Quantum

Prelocality No-Go Theorems and Exactly Solvable Relational History Hamiltonians

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Briefings

Research presents two exactly solvable relational history Hamiltonians: a cyclic clock with a unique ground state and a fixed-weight control penalty architecture. The clock model yields a gap scaling as $\Delta_{N,L}/E_* = \pi^2/(2L^2)-2\pi^2/L^3+O(L^{-4})$ independent of vertex count, while the alternative architecture achieves a size-independent gap of $E_*(1-1/\sqrt{2})$. These results establish finite resource tradeoffs between locality and clock resources but do not prove an emergent geometric phase.

Key points

• A cyclic clock with a reversible incidence update has a unique ground state.
• The gap of the clock model scales as $\Delta_{N,L}/E_* = \pi^2/(2L^2)-2\pi^2/L^3+O(L^{-4})$ independent of vertex count.

Context

This study provides exact finite results for relational history Hamiltonians, contrasting a cyclic clock model with a fixed-weight control penalty architecture. The findings offer quantitative tradeoffs between locality and clock resources without establishing an emergent geometric phase.

Limitations

The evidence is limited to two specific Hamiltonian models; no additional background on the broader field is supplied. Results are preprint status (arXiv:2610.14081v1) with uncertainty regarding the exact nature of the gap scaling and the absence of a geometric phase.

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