Bounded Markov Order Nonunital Extension Candidate¶
Claim/Theorem¶
Current proof candidate for the minimal Conjecture-1 target: restrict to a depth-independent process tensor of bounded Markov order relative to the intervention class generated by the local random circuit ensemble. Represent the noise by a bounded-memory process tensor on a sliding time window, or equivalently by a finite memory register coupled to the system over consecutive layers. Then attempt to adapt the PRX Quantum 2024 Markovian nonunital moment argument to the enlarged system-plus-memory evolution. If the nonunital contraction survives as a depth-uniform bias on this enlarged transfer operator, one expects a second-moment lower bound of the form
\[
\mathbb E_{U}\!\left[q^n \sum_x p_{U,\Upsilon}^{(D)}(x)^2\right] \ge 1 + c_{\mathrm{ac}}
\]
with a constant depending on the memory parameters but not on \(D\).
Dependencies¶
- [[markovian-nonunital-anticoncentration-obstruction.md]]
- [[process-tensor-framework.md]]
- [[instrument-specific-finite-quantum-markov-order.md]]
- [[unital-noisy-random-circuit-uniformization.md]]
Conflicts/Gaps¶
- The missing step is a rigorous transfer from an instrument-specific bounded-memory description to the uniform-over-histories formulation used in Conjecture 1.
- It is not yet proved that the enlarged memory register can be chosen with bounded effective dimension under the exact assumptions of the conjecture, rather than under a more model-specific collision or hidden-Markov construction.
- The current cycle does not yet bridge spatial locality of the process tensor to the moment calculation; it only identifies the most promising temporal-memory reduction.
Sources¶
10.1103/PRXQuantum.5.03031710.1103/PhysRevA.97.01212710.1103/PhysRevLett.122.14040110.1103/PhysRevA.99.04210810.1103/PRXQuantum.3.040329