Add (fast) full update

[Yue-Zhengyuan]

This PR tries again to add the fast full update algorithm (arXiv 1503.05345) (see also discussion in the old PR #106).

The performance of FU is benchmarked with the real-time evolution of the transverse field Ising model (TFIM) (see test/timeevol/tf_ising_fu.jl), which calculates the magnetization in the x-direction after the infinite transverse field is reduced to a finite value.

In addition, I added the option to use pseudo-inverse to solve linear equations in ALSTruncation. Otherwise the test on TFIM will fail (but FullEnvTruncation is still fine with KrylovKit.linsolve).

Limitations:

• The pipelines for imaginary time evolution (ground state search), real time evolution and finite temperature calculation are quite different. In addition, it's difficult to set a convergence criterion for ground state search for FU. Therefore only the iteration functions fu_iter, fu_iter2 are exported, and I deliberately did not add a fullupdate function that wraps up everything. It's left to the users to decide how to assemble a complete FU program.
• Current implementation does not support bipartite unit cell structure. We can try it later after proposals in [ENH] CTMRG arbitrary unit cell #207 are implemented.
• Only nearest neighbor Hamiltonians are supported (in principle it can be done; see arXiv 1711.07584).

[Yue-Zhengyuan]

@ogauthe You can now try the full update on models you are interested in. You can know the basic usage of the related functions from the test file test/timeevol/tf_ising_fu.jl.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Files with missing lines	Coverage Δ
src/PEPSKit.jl	100.00% <ø> (ø)
src/algorithms/contractions/bondenv/als_solve.jl	100.00% <100.00%> (ø)
src/algorithms/ctmrg/sequential.jl	98.52% <100.00%> (+0.11%)	⬆️
src/algorithms/time_evolution/evoltools.jl	95.52% <ø> (-0.07%)	⬇️
src/algorithms/time_evolution/fullupdate.jl	100.00% <100.00%> (ø)
src/algorithms/truncation/bond_truncation.jl	97.22% <100.00%> (+0.20%)	⬆️
src/utility/svd.jl	87.12% <100.00%> (+0.16%)	⬆️

... and 4 files with indirect coverage changes

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[ogauthe]

Thank you! I will have a look.

[lkdvos]

@Yue-Zhengyuan , can you let me know what the status is on this (or other) PRs, and which ones should be reviewed/merged? I would like to consider releasing a new version of PEPSKit somewhere in the week of July 7th.

[Yue-Zhengyuan]

@lkdvos Right now we only have Sander's #184 and this one. #184 has been stagnated for quite a while, and maybe needs to be addressed first.

Fast full update is also ready for review actually. I'm just waiting for @ogauthe's feedback on whether it can provide enough improvement over SU for finite-T calculations.

[ogauthe]

I checked the finite temperature results obtained with full update as compared to simple update. See https://github.com/ogauthe/BenchmarkPEPS/blob/main/run/run_heisenberg_fullupdate.jl for the script.

I confirm I get the correct results with the full update. I find a slightly lower energy with the full update as compared to the simple update. Assuming energy(FU) = energy(SU) for beta < beta1 then energy(FU) < energy(SU) for beta > beta1 corresponds to a lower free energy, therefore a better approximation of the exact thermal density matrix. Congrat @Yue-Zhengyuan !

In terms of interface, I think ψ, env, = fu_iter2(fu_hamilt, ψ, env, fu_alg) can be improved. The function name could be more explicit and the interface (argument order) closer to simpleupdate. What about

simpleupdate(wpeps, hamilt, su_alg)
fullupdate(ψ, env, hamilt, fu_alg)

or

evolve_simpleupdate(wpeps, hamilt, su_alg)
evolve_fullupdate(ψ, env, hamilt, fu_alg)

or maybe

time_evolve(wpeps, hamilt, su_alg)
time_evolve(ψ, env, hamilt, fu_alg)

using dispatch on algorithm?

[Yue-Zhengyuan]

TL;DR: the name fu_iter2 can be improved, but I don't have the energy to unify the interface of FU, SU right now. SU can be completely revamped after belief propagation is implemented.

---

time_evolve(wpeps, hamilt, su_alg)
time_evolve(ψ, env, hamilt, fu_alg)

This can be a good improvement for future work. YASTN started on time evolution earlier than us and they have a unified interface for different algorithms. I didn't use it mainly because I'm more familiar with how TensorKit handles fermions. But currently I really don't have the energy for interface revamps to catch up with them...

In particular, fu_iter2 does not have an SU counterpart: it packs several FU steps together, because in fast FU we don't want to reconverge the CTMRG environment after every fu_iter.

Also, FU uses the usual InfinitePEPS, but SU uses InfiniteWeightPEPS. I actually regret writing SU in this way. Looking back on it, I only need to introduce SUWeight, which can later be generalized to BPEnv (#223). SU is actually FU with the bond environment given by the weights instead of the CTM tensors. After BPEnv is finished, we can unify SU, FU interface to

time_evolve(ψ::InfinitePEPS, env::BPEnv, H, alg::SimpleUpdate)
time_evolve(ψ::InfinitePEPS, env::CTMRGEnv, H, alg::FullUpdate)