open_system_hardware.md

Open-System Hardware Circuits

Hardware-executable methods for simulating open quantum systems on NISQ devices. Two complementary approaches:

1. Single-ancilla Lindblad circuit (phase/ancilla_lindblad.py) — deterministic circuit with mid-circuit measurement and reset
2. Monte Carlo Wave Function (phase/tensor_jump.py) — stochastic trajectory sampling, scales better than full density matrix

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Part 1: Single-Ancilla Lindblad Circuit

scpn_quantum_control.phase.ancilla_lindblad

Theory

Full Lindblad evolution requires the density matrix ($4^n$ elements) — impossible on quantum hardware. The single-ancilla method avoids this by simulating dissipation through repeated system-environment interactions:

1. Coherent step: Trotterised XY evolution on system qubits
2. Dissipation step: Controlled-$R_y$ from each system qubit to one ancilla qubit, with rotation angle $\theta = 2\arcsin(\sqrt{\gamma \cdot dt})$
3. Reset: Measure and reset the ancilla

After tracing out the ancilla, the system state has undergone amplitude damping at rate $\gamma$ per qubit per step. Repeating for $n_\text{steps}$ dissipation rounds approximates continuous Lindblad dynamics.

Limitation: This implements amplitude damping only. Pure dephasing requires a different ancilla protocol (not implemented).

API Reference

build_ancilla_lindblad_circuit

from scpn_quantum_control.phase.ancilla_lindblad import build_ancilla_lindblad_circuit

qc = build_ancilla_lindblad_circuit(
    K: np.ndarray,              # float64 (n, n) coupling matrix
    omega: np.ndarray,          # float64 (n,) natural frequencies
    t: float = 0.1,             # evolution time
    trotter_reps: int = 5,      # Trotter steps per coherent block
    gamma: float = 0.05,        # amplitude damping rate
    n_dissipation_steps: int = 3,  # ancilla reset cycles
) -> QuantumCircuit

Returns: Qiskit QuantumCircuit with $n+1$ qubits ($n$ system + 1 ancilla).

ancilla_circuit_stats

from scpn_quantum_control.phase.ancilla_lindblad import AncillaCircuitStats, ancilla_circuit_stats

stats = ancilla_circuit_stats(K, omega, t=0.1, trotter_reps=5,
                               gamma=0.05, n_dissipation_steps=3)

Returns:

{
    "n_qubits": int,          # total qubits (system + ancilla)
    "n_system": int,           # system qubits
    "n_ancilla": int,          # always 1
    "n_cx_gates": int,         # CNOT count
    "n_resets": int,           # ancilla reset count
    "total_gates": int,        # analytic gate-count estimate
    "n_dissipation_steps": int,
    "gamma": float,
}

The stats payload is typed as AncillaCircuitStats and uses integer counts plus the validated real-valued damping rate.

Tutorial

import numpy as np
from scpn_quantum_control.phase.ancilla_lindblad import (
    build_ancilla_lindblad_circuit, ancilla_circuit_stats
)

n = 4
K = 0.45 * np.exp(-0.3 * np.abs(np.subtract.outer(range(n), range(n))))
np.fill_diagonal(K, 0.0)
omega = np.linspace(0.8, 1.2, n)

# Build circuit
qc = build_ancilla_lindblad_circuit(K, omega, t=0.5, gamma=0.05,
                                     n_dissipation_steps=5)
print(f"Qubits: {qc.num_qubits} ({n} system + 1 ancilla)")
print(f"Depth: {qc.depth()}, Gates: {qc.size()}")

# Check resource costs before submitting
stats = ancilla_circuit_stats(K, omega, t=0.5, gamma=0.05,
                               n_dissipation_steps=5)
print(f"CX gates: {stats['n_cx_gates']}")
print(f"Resets: {stats['n_resets']}")

Comparison

Feature	This module	Cattaneo et al. (2024)	Hu et al. (2023)
Ancilla count	1	1	1
Reset protocol	Mid-circuit reset	Mid-circuit reset	Post-selection
Hamiltonian	Kuramoto-XY	Generic	Heisenberg
Hardware tested	IBM (via scpn-qc)	IBM	Simulation only

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Part 2: Monte Carlo Wave Function (MCWF)

scpn_quantum_control.phase.tensor_jump

Theory

The MCWF method (also called Quantum Jump method) replaces the density matrix with an ensemble of stochastic pure-state trajectories. Each trajectory evolves under the effective non-Hermitian Hamiltonian:

$$H_\text{eff} = H - \frac{i}{2}\sum_k L_k^\dagger L_k$$

At each time step $dt$:

1. Evolve: $|\psi(t+dt)\rangle = e^{-iH_\text{eff},dt}|\psi(t)\rangle$
2. Compute jump probability: $dp = 1 - \langle\psi|\psi\rangle$
3. With probability $dp$, apply a quantum jump: $|\psi\rangle \to L_k|\psi\rangle / |L_k|\psi\rangle|$
4. Otherwise, renormalise: $|\psi\rangle \to |\psi\rangle / |\psi|$

Averaging $R(t)$ over many trajectories recovers the Lindblad result.

Advantage over Lindblad: Memory scales as $O(2^n)$ (state vector) instead of $O(4^n)$ (density matrix). Each trajectory is independent → trivially parallelisable.

Disadvantage: Statistical noise from finite trajectory count. Typically 50–500 trajectories needed for smooth $R(t)$.

API Reference

mcwf_trajectory

from scpn_quantum_control.phase.tensor_jump import mcwf_trajectory

result = mcwf_trajectory(
    K: np.ndarray,              # (n, n) coupling matrix
    omega: np.ndarray,          # (n,) frequencies
    gamma_amp: float = 0.05,    # amplitude damping rate
    gamma_deph: float = 0.0,    # dephasing rate
    t_max: float = 1.0,         # total time
    dt: float = 0.05,           # time step
    seed: int | None = None,    # RNG seed
) -> dict

Returns:

{
    "times": np.ndarray,      # shape (n_steps,)
    "R": np.ndarray,          # order parameter at each step
    "psi_final": np.ndarray,  # final state vector, shape (2^n,)
    "n_jumps": int,            # total quantum jumps in this trajectory
}

mcwf_ensemble

from scpn_quantum_control.phase.tensor_jump import mcwf_ensemble

result = mcwf_ensemble(
    K: np.ndarray,
    omega: np.ndarray,
    gamma_amp: float = 0.05,
    gamma_deph: float = 0.0,
    t_max: float = 1.0,
    dt: float = 0.05,
    n_trajectories: int = 50,
    seed: int | None = None,
) -> dict

Returns:

{
    "times": np.ndarray,           # shape (n_steps,)
    "R_mean": np.ndarray,          # ensemble-averaged R(t)
    "R_std": np.ndarray,           # standard deviation of R(t)
    "R_trajectories": np.ndarray,  # shape (n_trajectories, n_steps)
    "total_jumps": int,             # total jumps across all trajectories
}

Seeded Objective Evidence

run_open_system_objective_suite() uses the production mcwf_ensemble trajectory-batch path for bounded objective rows:

from scpn_quantum_control.phase import run_open_system_objective_suite


suite = run_open_system_objective_suite(backends=("mcwf_ensemble",))
record = suite.records[0]
print(record.gradient)
print(record.reproducibility_certificate.same_seed_max_abs_diff)

The MCWF row records the trajectory-batch shape, seed, final ensemble mean, standard deviation, total jumps, and same-seed replay difference. The gradient is a central finite difference over coupling and damping scales with the same seed replayed for each shifted evaluation. That makes the row deterministic and reviewable; it does not claim an unbiased stochastic-gradient estimator, hardware gradient, provider execution, or isolated performance evidence.

Rust Acceleration

The inner loop of _order_param_vec (computing $R$ from a state vector) uses the Rust function order_param_from_statevector when the engine is installed. Measured speedup: 851× at $n=8$ (0.008 ms vs 6.47 ms). Falls back to NumPy automatically.

Tutorial

import numpy as np
from scpn_quantum_control.phase.tensor_jump import mcwf_ensemble
from scpn_quantum_control.phase.lindblad import LindbladKuramotoSolver

n = 4
K = 0.45 * np.exp(-0.3 * np.abs(np.subtract.outer(range(n), range(n))))
np.fill_diagonal(K, 0.0)
omega = np.linspace(0.8, 1.2, n)

# MCWF ensemble
mcwf = mcwf_ensemble(K, omega, gamma_amp=0.1, t_max=1.0, dt=0.05,
                      n_trajectories=100, seed=42)
print(f"MCWF R(T) = {mcwf['R_mean'][-1]:.3f} ± {mcwf['R_std'][-1]:.3f}")
print(f"Total jumps: {mcwf['total_jumps']}")

# Compare with exact Lindblad
solver = LindbladKuramotoSolver(n, K, omega, gamma_amp=0.1)
lindblad = solver.run(t_max=1.0, dt=0.05)
print(f"Lindblad R(T) = {lindblad['R'][-1]:.3f}")

# They should agree within statistical error

Scaling Comparison: Lindblad vs MCWF

$n$	Lindblad memory	MCWF memory (1 traj)	MCWF advantage
8	1 MB	4 KB	256×
10	16 MB	16 KB	1,024×
12	256 MB	64 KB	4,096×
14	4 GB	256 KB	16,384×

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Choosing Between Methods

Criterion	Lindblad	MCWF	Ancilla circuit
Accuracy	Exact	Statistical ($1/\sqrt{N_\text{traj}}$)	Trotter error
Memory	$O(4^n)$	$O(2^n)$	$O(n)$ qubits
Max $n$ (32 GB)	~12	~16	~20 (hardware limit)
Hardware-executable	No	No	Yes (IBM, etc.)
Parallelisable	No	Yes (per trajectory)	Yes (shots)

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References

1. Dalibard, J., Castin, Y. & Mølmer, K. "Wave-function approach to dissipative processes in quantum optics." PRL 68, 580 (1992).
2. Causer, L. et al. "Tensor jump method." Nature Comms (2025).
3. Cattaneo, M. et al. "Quantum collision models." PRR 6, 043321 (2024).
4. Hu, Z. et al. "Open-system simulation on quantum hardware." arXiv:2312.05371 (2023).

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See Also

• Lindblad Solver — exact density matrix evolution
• Error Mitigation — ZNE/DDD for noisy hardware circuits
• Multi-Platform Export — export ancilla circuits to other platforms