NNQA: Neural-Native Quantum Arithmetic for End-to-End Polynomial Synthesis

Abstract

Hybrid classical quantum learning is often bottlenecked by communication overhead and approximation error from generic variational ansatzes. In this study, we introduce Neural Native Quantum Arithmetic (NNQA), which compiles classically learned nonlinear representations into precise quantum arithmetic composed of native unitary blocks. Theoretically, we prove that the universal approximation of quantum polynomial arithmetic can be realized by transforming a classical neural network into a quantum circuit, with the resulting error arising solely from measurement shot noise, thereby extending classical operator-level estimation guarantees into the quantum regime. Empirical validation on IBM Quantum Heron3 and IonQ Forte processors shows performance limited primarily by device noise without variational fine tuning: we achieve over 99.5% accuracy for polynomials up to degree 35 and demonstrate scalability on IonQ hardware up to 36 qubits and circuit depths of 70, reaching a negligible RMSE of 0.005. Overall, NNQA establishes a new paradigm of synthesizing quantum arithmetic for native quantum computation.

Figures (8)

• Figure 1: Error scaling of NNQA for quantum simulation, where the noise is controlled by degree $d$ and shots. The error is decreasing as $O(1/\sqrt{N})$.
• Figure 2: Classical training produces polynomial coefficients $\{a_k\}$, which are compiled to rotation angles $\{\alpha_k\}$ via \ref{['thm:coeff-to-weight']}, then composed into quantum polynomial circuits.
• Figure 3: Degree-3 polynomial circuit compiled by NNQA. The circuit encodes input $x$, computes powers $x^2, x^3$ via chained multiplications, and aggregates terms with weighted sums and parity corrections. Measurement yields the polynomial evaluation.
• Figure 4: The degree-independence of recovery error on IonQ is demonstrated as follows: (a) The root mean square error (RMSE) remains consistent across degrees 1 to 35, with an average value of approximately 0.0155, which aligns with shot noise and hardware error. (b) The correlation exceeds 0.994 for all degrees, with an average of 0.997. Error bars represent the standard deviation over 10 trials.
• Figure 5: The polynomial recovery on the IBM Heron3 QPU is depicted as follows: dashed curves represent the theoretical polynomials, squares denote the predictions made by classical neural networks, and diamonds with error bars illustrate the quantum measurements residual error representing each independent five trials. All results are contained within the shot-noise uncertainty bands.

Theorems & Definitions (12)

• Theorem 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 3.1
• proof
• Corollary 3.2
• Lemma 1.1
• proof