Theory and common conventions¶
This reference defines the mathematical language shared by the user and maintainer guides. The implemented model domains are listed in Choose and define a model; a mathematical definition alone is not an implementation support claim.
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Content |
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Cells, chains, qudits, stabilizer projectors, coherent sources and canonical thermal equilibrium. |
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Ordered finite-imaginary-time trace, exact binary local weights, reduced couplings, constants and numerical limits. |
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Local/string/closed-cycle operators, primal/dual measurements, connected functionals and deterministic decoded operators. |
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Exact integration over spatial logical shifts and conditional joint estimators in the higher-degree production ensemble. |
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The circle code, its two-dimensional spin image, spacetime insertions, magnetization moments and boundary twists. |
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Shared symbols and terms. |
The thermal system and the sampler¶
For a Hamiltonian H and inverse temperature \(\beta=1/T\), equilibrium means the canonical state
Temperature specifies an ensemble. It is not an additional unitary term in H. The source coefficients in this package are coherent Hamiltonian parameters; they are not stochastic error probabilities or bath noise rates. The package does not implement open-system real-time evolution.
Monte Carlo samples a classical representation of a thermal trace. A Markov update is an algorithmic transition, not an interval of quantum physical time. The extra lattice axis is imaginary time. Its spacing, a Monte Carlo sweep and wall-clock time are three distinct notions.
Exact identities and approximations¶
The binary local-weight and logical-marginalization identities below are finite-lattice algebraic identities in exact arithmetic. The finite-M trace generally approximates the full quantum Gibbs trace. The implementation uses finite floating-point arithmetic, resource-limited histories and numerical diagnostics. Statistical error, initialization/mixing error, finite volume, finite M and any interpolation/decoder dependence must be assessed separately.
The definitions follow the homological-code spacetime construction cited in References, with the projector and binary-source normalizations fixed explicitly here. They do not import every claim of that reference as a theorem.