Numerical simulation of M=2 Virtual Distillation (Huggins et al., 2021) for quantum error mitigation on neutral-atom Rydberg platforms. The framework spans four research phases — from gate-based VD infrastructure through topological order detection, entanglement entropy measurement, and Kibble-Zurek dynamic scaling — all built on an analog-digital hybrid architecture using Pulser/Bloqade for state preparation and Qiskit Aer for the digital measurement layer.

The M=2 protocol estimates error-mitigated expectation values via

$$\langle O \rangle_{\text{mit}} = \frac{\text{Tr}(O,\rho^2)}{\text{Tr}(\rho^2)}$$

by performing joint measurements on two copies of a noisy quantum state.

Requires Python >= 3.10. Uses uv for environment management:

uv sync

Four high-level functions cover the main workflows:

from src.api import mitigate, topological_order, entanglement_entropy, kibble_zurek

result = mitigate(n_qubits=5, noise=(0.01, 0.02))
print(f"Unmitigated <Z_0>: {result['unmitigated_z']:+.4f}")
print(f"Mitigated   <Z_0>: {result['mitigated_z']:+.4f}")
print(f"Tr(rho^2):         {result['tr_rho_sq']:.4f}")

# Single noiseless evaluation
result = topological_order(lattice_size=(2, 2))
print(f"<W_hex> mitigated: {result['mitigated_w']:+.4f}")

# Noise sweep benchmark
bench = topological_order(noise_sweep=[0.0, 0.01, 0.05, 0.10])

result = entanglement_entropy(n_atoms=8)
print(f"Central charge c = {result.fitted_central_charge:.3f}")

result = kibble_zurek(n_atoms=6, quench_durations=[500, 1000, 2000])
print(f"KZ exponent (mitigated): {result.kz_exponent_mitigated:.3f}")
print(f"KZ prediction (Ising):   {result.kz_prediction:.3f}")

For finer control, import directly from the submodules. All public symbols are re-exported from src.

uv run python src/benchmark.py

uv run pytest tests/ -v

117 tests across all modules, ~17 seconds total.

src/
  api.py               # High-level user-friendly API (mitigate, topological_order, ...)
  __init__.py           # Public API exports (45+ functions, 6 dataclasses)
  utils.py              # B^(2) gate, noise models, post-processing
  simulation.py         # Circuit builders, transpilation, simulation runners
  analog.py             # Pulser/Bloqade integration
  neutral_atom_noise.py # Physical noise parameters for neutral atoms
  topological.py        # Kagome lattice, Z2 QSL, string operators
  entanglement.py       # Subsystem purity, Renyi entropy, CFT fitting
  kibble_zurek.py       # Correlation functions, KZ scaling
  spam_analysis.py      # SPAM error analysis scripts
  validation.py         # Cross-validation checks
  benchmark.py          # N=5, N=20 benchmarks
tests/
  test_analog.py        # Pulser/Bloqade backend tests
  test_handoff.py       # Analog-digital handoff tests
  test_noise.py         # Noise model tests
  test_topological.py   # Kagome lattice & string operator tests
  test_entanglement.py  # Subsystem entropy tests
  test_kibble_zurek.py  # Correlation & KZ scaling tests

• B^(2) beamsplitter gate: Decomposes into CX → CRy(+π/2) → CX (the inverse transform B^(2)†). Each gate contributes ~2 non-Clifford gates in Clifford+T basis.
• Pauli noise only: All error channels use Pauli operators exclusively, enabling Monte Carlo trajectory sampling on MPS and extended_stabilizer backends without density-matrix fallback.
• Qiskit bit string convention: Big-endian — bitstring[0] = highest qubit. Copy 1 bits: b1[j] = bitstring[2N-1-j], Copy 2 bits: b2[j] = bitstring[N-1-j].
• SWAP eigenvalue: s_j = -1 if (b1[j]==0, b2[j]==1), else +1. Mitigated value is ratio of sample means.
• Extended stabilizer limitation: B^(2) layer adds O(N) non-Clifford gates (~2N T-gates). At N=20: ~44 T-gates, impractical for extended_stabilizer. Use MPS instead.
• Pulser normalization: Statevectors from Pulser numerical integration may have norm ≠ 1.0; always renormalized before initialize().

• W. J. Huggins et al., "Virtual Distillation for Quantum Error Mitigation," Phys. Rev. X 11, 041036 (2021).
• B. Koczor, "Exponential Error Suppression for Near-Term Quantum Devices," Phys. Rev. X 11, 031057 (2021).
• Semeghini et al., "Probing topological spin liquids on a programmable quantum simulator," Science 374, 1242 (2021).
• Calabrese & Cardy, "Entanglement entropy and quantum field theory," JSTAT P06002 (2004).
• Keesling et al., "Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator," Nature 568, 207 (2019).