Holomorphic representation of quantum computations

Abstract

We study bosonic quantum computations using the Segal-Bargmann representation of quantum states. We argue that this holomorphic representation is a natural one which not only gives a canonical description of bosonic quantum computing using basic elements of complex analysis but also provides a unifying picture which delineates the boundary between discrete- and continuous-variable quantum information theory. Using this representation, we show that the evolution of a single bosonic mode under a Gaussian Hamiltonian can be described as an integrable dynamical system of classical Calogero-Moser particles corresponding to the zeros of the holomorphic function, together with a conformal evolution of Gaussian parameters. We explain that the Calogero-Moser dynamics is due to unique features of bosonic Hilbert spaces such as squeezing. We then generalize the properties of this holomorphic representation to the multimode case, deriving a non-Gaussian hierarchy of quantum states and relating entanglement to factorization properties of holomorphic functions. Finally, we apply this formalism to discrete- and continuous- variable quantum measurements and obtain a classification of subuniversal models that are generalizations of Boson Sampling and Gaussian quantum computing.

60 + 22 pages. Version accepted in Quantum. Comments are welcome!

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• Quantum Kernel Machine Learning With Continuous Variables
• Classical simulation and quantum resource theory of non-Gaussian optics
• Clifford operations and homological codes for rotors and oscillators
• Topological obstructions to quantum computation with unitary oracles
• The stellar decomposition of Gaussian quantum states
• Characterizing the Multipartite Entanglement Structure of Non-Gaussian Continuous-Variable States with a Single Evolution Operator
• Complexity of quantum tomography from genuine non-Gaussian entanglement
• Adding or Subtracting a single Photon is the same for Pure Squeezed Vacuum States
• Qubit Geometry through Holomorphic Quantization
• The symplectic rank of non-Gaussian quantum states
• Non-Gaussianity from superselection rules