Classical output, quantum-state output, and access costs

An algorithm result contract declares exactly one output type:

• Classical string: a finite bit string in a stated encoding. Its cost includes every output bit written and any required formatting.
• Classical sample: an element of a stated finite set, with the output law and the cost of writing the sample encoding.
• Quantum state: a density operator, as in Density operator, on an explicitly named finite register, produced by a declared state-preparation procedure. This does not mean that its amplitude list or a classical description of the state is output.

The accompanying cost record keeps separate: input encoding length and preprocessing; each data-structure or oracle operation; state preparation and its inverse or controlled forms; ordinary gates, controlled gates, and inverse gates; register width and clean/dirty/other workspace; measurement and classical postprocessing; finite-gate approximation precision; and classical output bits. A coordinate that the model does not use is marked not applicable. No state-preparation oracle, QRAM, controlled access, inverse access, or full classical state description is assigned zero cost without an explicit interface.

For the exact continuous-rotation table model, QC24 State Preparation Cost Model and QC24 State Preparation specify the table input and circuit procedure; QC24 Explicit State-Preparation Cost Guarantee gives the corresponding $D=2^n$ gate/ancilla upper bound. That bound uses exact continuous rotations and gives no finite-gate or uniform bit-time claim. For a fixed finite gate set, Solovay–Kitaev Unitary Compiler for SU(2) and Solovay–Kitaev Error and Resource Bound supply a separate approximation and resource contract. The circuit-family coordinates are those of Circuit-family and computation cost model.