Energy-dependent barren plateau in bosonic variational quantum circuits

Variational quantum circuits (VQCs) [1] are candidates for achieving practical quantum advantages in the noisy intermediate-scale quantum (NISQ) era [2], when scalable error-corrected quantum computers are not yet available. VQCs utilize classical control to optimize a quantum circuit to solve computation problems, including optimization [3], eigen-system problem [4-10], partial-differential equations [11], quantum simulation [12-14] and machine learning [15-23]. As a general approach of designing quantum circuits, it has also found applications in the approximation [24], preparation [25, 26], classification [27-31] and tomography [32] of quantum states. Initial works on VQCs concern discrete-variable (DV) finite-dimensional systems of qubits, which are natural for computation; while continous-variable (CV) systems of bosonic qumodes are less explored. Yet, many important quantum systems are modelled by qumodes. For example, quantum communication and networking [33-37] rely on photons--the only flying quantum information carrier. In this regard, quantum transduction and entanglement distillation are shown to be enhanced by CV VQCs [38]; Photonic quantum computers [39, 40] are also relying on bosonic encoding such as the cat code and Gottesman-Kitaev-Preskill (GKP) code [41], which has shown great promise [42, 43]. The engineering of such code states are greatly boosted by CV VQCs [44-47]; Finally, distributed entangled sensor networks ubiquitously rely on CV VQCs to achieve quantum advantages in sensing [48-51] and data classification [52, 53]. Different from traditional algorithms, the runtime of VQCs is characterized by the time necessary to train the variational parameters to optimize a cost function.