EHands: Quantum Protocol for Polynomial Computation on Real-Valued Encoded States

The novel constructive EHands protocol defines a universal set of quantum operations for multivariable polynomial transformations on quantum processors by introducing four basic subcircuits: multiplication, addition, negation, and parity flip, and by using expectation-value encoding (EVEN) to represent real numbers in quantum states. These elementary arithmetic operations can be systematically composed to compute degree-$d$ polynomials, $P_d(x)$, on a QPU. The resulting quantum circuit structure closely mirrors the stepwise evaluation of polynomials on a classical calculator, providing an intuitive and efficient approach to polynomial computation on quantum hardware. By enabling direct, predictable polynomial and nonlinear data transformations on a QPU, our method reduces dependence on classical post-processing in hybrid quantum-classical algorithms, enabling advancements in many quantum algorithms. The EHands quantum circuits are compact enough to deliver meaningful and accurate results on today's noisy quantum processors. We present a detailed implementation of $P_4(x)$ and report experimental results for polynomial approximations of common functions, obtained using IBM's Heron-class quantum processors and an ideal Qiskit simulator.