C-T01: H₂ Chemistry Experiment - Setup & Methods¶

Experiment ID: C-T01 / S-CHEM Manifest ID: 2a89df46-3c81-4638-9ff4-2f60ecf3325d Executable: C:\Users\User\Desktop\Projects\QuartumSE\experiments\shadows\h2_energy\run_h2_energy.py

Circuit Description¶

H₂ Molecular Ansatz (4 Qubits)¶

Molecular Parameters: - Molecule: H₂ (hydrogen dimer) - Geometry: R = 0.74 Å (equilibrium bond length) - Basis Set: STO-3G (minimal basis) - Active Space: 2 orbitals × 2 spins = 4 qubits

Ansatz Circuit:

Circuit Depth: 5
Gates:
  - Hadamard (h): 1
  - CNOT (cx): 3
  - RY rotations: 3
  - RZ rotations: 3

Circuit Hash: 4d5f8436e8e437af

State Preparation: Hardware-efficient ansatz (not chemically-inspired): 1. Initial state: |0000⟩ 2. Apply Hadamard and rotation layers 3. Entangle with CNOT ladder 4. Final rotations for variational parameters

[NOTE: Circuit parameters not optimized in this run, using placeholder values]

Hamiltonian Observable Set¶

H₂ Hamiltonian (12 Pauli Terms):

Observable	Coefficient	Physical Meaning
IIII	-1.05	Nuclear repulsion + constant
ZIII	0.39	Orbital 0 occupation
IZII	-0.39	Orbital 1 occupation
ZZII	-0.01	Orbital 0-1 correlation
IIZI	0.39	Orbital 2 occupation
IIIZ	-0.39	Orbital 3 occupation
IIZZ	-0.01	Orbital 2-3 correlation
ZIZI	0.03	Cross-orbital correlation
IZIZ	0.03	Cross-orbital correlation
XXXX	0.06	Hopping term (4-body)
YYXX	-0.02	Hopping term (4-body)
XXYY	-0.02	Hopping term (4-body)

[NOTE: These are placeholder coefficients for demonstration. Real H₂@STO-3G coefficients should be loaded from qiskit-nature.]

Total Energy:

E = Σ coefficient_i × ⟨Pauli_i⟩

Backend Configuration¶

IBM Quantum Hardware: ibm_fez¶

Device Specifications: - Processor: 156-qubit superconducting transmon - Topology: Heavy-hex lattice - Qubits Used: 0, 1, 2, 3 (linear connectivity) - Calibration: 2025-11-03T13:17:32Z (< 1 hour before experiment)

Qubit Quality (Used Qubits): | Qubit | T1 (μs) | T2 (μs) | Readout Error | Gate Error (SX) | |-------|---------|---------|---------------|-----------------| | 0 | 63.6 | 49.7 | 0.98% | 0.0364% | | 1 | 174.8 | 199.1 | 2.22% | 0.0364% | | 2 | 208.9 | 178.7 | 0.77% | 0.0364% | | 3 | 126.5 | 143.8 | 2.10% | 0.0364% |

Two-Qubit Gates: - Average CZ Error: 1.083% - Circuit depth 5 → Total gate error budget: ~3-5%

Assessment: Excellent qubit quality for 4-qubit chemistry experiment.

Classical Shadows Configuration¶

v1 Noise-Aware Configuration¶

shadow_config = ShadowConfig(
    shadow_size=300,                          # Conservative for 12 observables
    random_seed=77,                           # Reproducibility
    confidence_level=0.95,                    # 95% CIs
    version=ShadowVersion.V1_NOISE_AWARE,    # Use inverse channel
    apply_inverse_channel=True               # Noise correction enabled
)

Key Parameters: - Shadow Size: 300 snapshots (25 shots/observable average) - Version: v1 (noise-aware with inverse channel correction) - Bootstrap Samples: 1000 (for CI estimation) - Random Seed: 77 (different from SMOKE-SIM/HW for independence)

Theoretical Shot Budget¶

Naive Approach (Direct Measurement): - 12 terms × 100 shots/term = 1,200 shots minimum

Grouped Pauli Measurement: - Typical grouping: 3-5 groups for H₂ Hamiltonian - 3 groups × 400 shots/group = 1,200 shots

Classical Shadows (This Experiment): - 300 shadows estimate ALL 12 terms - Expected SSR: 1,200 / 300 = 4×

Mitigation Strategy¶

Measurement Error Mitigation (MEM)¶

Calibration: - Technique: Confusion matrix measurement - Shots per State: 128 shots × 2^4 = 2,048 total calibration shots - Qubits Calibrated: [0, 1, 2, 3] - Confusion Matrix Path: data/mem/2a89df46-3c81-4638-9ff4-2f60ecf3325d.npz - Checksum: 69dced449ce1479211404c31e77abafa...

Procedure: 1. Prepare computational basis state |b⟩ for b ∈ {0, 1}^4 2. Measure 128 times without rotation 3. Construct 16×16 confusion matrix C[measured|prepared] 4. Invert matrix: C^{-1} corrects readout errors

Expected Improvement: - Readout errors: 0.77-2.22% per qubit - Uncorrelated assumption → ~5-10% total bitstring error - MEM expected to reduce readout variance by 20-30%

v1 Inverse Channel¶

Noise Model: Locally-biased inverse channel accounts for: - Single-qubit depolarizing noise - Readout errors (already corrected by MEM) - Approximate gate errors via effective noise parameter

Application: Each shadow snapshot ρ̂_i corrected:

ρ̂_corrected = Λ^{-1}(ρ̂_measured)

where Λ is learned noise channel.

[NOTE: In this run, inverse channel parameters derived from backend calibration data.]

Execution Workflow¶

Step 1: Backend Validation¶

quartumse runtime-status --backend ibm:ibm_fez

Confirm: - Queue depth < 200 jobs - Calibration < 24 hours old - Target qubits (0-3) have readout error < 3%

Step 2: MEM Calibration¶

# Automatic in run_h2_energy.py
# Generates confusion matrix for qubits 0-3
# Saves to data/mem/{experiment_id}.npz

Step 3: Shadow Measurement¶

cd C:\Users\User\Desktop\Projects\QuartumSE

python experiments/shadows/h2_energy/run_h2_energy.py \
    --backend ibm:ibm_fez \
    --shadow-size 300 \
    --seed 77 \
    --data-dir ./data

Execution Flow: 1. Transpile H₂ ansatz circuit for ibm_fez topology 2. Submit MEM calibration job (2,048 shots) 3. Submit shadow measurement job (300 shots) 4. Retrieve results and apply MEM correction 5. Estimate all 12 Hamiltonian terms with bootstrap CIs 6. Save manifest + shot data

Step 4: Post-Processing¶

Automatic: - Observable expectation values computed - 95% confidence intervals via bootstrap (1000 samples) - Total energy = Σ coefficient_i × ⟨Pauli_i⟩ - Manifest saved with full provenance

Key Code Snippets¶

Hamiltonian Definition¶

from quartumse.shadows.core import Observable

hamiltonian_terms = [
    Observable("IIII", coefficient=-1.05),
    Observable("ZIII", coefficient=0.39),
    Observable("IZII", coefficient=-0.39),
    Observable("ZZII", coefficient=-0.01),
    Observable("IIZI", coefficient=0.39),
    Observable("IIIZ", coefficient=-0.39),
    Observable("IIZZ", coefficient=-0.01),
    Observable("ZIZI", coefficient=0.03),
    Observable("IZIZ", coefficient=0.03),
    Observable("XXXX", coefficient=0.06),
    Observable("YYXX", coefficient=-0.02),
    Observable("XXYY", coefficient=-0.02),
]

Shadow Estimation with MEM¶

from quartumse import ShadowEstimator
from quartumse.reporting.manifest import MitigationConfig

mitigation_config = MitigationConfig(
    techniques=["MEM"],
    parameters={"mem_shots": 128}
)

estimator = ShadowEstimator(
    backend="ibm:ibm_fez",
    shadow_config=shadow_config,
    mitigation_config=mitigation_config,
    data_dir="./data"
)

result = estimator.estimate(
    circuit=h2_ansatz,
    observables=hamiltonian_terms,
    save_manifest=True
)

total_energy = sum(
    term.coefficient * result.observables[str(term)]["expectation_value"]
    for term in hamiltonian_terms
)

Replay from Manifest¶

# Estimate NEW observables without re-running on hardware
estimator = ShadowEstimator.replay_from_manifest(
    "data/manifests/2a89df46-3c81-4638-9ff4-2f60ecf3325d.json"
)

new_observables = [
    Observable("ZZZZ"),  # All-Z correlation
    Observable("XXII"),  # X-X hopping
]

new_result = estimator.estimate_from_replay(new_observables)

Data Storage¶

Artifacts Generated¶

Manifest: data/manifests/2a89df46-3c81-4638-9ff4-2f60ecf3325d.json
2,136 lines of JSON
Complete circuit QASM3 representation
All 12 observable estimates with CIs
Backend calibration snapshot
Software versions (Qiskit 2.2.1, QuartumSE 0.1.0)
Shot Data: data/shots/2a89df46-3c81-4638-9ff4-2f60ecf3325d.parquet
300 shadow measurement outcomes
Basis choices (random Clifford per qubit)
Raw bitstrings before MEM correction
Confusion Matrix: data/mem/2a89df46-3c81-4638-9ff4-2f60ecf3325d.npz
16×16 matrix for 4-qubit system
Calibration metadata (shots, timestamp)

Validation Checks¶

Automated in script: 1. ✓ All 12 observables estimated 2. ✓ Confidence intervals non-degenerate 3. ✓ Identity term (IIII) ≈ coefficient (constant check) 4. ✓ Manifest saved with all required fields 5. ✓ Total energy within physical bounds

Reproducibility¶

Full Reproduction:

from quartumse import ShadowEstimator

# Exact replication using manifest + shot data
result = ShadowEstimator.replay_from_manifest(
    "data/manifests/2a89df46-3c81-4638-9ff4-2f60ecf3325d.json"
)

# Results identical to original run (deterministic replay)

Random Seed Control: - Shadow sampling: seed=77 - Bootstrap: seed=77 + offset - NumPy RNG: seeded in estimator

Link to Executable Notebook¶

Interactive notebook: notebooks/experiments/chemistry/h2_energy_shadows.ipynb [TBD]

For now, use standalone script with --help for options.

Next Experiments¶

C-T01 Re-Run with Real Hamiltonian: Update coefficients from qiskit-nature
C-T01 Baseline: Run grouped Pauli measurement for SSR validation
C-T02 (LiH): Scale to 6-qubit molecule
S-T03 (Fermionic Shadows): Direct 2-RDM estimation