Hadamard random forest for reconstructing real valued pure quantum states
Zhixin Song, Hang Ren, M.W. Lee, Bryan Gard, Nicolas Renaud, Spencer H. Bryngelson · Communications Physics · 2026
Many potential quantum computing applications, such as linear system solvers, require efficient readout to preserve the algorithmic speedup, but only involve real amplitudes. Conventional tomography methods demand exponential resources for general complex-valued states. We introduce a readout method for real-valued pure quantum states based on amplitude superposition with Hadamard operations and a random forest algorithm for sign determination. Our method reduces the measurement settings required for state vector reconstruction to $${{{\mathcal{O}}}}({N}_{{{{\rm{q}}}}})$$ for an Nq qubit system; the post-processing cost remains exponential $$\Omega ({2}^{{N}_{{{{\rm{q}}}}}})$$. We experimentally validate our method on up to 10 qubits using the latest available IBM quantum processor and demonstrate that it accurately extracts key properties, such as magic. Our method also outperforms the standard SWAP test for state overlap estimation. This calculation resembles numerical integration in certain cases and can be applied to extract nonlinear properties, which are important in application fields. We further implement the method to read out the solution from a quantum linear solver. Quantum computing applications and system characterization often require efficient state reconstruction, yet conventional methods demand extensive resources. Here, the authors introduce a readout method for real-valued quantum states, significantly reducing measurement settings and outperforming existing techniques like compressed sensing tomography, with implications for enhancing quantum linear solvers and nonlinear property extraction.