Holomorphic representation of quantum computations

Ulysse Chabaud1 and Saeed Mehraban1,2

1Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
2Computer Science, Tufts University, Medford, MA 02155, USA

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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.

Information processing involves encoding information using the properties of physical systems. Quantum properties process information in a fundamentally different way than classical ones, leading to dramatic advantages in computation, communication, cryptography and sensing. Various quantum properties, such as electron spin or light polarization, are discrete in nature and are mathematically captured by quantum bits, or qubits. Other quantum properties, such as position or momentum, are on the other hand continuous in nature and captured by quantum modes.

Quantum information processing with discrete variables is a well studied field and has been defined as the backbone of the standard model of quantum computing. However, continuous variable quantum information has attracted increasing interest in recent years due to its appealing features in terms of bosonic quantum error correction, efficient measurements, and large entanglement generation.

Although discrete and continuous variable quantum models appear to be qualitatively incomparable, they share remarkable similarities. These distinct differences and similarities motivate the need for a representation of quantum mechanics which puts discrete and continuous models of quantum computing on equal footing and delineates the boundary between them.

In our work, we present such a unifying formalism, in which quantum states are represented by analytic functions. Discrete representations are given by polynomials whereas continuous ones are given by holomorphic functions. Using this representation, we generalise the so-called stellar representation of non-Gaussian states, we show that a large class continuous variable quantum dynamics can be represented elegantly using trajectories of classical particles, and we provide a classification of continuous variable quantum computing models based on their complexity.

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[2] Ulysse Chabaud and Mattia Walschaers, "Resources for bosonic quantum computational advantage", arXiv:2207.11781.

[3] Ulysse Chabaud, Abhinav Deshpande, and Saeed Mehraban, "Quantum-inspired permanent identities", arXiv:2208.00327.

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