QC24 State Preparation Cost Model

Fix an integer $n\ge 0$ and $D=2^n$. An explicit amplitude-table input is a list of $D$ exact complex numbers $(a_x{)}_{x\in \{0,1{\}}^n}$ in the ordered basis convention of Big-endian encoding of ordered bit strings; it is valid when ${\sum }_x\mathrm{\mid }{a}_x{\mathrm{\mid }}^2=1$. Its target is the pure-state ray $[a]$ under Pure state and ray: nonzero scalar multiples represent the same ray, and $a$ is a unit representative when the table is normalized. A claim must state whether its output error is projective chord distance QC24 Projective Chord Distance or a representative-level vector error.

The cost record keeps these quantities separate: (1) the classical encoding length and table-reading/preprocessing work; (2) continuous-angle gate occurrences, or finite-alphabet gate occurrences when that model is specified; (3) width, clean-workspace and serial-depth usage; and (4) the declared output-error metric. If an encoding assigns exactly $b$ bits to each real coordinate and $b$ bits to each imaginary coordinate, with basis order implicit, the dense table contains exactly $2Db$ input bits. Those bits must decode to the input table in an exact-table claim; finite-precision approximations are separate inputs with separate error hypotheses. An explicit table is classical input: this definition grants no coherent table oracle, QRAM, or free loading operation. Ideal exact complex arithmetic and finite-precision bit cost are distinct models.