๐ŸŒŒ Quantum Grand Challenges

Systematic exploration of 20 of the world's most challenging scientific problems using quantum computing

๐ŸŽฏ What Success Looks Like

Every challenge follows the same reproducible pipeline, from Python baselines to fault-tolerant quantum resource forecasts.

Stage A - Problem Framing

Define the objective, baseline metrics, and evidence plan before any quantum claim is made.

Stage B - Baseline + Scaffold

Deliver reproducible classical baselines and Q# scaffolds with aligned interfaces and testable outputs.

Stage C - Implemented Kernel

Ship a working quantum kernel with benchmarked behavior, bounded uncertainty, explicit evidence links, and DiVincenzo readiness coverage.

Stage D - Advantage Claim

Permit advantage claims only when calibrated, stress-tested, and policy-gated in CI.

๐Ÿ“š Featured Case Studies

Stage C implementations now anchor the portfolio while Stage B tracks continue progressing under policy-gated evidence requirements.

QAE Risk Analysis (Stage C Complete)

Monte Carlo: 16.1% ยฑ 0.37% (10k samples)

Classical baseline achieves 0% error vs theoretical for tail risk P(Loss > 2.5) on log-normal(0,1) distribution. Log-normal PDF corrected March 2026.

Canonical QAE: 594k qubits, 6.4s, 965k T-states

QAE kernel recalibrated with corrected distribution encoding. Emulator jobs queued on Quantinuum H2-1E for full simulation validation.

Explore the repository โ†’

Database Search (Stage C Complete)

Search baselines: 16, 32, and 4096-item instances

Classical search and verification harnesses provide reproducible reference outputs for Grover comparisons.

Canonical Grover: quadratic search behavior

Multi-target oracle and measurement validation support the Stage C evidence bundle for database search.

Explore the repository โ†’

Hubbard Model (Azure Validated)

Exact diagonalization: ฮ”c = 5.66 at U/t = 4

Two-site half-filled Hubbard model with classical baseline achieving perfect accuracy for ground state energy and Mott gap.

VQE circuit validated on Quantinuum H2 simulator

Real VQE ansatz (2-qubit, Ry+CNOT+Rz) submitted to Azure Quantum. Resource estimates generated for surface_code and gate_ns_e3 architectures. Emulator results pending.

Explore the repository โ†’

๐ŸŽฏ Algorithm Comparison Dashboard

Comprehensive resource analysis across VQE, HHL, and QAE showing physical qubits, runtime, T-states, and quantum advantage assessment.

VQE (Hubbard)

Physical Qubits:79k
Runtime:114 ฮผs
T-States:1.6k
Error-resilient optimization

HHL (Linear Solver)

Physical Qubits:18.7k
Runtime:52 ms
T-States:185k
Most qubit-efficient

QAE (Risk Analysis)

Physical Qubits:594k
Runtime:6.4 s
T-States:965k
Quadratic speedup O(1/ฮต)

๐Ÿ“Š Visualization Gallery

Publication-quality comparisons showing resource requirements, scaling predictions, and quantum advantage assessment.

Physical Qubit Requirements

Physical Qubit Requirements

HHL is most efficient (18.7k), QAE requires 31.8ร— more qubits (594k). T-state factories dominate 93-97% of all qubits.

Runtime Comparison

Runtime Comparison

VQE is fastest (114ฮผs), HHL takes 52ms, QAE requires 6.4s. Shows 3 orders of magnitude range across algorithms.

T-State Analysis

T-State Analysis

QAE needs 965k T-states (5.2ร— more than HHL). Rotation gates dominate costs at 20 T-states per rotation.

Quantum Advantage Map

Quantum Advantage Map

Heatmap assessment: HHL wins near-term viability (2027-2029), VQE wins runtime, QAE wins advantage potential.

Scaling Predictions

Scaling Predictions

VQE scales linearly, HHL logarithmically, QAE exponentially. Shows practical limits for each algorithm.

Technology Timeline

Technology Timeline

HHL ready by 2027-2029 (50k qubits), VQE by 2028-2030 (100k qubits), QAE by 2033-2035 (1M qubits).

๐Ÿ“ Real Gate Counts (QASM-Derived)

Actual gate decomposition from OpenQASM circuits validated on Quantinuum H2 trapped-ion simulator. These replace mock estimates with real circuit-level resource data.

CircuitQubitsTotal GatesT-gatesCXRotations
Hubbard VQE26023
QAOA MaxCut315066
Grover Key Search45911240
QAOA Scheduling48061
Grover DB Search421278610
QEC Repetition510080

Key insight: Grover circuits are T-gate heavy (needed for multi-controlled Z decomposition), while VQE and QAOA use only rotation gates โ€” critical for hardware target selection.

๐Ÿ“„ Research

Methodology Paper

"A Systematic Framework for Developing, Validating, and Benchmarking Quantum Algorithms Across 20 Application Domains"

Covers: maturity gate model, 9 algorithm families, automated CI validation, Azure Quantum integration, and lessons learned from building real quantum circuits.

Read Draft โ†’

Cite This Work

CITATION.cff is available in the repository root. GitHub shows a "Cite this repository" button automatically.

@software{rall2026quantum,
  author = {Rall, Werner},
  title = {Quantum Grand Challenges},
  year = {2026},
  url = {https://github.com/WernerRall147/quantum-grand-challenges}
}

๐Ÿ“Š Current Status

  • โœ… Stage Coverage: 20/20 problems include objective maturity gates and advantage-claim contracts
  • โœ… Quantum Implementations: 20/20 problems have real quantum gate operations (VQE, QAOA, Grover, HHL, Shor, swap test, QEC, quantum walk, Trotter)
  • โœ… KPI Coverage: 20/20 estimator summaries, 20/20 backend assumptions, 20/20 advantage contracts
  • โœ… Azure Validated: 20/20 QASM circuits validated on Quantinuum H2-1SC trapped-ion simulator
  • โœ… Calibration: 19/20 problems have multi-run uncertainty bounds (3-5 runs each)
  • โœ… DOI: 10.5281/zenodo.19222021
  • โœ… Research: Methodology paper with 20-circuit validation table, calibration data, and 9 sections
  • โœ… Azure Successful Runs: `27` successful live runs recorded across `2` quantum systems
  • โœ… Runnable/Correctness Audit: `20/20` problems pass executable correctness checks (100%)
  • โœ… CI/CD: 7 automated checks on every PR: build, correctness, maturity, integrity, secrets, security

Latest: 20/20 Azure Validated + 19/20 Calibrated

All 20 QASM circuits validated on Quantinuum H2 trapped-ion simulator (including 8-qubit Shor and 5-qubit HHL). 19 problems have multi-run calibration evidence with uncertainty bounds. DOI: 10.5281/zenodo.19222021. Methodology paper updated with full 20-circuit validation table and calibration results.

๐Ÿšง Active Work Queue

Current promotion queue for moving Stage B tracks into Stage C with benchmarked kernel evidence.

Stage C Promotion

All 20 problems now have real quantum circuits. Next: generate resource estimates, backend assumptions, and calibration evidence to promote from Stage B to Stage C.

Azure Emulator Results

4 circuits executing on Quantinuum H2-1E emulator (Hubbard VQE, QAOA MaxCut, Grover, QAE). Collect results and validate against classical baselines.

Resource Estimation Coverage

Expand estimator pipeline to all 20 problems. Currently 9 have estimator summaries; target is 20/20.

Stage D Evidence Hardening

Lock uncertainty thresholds and complete advantage claim contracts for QAE, QAOA, and Grover (current Stage C leaders).

๐Ÿงญ Stage D Readiness

Continuous readiness audit for current Stage D candidates and next expansion queue.

Fully Ready
0/3
Average Readiness
83.3%
Open Checklist Items
0
Artifact Issues
8
ProblemClaimReadinessOpen ChecklistArtifact Issues
03_qae_riskprojected5/6 (83.3%)03
05_qaoa_maxcuttheoretical5/6 (83.3%)02
15_database_searchprojected5/6 (83.3%)03

Next Candidate Expansion Queue

  • 01_hubbard: VQE circuit validated on Azure Quantum with resource estimates. Nearest Stage C to Stage D candidate after classical optimizer integration.
  • 04_linear_solvers: Full HHL circuit with resource estimates is ready for Stage C promotion and subsequent Stage D evidence hardening.
  • 15_database_search: Grover circuit submitted to Azure emulator with resource estimates. Stage D advantage claim pending oracle cost accounting.

โ˜๏ธ Azure Run Ledger

Automatic record of successful Azure Quantum executions and the systems used.

Provider Families

  • Quantinuum: 25
  • Rigetti: 2

Run Trend (UTC Day)

2026-03-04
3
2026-03-05
5
2026-03-06
15
2026-03-24
4
ProblemInstance/DepthSystemRecorded UTC
15_database_searchsmall / d1quantinuum.sim.h2-1sc2026-03-24T13:03:56Z
05_qaoa_maxcutsmall / d1quantinuum.sim.h2-1sc2026-03-24T13:03:39Z
01_hubbardsmall / d1quantinuum.sim.h2-1sc2026-03-24T13:03:22Z
01_hubbardsmall / d1quantinuum.sim.h2-1sc2026-03-24T13:03:10Z
19_quantum_chromodynamicssmall / d1quantinuum.sim.h2-1sc2026-03-06T14:06:00Z
18_photovoltaicssmall / d1quantinuum.sim.h2-1sc2026-03-06T14:05:57Z

โœ… Runnable And Correctness

Nightly-style execution audit tracking classical runnability and baseline output validity across all registered problems.

Pass Count20/20
Pass Rate100%
Audit Timestamp2026-03-10T14:30:39Z

No failing problems in the latest audit.

Hubbard Model

Stage B

Stage B complete + Azure validated

VQE ansatz (2-qubit) submitted to Quantinuum H2 simulator and emulator. Resource estimates now available for surface_code and gate_ns_e3 targets.

Open problem folder โ†’

QAE Risk Analysis

Stage C

Stage C complete

Log-normal PDF corrected and QAE recalibrated. Theoretical tail probability now matches classical baseline exactly (16.1%). Emulator jobs queued on Quantinuum H2-1E.

Open problem folder โ†’

Catalysis Simulation

Implemented

Quantum circuit implemented

VQE for H2 molecular energy with 2-qubit Pauli decomposition. Estimates H2 ground state energy via variational optimization.

Open problem folder โ†’

Linear Solvers

Stage B

Stage B complete + estimates

Full HHL circuit implemented (6 qubits). Resource estimates generated for surface_code and gate_ns_e3 targets. Ready for Stage C promotion.

Open problem folder โ†’

QAOA MaxCut

Stage C

Stage C complete

Depth/noise sweeps, uncertainty reports, estimator routing, and env-gated Azure job-manifest validation are complete; next target is Azure submit/collect hardening and portability adapters

Open problem folder โ†’

High-Frequency Trading

Implemented

Quantum circuit implemented

Quantum VaR estimation via amplitude encoding and oracle marking for loss state detection

Open problem folder โ†’

Drug Discovery

Implemented

Quantum circuit implemented

VQE molecular binding energy estimation with 2-qubit Pauli Hamiltonian decomposition

Open problem folder โ†’

Protein Folding

Implemented

Quantum circuit implemented

QAOA lattice conformation search optimizing pairwise contact energies

Open problem folder โ†’

Factorization

Implemented

Quantum circuit implemented

Full Shor period-finding with QPE + controlled modular multiply. Factors 15 = 3 x 5.

Open problem folder โ†’

Post-Quantum Cryptography

Implemented

Quantum circuit implemented

Grover key search attack simulation. 80-92% success on 3-5 qubit keyspaces.

Open problem folder โ†’

Quantum Machine Learning

Implemented

Quantum circuit implemented

Swap test kernel estimation (5 qubits). Quantum kernel matrix matches classical.

Open problem folder โ†’

Quantum Optimization

Implemented

Quantum circuit implemented

QAOA for weighted job scheduling. Found exact optimal with approximation ratio 1.0.

Open problem folder โ†’

Climate Modeling

Implemented

Quantum circuit implemented

HHL algorithm for thermal diffusion PDE with QPE and eigenvalue inversion

Open problem folder โ†’

Materials Discovery

Implemented

Quantum circuit implemented

VQE band gap estimation for tight-binding models across 3 material types

Open problem folder โ†’

Database Search

Stage C

Stage C complete + Azure validated

Grover 4-qubit circuit submitted to Quantinuum H2 simulator and emulator. Resource estimates now generated for two architectures.

Open problem folder โ†’

Error Correction

Implemented

Quantum circuit implemented

3-qubit bit-flip repetition code with syndrome extraction. 100% correction rate.

Open problem folder โ†’

Nuclear Physics

Implemented

Quantum circuit implemented

VQE for deuteron binding energy with pionless EFT Hamiltonian

Open problem folder โ†’

Photovoltaics

Implemented

Quantum circuit implemented

Discrete quantum walk for exciton transport in organic photovoltaic networks

Open problem folder โ†’

Quantum Chromodynamics

Implemented

Quantum circuit implemented

Trotter simulation of 1D lattice gauge theory showing confinement transition

Open problem folder โ†’

Space Mission Planning

Implemented

Quantum circuit implemented

QAOA trajectory optimization for multi-leg missions. Found exact optimal.

Open problem folder โ†’

๐Ÿงช Reproduce It Yourself

Spin up a Codespace or local clone and run the same pipelines that feed this dashboard.

Setup

git clone https://github.com/WernerRall147/quantum-grand-challenges.git cd quantum-grand-challenges dotnet --list-runtimes | findstr 6.0

Run QAE Risk

cd problems/03_qae_risk make classical make analyze

Run Hubbard Benchmark

cd problems/01_hubbard make classical make build