Quantum Information Processing, Vol. 4, No. 6, December 2005 (© 2006)
Lattice Quantum Algorithm for the Schr
Equation in 2+1 Dimensions with a Demonstration by
Modeling Soliton Instabilities
and Linda Vahala
Received July 8, 2005; accepted December 5, 2005; publishedonline February 24, 2006
A lattice-based quantum algorithm is presented to model the non-linear
odinger-like equations in2+1dimensions. In this lattice-based model, using
only 2 qubits per node, a sequence of unitary collide (qubit–qubit interaction)
and stream (qubit translation) operators locally evolve a discrete ﬁeld of prob-
ability amplitudes that in the long-wavelength limit accurately approximates a
non-relativistic scalar wave function. The collision operator locally entangles pairs
of qubits followed by a streaming operator that spreads the entanglement throughout
the two dimensional lattice. The quantum algorithmic scheme employs a non-linear
potential that is proportional to the moduli square of the wave function. The model
is tested on the transverse modulation instability of a one dimensional soliton wave
train, both in its linear and non-linear stages. In the integrable cases where ana-
lytical solutions are available, the numerical predictions are in excellent agreement
with the theory.
KEY WORDS: Non-linear Schr
odinger wave equation; quantum algorithm;
soliton dynamics; non-linear quantum mechanical instability; quantum
computing; computational physics.
PACS: 03.67.Lx; 05.45.Yv; 02.60.Cb.
The non-linear Schr
odinger (NLS) equation is one of the most basic equa-
tions of non-linear physics. Its salient feature is that it emits soliton solutions
by exact integration. Hence, it plays a vital role in weakly non-linear systems
Air Force Research Laboratory, Hanscom Field, Bedford, MA 01731, USA.
Department of Physics, William & Mary, Williamsburg, VA 23187, USA.
College of Engineering & Technology, Old Dominion University, Norfolk, VA 23529, USA.
To whom correspondence should be addressed. E-mail: firstname.lastname@example.org,
1570-0755/05/1200-0457/0 © 2006 Springer Science+Business Media, Inc.