sweep, otherwise false
• half_sweep: the half-sweep number, equal to 1 for a left-to-right, first half sweep, or 2 for the second, right-to-left half sweep
• spec: the Spectrum object returned from factorizing the local superblock wavefunction tensor in two-site DMRG
• outputlevel: an integer specifying the amount of output to show
• projected_operator: projection of the linear operator into the current MPS basis

For our minimal DemoObserver example here, we will just make a measure! function that prints out some of the information above, but in a more realistic setting one could use the MPS psi to perform essentially arbitrary measurements.

function ITensorMPS.measure!(o::DemoObserver; kwargs...)
  energy = kwargs[:energy]
  sweep = kwargs[:sweep]
  bond = kwargs[:bond]
  outputlevel = kwargs[:outputlevel]

  if outputlevel > 0
    println("Sweep $sweep at bond $bond, the energy is $energy")
  end
end

Calling DMRG with the Custom Observer

After defining an observer type and overloading at least one of the methods checkdone! or measure! for it, one can construct an object of this type and pass it to the ITensor dmrg function using the observer keyword argument.

Continuing with our DemoObserver example above:

obs = DemoObserver(1E-4) # use an energy tolerance of 1E-4
energy, psi = dmrg(H,psi0,sweeps; observer=obs, outputlevel=1)

Complete Sample Code

using ITensors, ITensorMPS

mutable struct DemoObserver <: AbstractObserver
   energy_tol::Float64
   last_energy::Float64

   DemoObserver(energy_tol=0.0) = new(energy_tol,1000.0)
end

function ITensorMPS.checkdone!(o::DemoObserver;kwargs...)
  sw = kwargs[:sweep]
  energy = kwargs[:energy]
  if abs(energy-o.last_energy)/abs(energy) < o.energy_tol
    println("Stopping DMRG after sweep $sw")
    return true
  end
  # Otherwise, update last_energy and keep going
  o.last_energy = energy
  return false
end

function ITensorMPS.measure!(o::DemoObserver; kwargs...)
  energy = kwargs[:energy]
  sweep = kwargs[:sweep]
  bond = kwargs[:bond]
  outputlevel = kwargs[:outputlevel]

  if outputlevel > 0
    println("Sweep $sweep at bond $bond, the energy is $energy")
  end
end

let
  N = 10
  etol = 1E-4

  s = siteinds("S=1/2",N)

  a = OpSum()
  for n=1:N-1
    a += "Sz",n,"Sz",n+1
    a += 0.5,"S+",n,"S-",n+1
    a += 0.5,"S-",n,"S+",n+1
  end
  H = MPO(a,s)
  psi0 = random_mps(s;linkdims=4)

  nsweeps = 5
  cutoff = 1E-8
  maxdim = [10,20,100]

  obs = DemoObserver(etol)

  println("Starting DMRG")
  energy, psi = dmrg(H,psi0; nsweeps, cutoff, maxdim, observer=obs, outputlevel=1)

  return
end