REVIEW
Quantum State Preparation for Complex-Valued Amplitudes
Quantum Physics · Submitted 27 Oct 2026 · 14 references
Pith's verdict: A well-organized quantum compilation subroutine that combines CORDIC and AMP for complex amplitude preparation, with a load-bearing but unproven corner in the division-based uncomputation for negative numbers.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
A state preparation procedure converts binary registers of real and imaginary parts into a quantum amplitude approximating the complex number, using CORDIC for polar conversion and reversible arithmetic with O(n^2) T gates.
The reading
What carries the argument
The central mechanism is the CORDIC algorithm (Coordinate Rotation Digital Computer), which iteratively rotates an (x,y) vector toward the x-axis using only shifts and additions, accumulating the argument in a separate register. A reversible in-place multiplier (Appendix A) that multiplies by 1+$2^{{-2j}}$ allows the y-coordinate update to be performed without extra storage. The AMP procedure [8] then encodes a binary fraction into an amplitude, and phase gates controlled by the argument register introduce the complex phase.
Extended reading notes
Core claim
The paper claims a coherent state preparation procedure (Eq. 1) that, for inputs |x> and |y> representing z = x + iy in n-bit two's-complement fixed-point, produces an output state where the zero-branch amplitude approximates z/α, with total error O($2^{{-(n-l)}}$/α + $2^{{-n}}$) and resource counts of O($n^{2}$) T gates and O(n) additional qubits. The key is to first compute polar coordinates (r', φ') via n CORDIC iterations, then encode r'/α using the AMP procedure, apply phase gates to introduce $e^{{iφ'}}$, and finally uncompute all arithmetic registers. The procedure acts coherently on superpositions and preserves the original input registers.
Load-bearing premise
The correctness of the in-place multiplication by 1+$2^{{-2j}}$ (Appendix A) rests on the claim that the original input can be recovered from the product by a specific division procedure, including handling of negative numbers, which is not fully proven for all two's-complement inputs at the chosen fixed-point precision.
Editorial extensions
If this is right
- When complex-valued amplitudes need to be prepared from binary arithmetic results inside a larger quantum algorithm, this method provides a concrete, stand-alone circuit with known T-gate depth and ancilla count.
- Because the normalization α can be chosen as the smallest power of two bounding the computed magnitude, the amplitude of the target state can be larger than in a direct LCU approach, reducing the number of amplitude-amplification rounds.
- The polar representation (r', φ') remains available in quantum registers after encoding, enabling further arithmetic operations such as reciprocal computation for fast inversion, without re-deriving the magnitude and argument.
- The procedure’s error scales as O(2^{-(n-l)}/α + 2^{-n}), so for a given input precision one can choose n and l to meet a target fidelity while balancing T-gate cost.
Reading between the lines
- The CORDIC-based approach is likely to be most resource-competitive when the expensive part of the parent algorithm is the repeated computation of the input registers; the extra O(n^2) T gates from the polar conversion may be paid off by reducing the number of times those expensive computations are repeated.
- The in-place multiplication by 1+2^{-2j} (Appendix A) is a self-contained routine that could be extracted and used in other quantum arithmetic circuits that require a cheap, reversible multiplication by a constant near unity.
- A concrete numerical comparison for a specific problem (e.g., preparing entries of a sparse Hamiltonian) would clarify whether the smaller normalization indeed leads to a net savings in realistic gate counts, since the O(n^2) overhead from CORDIC may dominate for small n.
Original 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 $z/\alpha$, where $\alpha$ 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 compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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 $z/\alpha$, where $\alpha$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $z/\alpha$, where $\alpha$ is a normalization constant. | contribution 0.95 | 0.93 | 0.34 | asserted 1.00 |
| s1 | The magnitude and argument are computed using the Coordinate Rotation Digital Computer (CORDIC) algorithm. | method 1.00 | 0.56 | 0.04 | asserted 1.00 |
| s2 | We then encode the magnitude into an amplitude, and apply phase gates to introduce the argument. | method 1.00 | 0.85 | 0.06 | asserted 1.00 |
| s3 | A reversible in-place multiplier allows the coordinate updates to use fixed precision while clearing their temporary workspace. | method 1.00 | 0.81 | 0.04 | asserted 1.00 |
| s4 | The procedure uses $O(n^2)$ $T$ gates and $O(n)$ additional qubits. | result 0.79 | 0.90 | 0.13 | asserted 1.00 |
| s5 | 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. | interpretation 0.57 | 0.73 | 0.26 | hedged possibility 1.00 |
| s6 | The computed polar representation also remains available for further arithmetic before amplitude-encoding. | interpretation 0.33 | 0.80 | 0.08 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v1
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- P abstract
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Figures
Figures reproduced from arXiv:2610.36680 by the authors.
References
Discussion
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