ALS truncation for 3-site MPS

[Yue-Zhengyuan]

This PR adds ALS truncation for 3-site MPS, which will find its use in FU with NNN gates.

For the first and the last (third) site, the gate acts on the rank-(2, 1) reduced bond tensor. At the middle (second) site, the gate acts on the full state tensor. Therefore, in NTU/FU applications, the MPS to be truncated looks like (for iPEPS)

       ╲ ╱      
----a===m===b----
    ↓   ↓   ↓   

= implies "thickened" bond to be truncated. The dimension of left leg of a, and right leg of b is $D d$. Therefore, the size of the 3-site cluster environment is $(D^4 d^2)^2 = D^8 d^4$.

We then directly perform ALS sweeps over a, m, b (used in PhysRevB.97.174408). Note that truncated m has a large size $D^4 d$ for iPEPS, or $D^4 d^2$ for iPEPO. In comparison, truncated a and b only have size $D^2 d^2$. Therefore, we solve a, b more frequently.

---

I wanted to add another approach. But I realized some subtleties and decide not to include it to make this PR more focused.

Details

To avoid directly solving m, I implement a second approach following PhysRevB.87.115144, but with some slight improvements. Here, we first find the projectors (p0, ..., p4) on each bond, each only having size $D^2 d^2$:

              ╲ ╱      
----a==p0--p1==m==p2--p3==b----
    ↓          ↓          ↓   

They can be initialized with (truncated) Vidal gauge. Then, we perform ALS sweeps over (a p0), p1, p2 and (p3 b). In tests with random tensors, I find it works slightly better than factoring out bond tensors with QR/LQ and optimizing the bond tensors. But the improved cost function is always higher than the first approach, which can be seen in the test test/bondenv/3site_truncate.jl. It is also known to converge slowly, first shown in PhysRevB.97.174408 (Fig. 6b).

[codecov]

Codecov Report

❌ Patch coverage is 94.94949% with 5 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/algorithms/truncation/3site_truncation.jl	93.44%	4 Missing ⚠️
...algorithms/contractions/bondenv/als_solve_3site.jl	95.23%	1 Missing ⚠️

Files with missing lines	Coverage Δ
src/algorithms/contractions/bondenv/als_solve.jl	86.88% <100.00%> (+1.97%)	⬆️
src/algorithms/time_evolution/apply_mpo.jl	100.00% <100.00%> (+0.79%)	⬆️
...algorithms/contractions/bondenv/als_solve_3site.jl	95.23% <95.23%> (ø)
src/algorithms/truncation/3site_truncation.jl	93.44% <93.44%> (ø)

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