Holomorphic representation of quantum computations

Abstract

We introduce a model of computing on holomorphic functions, which captures bosonic quantum computing through the Segal-Bargmann representation of quantum states. We argue that this holomorphic representation is a natural one which not only gives a canonical definition 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 introduce formally the model of Holomorphic Quantum Computing. Non-adaptive circuits in this model allow for a classification of subuniversal models that are generalisations of Boson Sampling and Gaussian quantum computing, while adaptive circuits capture existing universal bosonic quantum computing models. Our results leave open the possibility that continuous-variable quantum computations may outperform their discrete-variable counterpart.

Resources for bosonic quantum computational advantage

“…For example, it has been shown that negativity of the Wigner function is one of such necessary resources [30,31], albeit not sufficient [32]. More recently, it became clear that squeezing and entanglement also play an important role in the hardness of some sampling problems, but only when combined in the right way [33,34]. In Gaussian Boson Sampling, for example, the state at hand is an entangled Gaussian state, which can be described using a positive Wigner function, while negativity of the Wigner function is provided by the non-Gaussian photon detectors.…”

“…The stellar rank is a faithful and operational non-Gaussian measure [34], as it is invariant under Gaussian unitaries, nonincreasing under Gaussian maps, and it lower bounds the minimal number of non-Gaussian operations (such as photon additions or photon subtractions) necessary to prepare a bosonic state from the vacuum, together with Gaussian unitary operations. Moreover, any state can be approximated arbitrarily well in trace distance by states of finite stellar rank, and an optimal approximating state of a given stellar rank can be found efficiently [37].…”

“…We note that Û † k; j (ξ) is the two-mode squeezer that connects the k th detection mode to the auxiliary mode that implements the j th photon-subtraction operation associated with it and thus ( Ŝ † ⊗ 1aux ) Û † is a Gaussian unitary. In particular, r ( ρtotal ) = r ( ρ ⊗ |1 1| ⊗n ) = r ( ρ) + m k=1 r k since the stellar rank is fully additive for pure states [34].…”

“…However, it turns out that not all forms of entanglement are equally suitable. In previous works [34,47], we have discussed the concept of passive separability: a quantum state is said to be passively separable if at least one mode-basis exists in which the state is separable. In other words, for a passively separable state, any entanglement can be undone by an interferometer built with beam-splitters and phase-shifters.…”

“…The stellar function, stellar rank and stellar hierarchy possess various remarkable properties [34,37,50]:…”