Potential directions to go

[GiggleLiu]

1. Tropical approach for quantum error correction, key innovation: the tropical tensor factorization algorithm. Ref: https://github.com/GiggleLiu/TropicalFactorization.jl (private)
2. TreeContractor based approach, key innovation: a tensor network contraction-compression scheme with complexity upper bounded by tree-width, but can be much lower. https://github.com/nzy1997/TreeContractor.jl
3. Branching based QEC for atom loss error, key innovation: incoporate atom loss in branching scheme. I think it is straight forward, can it can be potentially very fast.

[GiggleLiu]

https://github.com/quantumlib/tesseract-decoder

[GiggleLiu]

What happened when applying a two qubit gate on two qubits with one atom loss?

[ChanceSiyuan]

Comment Proposal
Regarding the points discussed in 1 and 3, I agree that we should proceed sequentially. Here are my thoughts on the next steps:

1. Addressing Memory Constraints in Tropical Contraction
   The memory requirements for current tropical tensor network contractions are becoming a significant bottleneck. Moving toward approximate algorithms is a natural progression.

Research Task: I need to investigate how approximation in the tropical semiring differs from standard MPS (Matrix Product State) approaches using SVD and truncation.

Output: I will study the specific mechanics of tropical approximation and document my findings in a technical note.

2. Leakage/Loss Detection and Error Correction
   Incorporating loss detection and leakage reduction/refill layers is critical.

Threshold Concerns: Without these interleaved layers, the error correction protocol likely won't reach the necessary threshold for fault tolerance.

Decoding Strategy: I am looking into how to feed loss detection information back into the decoder.

References
For the decoding approach, this paper provides a highly relevant framework for handling detectable errors:

arXiv:2302.03063: High threshold codes for neutral atom qubits with biased erasure errors (Sahay et al.)

[GiggleLiu]

I need to investigate how approximation in the tropical semiring differs from standard MPS (Matrix Product State) approaches using SVD and truncation.

Very challenging, NP-hard. It is a very important open question itself, if you can find a method better than greedy, then itself is a great acompilishment. Even for boolean tropical algebra, it is NP-hard, it is equivalent to biclique-cover.

[GiggleLiu]

I think your second proposal is promising, which is directly useful to experimentalists. Tropical tensor network based formalism atom loss error + clever branching strategy (or searching) may be a reasonable direction to explore.

@ChanceSiyuan