Linear optics

As an example, and using the Dirac bracket notations (see bra-ket notations), the transformation ${\displaystyle |k\rangle \rightarrow e^{ik\theta }|k\rangle }$ is linear, while the transformation ${\displaystyle \alpha _{0}|0\rangle +\alpha _{1}|1\rangle +\alpha _{2}|2\rangle \rightarrow \alpha _{0}|0\rangle +\alpha _{1}|1\rangle -\alpha _{2}|2\rangle }$ is non-linear. In the above examples, ${\displaystyle k=0,1,\ldots }$ is an integer representing the number of photons. The transformation in the first example is linear in the number of photons, while in the second example it is not. This specific nonlinear transformation plays an important role in optical quantum computing.

Phase shifters and beam splitters are examples of devices commonly used in linear optics.

In contrast, frequency-mixing processes, the optical Kerr effect, cross-phase modulation, and Raman amplification, are a few examples of nonlinear effects in optics.

One currently active field of research is the use of linear optics versus the use of nonlinear optics in quantum computing. For example, one model of linear optical quantum computing, the KLM model, is universal for quantum computing, and another model, the boson sampling-based model, is believed to be non-universal (for quantum computing) yet still seems to be able to solve some problems exponentially faster than a classical computer.

The specific nonlinear transformation ${\displaystyle \alpha _{0}|0\rangle +\alpha _{1}|1\rangle +\alpha _{2}|2\rangle \rightarrow \alpha _{0}|0\rangle +\alpha _{1}|1\rangle -\alpha _{2}|2\rangle }$, (called "a gate" when using computer science terminology) presented above, plays an important role in optical quantum computing: on the one hand, it is useful for deriving a universal set of gates, and on the other hand, with (only) linear-optical devices and post-selection of specific outcomes plus a feed-forward process, it can be applied with high success probability, and be used for obtaining universal linear-optical quantum computing, as done in the KLM model.

• Optics
• Quantum optics
• Nonlinear optics
• Linear optical quantum computing (LOQC)
• KLM model for LOQC
• Optical phase space
• Optical physics
• Nonclassical light