Bifurcation in Entanglement Renormalization Group Flow of a Gapped Spin Model

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A constant-depth quantum circuit (depth-32 circuit) maps the cubic-code ground state to a tensor product of two stabilizer ground states: one is the original cubic code coarse-grained by factor 2 and the other is a new commuting Pauli code that itself splits under a ≈100-layer Clifford circuit into two self-similar copies, producing a branching MERA. The derivation uses a polynomial translation-invariant stabilizer encoding and an explicit Mathematica CNOT script to perform row operations; degeneracy obeys a gcd-of-polynomials formula (power-of-two sizes give K ~ 4L envelope) and area-law entanglement persists in 3D.

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1. 0:00Intro
2. 0:58CUBIC CODE MODEL
3. 4:31LOCALLY INDISTINGUISHABLE GROUND STATES
4. 5:33ISOLATING EXCITATION
5. 7:42AS A QUANTUM MEMORY
6. 10:34DEGENERACY
7. 12:46RENORMALIZATION GROUP
8. 13:45ENTANGLEMENT RG
9. 15:202D TORIC CODE STATE
10. 18:47OF CUBIC CODE STATE
11. 19:44OF THE NEW STATE
12. 20:06A NEW STATE
13. 22:15BRANCHING MERA
14. 23:20ENTANGLEMENT ENTROPY
15. 26:29METHODS
16. 28:13CLASSICAL CYCLIC CODES
17. 29:42POLYNOMIAL DESCRIPTION
18. 33:38CLIFFORD CIRCUIT = ROW OPERATION
19. 37:56CALCULATION
20. 38:48SUMMARY