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Quantum Bootstrapping Protocol (QBP) (v1.2)

Autonomous Initialization for Phase-Coordinated Quantum Networks

Status: Official Spec (v1.2)
DOI: 10.5281/zenodo.18064435

Abstract

This paper specifies the Quantum Bootstrapping Protocol (QBP) as a hybrid classical–quantum initialization method for establishing sufficiently consistent cycle-anchored phase estimates across distributed quantum-network nodes without relying on continuous external timing infrastructure. The bootstrapping goal is operational: nodes must estimate relative clock offsets (and optionally drifts) to compute consistent local estimates of a shared cyclic phase variable and cycle index, enabling window-triggered coordination. QBP combines (i) coarse authenticated two-way exchange for initial alignment and ambiguity removal, with (ii) quantum refinement using temporal correlations from energy–time entangled photon pairs (coincidence timing), fused via a simple state-space estimator (e.g. Kalman filtering). Achievable precision is parameter-dependent and must be reported under explicit hardware timing resolution, detector jitter, link dispersion, and usable coincidence counts.

1. Problem statement and scope

Distributed quantum networking requires coordinated actions (emissions, pulses, measurements) across nodes. Phase-window coordination replaces long-horizon absolute timestamp targeting by triggering actions when a shared cyclic phase condition is met. This requires an initialization layer: nodes must compute sufficiently consistent local phase and cycle index estimates relative to shared conventions.

Remark 1.1 (Scope): QBP is an initialization and tracking procedure. It does not define routing, authentication protocols, or a complete network stack. It assumes authenticated classical messaging is available for coordination messages.

2. Canonical conventions and local estimates

Definition 2.1 (Protocol conventions, phase, and cycle index): A deployment fixes protocol conventions:

  • reference epoch t0,
  • cycle period Tcycle > 0.
For an ideal reference time variable t ∈ R, define the cycle index and canonical normalized phase:

𝑛(𝑡) := ⌊(𝑡 − 𝑡0) / 𝑇cycle⌋ ∈ Z,

𝜙(𝑡) := ((𝑡 − 𝑡0) / 𝑇cycle) mod 1 ∈ [0, 1) ≅ S1.

Remark 2.1 (Conventions vs calibration): The parameters (t0, Tcycle) are protocol conventions. QBP does not define them. QBP estimates node-specific clock parameters (offset/drift) so that nodes can compute accurate local estimates of 𝑛(𝑡) and 𝜙(𝑡) with respect to the already-chosen conventions.

Definition 2.2 (Local timebase and local phase estimate): Node i forms a locally available corrected time variable t̂i(t) (implementation-dependent), and computes local estimates:

𝑛̂i(t) = 𝑛(t̂i(t)), 𝜙̂i(t) = 𝜙(t̂i(t)).

3. Clock model

We model node i’s local clock reading as

𝐶i(𝑡) = 𝑡 + 𝛿i + 𝛽i𝑡 + 𝑋i(𝑡) + 𝜂i(𝑡),

where 𝛿i is offset, 𝛽i is (small) fractional frequency offset (drift), 𝑋i(𝑡) is integrated stochastic phase noise, and 𝜂i(𝑡) is bounded short-term timestamping/electronics jitter.

4. Quantum correlation measurement model

QBP assumes availability of correlated detection events from an energy–time entangled photon source (e.g. Franson-type energy–time entanglement) [6]. Each node time-tags detections using its local clock. By exchanging event identifiers and local time tags over a chosen interval, nodes build coincidence histograms. A coincidence peak shift yields an offset measurement.

Remark 4.1 (Local time-tags vs a global timing service): QBP does not require a globally synchronized absolute clock service. However, it exchanges local time tags to estimate relative offsets. This is consistent with the goal of avoiding continuous external timing infrastructure.

5. Quantum Bootstrapping Protocol (QBP)

Protocol 5.1 (QBP (outline))

Phase 0: Coarse classical alignment (practical)

  1. Perform repeated authenticated two-way exchange to obtain an initial offset estimate δ̂(0) and reduce ambiguity in cycle anchoring (select the intended cycle index).
  2. Use robust aggregation (median / trimmed mean) to mitigate outliers.

Phase 1: Quantum refinement from temporal correlations

  1. Distribute correlated photons to participating nodes; each node time-tags detections locally.
  2. Exchange detection identifiers and local time tags for a selected window of operation.
  3. Build coincidence histograms and extract offset measurements zk from peak shifts.

Phase 2: Estimation and tracking

  1. Fuse measurements {zk} to estimate relative offset δ and drift β.
  2. Repeat periodically (as needed) to maintain short-horizon phase consistency for window-triggered coordination.

6. A simple estimator model (example)

Definition 6.1 (Offset–drift state model (discrete time)): Let the state be xk = [δk, βk]T. For step duration Δt:

xk+1 = [[1, Δt], [0, 1]] xk + wk,

zk = δk + vk,

where wk models drift wandering and vk measurement noise (jitter + peak estimation error).

Remark 6.1: A Kalman filter is a standard choice under Gaussian assumptions. Robust filters can be used under heavy-tailed noise.

7. Performance notes and integration with phase windows

Let 𝜎meas denote the RMS uncertainty of one offset measurement extracted from coincidence data. If Neff usable coincidences contribute independently, a crude scaling is:

𝜎𝛿 ∼ 𝜎meas / √Neff,

up to implementation-specific constants and estimator structure.

Remark 7.1: Achievable precision depends strongly on hardware time-tag resolution, detector jitter, link dispersion, losses, and coincidence identification. QBP is best interpreted as an architectural procedure whose realized precision must be reported with explicit experimental parameters.

7.1 Engineering condition for window-triggered coordination

Assume an execution layer triggers within a phase acceptance window of half-width ΔΦ. If residual offset uncertainty is characterized by 𝜎𝛿, a common engineering requirement is:

(𝜎𝛿 / 𝑇cycle) ≪ ΔΦ,

so that phase errors are small compared with the acceptance radius of the window.

8. Security scope

Remark 8.1 (Security scope): QBP requires authenticated classical messaging for coordination messages and time-tag exchanges. Time-tags and coincidence identifiers are not assumed secret. Security guarantees (integrity, replay protection, confidentiality) require explicit cryptographic protocols and threat models, which are out of scope here.

9. Conclusion

QBP provides a hybrid classical–quantum bootstrapping layer for distributed systems coordinating on a shared cyclic phase. It estimates relative offsets/drifts so nodes can compute consistent cycle-anchored phase estimates for window-triggered execution without relying on continuous external timing infrastructure. Precision is parameter-dependent and should be reported under explicit hardware and link conditions.

References

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