An Explicit Universal Gate-set for Exchange-only
and K. B. Whaley
Received December 18, 2003; accepted December 16, 2003
A single physical interaction might not be universal for quantum computation in
general. It has been shown, however, that in some cases it can achieve universal
quantum computation over a subspace. For example, by encoding logical qubits into
arrays of multiple physical qubits, a single isotropic or anisotropic exchange
interaction can generate a universal logical gate-set. Recently, encoded universality
for the exchange interaction was explicitly demonstrated on three-qubit arrays, the
smallest nontrivial encoding. We now present the exact speciﬁcation of a discrete
universal logical gate-set on four-qubit arrays. We show how to implement the single
qubit operations exactly with at most 3 nearest neighbor exchange operations and
how to generate the encoded controlled-NOT with 27 parallel nearest neighbor
exchange interactions or 50 serial gates, obtained from extensive numerical
optimization using genetic algorithms and Nelder–Mead searches. We also give
gate-switching times for the three-qubit encoding to much higher accuracy than
previously and provide the full speciﬁcation for exact CNOT for this encoding. Our
gate-sequences are immediately applicable to implementations of quantum circuits
with the exchange interaction.
KEY WORDS: Quantum computation; quantum information theory.
PACS: 03.67.Lx, 03.65.Ta, 03.65.Fd, 89.70.þc:
To implement universal computation in the quantum regime, one must be
able to generate any unitary transformation on the logical qubit states. By
now it has become part of the quantum computation folklore that the group
SUð2Þ of single-qubit operations and an entangling two-qubit operation
such as the controlled-NOT (CNOT) can generate any unitary trans-
Department of Chemistry, University of California, Berkeley, California, USA.
Computer Science Division, University of California, Berkeley, California, USA.
CNRS-LRI, UMR 8623, Universite
de Paris-Sud, 91405 Orsay, France.
Quantum Information Processing, Vol. 2, No. 4, August 2003 (# 2004)
1570-0755/03/0100–0289/0 # 2004 Plenum Publishing Corporation