Quantum Information Processing, Vol. 4, No. 6, December 2005 (© 2006)
Relativistic Path Integral as a Lattice-based Quantum
Received August 22, 2005; accepted December 13, 2005; published online February 16, 2006
We demonstrate the equivalence of two representations of many-body relativis-
tic quantum mechanics: the quantum lattice-gas method and the path integral
method. The former serves as an efﬁcient lattice-based quantum algorithm to sim-
ulate the space-time dynamics of a system of Dirac particles.
KEY WORDS: Dirac equation; quantum computing; quantum lattice gas;
many-body relativistic quantum mechanics; path integrals.
PACS: 03.67.Lx; 03.65.Pm; 04.25.Dm; 05.30.Fk.
Finding a simple rule to represent the spacetime quantum mechanical
dynamics of a system of Dirac particles in 1+1 dimensions as a discrete
path integral, or more accurately as a path summation, is known as the
Feynman chessboard problem.
In Feynman’s notes we see that he ﬁrst
solved this problem in 1946.
A proof by Jacobson and Schulman of Fe-
ynman’s solution to this chessboard problem relies on a deep isomorphism
between the discrete path integral and the partition function in statistical
mechanics of an Ising spin system with nearest-neighbor spin–spin interac-
The 1+1 dimensional chessboard is a square spacetime lattice with
This discrete path integral formalism, included in the beginning of this paper, was pre-
sented on August 20, 2004 as an invited talk entitled “Lattice-based quantum algorithms
for computational phsyics” at the 13th International Conference on the Discrete Simulation
of Fluid Dynamics, hosted by Tufts University in Cambridge, Massachusetts. The quantum
algorithm for the Dirac system in 3+1 dimensions, included at the end of this paper, was
presented on May 9, 2002 at the Quantum Computation for Physical Modeling Workshop
2002, hosted by the Air Force Research Laboratory in Edgartown, Massachusetts.
Air Force Research Laboratory, 29 Randolph Road, Hanscom Field, Massachusetts 01731.
E-mail: Jeffrey.Yepez@hanscom.af.mil; URL: http://qubit.plh.af.mil
1570-0755/05/1200-0471/0 © 2006 Springer Science+Business Media, Inc.