I.1 The quantum seed realization
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Conventions.
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Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying
All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write \([A,B]=AB-BA\) and use units with \(\hbar=1\). Parameter derivatives are real derivatives. A matrix \(X\) is supported on a code projector \(P\) when \(PXP=X\).
The common operator sequence of the series becomes
Text description
Seven nodes are connected from left to right: seed, gates, project, transfer, refine, constrain, and reconstruct. The diagram organizes the finite operator language used across the chapter; it is not an experimental process diagram.
Definition I.1.1: Finite quantum seed system
A finite quantum seed system is a tuple
where \({\mathcal H}\) is a finite-dimensional Hilbert space, \(\psi_0\in{\mathcal H}\) is a nonzero seed, \({\mathcal O}\) is the declared operator grammar, \({\mathcal R}\) is a collection of algebraic or physical constraints, \(\{P_{\ell,\alpha}\}\) is an exact packet system at each refinement level, \(H=H^*\) is a Hamiltonian when dynamics are present, and \(\tau\) specifies the topology used in a later completion.
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The generated state space is
The quantum seed is cyclic for the grammar when
This is a statement about the pair \((\psi_0,{\mathcal O})\), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.