Quantum State Preparation for Complex-Valued Amplitudes

Abstract

We present a state preparation procedure that takes binary registers containing the real and imaginary parts of a complex number $z=x+iy$ and prepares an amplitude approximating $z/α$, where $α$ is a normalization constant. The magnitude and argument are computed using the Coordinate Rotation Digital Computer (CORDIC) algorithm. We then encode the magnitude into an amplitude, and apply phase gates to introduce the argument. A reversible in-place multiplier allows the coordinate updates to use fixed precision while clearing their temporary workspace. The procedure uses $O(n^2)$ $T$ gates and $O(n)$ additional qubits. Compared with separately encoding the real and imaginary parts and combining them through LCU, the construction can allow a smaller normalization and reduce amplitude-amplification costs. The computed polar representation also remains available for further arithmetic before amplitude-encoding.

The paper considers coherent conversion of binary-encoded real and imaginary parts of $z=x+iy$ into a complex amplitude, while preserving the inputs and clearing arithmetic workspace. Its target transformation has a designated output amplitude $z'/\alpha$, where $z'$ approximates $z$ and $\alpha$ is a power-of-two normalization bound. The construction computes magnitude and argument using $n$ reversible CORDIC iterations, encodes the magnitude with an existing binary-to-amplitude procedure, applies phase gates representing the argument, and reverses the arithmetic. A reversible in-place multiplication by $1+2^{-2j}$ enables sequential coordinate updates without retaining a separate coordinate copy at every iteration. The appendix constructs this multiplier by recovering the original input exactly from the rounded product using structured division. For $n$-bit inputs with $l$ integer bits and one sign bit, the procedure uses $O(n^2)$ $T$ gates and $O(n)$ additional qubits. With $O(\log n)$ extra fractional working bits, the rounding analysis gives $|z'-z|=O(2^{-(n-l)})$. Including phase-gate synthesis, the designated-amplitude error is $O(2^{-(n-l)}/\alpha+2^{-n})$. Separately encoding the real and imaginary parts and combining them through LCU has asymptotically lower Toffoli cost. The proposed construction can instead permit a smaller normalization, potentially reducing amplitude-amplification calls when input computation is expensive; it also exposes the polar representation for further arithmetic. Application-specific resource comparisons and whether these savings outweigh the additional CORDIC cost are left for future work.