Thread by @fgrosshans

Seated for a seminar by Francesco Arzani on his work with Nicolas Treps and Giulia Ferrini on
Polynomial approximation of non-Gaussian unitaries by counting one photon at a time (arXiv:/1703.06693 arxiv.org/abs/1703.06693 / PRA 95 052352 dx.doi.org/10.1103/PhysRe… )
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Francesco Arzani: It’s difficult to define computation in a continuous variable (CV) set-up. People usually chose specific encoding of qubits. But Francesco (and myself!) finds encoding independent definitions more interesting.
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Francesco Arzani: we are interested in the transformation exp(i H(q,p)).
A universal state should be bale to approximate any polynomial hamiltonian. arXiv:quant-ph/9810082 arxiv.org/abs/quant-ph/9… / PRL 82 1784 doi.org/10.1103/PhysRe… gave an example with quadratic gates + 1 cubic

Francesco Arzani: To be able to have a polynomial of unbounded degree, one needs one non-Gaussian/non-quadratic gate.
For the paper above, the it was exp(iq³s).
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Francesco Arzani: The idea is to use here cluster states, CV analogue of graph states, with nulliffiers instead of stabilizers. Finite squuezing approximation of them are Gaussian states and realistic / actually done in the lab (e.g. @lkb_lab )
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@lkb_lab Francesco Arzani: For quantum advantage, One needs a non-Gaussian operation. Here, it is photon subtraction, either done with a beamsplitter and a photon counter or a χ⁽²⁾ crystal + photon detection.

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@lkb_lab Francesco Arzani: That, chained with homodyne measurement and conditioned unitary transformation gives an effective operator corresponding to a monomial (q—λ), chaining them allows to apply a higher degree polynomial (3 to approximate a cubic gate)
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@lkb_lab Francesco Arzani: This being probabilist, chaining too many operations leads to low probabilities.
The fidelity is not ideal too, but is OK for low photn numbers.
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@lkb_lab Francesco Arzani: The previous gates included post selection on continuous parameters. Another set-up without such post selection (only dicrete) has a slightly higher success probablities 10⁻² – 10⁻³.
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