Application of Impedance Measurement Technique for Adiabatic Quantum Algorithm Demonstration

1. Application of the impedance measurement technique for demonstration of an adiabatic quantum algorithm. M. Grajcar, Institute for Physical High Technology, P.O. Box 100239, D-07702 Jena, Germany and Department of Solid State Physics, Comenius University,SK-842 48 Bratislava, Slovakia A. Izmalkov, E. Il’ichev, H.-G. Meyer Institute for Physical High Technology, P.O. Box 100239, D-07702 Jena, Germany We are grateful to our coauthors N. Oukhanski, D. Born, U. Hübner, T. May, Th. Wagner, I. Zhylaev, Ya. S. Greenberg, H. E. Hoenig, W. Krech, M. H. S. Amin, A. Smirnov, Alec Maassen van den Brink, A. M. Zagoskin for their help and contribution to this work on different stages.

2. Contents • Introduction • A. Adiabatic quantum computation • B. Measurements by parametric transducer • Control of the evolution for: • A. Single qubit • a. “Classical” regime. • b. Landau-Zener transitions. • c. Adiabatic response • B. Two coupled qubits – adiabatic behaviour • C. Three coupled qubits - MAXCUT problem • Readout of the adiabatic quantum computing for 3 coupled qubits • Conclusions

3. Adiabatic quantum computing There are a set of problems, which can be solved by quantum algorithm more efficiently than by conventional one. • Start with initial Hamiltonian HI with known ground state |I> • Make adiabatic evolution from |I> to the unknown ground state |g>of HP • Readout the ground state of HP Realization for superconducting flux qubits 1) For flux qubits we choose initial Hamiltonian HI with trivial ground state |0> 2) Changing the bias of individual qubits adiabatically, Hamiltonian Hi is transformed to HP.

4. Parametric transducer LT CT • Mechanical displacement x   x = 6 10-17 cm Braginski et. al., JETP Lett., 33, 404, 1982.

5. Resonator  • Quantum nondemolution measurements of a resonator‘s energy  =L/v(E) Dielectric sucseptibility • No perturbation of the measured observable • The canonically conjugate to the measured observable is perturbed according to uncertainty principle.

6. V L T T L C T F i I b J - phase shift between Ib and VT Experimental method i=eLI(i) L0 F=i e A Golubov, M. Kupriyanov and E. Il‘ichev, Rev. Mod. Phys., to be publ in April, 2004 E. Il‘ichev et al. Rev. Sci. Instr., 72., 1883, 2001 E. Il‘ichev et al. Cond-mat 0402559

7. Experimental setup Tmix. chamber=10mK T1K pot1.8 K Room temperature amplifier TN200 mK Advatages: 1) high quality PRC - narrow bandwidth filter 2) high sensitivity Q ~103 3) small coupling coefficient k~10-2  small back-aktion of the amplifier HEMT

8. Classical and Quantum regime 2

9. Nb persistent current ‘qubit’ in classical regime T=800 mK T=500 mK T=100 mK Al-qubit T=20 mK Junction area 3x3 mm2 EJ/Ec104 E. Il´ichev et al., APL 80 (2002) 4184

10. Al persistent current qubit placed in Nb coil • Nb coil is prepared on oxidized Si substrates by optical lithography. • The line width of the coil windings was 2 m, with a 2 m spacing. • Various square-shape coils with between 20 and 150 m windings were designed. • We use an external capacitance CT.

11. Al persistent current qubit • Material: Aluminium, Shadow-evaporation tecnique • Two contacts ~600x200nm, (IC 600 nA), the third is smaller, so that a=EJ1 /EJ2,3 ~0.8-0.9 • Inductance L 20 pH • J.E. Mooij et al., Science285, 1036, 1999.

12. Idea of measurements – Landau-Zener tunneling  C B A F E D e External flux rf + dc dc V dc Voltage across the tank vs dc energy at different external fluxes

13. Idea of measurements – Landau-Zener tunneling  C B A F E D e External flux rf + dc dc energy at different external fluxes V dc Voltage across the tank vs dc

14. Idea of measurements – Landau-Zener tunneling  C B A F E D e External flux rf + dc dc energy at different external fluxes V dc Voltage across the tank vs dc

15. Tank voltage vs external flux near f=0.5 Tmix. Chamber= 10 mK A. Izmalkov et al., EPL, 65, 844, 2004

16. Classical and Quantum regime 2

17. Phase shift  vs F near degeneracy point f=0.5 Ya. S. Greenberg et al., PRB 66, 214525, 2002 M. Grajcar et al., PRB 69, 060501(R), 2004

18. Phase shift vs Fat different temperatures

19. Temperature dependence of the dip height and width T3 Th=200 mK

20. Two coupled flux qubits Idc1+Ibias(t) CT RT LT A Vout(t) Ma/b,T qubit a qubit b Idc2 Mab

21. Determination of device parameters T=160 mK 90 mK 50 mK 10 mK T=160 mK 90 mK 50 mK IMT deficit Theory Experiment Two-qubit Hamiltonian • the width of one-qubit dips gives Da=450 MHz, Db=550 MHz • The height of one-qubit dips gives persistent currents Ia=Ib= 320 nA and J= MabIaIb =410 MHz.

22. Temperature dependence of the IMT dip amplitudes The measure of entanglement is Concurrence C For pure state: The concurrences of eigenstates of our two-qubit system are: C1=C4=0.39, C2=C3=0.97 • We substitute Da, Db Ia, Ib, J, determined from Low-T measurements. The T-dependence of IMT dips amplitudes agrees with these values. • Ratio of the dips grows with T, because the thermal excitations tend to destroy coherent correlations between the qubits. But the equilibrium concurrence Ceq (10 mK)=0.33 In our experiment the minimal temperature of the sample was about 40 mK, where Ceq=0.

23. Payoff function MAXCUT problem • The MAXCUT problem is part of the core NP-complete problems • MAXCUT adiabatic quantum algorithm already demonstrated by NMR • M. Stephen et al., quant-ph/0302057 Simple example for 4 nodes S3=0 S4=0 w3 w34 w4 w24 w23 w2 0 S2=1 w14 w13 1 w12 w1 S1=1

24. Hamiltonian of N inductively coupled flux qubits Payoff function is encoded in Hamiltonian HP if i<<Ji,jand HP – The MAXCUT problem Hamiltonian

25. Q1 Q2 B1 B2 2 Q3 B3 Readout by parametric transducer |1> |0>

26. First three energy levels of the three qubit system during readout Parameters

27. Measured quantity – d2E/df2 S2=1 S1=0 S3=0

28. Conclusions • By making use of the Parametric Transducer measurements the flux qubits can be completely characterized • Adiabatic quantum computing is operated even if the system is in mixed entangled states. • Parametric Transducer can effectively readout the result of the adiabatic quantum computing leaving the system in the ground state during and after the measurement. Such ‘non-demolition’ measurement naturally follows the idea of adiabatic quantum computing.