Implementation of the Deutsch- Josza Algorithm on an ion-trap quantum computer

Transcription

1 Implementation o the Deutsch- Josza Algorithm on an ion-trap quantum computer S. Gulde et Al. Przemyslaw Kotara HU Berlin (Ger) Tijl Schmelzer - U Ghent (Be)

2 Outline Theory: Problem/Motivation The algorithm Quantum Circuit Deutsch algorithm Deutsch-Jozsa algorithm Eperiment: Eperimental setup Error sources Results Reerences

3 Problem/Motivation Deutsch-Jozsa algorithm is a possibility or computing global properties o certain unctions in ep. less time than any class. algorithm goal determine the global property i a unction is constant or balanced conventional deterministic algorithm takes 2 n- + evaluations o in the worst case Deutsch-Jozsa quantum algorithm produces an answer that is always correct with just evaluation o Implementation serves to demonstrate the potential o ion traps or quantum computing

4 The Algorithm Quantum Circuit Upper qubit (upper line) gives inormation which side o the coin Lower qubit (lower line) is an auiliary working qubit R are rotations which create the superposition s U is an unitary operation Measurement o < a> 3 2 yields inormation i is balanced or constant

5 The Algorithm Deutsch Algorithm ) Input : a, w 2) ( H H ): a, w 2 ( + )( ) 3) U : a, w 2 ± 2 ± 2 ( + )( ) or ( ) ( ) ( )( ) or ( ) ( ) 4) ( H ): 5) a, w 3 ± 2 ± 2 ± 2 ( ) or ( ) ( ) ( ) or ( ) ( ) ( ) ( ) ( ) The state o the irst qubit shows i is constant or balanced

6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) or or I l> itselve is a superpostiton, we have: ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ± ± : : : : : : Eplanation o step 3: The Algorithm Deutsch Algorithm

7 The Algorithm Deutsch-Josza Algorithm It s easy to epand the algorithm to n qubit s: Initial state with n qubits is: a r, w, K 2 n n n + Algorithm is very similar to Deutsch Algorithm But applying the n-qubit Hadamard transormation to initial state: n H r H i i Final state o the n-qubit algorithm a w, 3 2 n r z r r r z + ( ) ( ) Decide i constant or balanced measure population o ground state l> r z 2 r ( ) 2 n ( ) r ± or or const. blanced

8 Outline Theory: Problem/Motivation The algorithms Quantum Circuit Deutsch algorithm Deutsch-Jozsa algorithm Eperiment: Eperimental setup Error sources Results Reerences

9 Eperimental Setup Overview 4 Ca+ ion in Linear Pauli-trap Lasercooling Ti-Sa-Laser or qubit-manipulations Wavelength 729nm (linewidth<hz) Acousto-optical modulator or req-change and phaseshit Electron Shelving or electronic state Detection 99,9% idelity 3ms detection time

10 Eperimental Setup Linear Pauli-trap Combination o static and alternating EM-ields conine ions in an eective potential Field o ion trap quadropole vanishes at center & increases in all directions any deviations results in a net restoring orce Linear ion traps allows to assemble many ions in a linear chain, thus: can be addressed by laser beams equilibrium position is ield ree in contrast to classical non-linear Paul trap where trough coulomb repulsion ions are pushed away rom ield ree point micro motion

11 Eperimental Setup Linear Pauli-trap 4 Ca+ ions in Linear Pauli-trap ω z 2π*,7MHz

12 Eperimental Setup Doppler Laser Cooling Laser cooling relies on the transer o momentum rom photos arrangement so that that orces push atoms in direction o the laser beam Momentum transered photo is absorbed Emission in contrast o the absorption process is not directed average eect o all emmission processes vanishes One need high scattering rate because otherwise the change in velocity is too small Using lasers scatter up to 8 photons per second atom can be stopped over short distance

13 Eperimental Setup Sideband Laser Cooling Doppler cooling yield average vibrational quantum numbers n z 2 urther cooling is achieved by sideband cooling Eicient laser cooling occurs when the requency o the laser beam is tuned to the red sideband In this case the atom undergoes the transition: lg,n> le,n-> spontaneous emission occurs predominantly at the carrier requency: le,n-> lg,n->

14 Eperimental Setup QM energy levels a. st Qubit a (Optical energylevels S½, D5/2) b. 2 nd Qubit w (Vibrational energylevels in ion trap z, z ) c. Combination law>

15 Eperimental Setup Qubit encoding Single-Qubit rotations R: Carrier rotation ls n z > ld n z > (729nm) Laser puls Double-Qubit rotations R+: Transitions on the blue sideband ls n z > ld n z +> (729nm + ω z ) Laser puls

16 Eperimental Setup Qubit encoding σ transitions between ls> and ld> b transitions between l z > and l z > θ ~ pulse duration φ phase between pulse and atomic polarization 2 important Rotations R y R(π/2, ) R y R(π/2, π)

17 Eperimental Setup Algorithm implementation

18 Eperimental Setup Startup Doppler lasercooling 2ms on S½ P½ Result Vibrational quantumnumber n z 2 Sideband Cooling 2ms Result Vibrational Groundstate z 99% Initialization by optically pump ion to S½ Result l> ls½ z >

19 Eperimental Setup Case 3 as eample 2 µs to 22 µs: R ya carrier pulse 54 µs to 22 µs: R U y w n R blue sideband yw pulse on law> The phase is switched at 87, 33 and 66 µs 24 to 25 µs: R ya carrier pulse

20 Error sources Phaseshit compensation algorithm several/many pulses Control relative phases precisely Unwanted shit has to be compensated

21 Error sources Computational subspace Subspace {ls z >, ld z >, ls z >, ld z >} Transitions on the blue sideband lsn z > ld n z +> ls z > ld 2 z > outside subspace Composite Pulses Sequence o carrier and blue sideband pulses that constrain the system to the subspace

22 Fidelity l<la>l² Case,3,4 >97% Case 2 >9% Error sources Results Decoherence laser-atom phase Mostly caused by ambient magnetic ield luctuations Case 2 most comple pulse sequence Measurements: Outcome Higher laser power to speed up algorithm This reduces sensitivity to phase decoherence This causes o-resonant carrier ecitation that limits idelity

23 Results Measurements: Evolution Follow evolution o l<la>l² Stop Pulse sequence anytime l<la(t)>l² Probability o inding ion in D5/2 state Very small deviation o normal calculated ideal values (solid lines)

24 Results Summary/Outlook High degree control over all relevant eperimental parameters over long pulsesequences Laser req. and intensity, optical phases, and trap requency ω z Good procedure or the uture More comple algorithms Scaling to multiple qubits Light shit compensation important or scaling Ion heavier higher laser intensities or sideband transitions which increases light shits All gate operations possible Possible 43 Ca+ instead o 4 Ca+ with potentially longer coherence time

25 Reerences Quantum Computing: A Short Courses orm Theory to Eperiment, J.Stolze and D.Suter Verschränkte Systeme: Die Quantenphysik au neuen Wegen, J. Andretsch Seminar Quanteninormationsverarbeitung Vortrag Nr.4 Quantengatter & Algorithmen WS 2/3, S. Bauer, TU München Quantum Computation and Quantum Inormation, M. Nielsen and I. Chuang QSIT Lectures