Quantum Circuit

Outputs

Save to data/circuit/{model}_dynamics.jld2 with keys:

• t — time array
• density_matrices — Vector of density matrices at each time step
• expectation_values — Dict of observable traces
• circuit_params — variational parameters θ at each step

Sub-task 1: Stinespring dilation

Embed Kraus operators into a unitary on system + ancilla qubits.

Construction: Given r Kraus operators {K_1, ..., K_r} each d×d:

1. Ancilla size: n_anc = ceil(Int, log2(r)) qubits. Pad r to next power of 2 if needed.
2. Build the isometry V = [K_1; K_2; ...; K_r; zeros...] of size (d * 2^n_anc) × d.
3. Extend V to a full unitary U (e.g., via QR completion or SVD).
4. Embed U as a gate in Yao.jl using matblock(U).

Two modes:

• (a) Mid-circuit measurement (default, ~11 qubits for FMO): After applying U_j for bath j, measure ancilla qubits, reset to |0⟩, reuse for bath j+1. Use measure! + register reset.
• (b) Deterministic dilation (~35 qubits for FMO): Separate ancilla registers per bath, no measurement. Trace out all ancillas at the end. Use when mid-circuit measurement is unavailable or noisy.

Implement both. Default to (a).

Sub-task 2: Variational ansatz (p-VQD)

Parameterized circuit V(θ_n) for system Hamiltonian evolution at each time step.

Method: Projected Variational Quantum Dynamics (Barison et al. 2021)

• Cost function: Fidelity F = |⟨ψ(θ_{n+1})|ψ_target⟩|² where ψ_target is the state after exact one-step evolution
• Gradients: Parameter-shift rule. For each parameter θ_i: ∂F/∂θ_i = [F(θ_i + π/2) - F(θ_i - π/2)] / 2
• Convergence: F > 1 - 1e-6
• Fallback: If p-VQD oscillates or fails to converge after 100 iterations, switch to direct fidelity maximization with BFGS via Optim.jl

Time-stepping loop

At each time step t_n → t_{n+1}:

1. Apply variational circuit V(θ_n) for system Hamiltonian
2. For each bath j = 1, ..., N_sites: apply Stinespring circuit for Kraus operators at step n
3. Measure/trace-out ancillas
4. Record density matrix and observables
5. Optimize θ_{n+1} via p-VQD

Validation criteria

• Stinespring output matches direct Kraus application: trace distance < 1e-8 (statevector sim)
• Ancilla in separable state after measurement/reset (no entanglement leakage)
• Full time-stepping loop reproduces classical-baselines dynamics for spin-boson and dimer
• Non-physical state check: after each step verify tr(ρ) = 1 and eigenvalues(ρ) ≥ 0. Apply trace renormalization and eigenvalue clamping if needed.

Visualization

• Circuit fidelity F vs time step index
• Variational parameter θ convergence curves
• Ancilla measurement outcome statistics (should be dominated by |0⟩ for weak coupling)

Common pitfalls

• Qubit ordering: Yao.jl uses LSB convention (qubit 1 is least significant bit). When mapping site indices to qubit indices, be explicit.
• In-place measurement: measure! collapses the register. Use measure (no bang) if you need the pre-measurement state. For density matrix simulation, use DensityMatrix register type.
• Padding Kraus operators: When r is not a power of 2, pad with zero matrices to fill the isometry. The unitary extension must still be valid (check unitarity: U†U = I).
• Optimizer stalling: Variational optimization can get stuck in local minima. Use multiple random initial θ (at least 3) and take the best. For early time steps, initialize θ near the identity circuit.

Pipeline context

Stage 4 of 5. Depends on: kraus-extraction (Kraus operator inputs), classical-baselines (validation targets). Feeds into: benchmark-scaling (full pipeline results).