The paper details a new procedure for preparing quantum states that encodes complex numbers into amplitudes using an algorithm called CORDIC; this computes magnitude and argument from binary registers representing real and imaginary parts. This approach allows smaller normalization than existing methods like LCU, reducing costs during amplitude amplification, and also calculates polar representation enabling further arithmetic before encoding. These underpin many quantum algorithms.

By utilising Coordinate Rotation Digital Computer (CORDIC) to calculate key properties like magnitude and argument, this procedure reduces computational demands compared with previous techniques. Calculations can be performed on these values *before* encoding them into a quantum state, offering greater flexibility in certain applications while potentially reducing resource requirements during computation. The team at softwareQ inc., and University of Waterloo have developed an improved method for preparing quantum states representing complex numbers; these underpin many quantum algorithms.

The procedure uses the CORDIC algorithm, akin to a series of rotations used in calculators to efficiently calculate trigonometric functions, to determine key properties like magnitude and argument from binary registers holding real and imaginary parts. This approach differs from existing methods such as Linear Combination of Units (LCU), which require more resources.

Calculations are performed *before* encoding into a quantum state using T gates, the basic building blocks of circuits, potentially reducing overall computational demands. The method allows for smaller normalization values than previous techniques; it will sharply reduce the number of steps needed during amplitude amplification.

CORDIC-based quantum state preparation lowers normalization constants and qubit overhead

A sharp reduction in normalization constants during quantum state preparation was achieved by scientists and softwareQ inc.; their new method allows for values smaller than those achievable with Linear Combination of Unitaries (LCU), previously considered standard. The breakthrough unlocks amplitude amplification procedures that were computationally prohibitive due to excessive resource demands, particularly scenarios requiring extremely high precision or large numbers of qubits.

Utilising the Coordinate Rotation Digital Computer (CORDIC) algorithm, they compute magnitude and argument from binary registers representing complex number components *before* encoding into an amplitude, preserving access to this polar representation for further arithmetic operations while using O(n²) T gates alongside only O(n) additional qubits.

The technique offers a potentially streamlined approach to preparing quantum states representing complex numbers, reducing computational demands compared with existing techniques; however, it delivers an approximation rather than exact value encoding. How much error can applications tolerate before accuracy suffers remains a key question. Despite yielding approximations instead of precise values, practical applications often accept some degree of error in exchange for reduced computational burden, a trade-off that enhances feasibility and speed.

Retaining information about both magnitude and argument opens avenues for more flexible arithmetic operations prior to final encoding. Efficient preparation of complex number states allows versatile calculations before the final step, as demonstrated by its application to fast-inversion procedures using reciprocals computed via sign changes and scaling. By performing this initial computation *before* creating the final state, they retain access to both magnitude and argument, enabling further arithmetic on encoded data, and potentially unlocking faster algorithms in the coming decade, offering a valuable first step towards more powerful simulations.

This research demonstrated a procedure for preparing quantum states representing complex numbers using the Coordinate Rotation Digital Computer (CORDIC) algorithm to compute magnitude and argument. This approach approximates encoding of complex numbers while requiring O(n²) T gates and O(n) additional qubits; it offers an alternative that may reduce computational demands compared to other methods by allowing smaller normalization values. The method also preserves information about the number’s magnitude and direction for further calculations before final state preparation. Researchers note this approximation introduces some error, but practical applications often accept trade-offs between accuracy and efficiency.

👉 More information
🗞 Quantum State Preparation for Complex-Valued Amplitudes
✍️ Fred Sun (University of Waterloo)
🧠 ArXiv: https://arxiv.org/abs/2610.36680

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