Efficient multimode Wigner tomography

Abstract: Advancements in quantum system lifetimes and control have enabled the creation of increasingly complex quantum states, such as those on multiple bosonic cavity modes. When characterizing these states, traditional tomography scales exponentially with the number of modes in both computational and experimental measurement requirement, which becomes prohibitive as the system size increases. Here, we implement a state reconstruction method whose sampling requirement instead scales polynomially with system size, and thus mode number, for states that can be represented within such a polynomial subspace. We demonstrate this improved scaling with Wigner tomography of multimode entangled W states of up to 4 modes on a 3D circuit quantum electrodynamics (cQED) system. This approach performs similarly in efficiency to existing matrix inversion methods for 2 modes, and demonstrates a noticeable improvement for 3 and 4 modes, with even greater theoretical gains at higher mode numbers.

Conclusion: In summary, we have applied the DEMESST sampling method to characterize multimode cavity states with Wigner tomography. DEMESST is most appropriate for multimode states that have population contained in a subspace of DFE-efficient elements of an overall Hilbert space, and outperforms traditional optimized inversion-based methods by scaling polynomially rather than exponentially with mode number. We observe this improvement for W states on 3 and 4 modes. Here, we have presented comparisons using the multimode Fock basis on multimode W states, but DEMESST also applies to different bases that more readily support other states; this tomography method can even be used for DFE by choosing as a basis the intended target state. While Wigner tomography was presented in this work, the method also operates beyond the bosonic Wigner function, and works for both continuous and discrete systems. This approach can in principle operate without coupling gates between modes, such as when each mode has its own transmon for performing parity measurements, which would be useful for calibrating entangled states over distributed quantum networks. Ultimately, the DEMESST sampling method enables efficient reconstruction of certain large multiqubit or multimode states, which will be advantageous as the size of quantum hardware increases and more complicated states are generated and applied for quantum simulation, bosonic logical state encoding, and error correction.

文章信息：https://doi.org/10.1038/s41467-024-48573-x