Exponential

[sanderdemeyer]

This implements the exponential of a matrix for both BLASFloats and BigFloats.

I have named these functions exponential and exponential!, instead of the usual exp and exp! from LinearAlgebra. Extending these methods while keeping the current structure using @algdef and @ functiondef results in some naming conflicts. The default for BLASFloats is to use LinearAlgebra.exp!. In TensorKit, we can still stick to the exp naming convention.

[codecov]

Codecov Report

❌ Patch coverage is 0% with 109 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/implementations/exponential.jl	0.00%	87 Missing ⚠️
src/interface/exponential.jl	0.00%	9 Missing ⚠️
src/interface/matrixfunctions.jl	0.00%	8 Missing ⚠️
ext/MatrixAlgebraKitGenericSchurExt.jl	0.00%	3 Missing ⚠️
src/common/view.jl	0.00%	2 Missing ⚠️

Files with missing lines	Coverage Δ
src/MatrixAlgebraKit.jl	100.00% <ø> (ø)
src/common/view.jl	7.40% <0.00%> (-92.60%)	⬇️
ext/MatrixAlgebraKitGenericSchurExt.jl	0.00% <0.00%> (-100.00%)	⬇️
src/interface/matrixfunctions.jl	0.00% <0.00%> (ø)
src/interface/exponential.jl	0.00% <0.00%> (ø)
src/implementations/exponential.jl	0.00% <0.00%> (ø)

... and 36 files with indirect coverage changes

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[Jutho]

Reduced allocation strategy:

Suggested change

	copyto!(expA, V * Diagonal(exp.(diagview(D))) * inv(V))
	iV = inv(V)
	map!(exp, diagview(D))
	mul!(expA, rmul!(V, D), iV)

[sanderdemeyer]

It has to be map!(exp, diagview(D), diagview(D)) instead of map!(exp, diagview(D)), but good suggestion otherwise. I have also added it for the ExponentialViaEig.
EDIT: the suggested change works only for Julia 1.12 onwards. That's why I will keep the version with
3 arguments.

[lkdvos]

Why not just diagview(D) .= exp.(diagview(D))?

[sanderdemeyer]

Is that more efficient than the current code? If not, I'd prefer to keep it that way, since it feels a bit more natural to me.

[lkdvos]

Overall I'm not fully convinced by the interface of exponential(!), especially in its current form and implementation this looks slightly strange.

LinearAlgebra uses an in-place version, i.e. it reuses the input array to return exp!, and looking at the different implementations you have here, it is not obvious that trying to fit this into a exponentiate!(A, expA, alg) signature is really helping us - on the contrary, all this is really doing is creating an additional copy at the end just to make sure that it is allocated in the provided output.
As we discussed for your previous PR, this really is not the purpose of being able to provide the output argument.

For the algorithms, thinking a bit ahead, it might be appropriate to just call these something along the lines of matrix functions via eig, since presumably these approaches are actually generic for all of these implementations.

[lkdvos]

Why not just diagview(D) .= exp.(diagview(D))?

[sanderdemeyer]

Overall I'm not fully convinced by the interface of exponential(!), especially in its current form and implementation this looks slightly strange.

LinearAlgebra uses an in-place version, i.e. it reuses the input array to return exp!, and looking at the different implementations you have here, it is not obvious that trying to fit this into a exponentiate!(A, expA, alg) signature is really helping us - on the contrary, all this is really doing is creating an additional copy at the end just to make sure that it is allocated in the provided output. As we discussed for your previous PR, this really is not the purpose of being able to provide the output argument.

The idea of putting it in this framework is to allow for different sorts of algorithms, i.e. ones that would also work for BigFloats. I get that in the BLASFloat case, we should avoid extra allocations, but could you elaborate on your suggestion for ExponentialViaLA? Do you just want to get rid of the preallocated output?

For the algorithms, thinking a bit ahead, it might be appropriate to just call these something along the lines of matrix functions via eig, since presumably these approaches are actually generic for all of these implementations.

I may be missing your point here, but that was the idea behind the ExponentialViaEigh and ExponentialViaEig naming conventions? Or do you really want separate names for each eig_alg?

[lkdvos]

The idea of putting it in this framework is to allow for different sorts of algorithms, i.e. ones that would also work for BigFloats. I get that in the BLASFloat case, we should avoid extra allocations, but could you elaborate on your suggestion for ExponentialViaLA? Do you just want to get rid of the preallocated output?

I am indeed referring to the preallocated output, not the algorithms part. It really only makes sense to have the option of giving a preallocated output if we are actually able to use this, and for the current implementations you have this is not saving us any work, rather it is increasing it because you add an extra allocation at the beginning and an extra copy at the end.

While it is definitely possible to have initialize_output(exponentiate!, A) = A to just allocate the result in-place, I still wonder if it is then useful to have to implement the boiler plate for fitting it in a framework that we are mostly going to bypass anyways.

I may be missing your point here, but that was the idea behind the ExponentialViaEigh and ExponentialViaEig naming conventions? Or do you really want separate names for each eig_alg?

Sorry I should have explained that better, I meant that I would like to avoid having to also define SqrtViaEig, SinViaEig, ..., and simply have some algorithm that signifies "solve this by doing an eigenvalue decomposition".

[sanderdemeyer]

I am indeed referring to the preallocated output, not the algorithms part. It really only makes sense to have the option of giving a preallocated output if we are actually able to use this, and for the current implementations you have this is not saving us any work, rather it is increasing it because you add an extra allocation at the beginning and an extra copy at the end.

While it is definitely possible to have initialize_output(exponentiate!, A) = A to just allocate the result in-place, I still wonder if it is then useful to have to implement the boiler plate for fitting it in a framework that we are mostly going to bypass anyways.

Is your suggestion then to just skip the whole @ functiondef and other general frameworks we have to define exponentiate and exponentiate!, or to keep the current framework somewhat, but just remove the preallocated output argument?

Sorry I should have explained that better, I meant that I would like to avoid having to also define SqrtViaEig, SinViaEig, ..., and simply have some algorithm that signifies "solve this by doing an eigenvalue decomposition".

Okay, I see. I agree and will change this.

[Jutho]

Regarding the algorithm names, I agree with Lukas and also think we want to have a general
MatrixFunctionViaEig, though it is useful to know about which function exactly, because some functions map real matrices to real matrices (even if complex eigenvalues are involved), and others do not. I think these are the two major categories, so we could just have two different algorithms for that, but I don't know good names for that.

Regarding the role of the output arguments, I only partially agree. The whole point of why we started MatrixAlgebraKit.jl, is because in TensorKit we first want to define the output tensor, and then compute block per block the result, where we want to store the result in the corresponding block of the output tensor. Ideally, yes, the computation is such that we also use that output data as storage during the computation, in such a way that the end result "naturally" ends up there, but if that is difficult, a final copy! can still be useful. We can then always try to improve this later on behind the scenes, but at least TensorKit can be agnostic about this.

Note that the LinearAlgebra exp! is also cheating, and typically also ends up just copying the final result back into A. There is no way to actually compute exp(A) fully in-place. The typical approach via Pade approximations has a ton of allocations. I think I actually have a slightly allocation-friendlier implementation lying around for some things we did in CMPSKit.jl

[Jutho]

Regardless of the comment about general matrix functions, it is a fact that the exponential is by far the most useful and common one that we need, so I am also not opposed to first thinking carefully about this one, and having some part of the implementation be specific for matrix exponentials.

In particular, one important consideration that we might want to include in this design, that is specific to our use case, is that we also might be interested in computing exp(-im* δt*H) for real time evolution, where it is probably useful to use eigh for the Hermitian matrix H, but still the end result will be complex. Julia has a cis function for cis(x) = exp(im * x), but unfortunately that comes without a minus sign in the argument. But maybe cis(-δt * H) is not too bad.

[lkdvos]

To comment on the TensorKit interaction, I definitely agree with the purpose, but this is not actually currently the design we ended up with.
Our interface very clearly states that we may use the provided output, but are in no way bound to do so.
This is reflected in the TensorKit implementation, see e.g. https://github.com/QuantumKitHub/TensorKit.jl/blob/feab2072b2909b9a127d871ef58c36e83623c2fc/src/factorizations/matrixalgebrakit.jl#L39
I'm of course happy to rediscuss that design decision, but I'm actually quite happy with how that is shaping out.

So basically there are two comments I have:

On the one hand, there is the question about whether or not there are implementations that benefit from providing an additional output array.
From looking at the implementations here, this is not obvious at all, actually it seems like instead we are simply allocating an additional array at the end to copy the results into, which we might as well have reused the A for, since we already "destroyed" that data anyways.
I am not saying that the implementations have to be fully allocation-free (the LAPACK routines still have workspaces too), I just think that if providing the output is a way to avoid having to do an allocation, it really shouldn't be increasing the allocations instead.
If we can't come up with a case where having this additional array actually helps us out, it might be reasonable to simplify define exponentiate!(A, alg) -> expA where expA === A instead, which would still work for TensorKit in the exact same way.
I definitely agree that this can be achieved by hacking it into the current interface as well, by simply having initialize_output(exponentiate!, A) = A, I just wanted to open up the discussion about whether or not that is something we like, or think this is just boilerplate we aren't using.
Giving it some thought, I'm definitely okay with the initialize_output(...) = A default and going from there, i.e. keeping @functiondef and "hacking" this into that.

On the other hand, given that interface, if there is no way of naturally making the output end up in the provided destination, I would really like to avoid ending up with a final copy!(provided_dest, computed_exp) at the end, even as a "for now" implementation that we can improve on later.
Given that we have to check for in-placeness in TensorKit anyways, I would in this case much rather have initialize_output() = nothing, and simply return the computed_exp instead.
This is the exact same comment and solution as for the generic linear algebra wrappers.

[Jutho]

I guess I am a bit confused, because most of the implementations now do actually perform the final step in the calculation in such a way that the result is directly stored in the output array, no? It is only the algorithm that goes via Base/LinearAlgebra that requires the extra copy! step, and that implementation I will happily replace with my own Pade implementation.

But it is also true that, by the time the final step of the calculation is reached; the memory of A, which has already been destroyed and is no longer containing active information, can be reclaimed to store the result, so initialize_output(...) = A could indeed be the right design choice.

[sanderdemeyer]

I don't get why you say the current code would not work with a real scalar. The argument τ is restricted to be a Number, not per se a real or imaginary one. Hence, it just corresponds to an irrelevant convention choice. If you really want to use it with a real argument (which would be useless, since you can just use the exponential function), you can still call exponentiali(-iτ, A). This would work perfectly fine. Indeed, this feels a bit unintuitive, so that's why we can just use the exponential function. I also think that the current implementation allows for both imaginary_evolution and real_evolution.

Apart from the fact that I don't see why the changes would make the difference between exponential! and exponentiali smaller (the fundamental problem remains), I feel like the proposed changes make everything more difficult than they need to be. You have always been an advocate for making an interface feel intuitive. I think that exp(A) = exponential(A) and exp(itA) = exponentiali(t, A)feel more intuitive than eitherexponential(ScaledOperator(it, A))orexp(A) = exponential(1, A) = ...`.

In terms of differentiating between eigh and eig during runtime: Then why do we not do this for eigh and eig themselves?

If you really feel strongly about any of this, feel free to tailor the implementation to how you see things. The only thing that I really care about is an implementation that works, preferably as generic as possible (again, because I would like to have this in the next TensorKit version). As long as the tests pass, this is all fine by me. I might not be a big fan of the choice of implementation, but naming conventions and interface choices are way less important to me than having either implementation.

[lkdvos]

I don't get why you say the current code would not work with a real scalar.

The difference is a bit subtle, but as I mentioned before this is about type stability.
(It is easy to detect real vs complex scalars in a type stable manner, but less so to detect purely imaginary vs complex.)
Maybe it is more obvious with an example:

function exponentialr(tau, A)
    expA = tau isa Complex ? complex(A) : A
    return exponentialr!(tau, A, expA) # this computes expA .= exp(A * tau) in whatever way
end
function exponentiali(t, A)
    expA = iszero(real(t)) ? A : complex(A)
    return exponentiali!(t, A, expA) # this computes expA .= exp(A * t * im) in whatever way
end

This first implementation is type stable for both purely real, purely imaginary and complex values of tau, while the second isn't.

---

In terms of differentiating between eigh and eig during runtime: Then why do we not do this for eigh and eig themselves?

The reason we opted for eigh and eig is that, unlike LinearAlgebra.eigen, this can be implemented in a type-stable manner for real input matrices. eigh will return real D and V, while eig will always return complex D and V.

For exponential we can revisit this, since both hermitian and non-hermitian real inputs get real outputs.

---

(again, because I would like to have this in the next TensorKit version)

I’m afraid that might be difficult: we’re trying to get a version released this week, so I don’t think integrating and polishing this in time is very realistic.

That said, if your main goal is to get something working for your use case in the next TensorKit release, you could already move ahead with something along the following lines:

function myexp(A::AbstractTensorMap)
    D, V = eig_full(A)
    D.data .= exp.(D.data)
    return V * D * inv(V)
end

From my side, I think it’s important that we take a bit of time here to come up with a good, consistent interface and to think through allocations and long-term maintenance, especially since this will likely influence all of the matrix functions. I’d really like to avoid having to introduce multiple breaking changes later.

But that shouldn’t block you from experimenting or using a simpler version for your immediate needs.

[sanderdemeyer]

Revisiting this after quite some time. I think the last main issue was about the type stability of the different methods. Indeed, we can rewrite exponential in a type-stable manner, as shown by @lkdvos above. Since this is not possible for exponentiali (or whatever name we will give the function that computes exp(im*tau*A)), I guess we have two options

• We remove exponentiali.
• We define exponentiali in such a way that it always returns a complex matrix, even in the case where tau is purely imaginary. This means we could still use eigh to compute the exponential of skew-hermitian matrices. It's indeed suboptimal when tau purely imaginary, but maybe then you should use exponential(im*tau, A) anyway.

Let me know which of the two you prefer, or whether there is another option that I'm missing here. Additionally, should there be any changes to this PR in light of #QuantumKitHub/TensorKit.jl#342?

[Jutho]

I have been reading a bit about matrix exponentials recently, and the conclusion seems to be that, for higher precision, you basically want to evaluate it as a Taylor series (potentially after some rescaling, followed by squaring the result, i.e.

exp(A) = (exp_via_taylor_up_to_order_m(A/2^n))^(2^n)

where order m and scaling exponent n is chosen according to some cost function. I hope to implement this soon.

[lkdvos]

I think you left out the third option here, which is to add an exponentialr where you give an imaginary factor, which would solve the issues of type stability while keeping the features.
However, in the interest of keeping things moving, it probably makes sense to just stick with exponential for now, and deal with the inefficiency for now, especially since I think this does not apply as much to @Jutho's Taylor suggestion?

[sanderdemeyer]

@lkdvos I guess this is equivalent to supporting exponential(A) and exponential(tau, A), where we dispatch on whether tau is real or not (maybe this is what you meant?). Then we would not have to support exponentiali, and everything should be type stable. Is that an approach you would be okay with?

[lkdvos]

yes, this is exactly what I meant! :)

[sanderdemeyer]

I finally got around to working on this. Some comments

• Instead of having exponentiali encoding exp(im*tau*A), we now have exponentialr encoding exp(tau*A). I don't know whether we want to keep this name, but changing it to just exponential is not possible withing the current framework of n_args.
• I added the cases where the eltype of tau and A can be different to the tests.
• One previous commit trying to avoid an allocation gave incorrect results, so I changed this back (see below). I don't know if there is a way to avoid this.

[sanderdemeyer]

In the complex case, V was overwritten, such that V' was not the 'original' V', but also contained the exponential. This could be resolved where tau is real, where we could absorb a factor exp(tau * D / 2). Is this something that would be preferred? Then we would have a case where tau is complex and a case where tau is real, where the latter will be further divided based on the eltype of A.

[lkdvos]

Overall this is looking quite nice! Apologies if this is a premature review, just showed up in my notifications and figured I'd have a look.

The only thing I'm still wondering is if we want to just replace exponentialr(t, A) = exponential((t, A)) and not have the two functions. I know we had this discussion before, I'm not sure what came out of that to be honest.

[lkdvos]

I know we had a lot of discussions about this in the past, but I think we now settled on actually copying over the data (see also the GLA extension, I think), so this should probably become:

Suggested change

	return LinearAlgebra.exp!(A)
	LinearAlgebra.exp!(A)
	A === expA || copy!(expA, A)
	return expA

[sanderdemeyer]

I know we had a lot of discussions about this in the past, but I think we now settled on actually copying over the data (see also the GLA extension, I think), so this should probably become:

This does not work, for the very bizarre reason that LinearAlgebra.exp!(A) does not guarantee that the result is directly stored in the matrix A (at least when I test it locally with version 1.12.0). Everything is fine if I replace line 74 with A = LinearAlgebra.exp!(A), but probably there is a better way to get around this?

[sanderdemeyer]

The only thing I'm still wondering is if we want to just replace exponentialr(t, A) = exponential((t, A)) and not have the two functions. I know we had this discussion before, I'm not sure what came out of that to be honest.

I also don't know whether there was a consensus. I would be in favour of indeed just using exponential((t,A)), since that seems to be the most convenient/logical option. Especially if we don't support exponentiali, it feels weird to call it 'exponentialr'. So if you agree with this, I will make the necessary changes.

[lkdvos]

Yes, let's do this and then push this PR over the edge 🎉

[sanderdemeyer]

I added these lines to be able to differentiate the check_input test between exponential(A) and exponential(tau,A), since the latter does not require @check_scalar. Feel free to make suggestions to do this in a different way.

[lkdvos]

Why does the latter not require that? If A is real and tau is complex, expA has to be complex right?

[sanderdemeyer]

You're right, I forgot about that. I added a method check_input for exponential((tau,A)), where I dispatch on whether tau is real.