[6][7][8] By modeling each mode of the electromagnetic field as a quantum harmonic oscillator with its associated creation and annihilation operators, one defines a canonically conjugate pair of variables for each mode, the so-called "quadratures", which play the role of position and momentum observables.
[11]) In 2013, quantum-optics techniques were used to create a "cluster state", a type of preparation essential to one-way (measurement-based) quantum computation, involving over 10,000 entangled temporal modes, available two at a time.
[1] The first method, proposed by Seth Lloyd and Samuel L. Braunstein in 1999, was in the tradition of the circuit model: quantum logic gates are created by Hamiltonians that, in this case, are quadratic functions of the harmonic-oscillator quadratures.
[15][16] Yet a third model of continuous-variable quantum computation encodes finite-dimensional systems (collections of qubits) into infinite-dimensional ones.
An algorithm might be described in the language of quantum mechanics, but upon closer analysis, revealed to be implementable using only classical resources.
Such an algorithm would not be taking full advantage of the extra possibilities made available by quantum physics.
[20] A second motivation is to explore and understand the ways in which quantum computers can be more capable or powerful than classical ones.
One example of a scientific problem that is naturally expressed in continuous terms is path integration.
In this context, it is known that quantum algorithms can outperform their classical counterparts, and the computational complexity of path integration, as measured by the number of times one would expect to have to query a quantum computer to get a good answer, grows as the inverse of ε.