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Conjecture 1 Research Map

Objective

Attack the minimal theorem-sized target for Conjecture 1: extend the known Markovian nonunital anticoncentration obstruction to a depth-independent process tensor of bounded Markov order, ideally first at the level of the second-moment inequality.

Local Resources

  • Use 10_conjectures.bib as the current local bibliography snapshot mirrored from the Zotero 10_conjectures collection. Future cycles should check this file before assuming a paper is absent.
  • The active PRX Quantum comparison papers behind [[unital-noisy-random-circuit-uniformization.md]] were validated from Zotero local attachments in prior source-fidelity passes; they are not currently mirrored in 10_conjectures.bib.

Current Status

  • The graph now contains the exact Markovian anchor theorem [[markovian-nonunital-anticoncentration-obstruction.md]].
  • The strongest current reduction route is to formulate the correlated noise through the process-tensor language [[process-tensor-framework.md]] and then isolate a finite-memory regime through [[instrument-specific-finite-quantum-markov-order.md]].
  • The baseline comparison is explicit but architecture-sensitive: noiseless random circuits can anticoncentrate on logarithmic or near-logarithmic scales for specific architectures, while the heralded-dephasing unital-noise model of [[unital-noisy-random-circuit-uniformization.md]] drives outputs toward uniformity; the nonunital obstruction is mathematically different from either regime.
  • Main unresolved gap: convert the conjecture's uniform diamond-norm memory-decay assumption into a bounded effective memory object compatible with the PRX Quantum 2024 moment method.

Active Path

  1. Treat [[markovian-nonunital-anticoncentration-obstruction.md]] as the theorem to be generalized rather than rebuilding the Markovian case from scratch.
  2. Use [[process-tensor-framework.md]] as the non-Markovian representation layer.
  3. Use [[instrument-specific-finite-quantum-markov-order.md]] to understand what a bounded-memory reduction can actually mean in quantum-process language.
  4. Use [[bounded-markov-order-nonunital-extension-candidate.md]] as the current synthesis node and stop at the point where the remaining mismatch is explicit.

Nodes

  • Conjecture 2 is adjacent because any proof that realistic nonunital noise enforces a persistent low-entropy bias in random circuits strengthens the view that nonunitality is a resource rather than merely an error source.
  • Conjecture 4 may eventually connect through decoder-independent logical survival under temporally correlated noise, but it is not needed for the present bounded-memory reduction.

Open Questions

  • Which exact intervention class should be fixed so that bounded Markov order is both physically meaningful and strong enough to imply the conjecture's memory-decay condition?
  • Can the PRX Quantum 2024 second-moment argument be lifted to a system-plus-memory transfer operator without constants deteriorating with circuit depth?
  • Is a hidden-Markov or collision-model subclass the right first target before tackling the full process-tensor formulation?