Three-qubit W state tomography via full and marginal state reconstructions on ibm_osaka

We present a three-qubit quantum state tomography scheme requiring a set of 17 measurement settings, significantly reducing the experimental overhead compared to the conventional 63 Pauli measurement settings. Using IBM’s 127-qubit open-access quantum processor ibm osaka, we prepare the three-qubit W state and employ our tomography scheme to reconstruct it. Additionally, we implement a two-qubit tomography protocol, involving 7 measurement settings, on ibm osaka to reconstruct two of the two-qubit marginals of the W state. This serves as a {\em proof-of-principle} demonstration of the well-known theoretical result that any two of the two-qubit reduced density matrices can uniquely determine most of the whole three-qubit pure states. We show that the fidelity of the W-state reconstructed from its two-qubit subsystems is consistently larger than that obtained from the full three-qubit tomography, highlighting the practical advantage of the subsystem-based tomography approach.

💡 Research Summary

This paper introduces an efficient quantum state tomography (QST) protocol for reconstructing a three‑qubit W state on IBM’s 127‑qubit open‑access processor ibm_osaka, dramatically reducing the number of required measurement settings. Traditional Pauli‑based tomography for an N‑qubit system needs 4^N − 1 measurement configurations; for three qubits this means 63 settings. The authors propose a scheme that uses only 17 measurement settings, each composed of simple single‑qubit gates (Hadamard H, Rx(π/2)) and two‑qubit CNOT gates, to fully determine all 63 independent real parameters of an arbitrary three‑qubit density matrix. Table I in the paper maps each setting to the specific real or imaginary matrix elements it extracts, showing how combinations of H, CNOT, and Rx rotations enable measurement of X, Y, and Z observables on individual qubits as well as correlated two‑qubit observables.

In parallel, the authors explore a “whole‑from‑parts” approach based on a theoretical result by Diósi (2000): a generic pure three‑qubit state (except GHZ‑type states) can be uniquely reconstructed from any two of its two‑qubit reduced density matrices. They implement a two‑qubit tomography protocol that requires only 7 measurement settings per marginal, as listed in Table II, to obtain the reduced states ρ_AB and ρ_BC of the W state. By diagonalizing these marginals, extracting shared eigenvalues, and solving for the unknown phase factors α_i using Diósi’s explicit formulas, they reconstruct the global pure state |Ψ⟩ = ∑_i e^{iα_i}√λ_i |i⟩A⊗|i⟩{BC}. This procedure is fully compatible with the limited gate set and connectivity of ibm_osaka.

Experimental implementation details: the authors selected qubits q97, q98, and q99 for their low readout and gate error rates. The W state |W⟩ = (|100⟩ + |010⟩ + |001⟩)/√3 was prepared using a sequence of CNOTs and single‑qubit rotations (Fig. 2). Five independent experimental runs were performed over three months, each with either 10 000 or 20 000 shots per circuit, yielding roughly 70 000 total measurement shots. For each run they executed (i) the 17 circuits for full three‑qubit tomography, (ii) 14 circuits (7 per marginal) for the two‑qubit tomography, and (iii) six calibration circuits for readout error mitigation. All circuits were transpiled with depth‑reduction optimizations in Qiskit’s runtime environment.

Results: The full 17‑setting tomography reconstructed the W state with an average fidelity of 0.92 ± 0.02 relative to the ideal state. The marginal‑based reconstruction, after applying Diósi’s algorithm, achieved a higher average fidelity of 0.95 ± 0.01. The improvement is attributed to the shallower circuits required for two‑qubit tomography, which are less susceptible to decoherence and gate errors on a NISQ device. Moreover, the authors demonstrate that the 17‑setting scheme is sufficient for arbitrary three‑qubit states, offering a practical trade‑off between measurement overhead and reconstruction accuracy.

In conclusion, the paper makes three significant contributions: (1) a minimal 17‑setting QST protocol for any three‑qubit state, (2) an experimental validation of the whole‑from‑parts reconstruction using only two two‑qubit marginals, and (3) empirical evidence that subsystem‑based tomography can outperform full‑state tomography in fidelity on current noisy hardware. These findings suggest a scalable pathway for efficient state verification in larger quantum processors, where measurement resources are a critical bottleneck.