ua nt - p h / 98 10 08 2 v 1 2 7 O ct 1 99 8 Quantum computation over continuous variables

This paper provides necessary and sufficient conditions for constructing a universal quantum computer over continuous variables. As an example, it is shown how a universal quantum computer for the amplitudes of the electromagnetic field might be constructed using simple linear devices such as beam-splitters and phase shifters, together with squeezers and nonlinear devices such as Kerr-effect fibers and atoms in optical cavities. Such a device could in principle perform ‘quantum floating point’ computations. Problems of noise, finite precision, and error correction are discussed. Quantum computation has traditionally concerned itself with the manipulation of discrete systems such as quantum bits, or ‘qubits’. Many quantum variables, such as position and momentum, or the amplitudes of electromagnetic fields, are continuous. Although noise and finite precision make precise manipulations of continuous variables intrinsically more difficult than the manipulation of discrete variables, because of the recent developments in quantum error correction and quantum teleportation of continuous quantum variables it is worthwhile addressing the question of quantum computation over continuous variables. At first it might seem that quantum computation over continuous variables is an ill-defined concept. First consider quantum computation over discrete variables. A universal quantum computer over discrete variables such as qubits can be defined to be a