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

TL;DR

EHands is a quantum protocol that efficiently computes multivariable polynomials on quantum hardware using real-valued encoding and expectation measurements, enabling parallel evaluation with shallow circuits.

Contribution

The paper introduces EHands, a novel quantum-native method for polynomial computation that operates directly on real-valued states and uses expectation values for output, differing from prior discretization approaches.

Findings

Successfully implemented on IBM quantum processors

Accurately approximates functions like ReLU and arctan

Operates with linear depth in polynomial degree

Abstract

We present EHands, a quantum-native protocol for implementing multivariable polynomial transformations on quantum processors. The protocol introduces four fundamental, reversible operators: multiplication, addition, negation, and parity flip, and employs the Expectation Value ENcoding (EVEN) scheme to represent real numbers as quantum states. Unlike discretization or binary encoding methods, EHands operates directly on vectorized real-valued inputs prepared in the initial state and applies a shallow quantum circuit that depends only on the polynomial coefficients. The result is obtained from the expectation value measured on a single qubit, enabling efficient parallel evaluation of a polynomial across multiple data points using a single circuit. We introduce both a reversible implementation for degree-$d$ polynomials, requiring $3d$ qubits, and a non-reversible variant that uses qubit resets to reduce the requirements to $d+1$ qubits. Both implementations exhibit linear depth scaling in $d$ and are explicitly decomposed into one- and two-qubit gates for direct execution on current quantum processing units. The protocol's effectiveness is demonstrated through experimental validation on IBM's Heron-class quantum processors, showing reliable polynomial approximations of functions like ReLU and arctan.