1. FOCUS and MCTP, Department of Physics, University of Michigan, Ann Arbor, Michigan 481092. LQIT and ICMP, Department of Physics, South China Normal University, Guangzhou, China3. JQI, University of Maryland and NIST, College Park, MD 20742

Guin-Dar Lin1, S.-L. Zhu1,2, R. Islam3, K. Kim3, M.-S. Chang3, S. Korenblit3, C. Monroe3, and Luming Duan1

Large-scale quantum computation in an anharmonic linear ion trap * EPL 86, 60004 (2009)

Ion trap quantum computation

Trapped atomic ions are believed to be one of the most promising candidates for realization of a quantum computer, due to their long-lived internal qubit coherence and strong laser-mediated Coulomb interaction. Various quantum gate protocols have been proposed and many have been demonstrated with small numbers of ions. The main challenge now is to scale up the number of trapped ion qubits to a level where the system behavior becomes intractable for any classical means. The linear rf (Paul) trap has been the workhorse for ion trap quantum computing, with atomic ions laser-cooled and confined in a 1D geometry. (Other examples include Penning traps (2D), arrays of microtraps.) Qubits are encoded in atomic internal hyperfine states, coupled to collective motional modes which serve as auxiliary channels.

Working principles

Main challenges in a large-scale linear ion trap (cont.) High-fidelity two-qubit gate design

Architecture

[1] J. I. Cirac and P. Zoller, PRL 74, 4091 (1995).

[2] D. Windland et al., J. Res. NIST 103, 259 (1998).

[3] A. Sorensen and K. Molmer, PRL 82, 1971 (1999); PRA 62, 022311 (2000).

[4] G. J. Milburn, S. Schneider, and D. F. V. James, Fortschr. Physik 48, 801 (2000).

[5] J. J. Garcia-Ripoll, P. Zoller, and J. I. Cirac, PRL 91, 157901 (2003); PRA 71, 062309 (2005).

[6] S.-L. Zhu, C. Monroe, and L.-M. Duan, PRL 97, 050505 (2006); EPL 73, 485 (2006).

[7] K. Kim et al., quant-ph/0905.2005 (2009).

Other imperfection

Pulse shape

Infidelity

Paul trap(Monroe’s group)

|

•••

01

2

•••

01

2

|

P1/2

S1/2

12.64 GHz

369.5 nm

2S1/2

2P1/2

F=0

F=1

F=0F=1

eg

Qubit implementation (171Yb+) Coupling spins and phonons

Δk

Bichromatic Raman lasers create a spin-dependent force

laser detuning

Laser field

ion jgate time

desired phase

phase spacedisplacement

Hamiltonian

Evolution

= 0 system restored as a cycle completed

Controlled-phase flip (CPF)

2. Cooling issue

3. Control issue

Axial Transverse

N=120• Generally speaking, side-band cooling is complicated.

• Better involving Doppler cooling only.

• Unable to resolve individual side-bands.

• Gate protocols must take all excitation modes into account.

• Controlling complexity scaling (with N)

--- how the difficulty increases to design a gate?

Idea: directly resolve the main challenges

• Build a uniform ion chain by adding anharmonic corrections to the axial potential

• Couple to the transverse modes: More confined. Doppler temperature required only.

• Controlling complexity does not increase with N: “Local” motional modes dominate.

Quartic trap (lowest-order correction)

N=120

excluded from computing

Adjustable planar trap (Schematic)

A perfect gate requires:

• Eliminating phonon part at the end of each cycle (N modes, real and imaginary).

• Acquiring the target qubit phase .

2N+1 constraints

Ω1

Ω2

••• ΩM

Chop pulse shape into M segments

• M=2N+1

(only required for 100% accuracy,

not necessary for satisfactorily high fidelity)

• M=5, 6, 7…, a few (indep. of N)

• Systematic procedures (quadratic minimization) for gate design

TP

infidelity

pulse shape

maximum spatial displacement

(Green dotted: reduced scheme)

laser laser

Two-qubit gate

laser laser

Two-qubit gate

Reduced scheme

frozen in space

Significance of local modes

• Axial-motion-induced inhomogeneity of the laser field (Gaussian beam)

Ion spacing δzn ~ 10 μm

Width of Gaussian beam ω ~ 4 μm

• High-order local anharmonicity

• Deviation of the Lamb-Dicke Limit

Main challenges in a large-scale ion trap

1. Harmonic architecture• Structural instability → either increase transverse confinement or elongate axial size

limit on rf trapping ion addressability

problem

limit on trap size coupling strength

problem

• lack of translational symmetry → require site-dependent control

N=20

N=60

N=120

Max.

Min.

ratio

• Other proposals

2. Quantum networks

BS

D1 D2

i 'i j 'j

pump pump

CNOT

pump pump

CNOT

Duan, Blinov, Moehring, Monroe, 2004

Kielpinksi, Monroe, Wineland, Nature 417, 709 (2002)

1. Ion shuttling:

J. P. Home et al., Science Express (2009)

~Ω(t)

shaping electrodes