Single qubit Deutsch-Jozsa algorithm in a quantum dot P. Bianucci1 , A. Muller,1 C. K. Shih1 , Q. Q. Wang2,3 , Q. K. Xue3 , C. Piermarocchi4 1
Department of Physics, The University of Texas at Austin, Austin, Texas 78712 2
3
Department of Physics, Wuhan University, Wuhan 430072, P. R. China
International Center for Quantum Structures, Institute of Physics, The Chinese Academy of Sciences, Beijing 100080, P. R. China
4
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-2320 Support: NSF-NIRT (DMR-0210383), NSF-FRG (DMR-0306239), NSF-ITR
(DMR-0312491), Texas Advanced Technology program, W. M. Keck Foundation Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 1
Outline •
The Deutsch-Josza algorithm ◦ The 1-bit Deutsch problem and its solution
•
Experimental setup ◦ Semiconductor Quantum Dots ◦ Wavepacket interferometry
•
The 1-bit Deutsch-Jozsa algorithm in a Quantum Dot
•
Conclusions
Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 2
The one-bit Deutsch problem How do we find out if a coin is fair?
fc : {top, bottom} → {head, tail} Fake (Constant) Fair (Balanced) f1 (x) = 0 f3 (x) = x f4 (x) = 1 − x f2 (x) = 1 (Top,Head ≡ 0, Bottom,Tail ≡ 1).
Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 3
Solving the Deutsch-Problem
Coin
Input (Top = 0) (Bottom = 1)
•
x
fc (x)
Classical Coin−o−meter
Output (Head = 0) (Tail = 1)
Classical algorithm: 2 runs needed.
Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 4
The Deutsch-Jozsa algorithm •
We can do better! (Only 1 run!) not |0> if f is balanced
|0>
H
n
|x>
|x>
H
n
Uf
|1>
|0> if f is constant
|y> |x f(y)>
H
H
Original version of the Deutsch-Jozsa algorithm [Deutsch and Jozsa, Proc. Roy. Soc. London A 439, 553 (1992)] |0>
not |0> if f is balanced H
n
Uf
H
n
|0> if f is constant
ˆ = H
√1 2
h
1
1
1
−1
i
, Uˆf |xi = (−1)f (x) |xi
Streamlined version of the Deutsch-Jozsa algorithm. [Collins et al. Phys. Rev. A 58 , 1633 (1998)] Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 5
Semiconductor Quantum Dots •
Quantum dots are mesoscopic systems in which electrons are confined in 3D.
Conduction Band
Eg2
Eg1
Exciton ground state Valence Band
Schematic level structure for a semiconductor quantum dot.
X-STM image of self-assembled InGaAs/GaAs quantum dots. [Liu et al., Phys. Rev. Lett. 84, 334 (2000)] Exciton excited state Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 6
Wavepacket Interferometry Far−Field PL setup
Spectrometer
CCD Array Detector Interferometer Pulsed Ti:Sa laser
Sample
PL
LHe cryostat (min 4.2 K)
τf τd = τ c + τ f
tf (fs) 0
|1>
4
|1’>
τc
Intensity (a.u.)
Schematic diagram of the experimental setup.
|0>
PL
Experiment schematic [Bonadeo et al.,Science 282, 1473 (1998)] [Kamada et al., Phys. Rev. Lett.87, 246401 (2000)] [Htoon et al.,Phys. Rev. Lett.88, 087401 (2002)]
0
20
40 td (ps)
60
80
Wavefunction autocorrelation Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 7
DJ algorithm in a single Quantum Dot |0>
Rx(π/2)
Rz(τd)
|0> if f is balanced (τd= 2( n+1)π/ω 0)
Rx(π/2)
|1> if f is constant (τd= 2nπ/ω 0)
τd
π/2 pulse
ˆx( π ) = R 2
π/2 pulse
√1 2
Minimum if f balanced Maximum if f constant
Measure PL
h
1
−1
1
1
t
i
ˆ z (τd ) = , R
h
1
0
0
eiω0 τd
i
ˆ Uˆf = −I, ˆ Uˆf = σˆz , Uˆf = −σˆz . Uˆf1 = I, 2 3 4 Quantum dot version of the single qubit DJ algorithm |1>
|1>
|0>
|0>
Bloch vector evolution: Constant f
Bloch vector evolution: Balanced f Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 8
PL (arb.units)
Deutsch-Jozsa algorithm in a single Quantum Dot 200 150 100 50 0 200
(b) 2π
3π
4π
200 150 100 50 0
(c) π
2π
3π
200 150 100 50 0
(d)
200 150 100
2π
3π
50
(e) π
2π
3π
4π
PL (arb. units)
150
100
50
(a)
0
0
10
20
30 Coarse τd (ps)
40
50
Experimental results Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 9
Conclusions •
We used the ability to optically manipulate the excitonic state of a single semiconductor quantum dot to implement a 1-qubit version of the Deutsch-Jozsa algorithm. ◦ We implemented a simple quantum algorithm on a solid state system!
[cond-mat/0401226, to appear in Phys. Rev. B]
Single qubit Deutsch-Jozsa algorithm in a quantum dot – p. 10
