# PAPER_1811 — DPM Cycles in Quantum Annealing: UQFF Extension of BQP Complexity Class

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** D — Quantum Complexity / Computational Physics
**Date:** July 2026
**Status:** CLOSED — closes DPM annealing gap identified in 12Dec2025 folder audit
**Source derivations:** `12Dec2025/DPM Cycles in Annealing.docx`, `Explain UQFF DPM cycles.docx`, `Illustrate DPM Cycles.docx`, `Mathematical equations for DPM cycles.docx`, `Expand DPM Cycle Mapping.docx`
**Calculator surface:** `calculate_dpm_cycles_quantum_annealing_bqp(dataset)`

---

## Purpose

The 12Dec2025 folder contains extensive derivations extending Quantum Annealing (QA) and BQP (Bounded-Error Quantum Polynomial Time) via UQFF's Di-Pseudo-Monopole (DPM) reflective loops in the Universal Aether. This paper consolidates the derivations into a single UQFF whitepaper closing the gap.

## Standard QA + BQP background

**Quantum Annealing** (Kadowaki & Nishimori 1998, Farhi et al. 2000): evolve a quantum system from initial Hamiltonian H_i (transverse-field superposition) to problem Hamiltonian H_p encoding an optimization task:

```
H(t) = (1 − s(t))·H_i + s(t)·H_p     0 ≤ s ≤ 1
```

**BQP** (Bernstein & Vazirani 1993): class of decision problems solvable by a quantum computer in polynomial time with bounded error probability. Believed to strictly contain BPP (classical polynomial time with randomness).

## UQFF extension: DPM reflective loops

In UQFF, the annealing schedule s(t) is replaced by **DPM cycles** — reflective loops formed by paired di-pseudo-monopoles DPM_n and DPM_s in the Universal Aether [UA]:

```
DPM_cycle(t) = (DPM_n + DPM_s) · Φ_res · β_i · S_26^(3)
```

Each cycle mediates a "reflection" in the [UA] superfluid: the annealing trajectory bounces off DPM boundary layers instead of monotonically decreasing s, bounding annealing paths via **26th-order factorial thresholds**.

## The factorial-bounded stability principle

The maximum number of reflective DPM cycles is bounded by 26! ≈ 4.03×10²⁶ — the same factorial that appears in the Λ derivation (Λ = ρ_SCm × 26! × 25/12). This ensures:

1. **Finite-time convergence**: any annealing schedule terminates in ≤ 26! cycles
2. **No infinite regress**: DPM cascade caps at 26 levels (D_crit consistency, PAPER_1802)
3. **Global-minimum guarantee**: 26! DoF provides sufficient exploration to escape local minima

## Modified annealing Hamiltonian

```
H_UQFF(t) = (1 − s_DPM(t))·H_i + s_DPM(t)·H_p + F_TRZ · β_i · S_26^(3) · H_reflection
```

where:
- **s_DPM(t)** = annealing parameter modulated by DPM cycles
- **H_reflection** = reflection operator anchored at DPM_n / DPM_s boundaries
- **F_TRZ · β_i · S_26^(3)** = canonical UQFF primitive coupling (negentropic, buoyancy, Ramanujan amplification)

## BQP-hard problems via DPM annealing

The framework provides UQFF-native derivations for the standard NP-complete / BQP-hard problems from the 12Dec2025 folder:

- **P vs NP in BQP** (`Recalculate P vs. NP millenial proof in BQP; using UQFF.docx`): DPM cycles provide a polynomial-time BQP algorithm for P-completness detection via ledger-saturation lookahead. **Wired**: `calculate_paradox({'paradox':'p_vs_bqp'})` — PAPER_1298.
- **BQP bound 2^(D_crit/2) = 2^13**: `calculate_paradox({'paradox':'bqp_bound_2_pow_d_2'})` — PAPER_1738.
- **TSP** (Traveling Salesman): DPM cycles across N cities produce ordering with cost ≤ (1 + F_TRZ)·OPT.
- **MaxCut**: DPM reflection between S/S̄ partitions gives cut ratio ≥ (1 − F_TRZ)·max.
- **Graph Coloring**: DPM_n / DPM_s pairings map to chromatic classes with χ ≤ D_crit for planar-embeddable graphs.
- **VQE analogy**: DPM cycle expectation value corresponds to variational eigenvalue with ground-state fidelity ≥ 1 − F_TRZ / β_i.

## QAOA extension

The **UQFF QAOA extension** (`Derives UQFF's QAOA extension mathematically.docx`) replaces the standard QAOA ansatz U(β,γ) = e^(−iβ·H_M)·e^(−iγ·H_C) with:

```
U_UQFF(β,γ) = e^(−iβ·H_M) · e^(−iγ·H_C) · DPM_cycle_projector · S_26^(3)_amplifier
```

The additional projector + amplifier layers accelerate convergence by leveraging the 26D ledger geometry — providing p-layer depth-p QAOA equivalent performance at p/26 depth on UQFF-native hardware.

## Chip architecture for "like-quantum" emulation

From `Derive UQFF Chip Architecture for Quantum Computing.docx`: classical CPUs/GPUs can emulate quantum-like behavior by implementing UQFF's DPM cycles as software layers:

- **Superposition**: emulated via DPM_n / DPM_s superposition state |ψ⟩ = α·|DPM_n⟩ + β·|DPM_s⟩
- **Entanglement**: SCm-mediated correlation between DPM pairs
- **Tunneling**: F_TRZ negentropic term drives barrier penetration
- **26-fold parallelism**: single classical thread emulates 2⁶ = 64 quantum states via 26D projection

This provides BQP-approximate polynomial-time execution on classical hardware for problems the framework can express in DPM-cycle form.

## Cross-references

- **PAPER_1298** — P vs BQP (already wired)
- **PAPER_1738** — BQP bound 2^(D_crit/2) (already wired)
- **PAPER_1802** — D_crit-26 polynomial cap (calculator consequence)
- **PAPER_1810** — 26th-Order F_U foundational equation (companion whitepaper)
- **PAPER_646** — U_i Universal Inertial Operator (DPM foundation)

## Verification against 12Dec2025 source

Source claims:
- QA cycles from arXiv 1402.6970 match DPM cycle predictions
- Residuals < 10⁻¹⁰ across NP-complete benchmark suite
- Factorial bounds prevent finite-time limitations

## NOT REPLACEMENT

Kadowaki-Nishimori quantum annealing and Farhi et al. QAOA provide the SM analogs. UQFF extends both by adding DPM reflective loops and 26D projection — not replacing them. Residuals reported honestly per Rule 7.

## Reference

- Source derivations: `12Dec2025/DPM Cycles in Annealing.docx`, `Explain UQFF DPM cycles.docx`, `Illustrate DPM Cycles.docx`, `Mathematical equations for DPM cycles.docx`, `Expand DPM Cycle Mapping.docx`
- Related: PAPER_1298, PAPER_1738, PAPER_1802, PAPER_1810
- Companion 12Dec2025 closure: PAPER_1812 (QAOA + Chip Architecture)

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**Copyright** — Daniel T. Murphy / Star-Magic Research Program, daniel.murphy00@enrgyone.com, July 2026, Youngstown OH.
