Efficient Reconstruction for Real-Valued Quantum States with Applications to Quantum Linear System Algorithms

Readout has been one of the bottlenecks that limit the practical value of Quantum Linear System Algorithms (QLSA) and quantum differential-equation solvers. Full state tomography requires exponential resources, yet many QLSA workloads only require real-valued states. We introduce Hadamard Random Forest (HRF), a readout method for reconstructing quantum states with real amplitudes in linear measurement settings while retaining expensive post-processing. Utilizing only the real parts for quantum computation is not rare. Many canonical quantum algorithms belong to this category such as Grover’s search and the Bernstein–Vazirani algorithm. Building upon recent insights from resource theory of imaginarity, which establish that real amplitudes are easier to prepare, we further extend the conclusion to readout such states. The standard quantum state tomography method requires 6 hours to reconstruct a general complex-valued 10 qubit state. Employing our simple reconstruction scheme, we reconstructed random real-valued 10 qubit states with high fidelity (~90%) on IBM’s Heron processor within 5 minutes. Our approach not only offers a substantial speedup over conventional tomography but also maintains high accuracy in estimating important quantum state properties such as entanglement and nonstabilizerness. These results provide a simple, hardware-validated route to dramatically lower measurement overheads whenever real-valued states suffice.