Hadamard Random Forest: Reconstructing real-valued quantum states with exponential reduction in measurement settings

TL;DR

This paper introduces a quantum state reconstruction method that significantly reduces measurement settings for real-valued states, enabling efficient characterization of quantum systems with fewer measurements, validated on IBM quantum hardware.

Contribution

The authors present a novel Hadamard Random Forest approach that reduces measurement settings from exponential to linear for real-valued quantum states, with practical validation.

Findings

Accurately reconstructs quantum states with fewer measurements.

Successfully validated on up to 10 qubits on IBM quantum hardware.

Outperforms standard methods like SWAP test in state overlap estimation.

Abstract

Quantum tomography is a crucial tool for characterizing quantum states and devices and estimating nonlinear properties of the systems. Performing full quantum state tomography on an $N_\mathrm{q}$ qubit system requires an exponentially increasing overhead with $O(3^{N_\mathrm{q}})$ distinct Pauli measurement settings to resolve all complex phases and reconstruct the density matrix. However, many potential quantum computing applications, such as linear system solves, require only real-valued amplitudes. We introduce a readout method for real-valued quantum states that reduces measurement settings required for state vector reconstruction to $O(N_\mathrm{q})$; the post-processing cost remains exponential $\Omega(2^{N_\mathrm{q}})$. This approach offers a substantial speedup over conventional tomography. We experimentally validate our method up to 10 qubits on the latest available IBM quantum processor and demonstrate that it accurately extracts key properties such as entanglement and magic. Our method also outperforms the standard SWAP test for state overlap estimation. This calculation resembles a 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 readout the solution from a quantum linear solver.