One clean qubit¶
A restricted quantum-computation model whose input is one pure qubit tensored with an otherwise maximally mixed register, followed by a polynomial-size unitary circuit and a bounded-gap decision readout of a designated qubit.
Core Idea¶
The one clean qubit model starts with one pure qubit and an otherwise maximally mixed register, applies an efficient unitary circuit, and measures a designated output under a stated inverse-polynomial decision gap. Purity is the scarce resource. Purity is the scarce resource.
Scope of Application¶
The model is used in quantum complexity, trace estimation, quantum-circuit sampling, NMR-motivated resource studies, and tests of nonclassical advantage with little purity. Use DQC1 only with the input density matrix, total and clean qubit counts, permitted resets/ancillas, circuit family, measurement, sampling precision, promise gap, and decision-versus-sampling variant explicit.
- Complexity theory. Defines DQC1 decision and sampling tasks.
- Trace estimation. Encodes normalized unitary traces in clean-qubit observables.
- Resource theory. Studies computational value of small purity.
- Classical hardness. Examines consequences of efficient simulation.
- NMR context. Motivates computation from highly mixed ensembles.
Clarity¶
State total qubits, exact initial density matrix, circuit uniformity and size, allowed ancillas/resets, output qubits, number of shots, promise gap, and whether the claim concerns decision, expectation estimation, or sampling. The closest near miss sets the boundary: Few-clean-qubit variants are the nearest related models; equivalence results depend on the allowed number and measurement convention. A positive case must satisfy this test: A computation qualifies when its input purity, allowed circuit, measurement, runtime, and acceptance promise meet the DQC1 definition.
Manages Complexity¶
The model isolates purity from circuit size: one controlled degree of initialization can interrogate an exponentially large mixed space. That abstraction clarifies resources but hides experimental polarization, readout noise, repeated ensembles, and precision cost. The central minimal purity–readout precision tradeoff is this: Restricting initialization exposes quantum structure while small signals demand many samples. A second class definition–amplification tension matters because Inverse-polynomial bias enables decision while extra clean repetitions are not free. The abstract mixed state–physical preparation tension adds that A maximally mixed register is mathematically simple while experimental ensembles have calibration and noise.
Abstract Reasoning¶
Use three linked moves: write the full input density matrix and count every initialized ancilla; specify the uniform polynomial circuit and disallow unmodeled purification steps; derive the designated-qubit observable or acceptance probability. As a collapse test, the case exits when state preparation smuggles extra purity, nonunitary resets, postselection, or exponentially precise measurement into the model. A fourth check is to show an inverse-polynomial YES/NO gap and account for sampling precision. A final check is to compare classical or quantum classes only under matching variants and reduction notions.
Knowledge Transfer¶
The idea that a small ordered subsystem probes a disordered environment transfers to metrology and ensemble computation, but DQC1 transfers literally only with its density matrix, circuit restrictions, and complexity-theoretic output rule. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Input purity is explicitly scarce. A designated observable converts evolved state into a decision signal.
Relationships to Other Abstractions¶
Current abstraction One clean qubit Domain-specific
Parents (1) — more general patterns this builds on
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One clean qubit is a kind of Quantum-Computation Model Domain-specific
One clean qubit satisfies the defining boundary of Quantum-Computation Model: A quantum-computation model is a formal specification of quantum information carriers, admissible initial states, operations, spatial or circuit organization, resource bounds, noise assumptions, and measurement rules used to define computations and compare computational power.
Hierarchy path (1) — routes to 1 parentless root
- One clean qubit → Quantum-Computation Model → Representation → Abstraction
Neighborhood in Abstraction Space¶
One clean qubit sits in a moderately populated region (48th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Quantum-Computation Model — 0.88
- Quantum Illumination — 0.88
- Clifford gate — 0.86
- Quantum State — 0.86
- Logic Circuit — 0.85
Computed from structural-signature embeddings · 2026-10-08