Quantum dynamics, bath memory and tomography

These integrations execute scientific engines and preserve full numerical evidence. Their default inputs are declared simulations. A passing check supports its stated equation, reconstruction or convergence test; it does not establish a PF-specific advantage, predictive experimental accuracy or commercial superiority.

Running a study

Use the research-integration worker/environment described in R24_START_HERE.md. The modules expose run(config, output_directory) and are also dispatched through run_research_integrations.py and the research-integration workbench. Existing standalone simulation engines remain independent of these optional dependencies.

Missing engines raise an explicit dependency error. No synthetic substitute is used when an engine is unavailable. The tomography demonstrations deliberately generate synthetic counts through Aer and label them accordingly.

GKSL dynamics: conventions and scope

Time is in seconds, decay rates in inverse seconds, and coefficients of H/hbar in radians per second. Tensor order is B,A for two qubits and D,C,B,A for four; the rightmost bit varies fastest. No untrusted expression, Python function or executable code is accepted in configuration.

The equation is d rho/dt = -i[H/hbar,rho] + sum(C rho C-dagger - {C-dagger C,rho}/2). The configurable Hamiltonian is

H/hbar = sum(delta Z_i/2) + sum(J Z_i Z_(i+1)/2) + cos(w t) sum(Omega X_i/2).

Local amplitude damping uses C_i = sqrt(gamma_local) |0><1|_i. Collective dephasing uses C = sqrt(gamma/2) sum Z_i. In the pure collective-dephasing reference, a matrix element between sum-Z eigenvalues m and n is multiplied by exp[-gamma (m-n)^2 t/4]. A single-spin coherence decays at gamma in this convention; the two-qubit GHZ coherence decays at 4 gamma. These explicit factors prevent confusing single-arm and collective rates.

The numerical comparison integrates the direct matrix equation with SciPy DOP853, while QuTiP constructs its own solver representation. Both calculations share the explicitly assembled physical operators, so the exact damping/dephasing maps and independent constant-unitary test supply additional model-assembly checks. The delivered matrices use the tighter QuTiP solution. Refinement compares relative/absolute tolerances and maximum internal step; merely changing the output sampling grid is not used as a convergence claim.

Supported states include complex GHZ states, singlet pairs, computational states and complete imported density matrices. Two-qubit imports use the existing pf.quantum.state document and explicitly convert A,B documents to B,A. Four-qubit imports require pf.quantum.joint_state, schema 1, D,C,B,A, dimensions [2,2,2,2], binary basis labels and declared provenance. Invalid, nonfinite, nonphysical or mismatched documents are rejected without normalization or eigenvalue repair.

For four-qubit results, mission_pair_states.jsonl traces out D,C and preserves B,A. This makes a valid pair state available for a selected mission epoch, but tracing cannot retain correlations with the discarded pair. The archive keeps the complete four-qubit matrices. These studies do not automatically replace every mission epoch's channel evolution.

Finite bath memory: narrowly scoped HEOM model

The HEOM study is the commuting (zero tunnelling) special case of a spin-boson model: H/hbar = delta Z_A/2, Q = I_B tensor Z_A, with spectral density J(w) = 2 lambda gamma w/(gamma^2+w^2). The thermal parameter is k_B T/hbar in radians per second, not a temperature entered directly in Kelvin. Initial system-bath factorization and a thermal harmonic bath are assumed.

The Matsubara expansion has nu_0=gamma and nu_k=2pikT for positive k. Its real coefficients are c_0=lambdagammacot(gamma/(2T)) and c_k=4lambdagammaTnu_k/(nu_k^2-gamma^2). For each time, the independent calculation sums c_k (nu_k t - 1 + exp(-nu_k t))/nu_k^2. Multiplication by the squared difference in coupling-operator eigenvalues gives the Gaussian dephasing exponent. Since Q has eigenvalues plus/minus one, the imaginary influence term proportional to the difference of squared eigenvalues vanishes.

Three genuine HEOM calculations separate the convergence directions: base depth/expansion, depth increased by two, and then Matsubara count increased by two. The refined HEOM trajectory is compared with the exact integral for that finite expansion and with a 4096-term analytic reference; a 2048-to-4096-term change separately measures the remaining reference tail. No terminator is silently added.

Default acceptance limits are absolute matrix-entry discrepancies: 1e-5 for finite-bath HEOM versus exact influence functional; 1e-3 for hierarchy refinement, bath refinement and the 4096-term reference; 1e-6 for the final analytic tail; 1e-9 for trace. These tolerances are declared before execution. The model rejects parameter/resource requests beyond its bounded warm, weak-coupling domain. A parameter change may fail the convergence criteria and must not be described as qualified merely because the calculation finishes.

The preset reproduced the solvable published model's limit with finite-bath error about 2.8e-7 and full-bath error about 2.8e-4 in the initial execution. Consult the packaged result for the actual run. This is not a reproduction of a noncommuting figure, a measured bath spectrum, or a general non-Markovian mission model.

Tomography: raw observations first

State tomography uses all nine two-qubit Pauli measurement settings. Qiskit's measurement index order is A,B (q0,q1), while count bitstrings and matrices are B,A (q1,q0). Measurement indices 0,1,2 mean Z,X,Y. The complete complex state is reconstructed; the phase-Bell preset explicitly exercises imaginary coherences.

Process tomography uses all twelve combinations of four input preparations and three Pauli measurements for one qubit. Preparation indices 0,1,2,3 mean |0>, |1>, |+>, |+i>. The Choi matrix is ordered input tensor output, with trace 2. The independent design equation is p = Tr[(rho_in^T tensor E_out) J]. The exact synthetic reference is assembled from analytical Kraus matrices, independently of the Aer channel execution.

Two fits are preserved separately:

1. Raw Qiskit linear inversion, with positivity and trace rescaling disabled, compared to an independently assembled NumPy least-squares reconstruction.
2. Qiskit's explicitly requested positivity/trace adjustment, retaining the original raw fit, eigenvalues, adjustment norm and all counts.

The adjustment is not maximum likelihood. In process mode it need not enforce trace preservation; the partial-trace defect is reported. A process estimate is therefore not automatically exported as a mission Kraus channel. In state mode the adjusted physical state can be imported through the existing mission-state contract; its provenance names the estimator and count artifacts.

Every setting/outcome row retains the integer count, shots, observed probability, raw fitted probability and marginal Wilson 95% interval. Synthetic examples also include exact probabilities and a predeclared Hoeffding criterion at familywise alpha=0.001 across all probability comparisons. These checks concern the simulated shot model; Wilson intervals by themselves do not account for calibration or SPAM errors. The probability plot retains those distinctions.

To analyse recorded counts, set parameters to only raw_records and provenance. The rows are the records array in a saved tomography_raw_counts.json; each requires counts, shots and metadata.m_idx, with metadata.p_idx for process mode. All settings must occur exactly once and declared shots must equal count totals. A measured import must declare data_kind: measured_counts, a nonempty source ID, acquisition ID and calibration ID. Synthetic replay must retain synthetic_counts. Imported counts are never silently evaluated against the demonstration's preparation as if it were their known truth. User-declared provenance is retained but is not proof of an independent experiment.

Traceable sources and limitations

sources_r24/quantum.json records ten primary papers and official technical sources with URLs, DOIs where applicable, scope, limitations and test mappings. The principal HEOM source is Lambert et al., Physical Review Research 5, 013181 (2023), especially the spin-boson discussion and correlation expansion in Eqs. 17–19; the pure-dephasing analytical limit is referenced in the authors' accompanying examples. QuTiP 5, Lindblad's generator theory, Lidar–Chuang–Whaley's decoherence-free subspaces, James et al.'s state-reconstruction methods and Poyatos–Cirac–Zoller's process-characterization methods establish the relevant numerical models. Official documentation fixes the API and basis conventions used here.

The implementation reproduces explicitly scoped equations and limits rather than claiming all results of those publications. No new experimental acquisition, hardware run, independent external replication or customer trial is reported. QuTiP uses a BSD-style three-clause license; Qiskit Experiments is distributed under Apache 2.0. Dependencies are installed separately and their licenses remain applicable; no author endorsement is implied.

Reproduction tests

Run python -m pytest -q test_r24_quantum.py test_r24_quantum_tomography.py in the optional integration environment. Tests cover arbitrary full complex two/four-qubit density matrices, tensor ordering, singlet invariance, independent matrix-exponential evolution, HEOM convergence, actual state/process tomography, exact raw-count replay and malformed-input rejection. Missing optional engines produce explicit test skips rather than passing scientific checks. Inspect the recorded test output and the per-run manifests for the executed versions and actual outcome.