Errors for handling breakdowns, or near breakdowns in the lanczos spectrum method.

[LynBar]

Problem Description

It would be nice to know when a breakdown occurs in the lanczos method #463 . At the moment it checks for convergence, but the method can suffer a breakdown whenever the two bi-orthogonal basis vectors are orthogonal or nearly orthogonal. Might be a good idea to check if the two initial vectors are orthogonal as well.

Proposed Solution

Check whenever the two basis vectors are orthogonal or nearly orthogonal and either break the for loop, or error.
Just below

QuantumToolbox.jl/src/spectrum.jl

Line 272 in 3028fe0

δₖ = sqrt(abs(buf))

if abs(δₖ) < breakdown_tol; error("Breakdown: Two basis vectors are orthogonal") end

Alternate Solutions

No response

Additional Context

No response

[albertomercurio]

Hi think that this is already handled by the convergence criteria. If we reach orthogonality, the iteration stops.

Did you mean something different instead?

[LynBar]

If during any iteration step abs(dot(wₖ, vₖ))==0 the process suffers a breakdown, with βₖ = buf / δₖ = Inf. If the vectors are very nearly orthogonal abs(dot(wₖ, vₖ))<ε, this will contribute a lot of numerical noise.
For the initial vectors w and v, if they are orthogonal the greens function normalisation factor gfNorm=0 leading to immediate breakdown.
I think it would be nice to have some indicator for when this is the case to save time chasing bugs.

[albertomercurio]

Maybe @matteosecli can give some insights here.

[matteosecli]

Hi all!

The Lanczos continued fraction expansion, which looks like

$$f(\omega) \doteqdot \left( \omega - i\hat{\mathcal{L}} \right)^{-1}_{11} = \cfrac{1}{\omega-\alpha_1-\cfrac{\beta_2\delta_2}{\omega-a_2-\cfrac{\beta_3\delta_3}{\omega-\alpha_3-\cdots}}} \,.$$

should indeed auto-stop whenever $\braket{w_k|v_k} = \beta_k \delta_k = 0$, so the loss of orthogonality is not really a big deal for this specific application; it just stops the continued fraction expansion, so that the current estimate of the continued fraction becomes equal to the one at the previous step.

However, we are also calculating Euler sequences $A_k$ and $B_k$ at each step, in order to have very cheap estimates of the continued fraction itself. In calculating these at every step, we also need $\braket{w_k|\mathcal{L}|v_k} = \alpha_k$, which is ill-defined because $\ket{v_k}$ and $\bra{w_k}$ have just been renormalized by their own scalar product, which is zero. So yes, we should check for $\braket{w_k|v_k} = 0$ and manually stop the algorithm.

As for the initial vectors, adding a check costs nothing and it's always a good idea. However, if the initial vectors are orthogonal, it would mean that you are in some degenerate case or not using proper operators, so that's really something that the user should fix. The reason is that this Lanczos procedure doesn't start from random initial vectors for the left/right subspaces, but from physically meaning ones. Incidentally, I've also realized that the initial vectors were not properly normalized to 1, with a potential complex phase still lingering around, so I've fixed that as well.

I've drafted a tentative PR with these changes over at #502; @LynBar would these changes cover the points you've raised?

[LynBar]

Yes! This is perfect, thanks.