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

TL;DR

This work presents EHands, a quantum-native framework for polynomial evaluation on real-valued data encoded directly in quantum states via the EVEN scheme. By introducing four reversible arithmetic primitives, EHands enables efficient, vectorized construction and evaluation of multivariable polynomials with shallow circuits, and provides both reversible (3d qubits) and non-reversible (d+1 qubits) implementations with linear depth in degree. The approach is experimentally validated on IBM QPUs for polynomial approximations of functions like ReLU and arctan, demonstrating practical viability on current hardware with error mitigation. Overall, EHands offers a path to direct, bulk polynomial processing on near-term quantum devices, with potential impact on quantum machine learning, signal processing, and quantum image analysis through batch, real-valued computations on quantum data.

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.

Figures (17)

• Figure 1: (a) EVEN encoding of the real number $x$ into the quantum state $\ket{\phi}$ using $\theta=\arccos(x)$, along with its decoding, where the expectation value of the Pauli-$Z$ operator is the estimator $\hat{x}$. The four building blocks of the EHands protocol are represented as circuits : (b) multiplication, (c) addition, (d) negation, and (e) parity flip.
• Figure 2: EHands circuits for a degree-4 polynomial $P_4(x) = \frac{1}{5}(a_0+a_1x+a_2x^2+a_3x^3+a_4x^4)$: (a) Reversible 12-qubits EHands circuit for computing $P_4(x)$, where the 4 copies of the input $x$ are stored in qubits $d_i$, and the 5 polynomial coefficients are encoded in qubits $a_i$. The 3 ancilla qubits $\text{anc}_i$ are needed for concatenation of the 4 sum operators. Note that only qubit $a_0$ is measured. Unitaries labeled as $\Pi$ and $\Sigma_w$ represent the multiplication and addition circuits shown in \ref{['fig:eh-prod', 'fig:eh-sum']}, respectively. (b) Non-reversible 5-qubits EHands circuit for the same degree-4 polynomial using only 1 ancilla qubit and many resets allowing for qubit recycling.
• Figure 3: Hardware results from IBM's Aachen, Kingston, and Marrakesh for polynomial approximations of the functions listed in \ref{['tab:ibm']}, where we enabled Pauli twirling to mitigate hardware errors.
• Figure 4: (a) Encoding of 2 real numbers $\{x_0,x_1\}$ in a 2-qubit state $\ket{\phi}$. (b)--(c) Decoding is performed by measuring the expectation value of the Pauli-$Z$ operator on the corresponding qubit.
• Figure 5: Computing product $x_0x_1$ as the expectation value of $\sigma_z$ operator measured on $q_1$.