Quantenelektronik 1 Application of the impedance measurement technique for demonstration of an adiabatic quantum algorithm. M. Grajcar, Institute for Physical High Technology, P.O. Box , D Jena, Germany and Department of Solid State Physics, Comenius University,SK Bratislava, Slovakia A. Izmalkov, E. Il’ichev, H.-G. Meyer Institute for Physical High Technology, P.O. Box , D 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.

Quantenelektronik 2 Contents 1.Introduction A. Adiabatic quantum computation B. Measurements by parametric transducer 2.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 3.Readout of the adiabatic quantum computing for 3 coupled qubits 4.Conclusions

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

Quantenelektronik 4 Parametric transducer Mechanical displacement LTLT CTCT  x    x = cm Braginski et. al., JETP Lett., 33, 404, 1982.

Quantenelektronik 5 Resonator Quantum nondemolution measurements of a resonator‘s energy    =  L/v(E) 1.No perturbation of the measured observable 2.The canonically conjugate to the measured observable is perturbed according to uncertainty principle. Dielectric sucseptibility

Quantenelektronik 6  i =  e  LI(  i ) L  0  i  e Experimental method - phase shift between I b and V T  I b V L T C T T L i 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

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

Quantenelektronik 8 Classical and Quantum regime 2

Quantenelektronik 9 Nb persistent current ‘qubit’ in classical regime E. Il´ichev et al., APL 80 (2002) 4184 Junction area 3x3  m 2 E J /E c  10 4 T=20 mK T=100 mK T=500 mK T=800 mK Al-qubit

Quantenelektronik 10 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 C T. Al persistent current qubit placed in Nb coil

Quantenelektronik 11 Al persistent current qubit Material: Aluminium, Shadow-evaporation tecnique Two contacts ~600x200nm, (I C  600 nA), the third is smaller, so that  =E J1 /E J2,3 ~ Inductance L  20 pH J.E. Mooij et al., Science 285, 1036, 1999.

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

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

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

Quantenelektronik 15 Tank voltage vs external flux near f  T mix. Chamber = 10 mK A. Izmalkov et al., EPL, 65, 844, 2004

Quantenelektronik 16 Classical and Quantum regime 2

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

Quantenelektronik 18 Phase shift vs  at different temperatures

Quantenelektronik 19 Temperature dependence of the dip height and width   T h =200 mK

Quantenelektronik 20 I dc2 M ab I dc1 +I bias (t) qubit a qubit b M a/b,T V out (t) LTLT CTCT RTRT A Two coupled flux qubits

Quantenelektronik 21 Determination of device parameters Theory the width of one-qubit dips gives  a  450 MHz,  b =550 MHz The height of one-qubit dips gives persistent currents I a =I b = 320 nA and J= M ab I a I b =410 MHz. T=160 mK 90 mK 50 mK 10 mK T=160 mK 90 mK 50 mK IMT deficit Experiment Two-qubit Hamiltonian

Quantenelektronik 22 Temperature dependence of the IMT dip amplitudes We substitute  a  b I a, I b, 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. The measure of entanglement is Concurrence C For pure state: The concurrences of eigenstates of our two-qubit system are: C 1 =C 4 =0.39, C 2 =C 3 =0.97 But the equilibrium concurrence C eq (10 mK)=0.33 In our experiment the minimal temperature of the sample was about 40 mK, where C eq =0.

Quantenelektronik 23 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/ Simple example for 4 nodes w1w1 w2w2 w3w3 w4w4 w 12 w 23 w 34 w Payoff function w 24 w 13 S 4 =0 S 3 =0 S 2 =1 S 1 =1

Quantenelektronik 24 Hamiltonian of N inductively coupled flux qubits Payoff function is encoded in Hamiltonian H P if  i <<J i,j and H P – The MAXCUT problem Hamiltonian

Quantenelektronik 25 Readout by parametric transducer B3B3 Q2Q2 Q3 Q1Q1 B1B1 B2B2 2 |0>|1>

Quantenelektronik 26 First three energy levels of the three qubit system during readout Parameters |s> E(K)

Quantenelektronik 27 Measured quantity – d 2 E/df 2 S 1 =0S 3 =0 S 2 =1

Quantenelektronik 28 Conclusions 1.By making use of the Parametric Transducer measurements the flux qubits can be completely characterized 2.Adiabatic quantum computing is operated even if the system is in mixed entangled states. 3.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.