Purification Access

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Category C2 · Module oracq.algorithms.input_model.density · Stage V1

Overview

The core abstraction of the density matrix (DM) input model is purification access: access to a density matrix \(\rho\) is defined as the quantum operation \(U\) preparing its purification, \(U|0\rangle = |\psi_\rho\rangle\), acting on the two registers system and environment, such that after tracing out the environment

\[ \operatorname{Tr}_{\mathrm{env}} |\psi_\rho\rangle\langle\psi_\rho| = \rho . \]

Every density matrix admits a purification with environment as wide as system, so this view is a stable interface that the downstream algorithms (B2 quantum SDP, Gibbs state preparation) rely on. Following the repository’s three-layer paradigm, the module provides three constructions: an abstract open declaration, a gate witness for explicit small matrices, and a pure-state adapter.

Interface and input model

abstract_purification(name, width, environment_width=None, *, reversible=True)
gate_purification(rho, *, name=None)
maximally_mixed_purification(width, *, name=None)
PurificationAccess.from_state_preparation(preparation)

API entry points: abstract_purification, gate_purification, maximally_mixed_purification

• abstract_purification: openly declares a purification access slot (the abstract layer of the DM input model); environment_width defaults to width; the program becomes closed after a witness implementation is bound via linking.bind.

• gate_purification: a gate witness for an explicit small density matrix. rho must be a Hermitian positive semidefinite matrix of unit trace whose dimension is a power of two within 2..16.

• maximally_mixed_purification: a purification generator for the maximally mixed state \(I/2^n\) (\(n\) Bell pairs, width ≤ 32).

• from_state_preparation: a pure state is a trivial purification; this adapts a StatePreparation that cleans its work back to zero into a purification access (work plays the role of environment).

All four entry points return a PurificationAccess (oracle_kind = "purification_access", operation signature ("system", "environment")). Module attributes and attributes:

Attribute	Meaning
width / environment_width	system / environment register widths
operation	the underlying Operation; module attributes include density_model="purification_access", zero_input=True
as_state_preparation()	viewing the whole as a StatePreparation on system⊕environment (for B2 composition)
describe()	oracle description (main_qubit=width, anc_qubit=environment_width)

Accompanying classical small-matrix tools: partial_trace (partial trace), trace_distance (trace distance \(T(\rho,\sigma) = \lVert\rho-\sigma\rVert_1/2\)), and gibbs_state (classical reference Gibbs state).

Implementation notes

The gate witness first performs a cyclic Jacobi eigendecomposition \(\rho = \sum_j p_j |v_j\rangle\langle v_j|\) (for complex Hermitian matrices, a diagonal phase rotation precedes real Jacobi elimination; eigenvalues in descending order), then takes the purification \(|\psi_\rho\rangle = \sum_j \sqrt{p_j}\,|v_j\rangle_{\mathrm{s}} |j\rangle_{\mathrm{e}}\), whose amplitude vector is prepared on the concatenated register by the multi-rotation tree of gate_state_prep; the basis-state index convention is system | (environment << system_width). Positive semidefiniteness is checked at generation time; negative eigenvalues below \(-10^{-7}\) raise an INPUT_PROMISE failure.

maximally_mixed_purification applies H and CNOT bit by bit to produce \(n\) pairs of \(|\Phi^+\rangle\); from_state_preparation relies on the clean_work=True promise of the preparation contract and refuses the adaptation when it is not promised. Applicability boundary: the gate witness supports only small density matrices of dimension up to 16 (small instances for validation); large-scale DM input requires providing an implementation of the abstract_purification declaration yourself and binding it.

Validation approach

Category C2 (approximately continuous semantics; the acceptance criteria are in ../development/validation-plan.md §2): the reduced density matrix after the partial trace must agree with the target \(\rho\) within tolerance. Three layers of evidence:

• Structure: the width and attribute assertions of the cases in tests/core/test_density.py:PurificationTests (e.g. test_gate_purification_recovers_rho checks access.width == 1); GibbsTests.test_invalid_inputs_fail_at_generation covers a non-unit trace, non-Hermitian input, and an illegal dimension for gate_purification, as well as maximally_mixed_purification(0), mismatched partial_trace lengths, inconsistent trace_distance dimensions, and similar violations.

• Numerical: PurificationTests.test_gate_purification_recovers_rho traces out environment after simulation and cross-checks element by element against the 2×2 mixed state RHO (places = 7); test_maximally_mixed_purification cross-checks \(I/4\) and verifies the Schmidt structure (nonzero amplitudes appear only on system == environment basis states); test_pure_state_adapter cross-checks the reduced matrix of the pure state \(\sqrt{0.3}|0\rangle + \sqrt{0.7}|1\rangle\); test_classical_tools checks the analytic values of gibbs_state and trace_distance (places = 12).

• Binding: PurificationTests.test_abstract_purification_binds — after the abstract declaration is bound to the gate_purification witness via bind, the partial trace still returns to RHO, covering the consistency of the abstract / gate layers.

Known gaps and planned stages

No known gaps; the stage V1 witnesses are complete (partial-trace cross-check + Schmidt structure + binding consistency).

Related links

• Source: src/oracq/algorithms/input_model/density.py

• Algorithms in the same module: Gibbs state preparation

• API reference: Density-matrix input model and Gibbs states

• Validation matrix: Validation coverage matrix

Numerical validation

The results in this section were produced on real backends by tests/verification/verify_stateprep.py (reference, rir-pysparq, adapter-pysparq, originir-ext + UniQC — four independent paths).

Experiment design (5 cases): for the purification prepared on each path, the system reduced density matrix is computed with an independently implemented numpy partial trace (without calling this module’s partial_trace / trace_distance), and the eigenvalues of the difference matrix then give the trace distance \(T(\rho_{\text{sim}}, \rho)\) and the element-wise error. Instances: a 2×2 complex Hermitian mixed state (off-diagonal entries with imaginary parts), 4×4 and 16×16 Gaussian random density matrices (\(G^\dagger G\) normalized, deterministic seeds), the 3-Bell-pair purification of the maximally mixed state \(I/8\) (with a Schmidt structure check: nonzero amplitudes appear only on system == environment basis states), and the from_state_preparation pure-state adapter (the reduced matrix should equal \(|\psi\rangle\langle\psi|\)).

Key metrics:

Case	Scale (system+environment)	Paths	trace_distance	max_element_error
purification-gate-rho2	1+1 qubit	four paths	1.1e-16	1.1e-16
purification-gate-rho4	2+2 qubit	four paths	3.2e-11	1.9e-11
purification-gate-rho16	4+4 qubit	four paths	8.9e-11	1.4e-11
purification-bell-w3	3+3 qubit	four paths	2.2e-16	5.6e-17
purification-pure-adapter	1+0 qubit	four paths	8.3e-17	1.1e-16

The residuals (~1e-10) of the 4×4 and 16×16 cases come from the method error of the generation-time cyclic Jacobi eigendecomposition (tolerance 1e-13), not from execution error — pairwise deviations between backends are at machine-precision level; the Bell purification’s off_schmidt_probability is 0, confirming the Schmidt structure.

Reproduction command:

PYTHONPATH=src /path/to/backend/python tests/verification/verify_stateprep.py

Artifact path: out/verification/stateprep.json.