Choosing and checking the parameters

Choosing them

Three parameters set the error of a result, and the user chooses all three: the truncation of the states, limits, the time step of an evolution, nsteps or time_step, and the number of sweeps of a search, Dmrg(; nsweeps). No value is right in general; the way to choose them is the same everywhere:

Start small: a maxdim the machine handles in seconds, the default cutoff, which only drops what rounding leaves, a time step a tenth of the shortest time scale of the evolver, and for a search a few sweeps with maxdim growing along them, as Limits(maxdim = [10, 20, 50, 100]).
Measure along the run what tells the error, MaxLinkdim, EntanglementEntropy, and on a density matrix TraceDeviation, HermiticityDeviation and Purity, see below.
Improve one parameter at a time, doubling maxdim, halving the time step, adding sweeps, and run again: the digits that do not move are those the result has. When MaxLinkdim reaches maxdim, the truncation is what limits the result.
Keep the cheapest parameters giving the digits needed, and say which checks they passed.

A density matrix needs a larger bond dimension than a wave function, that of a pure state already the square of the bond dimension of its wave function, and a conserved quantity declared on the sites, see Conserving a quantity, makes a given bond dimension cheaper.

Checking the accuracy

Three main approximations set the error of a result, and the user chooses all three: the truncation of the states, given by limits, the time step of an evolution or of thermalize, and the convergence of a search, the sweeps of its Dmrg. A result can be trusted once it no longer moves when each of them is improved: larger limits, a smaller time step, more sweeps. The measurements taken along a run tell which of them limits the accuracy. Two more have defaults: the tolerance of the Krylov method of each local step, see Krylov, and the truncation mpo_limits of the MPO of the operator of a solver, $L^\dagger L$ for steady_state, none by default: truncated, a result is that of the truncated operator, and the value steady_state returns its residual.

Truncation

MaxLinkdim tells when the bond dimension reaches maxdim, from which point the results depend on it, and EntanglementEntropy(cut), the operator space entanglement entropy on a mixed state, shows the entanglement that makes the bond dimension grow. A truncation by cutoff shows in neither, and only running again tells it. Limits() discards the singular values below about 1e-13 of the norm, which keeps rounding from making the bond dimension grow and may leave an error of that order: the truncation that makes a computation affordable is a choice, checked by running again with a larger maxdim and a smaller cutoff.

Time step

An evolution is checked by halving its time step, and by comparing Tdvp with ApproxW, whose errors have different origins. ApproxW of order k makes an error of order $\tau^k$ in the time step $\tau$ when the evolver does not depend on time. When it does, Tdvp and ApproxW of order 1 or 2 take its functions at the middle of each step, which caps them at order 2, while ApproxW of order 3 or 4 keeps the order of its approximation, see Time dependent evolvers. Tdvp projects the evolution on the states of the bond dimension the state has, its steps on two sites letting that dimension grow, and it enlarges the bonds of a state whose bond dimension is below 5 by a global Krylov expansion before a step: from a state of small bond dimension, as a product state, or for thermalize, compare with expand_period = 1, which expands before every step. A thermal state has checks of its own, see Thermal states.

Indicators of a density matrix

The exact density matrix is hermitian under a Lindbladian, under gates and noisy gates, which act on both of its sides, and for a thermal or a steady state, and its trace stays one when the evolution preserves the trace, as a Lindbladian does. What deviates from that is error, which three measurements of the state follow along a run:

TraceDeviation, $1 - \mathrm{tr}\,\rho$, the drift of the trace. Expectation values are divided by the trace, which hides the drift but not the error it comes from.
HermiticityDeviation, the squared norm of the anti-hermitian part of $\rho$, relative to that of $\rho$. That part being error, the square root of HermiticityDeviation is a lower bound of the error relative to the norm of $\rho$, in the Hilbert-Schmidt norm: a value of 1e-6 proves an error of 1e-3 at least. It bounds the error from below only, an error without an anti-hermitian part going unseen. hermitianize_period removes the anti-hermitian part, and with it what HermiticityDeviation shows: measure it before deciding to hermitianize.
Purity, $\mathrm{tr}(\rho^\dagger \rho) / (\mathrm{tr}\,\rho)^2$, lies between $1/D$ and 1 for a density matrix of dimension $D$. Above 1, the state is no longer a density matrix: it has lost its positivity, which nothing in a matrix product state guarantees, unless HermiticityDeviation shows that it has lost its hermiticity.

Conserved quantities

A quantity declared conserved on the sites, see Conserving a quantity, keeps the errors out of the sectors the exact state never reaches: the tensors have no blocks there, so that neither truncation nor rounding can put anything in them. A pure state keeps its charge exactly, a mixed state under a weak symmetry keeps no coherence between two sectors, and under a strong one stays in its sector. A quantity the exact evolution conserves without its being declared, as the energy under a Hamiltonian, which no site can declare, is still a check: its drift along a run is error, though its constancy proves nothing.

Searches

A search runs the sweeps of its Dmrg, and stops earlier once the energy changes by less than its tol between two sweeps, which a search stuck in a metastable state does as well. Variance(hamiltonian), zero exactly for an eigenstate, checks the state found, see variance, but does not tell the ground state from another eigenstate, which searches from other states, or in other sectors, do. Extrapolating the energy to zero variance as maxdim grows estimates the exact energy, and the error of the last one.

The value steady_state returns, the residual $\|L\rho\|^2$ for $\rho$ of unit Hilbert-Schmidt norm, is zero for a steady state. The state it finds is hermitian when the steady state is unique and the search has converged, and it warns when its HermiticityDeviation exceeds 1e-6: a state that is not hermitian comes from a search that has not converged, or from a Lindbladian with several steady states, of which dmrg returns any combination: one in each sector of a quantity that its Hamiltonian and its jump operators all commute with, for instance. The one a system reaches depends on the state it starts from: it is the limit of its evolution in time, when that converges, which evolve gives. When the quantity is one a site can conserve, declaring it strong keeps steady_state in the sector of the starting state, see Conserving a quantity. A hermitian state does not prove the steady state unique.

Against an exact computation

On a small system, about a dozen qubits, the exact result is at hand: matrix(op, system) is the dense matrix of an operator, its Jordan-Wigner strings laid, Vector(ψ) the vector of a pure state on the same basis, and reduced_density_matrix(ρ, 1:n) the density matrix of a mixed one, which LinearAlgebra diagonalizes or exponentiates:

λ = eigvals(Hermitian(matrix(hamiltonian, System(10, Qubit()))))

A script checked so on a small system then runs at the sizes no exact computation reaches.