Laser Technology, Quantum Optics and Applications
Decoherence Suppression By Parallelism In A Trapped Ion System
code 4
*Kallol Roy, **R.Srikanth, 1T.Srinivas
*,1
Indian Institute of Science, Bangalore. **Poornaprajna Institute Of Scientific Research, Bangalore.
Email: {kallol, tsrinu}@ece.iisc.ernet.in, srik@poornaprajna.org
Contact Information: *Kallol Roy, Applied Photonics Lab, ECE Department, IISc Bangalore,
Bangalore-560012, Email: kallol@ece.iisc.ernet.in, Mobile: +91-9900494907
ABSTRACT: Decoherence as an obstacle in quantum computation is viewed as a struggle
between two forces [1]: the computation which uses the exponential dimension of Hilbert
space, and decoherence which destroys this entanglement by collapse. In this model of
decohered quantum computation, a sequential quantum computer loses the battle, because at
each time step, only a local operation is carried out but
number of gates collapse. With
quantum circuits computing in parallel way the situation is differentnumber of gates
can be applied at each time step and
number gates collapse because of decoherence. As
competition here is even [1]. Our paper improves on this model by slowing
down
by encoding the circuit in parallel computing architectures and running it in
Single Instruction Multiple Data (SIMD) paradigm. We have proposed a parallel ion trap
architecture for single-bit rotation of a qubit.
KEYWORDS: laser cooling, paul traps, ion trapping, parallel computing, mesh network
TRAPPING ATOMIC IONS: Atomic ions are confined by electromagnetic fields. In
Figure 1 a schematic diagram of a linear Paul Trap is shown. A potential
is
applied between diagonally opposite rods. A potential
near the axis of the trap is created
(
and
)
is the distance between the axis and the surface of the electrode. A static harmonic well
is produced in direction and is given by
[
]
[
]
⁄
is a geometric factor, m and q are the ion mass and charge, and
(
⁄ )
is the
oscillation frequency for a single ion or centre of mass oscillation frequency for a collection
of ions along z direction [2]. The equation of motion of ions along the
directions
because of the potentials
are given by Mathieu equation
⁄ ,
where
⁄
),
[
]
[
]
)( ⁄
( ⁄
⁄
),
⁄
)( ⁄
( ⁄
. When
, the
ion virtually is confined in a harmonic pseudopotential
where
⁄(
in the radial direction [2] given by
⁄
)
⁄
(
is the radial
)
frequency. Ions are initialized to known pure states. Using stimulated-Raman transitions
(Figure 2) we can accomplish the transition
where
are internal quantum states that form quantum bits or qubits and
are motional
quantum states.
Figure 1
QUANTUM LOGIC WITH TRAPPED IONS: Single-bit rotations on the ion , (whose
quantum state is |
|
) are done by the transformation
where
(|
|
)
[
( ⁄ )
( ⁄ )
⁄
( ⁄ )
][
]
Experimentally single-bit rotations on ion can be done by application of a magnetic field
⃗
and an electric field ⃗
on the ion trap system. A
{ }), can be realised with
CNOT gate (
and
spectroscopy experiments composing of four level quantum system.
PARALLELIZING QUANTUM OPERATIONS: Suppose an ion in a string of ions
∑
∑
(quantum state of the ion string is
and
) is to be
rotated and the rest of the ions are left undisturbed. The operator
is applied to the
qubit and operator is applied to all other qubits. The overall operator applied on the ion
string is given by
(
Operator
turns out to be a sparse matrix of dimension
of the form
(
and all
)
)
[ ] and given by
are same, and of dimension
( ⁄ )
( ⁄ )
( ⁄ )
( ⁄ )
⁄
( ⁄ )
⁄
(
( ⁄ )
The rotation of only ion in a string of ions is described as
decomposed as a sequence of submatrix-subvector multiplication [4].
(
)(
and
)
can be
)
Each Functional Unit in the above matrix equation consists of ion groups. In other words
each of the Functional Unit contains their own private set of ions (Figure 3). In an
experimental paradigm all the independent Functional Units (Figure 3) interact with the
electromagnetic field applied on them. Transformation of ion group in Functional Unit
1(because of applied electromagnetic field) will have no consequence on the ion group of any
other Functional Unit. Each of the Functional Unit set represents an independent processor in
parallel computing language and the whole set of ion-trap system can be viewed as a SIMD
parallel computer [3]. Each of the Functional Unit (parallel processor) will run the same
programme (matrix transformation) on their own set of ion group.
PROPOSED ARCHITECTURE: We propose an architecture for parallel processing with
an ion trap system by simultaneous interactions of ion group (in each Functional Unit) with
electromagnetic fields applied on them. To address each of the Functional Unit with
electromagnetic fields, Functional Units need to be spatially separated. The other constraint
is all the interactions between the ions (in each Functional Unit) and the electromagnetic field
needs to be synchronous. We have shown a design prototype consists of sixteen functional
units embedded in a mesh network [3].
Functional
Unit 2
Functional
Unit 3
Functional
Unit 0
Functional
Unit 1
Functional
Unit 4
Functional
Unit 5
Functional
Unit 6
Functional
Unit 7
Functional
Unit 8
Functional
Unit 9
Functional
Unit 10
Functional
Unit 11
Functional
Unit 12
Functional
Unit 13
Functional
Unit 14
Functional
Unit 15
RESULTS AND DISCUSSION: We have proposed a parallel computing architecture in an
ion trap system for a very simple single-bit rotation of an ion embedded in an ion string.
Extent of parallelism depends on the how much can we can sparse the matrix .
REFERRENCES
[1] D. Aharonov and M.Ben-Or, Polynomial simulation of Decohered Quantum Computer,
arXiv:quant-ph/9611029 v1 17 Nov 1996.
[2] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof,
Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions,
Journal of Research of the National Institute of Standards and Technology, Volume 103,
Number 3, May–June 1998.
[3] Ananth Grama, Anshul Gupta, George Karypis and Vipin Kumar, Introduction to Parallel
Computing, Pearson Education.
[4] Jumpei Niwa, Keiji Matsumoto and Hiroshi Imai, General-Purpose Parallel Simulator for
Quantum Computing, Review Article.