page_title: KLM protocol
section_title: Implementations of elementary quantum gates
hierarchical_section_title: KLM protocol / Elements of the KLM scheme / Implementations of elementary quantum gates
caption_attribution_description: English: Hadamar Linar optics Circuit
context_page_description: The KLM scheme or KLM protocol is an implementation of linear optical quantum computing, developed in 2000 by Knill, Laflamme and Milburn. This protocol makes it possible to create universal quantum computers solely with linear optical tools. The KLM protocol uses linear optical elements, single photon sources and photon detectors as resources to construct a quantum computation scheme involving only ancilla resources, quantum teleportations and error corrections.
context_section_description: Ignoring error correction and other issues, the basic principle in implementations of elementary quantum gates using only mirrors, beam splitters and phase shifters is that by using these linear optical elements, one can construct any arbitrary 1-qubit unitary operation; in other words, those linear optical elements support a complete set of operators on any single qubit.
  The unitary matrix associated with a beam splitter  is:
  ,
  where  and  are determined by the reflection amplitude  and the transmission amplitude  (relationship will be given later for a simpler case). For a symmetric beam splitter, which has a phase shift  under the unitary transformation condition  and , one can show that
  ,
  which is a rotation of the single qubit state about the -axis by  in the Bloch sphere.
  A mirror is a special case where the reflecting rate is 1, so that the corresponding unitary operator is a rotation matrix given by
  .
  For most cases of mirrors used in QIP, the incident angle .
  Similarly, a phase shifter operator  associates with a unitary operator described by , or, if written in a 2-mode format
  ,
  which is equivalent to a rotation of  about the -axis.
  Since any two  rotations along orthogonal rotating axes can generate arbitrary rotations in the Bloch sphere, one can use a set of symmetric beam splitters and mirrors to realize an arbitrary  operators for QIP. The figures below are examples of implementing a Hadamard gate and a Pauli-X-gate (NOT gate) by using beam splitters (illustrated as rectangles connecting two sets of crossing lines with parameters  and ) and mirrors (illustrated as rectangles connecting two sets of crossing lines with parameter ).
  In the above figures, a qubit is encoded using two mode channels (horizontal lines):  represents a photon in the top mode, and  represents a photon in the bottom mode.
  In the KLM scheme, qubit manipulations are realized via a series of non-deterministic operations with increasing probability of success. The first improvement to this implementation that will be discussed is the nondeterministic conditional sign flip gate.
caption: Implementation of a Hadamard gate with a beam splitter and a mirror. Quantum circuit is on the top part.
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