Climate PDEs are well-suited to HHL if the state preparation and readout problems can be solved. The exponential speedup applies to the linear algebra core but practical utility requires end-to-end advantage including I/O.

Problem 13 · Quantum-Accelerated Climate Modeling

Overview

Accurate climate projections require solving large sparse linear systems arising from discretized partial differential equations. This scaffold introduces a reproducible classical baseline based on an energy balance diffusion model while preparing a Q# project that will eventually host hybrid HHL-style solvers. The goal is to compare classical finite-difference climate propagation with future quantum linear solver accelerators.

Directory Layout

13_climate_modeling/
├── estimates/                        # JSON artifacts from classical / quantum workflows
├── instances/                        # Discretization grids and forcing scenarios (small/medium/large)
├── plots/                            # Generated figures from analyze.py
├── python/
│   ├── classical_baseline.py         # 1D energy balance diffusion solver
│   └── analyze.py                    # Visualization of temperature evolution and convergence
└── qsharp/
    ├── qsharp.json            # Modern QDK project file
    └── Program.qs                    # Stubbed quantum workflow

Quick Start

cd problems/13_climate_modeling

# Classical finite-difference baseline
python python/classical_baseline.py

# Plot temperature profiles and convergence diagnostics
python python/analyze.py

# Quantum placeholder (uses modern QDK  qsharp Python package)
python -c "import qsharp; qsharp.init(project_root='qsharp'); print('Build OK')"
 python tooling/run_all_qsharp.py  # runs via qsharp Python package

Next Quantum Milestones

1. Linear System Encoding – Translate diffusion operators into sparse Hermitian matrices suitable for HHL.

2. State Preparation – Implement physically motivated right-hand-side encodings for radiative forcing profiles.

3. Error Budgeting – Quantify precision requirements and map to logical qubit counts.

4. Hybrid Calibration – Integrate quantum solvers with classical refinement loops for multi-scale climate modeling.

This scaffold keeps the classical diffusion baseline reproducible while we iterate toward quantum-enhanced climate projections. 🌍⚛️

Objective Maturity Gate

• **Current gate**: **Stage B complete** (classical baseline and Q# scaffold/build path are in place).
• **Next gate target**: **Stage C** (hardware-aware validation with uncertainty-bounded comparisons).

Stage C exit criteria for this problem:

• Execute at least one non-placeholder quantum workflow path tied to the problem objective.
• Report uncertainty-bounded comparisons between classical and quantum outputs on `small` and `medium` instances.
• Document transpilation/connectivity and backend assumptions used for reported quantum runs.
• Add calibration/noise-sensitivity evidence for the reported quantum metrics.

DiVincenzo Readiness (Stage C/D Overlay)

| Criterion | Status | Evidence / Notes |

|---|---|---|

| Scalable qubit system | partial | Problem-scoped instance baselines are in place; full hardware-scale projections are tracked as Stage C work. |

| Initialization | partial | Input/state initialization path is defined for current workflows, with backend-ready loading fidelity still to be hardened. |

| Coherence vs gate time | not-yet | Backend-calibrated coherence-vs-depth evidence is pending and required for Stage C/D promotion. |

| Universal gate set | partial | Q# scaffold/build path exists; gate-basis decomposition and transpilation evidence remain Stage C tasks. |

| Qubit-specific measurement | partial | Measurement outputs are defined for current validation flows; hardware readout characterization is pending. |

Advantage Claim Contract

• **Claim category (current)**: `theoretical`.
• **Problem class and regime**: Problem-specific challenge instances defined in this directory.
• **Fair baseline**: Problem-local classical baseline in `python/` outputs.
• **Quantum resource scaling claim**: Expected asymptotic advantage depends on algorithm family and implementation assumptions; no hardware-demonstrated speedup claim yet.
• **Data-loading and I/O assumptions**: Must be documented alongside future advantage claims.
• **Noise/error model assumptions**: Backend-specific model and calibration assumptions to be added at Stage C.
• **Confidence/uncertainty method**: To be reported using shot-based confidence intervals or equivalent statistical bounds.
• **Residual risks**: Oracle/state-preparation/transpilation overhead may dominate for near-term instance sizes.