Summary of the available algorithms

Time Dependent Variational Principle (TDVP)

Emu-mps uses a second order 2-site TDVP to compute the time-evolution of the system (see here for details). Briefly, the algorithm repeatedly computes the time-evolution for 2 neighbouring qubits while truncating the resulting MPS to keep the state small. It does this by

evolving qubit 1 and 2 forwards in time by \(dt/2\)
evolving qubit 2 backwards by \(dt/2\)
evolving qubit 2 and 3 forwards in time by \(dt/2\)

...

evolving qubit \(n-1\) and \(n\) forward in time by \(dt\)
evolving qubit \(n-1\) backwards in time by \(dt/2\)
evolving qubit \(n-2\) and \(n-1\) forward in time by \(dt/2\)

...

evolving qubit 1 and 2 forwards in time by \(dt/2\)

The fact that we sweep left-right and the right-left with timesteps of \(dt/2\) makes this a second-order TDVP.

Density Matrix Renormalization Group (DMRG)

DMRG is a powerful variational method for finding the ground state and the first few excited states of strongly correlated \(1\)D and \(2\)D quantum many-body systems. More precisely, DMRG finds an MPS representation of the low-energy eigenstates by variationally optimizing MPS tensors to minimize the energy of the system. For a more detailed description of the algorithm, please refer to [1].

How DMRG can be useful in the emulators context

In the emu-mps context, we use DMRG to replicate the dynamics of an adiabatically long Pulser sequence, thereby computing the ground state of the time-dependent Hamiltonian along the pulse. This can be done because according to the adiabatic theorem, a sufficiently slow (adiabatic) drive keeps the system close to the ground state of the time-dependent Hamiltonian, therefore sweeping DMRG over the instantaneous Hamiltonian at successive times provides an efficient approximation to the adiabatic evolution. For a more detailed description of the adiabatic theorem please refer to [2] and [3].

The algorithm

Emu-mps implements a two-site DMRG. The algorithm repeatedly optimizes two neighbouring MPS tensors, while controlling the bond dimension by truncating the singular values up to some precision configured by the user. To be more precise:

At each two-site step the MPS tensors are contracted over the common index, and the effective Hamiltonian for that two-site block is minimized using the Lanczos eigensolver[4].
The optimized block is split back into two tensors via SVD. This is where the truncation procedure takes place.
The algorithm performs left-to-right and right-to-left sweeps until the ground-state energy converges.

Sweeping mechanism

minimize qubits \(1\) and \(2\) (left --> right)
minimize qubits \(2\) and \(3\) (left --> right)
...
minimize qubits \(l-1\) and \(l\) (left --> right)
minimize qubits \(l\) and \(l-1\) (right --> left)
...
minimize qubits \(3\) and \(2\) (right --> left)
minimize qubits \(2\) and \(1\) (right --> left)

------ End of one full sweep ------

At the end of the sweep (back and forth), check energy convergence:

If converged, move to the next time-step.
If not converged, restart the sweeping procedure.