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The main achievement of the KLM protocol is demonstrating that quantum computing can be done with only linear optical elements, i.e., beam splitters, phase shifters, single-photon sources, and photo-detectors. I wonder why it is important to restrict to linear optics.

The KLM protocol paper mentions an early implementation of a quantum phase gate, which uses a Kerr nonlinearity. I think the reason we are restricted to linear optics is that the Kerr nonlinearity is too weak. Since those papers are outdated, I wonder whether there is any stronger optical nonlinearity implemented at present.

Moreover, regarding other quantum computing platforms, Josephson junctions are very strong nonlinear elements. However, if a small amount of nonlinearity can satisfy all (or most) requirements, then having no strong nonlinearity may not be that bad. In other words, if quantum computing can be conveniently done with most linear components + some weak nonlinearity, then it seems not that important to develop an ALL linear platform.

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    $\begingroup$ the point is that linear operations are much easier to realise, at least in optical platforms. $\endgroup$
    – glS ♦
    Commented Sep 23, 2021 at 7:24
  • $\begingroup$ What kind of non-linear operators are you considering? Note that, for example, anti-linear gates break causality, see Are anti-unitary gates possible? $\endgroup$
    – Mauricio
    Commented Sep 23, 2021 at 13:01
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    $\begingroup$ @Mauricio in this context, the "linear" in "linear optics" doesn't refer to whether the operations act linearly on ket states. Rather, it refers to the type of terms allowed in the Hamiltonian. Roughly speaking, these can be understood as the class of operations which preserve the single-boson/photon subspace (and whose action in higher-boson-number spaces is induced from the single-boson one as usual). See e.g. en.wikipedia.org/wiki/Linear_optics and en.wikipedia.org/wiki/Linear_optical_quantum_computing. So here, both linear and nonlinear ops are standard unitary operations $\endgroup$
    – glS ♦
    Commented Sep 23, 2021 at 13:24
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    $\begingroup$ Continuing from @glS, linear here means that the field operators $a$ and $b$ transform into linear combinations of each other, whereas the easiest nonlinear operations have $a$ and $b$ transform into linear combinations of each other and $a^\dagger$ and $b^\dagger$ (squeezing, aka Bogoliubov transformations). Displacements (linear) essentially come from interaction Hamiltonians that yield $\dot{{a}}\propto {b}$ and squeezing operations (nonlinear) come from Hamiltonians with $\dot{{a}}\propto {b}^\dagger$ or $\dot{{a}}\propto {a}^\dagger$, etc. $\endgroup$
    – Quantum Mechanic
    Commented Sep 25, 2021 at 15:40

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