Optimizing State Preparation for Variational Quantum Regression on NISQ Hardware

TL;DR

This paper tackles the challenge of running a variational quantum regression algorithm on NISQ hardware by integrating a novel real-valued state preparation method with ZX-calculus-based circuit optimizations. The authors demonstrate a linear-gate-count realization of the state preparation using Pauli pushing, phase folding, and Hadamard pushing, enabling execution on a 20-qubit device with limited mid-circuit measurement support. They couple these circuit optimizations with classical post-processing (classical shadows and MThree mitigation) to improve loss estimation and regression performance, validating the approach on a Graduate Admission dataset. The work shows that, despite hardware noise, the optimized quantum regression remains competitive with classical baselines and provides a broadly applicable approach for arbitrary real-valued state preparation in quantum algorithms, with potential to extend to other variational and quantum learning tasks.

Abstract

The execution of quantum algorithms on modern hardware is often constrained by noise and qubit decoherence, limiting the circuit depth and the number of gates that can be executed. Circuit optimization techniques help mitigate these limitations, enhancing algorithm feasibility. In this work, we implement, optimize, and execute a variational quantum regression algorithm using a novel state preparation method. By leveraging ZX-calculus-based optimization techniques, such as Pauli pushing, phase folding, and Hadamard pushing, we achieve a more efficient circuit design. Our results demonstrate that these optimizations enable the successful execution of the quantum regression algorithm on current hardware. Furthermore, the techniques presented are broadly applicable to other quantum circuits requiring arbitrary real-valued state preparation, advancing the practical implementation of quantum algorithms.

Figures (13)

• Figure 1: Decomposing and optimizing multi-controlled $R_Z$ gates. a) Decomposition of a 3-controlled $R_Z$ gate. b) A simple example of phase folding.
• Figure 2: End-to-end example(s) for the optimization process for the quantum regression algorithm with the compact binary encoding. a) An initial, non-optimized circuit example with $K=4$ and $M=1$. Each of the multi-controlled $R_Z$ gates present decompose to an exponential amount of CNOT gates with respect to the amount of control qubits, similarly to the example in Fig. \ref{['fig:3-control']}a. b) An optimized version of the example circuit above. In this circuit, all-to-all qubit connectivity is assumed.
• Figure 3: Scaling of all elementary gates in a naive implementation, optimized version, and Qiskit built-in state preparation.
• Figure 4: The effects of the classical post-processing techniques on the R2 score. a) ADAM with and without MThree and the classical shadow. b) Nelder-Mead with and without MThree and the classical shadow.
• Figure 5: R2 score results for the Admission Predict dataset with the IQM Garnet quantum hardware, a noiseless simulator, and a purely classical regressor.