QUANTUM COMPUTATION IN SOLID STATE SYSTEMS
CONTENTS DECOHERENCE OF A JOSEPHSON QUANTUM BIT DURING ITS FREE EVOLUTION: THE QUANTRONIUM INTRODUCTION THE QUANTRONIUM CIRCUIT AND ITS DECOHERENCE SOURCES PRINCIPLE EXPERIMENTAL IMPLEMENTATION DECOHERENCE SOURCES THEORETICAL RELAXATION AND DECOHERENCE RATES EXPERIMENTAL CHARACTERIZATION OF DECOHERENCE DURING - FREE EVOLUTION RELAXATION TIME (T1) MEASUREMENT COHERENCE TIME (T2 AND TE) MEASUREMENTS DISCUSSION CONCLUSION CONDITIONAL GATE OPERATION IN SUPERCONDUCTING CHARGE QUBITS EXPERIMENTAL DETAILS DEVICE STRUCTURE OPERATION SCHEME EXPERIMENTAL SETUP RESULTS AND DISCUSSION OPERATION POINT CONDITIONAL GATE OPERATION CONCLUSIONS COUPLING AND DEPHASING IN JOSEPHSON CHARGE-PHASE QUBIT WITH RADIO FREQUENCY READOUT QUBIT PARAMETERS AND THE MODEL EFFECT OF FINITE INDUCTANCE OF THE RING QUBIT DEPHASING THE JOSEPHSON BIFURCATION AMPLI?ER - FOR QUANTUM MEASUREMENTS THEORY DEVICES AND SETUP RESULTS CURRENT-CONTROLLED COUPLING - OF SUPERCONDUCTING CHARGE QUBITS CURRENT-CONTROLLED COUPLING OF TWO QUBITS MAXIMUM COUPLING STRENGTH OPERATING THE SYSTEM COUPLING VIA MEASUREMENT JUNCTIONS EXTENSION TO ARBITRARY NUMBER OF QUBITS DIRECT MEASUREMENTS OF TUNABLE JOSEPHSON PLASMA RESONANCE IN THE L-SET THE L-SET CIRCUIT PLASMA OSCILLATIONS IN L-SET SIMULATION SCHEME EXPERIMENT CHARGE DETECTION HARMONIC OSCILLATIONS SWITCHING AND NONLINEAR OSCILLATIONS TIME DOMAIN ANALYSIS OF DYNAMICAL SWITCHING IN A JOSEPHSON JUNCTION THE EXPERIMENT THE MODEL SUMMARY COOPER PAIR TRANSISTOR IN A TUNABLE ENVIRONMENT SAMPLE FABRICATION AND MEASUREMENT TECHNIQUES SQUID ARRAYS CHARACTERIZATION MEASUREMENT OF THE CPT PHASE SLIP PHENOMENA IN ULTRA-THIN SUPERCONDUCTING WIRES THEORETICAL EXPERIMENTAL EXPERIMENTAL EVIDENCE OF QUANTUM PHASE SLIPS? 10 DYNAMICS OF A QUBIT COUPLED TO A HARMONIC OSCILLATOR 10 1 INTRODUCTION 10 2 THE SYSTEM 10 3 SAMPLE FABRICATION, SETUP AND CHARACTERIZATION 10 4 COUPLED DYNAMICS 10 4 1 SPECTROSCOPY 10 4 2 DYNAMICS 10 5 USING THE COUPLED DYNAMICS TO PROBE THE SYSTEM 10 5 1 MEASURING THE OSCILLATOR TEMPERATURE 10 5 2 PROBING THE QUBIT STATE 10 6 CONCLUSION 11 JOSEPHSON JUNCTION MATERIALS RESEARCH USING PHASE QUBITS 11 1 INTRODUCTION 11 2 JOSEPHSON PHASE QUBITS 11 3 JUNCTION FABRICATION PROCESSES 11 3 1 THE “ION MILL JUNCTION PROCESS” 11 3 2 THE “STANDARD TRILAYER JUNCTION PROCESS” 11 3 3 THE “EVAPORATED TRILAYER JUNCTION PROCESS” 11 4 JOSEPHSON JUNCTION AND QUBIT CHARACTERIZATION 11 4 2 MEASUREMENTS OF QUBIT SPECTROSCOPY 11 5 CONCLUDING REMARKS 12 ENERGY LEVEL SPECTROSCOPY OF A BOUND VORTEX-ANTIVORTEX PAIR 13 ADIABATIC QUANTUM COMPUTATION WITH FLUX QBITS 13 1 INTRODUCTION 13 2 ADIABATIC QUANTUM COMPUTATION 13 3 AQC WITH AN ARRAY OF FLUX QBITS 13 4 VERTICAL TWO JOSEPHSON JUNCTIONS INTERFEROMETER 13 5 CONCLUSIONS 14 ANOMALOUS THERMAL ESCAPE IN JOSEPHSON SYSTEMS PERTURBED BY MICROWAVES 14 1 INTRODUCTION 14 2 THEORY 14 3 EXPERIMENTS AND SIMULATIONS 14 4 CONCLUSIONS 15 REALIZATION AND CHARACTERIZATION OF A SQUID FLUX QUBIT WITH A DIRECT READOUT SCHEME 15 1 INTRODUCTION 15 2 THE RF SQUID QUBIT 15 3 THE DOUBLE SQUID QUBIT 15 4 GRADIOMETRIC CON?GURATION 15 5 DIRECT READOUT SCHEME 15 6 EXPERIMENTAL CHARACTERIZATION 15 7 CONCLUSIONS 16 A CRITIQUE OF THE TWO LEVEL APPROXIMATION 16 1 INTRODUCTION 16 2 SYSTEM AND ENVIRONMENT 16 3 DENSITY MATRIX SOLUTIONS 16 3 1 THE EFFECT OF SPECTRAL DISTRIBUTION ON DECOHERENCE 16 3 2 THE EFFECT OF NUMBER OF LEVELS ON DECOHERENCE 16 4 THE PHOTON NUMBER CALCULATIONS 16 5 CONCLUSIONS AND DISCUSSIONS 17 JOSEPHSON JUNCTION QUBITS WITH SYMMETRIZED COUPLINGS TO A RESONANT LC BUS 17 1 INTRODUCTION 17 2 JOSEPHSON JUNCTION TRIANGULAR PRISM QUBITS 17 3 COUPLING THE RESONANT LC BUS TO THE QUBITS 17 3 1 LOGICAL QUBITS FOR A DECOHERENCE FREE SUBSPACE 17 3 2 MØLMER–SØRENSEN GATE 17 4 CIRCULATING CURRENT PATTERNS FOR THE QUBIT STATES 17 4 1 INITIALIZING THE QUBITS AND THE EFFECT OF CRITICAL CURRENT AND GEOMETRIC DEFECTS 17 4 2 CONNECTING BUSES INTO A NETWORK 17 5 CONCLUSIONS 18 SPATIAL BOSE-EINSTEIN CONDENSATION IN JOSEPHSON JUNCTION ARRAYS 18 1 INTRODUCTION 18 2 PRELIMINARY STUDIES FOR ENGINEERING GRAPH-SHAPED NETWORKS 18 3 FABRICATION OF COMB-SHAPED JJNS 18 4 MEASUREMENTS 19 COOPER PAIR SHUTTLE: A JOSEPHSON QUANTUM KICKED ROTATOR 19 1 INTRODUCTION AND SYSTEM DESCRIPTION 19 2 CLASSICAL AND QUANTUM DYNAMICS 19 3 FIDELITY 19 4 THE EFFECT OF NOISE 19 5 FIDELITY MEASUREMENT 20 SIZE DEPENDENCE OF THE SUPERCONDUCTOR-INSULATOR TRANSITION IN JOSEPHSON JUNCTION ARRAYS 20 1 INTRODUCTION 20 2 FINITE-SIZE EFFECT IN THE 2DXY MODEL 20 3 THE EFFECTIVE HAMILTONIAN 20 4 DMRG 20 5 SPONTANEOUS SYMMETRY BREAKING 20 6 BKT AS A SSB 20 7 EXPERIMENTAL IMPLICATIONS 20 8 CONCLUSIONS 21 MONTE CARLO METHOD FOR A SUPERCONDUCTING COOPER-PAIR-BOX CHARGE QUBIT MEASURED BY A SINGLE-ELECTRON TRANSFER 21 1 INTRODUCTION 21 2 MODEL HAMILTONIAN 21 3 MEASUREMENT RECORDS AND CONDITIONAL DENSITY MATRIX 21 4 STOCHASTIC MASTER EQUATION 21 5 CONNECTION TO “PARTIALLY” REDUCED DENSITY MATRIX 21 6 CONCLUSION 22 ON THE CONVERSION OF ULTRACOLD FERMIONIC ATOMS TO BOSONIC MOLECULES VIA FESHBACH RESONANCES 22 1 INTRODUCTION 22 2 INITIAL STATE PREPARATION 22 3 GENERAL SYMMETRY ARGUMENTS 22 4 EXPLAINING THE EXPERIMENTAL LIMITED TRANSFER EF?CIENCY 22 5 TRANSFER EF?CIENCIES ABOVE 0 5 22 6 SUMMARY 23 REVEALING ANISOTROPY IN A PAUL TRAP THROUGH BERRY PHASE 23 1 INTRODUCTION 23 2 PHYSICAL SYSTEM AND HAMILTONIAN MODEL 23 3 CALCULATION OF BERRY PHASE 23 4 ANISOTROPY VS BERRY PHASE EFFECTS 23 5 CONCLUSIONS 24 DISTILLING ANGULAR MOMENTUM SCHR¨ODINGER CATS IN TRAPPED IONS 24 1 INTRODUCTION 24 2 DISTILLING ANGULAR MOMENTUM ‘CATS’ THROUGH VIBRONIC COUPLINGS 24 2 1 VIBRONIC COUPLINGS IN TRAPPED IONS 24 2 2 DISTILLING ANGULAR MOMENTUM EIGENSTATES 24 3 CONTROLLABLE CAT-LIKE SUPERPOSITION 24 4 CONCLUSIVE REMARKS 24 5 ACKNOWLEDGMENTS 25 LINEAR-RESPONSE CONDUCTANCE OF THE NORMAL CONDUCTING SINGLE-ELECTRON PUMP 25 1 INTRODUCTION 25 2 MOTIVATION FOR THE MEASUREMENTS 25 3 SEQUENTIAL MODEL 25 4 EXPERIMENTAL DETAILS AND RESULTS 25 5 DISCUSSIONS 25 6 SUMMARY 26 TRANSMISSION EIGENVALUES’ STATISTICS FOR A QUANTUM POINT CONTACT 26 1 INTRODUCTION 26 2 MODEL OF QPC WITH IMPURITIES 26 3 SCATTERING MATRIX ELEMENTS AND CORRECTION TO THE CONDUCTANCE 26 4 DISTRIBUTION FUNCTION 26 4 1 ONE OPEN CHANNEL 26 4 2 CIRCUIT THEORY 26 5 CONCLUSIONS 27 CREATING ENTANGLED STATES BETWEEN SQUID RINGS AND ELECTROMAGNETIC FIELDS 27 1 INTRODUCTION 27 2 BACKGROUND 27 3 THE HAMILTONIAN’S SPECTRUM 27 4 RAPID PASSAGE 27 5 TOWARDS THE ADIABATIC LIMIT 27 6 THE EFFECT OF DISSIPATION 27 7 CONCLUSIONS 28 FREQUENCY DOWN CONVERSION AND ENTANGLEMENT 28 1 INTRODUCTION 28 2 ENTANGLEMENT OF EM FIELDS VIA A SQUID RING 28 3 ENTANGLEMENT OF ?ELD MODES AT SIMILAR FREQUENCIES 28 3 1 ENTANGLEMENT OF DISSIMILAR INPUT/OUTPUT FREQUENCY ?ELD MODES 28 4 ENERGY DOWN CONVERSION AT HIGHER FREQUENCY RATIOS 28 4 1 FACTOR 5 DOWN CONVERSION 28 5 LARGER FREQUENCY RATIO (FACTOR OF 10) DOWN CONVERSION 28 6 CONCLUSIONS 29 ENTANGLEMENT OF DISTANT SQUID RINGS 29 1 INTRODUCTION 29 2 INTERACTION OF A SINGLE SQUID RING WITH NONCLASSICAL MICROWAVES 29 3 INTERACTION OF TWO DISTANT SQUID RINGS WITH ENTANGLED MICROWAVES 29 3 1 MICROWAVES IN NUMBER STATES 29 3 2 MICROWAVES IN COHERENT STATES 29 3 3 NUMERICAL RESULTS 29 4 DISCUSSIONS 30 TIME EVOLUTION OF TWO DISTANT SQUID RINGS IRRADIATED WITH ENTANGLED ELECTROMAGNETIC ?ELD 30 1 INTRODUCTION 30 2 INTERACTION OF TWO SQUID RINGS WITH NONCLASSICAL MICROWAVES 30 3 DISCUSSIONS 31 PHASE DIAGRAM OF DISSIPATIVE TWO-DIMENSIONAL JOSEPHSON JUNCTION ARRAYS 31 1 INTRODUCTION 31 2 PATH INTEGRAL MONTE CARLO FOR DISSIPATIVE SYSTEMS 31 3 JOSEPHSON JUNCTION ARRAYS 31 4 NUMERICAL SIMULATIONS AND PHASE DIAGRAM 31 5 DISCUSSION OF THE RESULTS 32 PERSISTENT CURRENTS IN A SUPERCONDUCTOR/NORMAL LOOP 32 1 INTRODUCTION 32 2 SNS JUNCTION 32 3 NS LOOP 33 JOSEPHSON JUNCTION LADDERS: A REALIZATION OF TOPOLOGICAL ORDER 33 1 INTRODUCTION 33 2 JOSEPHSON JUNCTION LADDERS 33 3 M-REDUCTION TECHNIQUE 33 4 TOPOLOGICAL ORDER AND “PROTECTED” QUBITS 34 SINGLE-ELECTRON CHARGE QUBIT IN A DOUBLE QUANTUM DOT 34 1 INTRODUCTION 34 2 CHARGE QUBIT IN A DOUBLE QUANTUM DOT 34 2 1 ROTATION GATE AND PHASE-SHIFT GATE OPERATION 34 2 2 DECOHERENCE OF THE SYSTEM 34 3 CHARGE DETECTION OF A DOUBLE QUANTUM DOT 34 4 SUMMARY 35 QUANTUM DOTS FOR SINGLE PHOTON AND PHOTON PAIR TECHNOLOGY 36 SEMICONDUCTOR FEW-ELECTRON QUANTUM DOTS AS SPIN QUBITS 36 1 QUBIT 36 2 READ-OUT 36 3 INITIALIZATION 36 4 COHERENCE TIMES 36 5 COHERENT SINGLE-SPIN MANIPULATION: ESR 36 6 COHERENT SPIN INTERACTIONS: ?SWAP 36 7 CONCLUSION 37 SPIN AMPLI?ER FOR SINGLE SPIN MEASUREMENT 37 1 INTRODUCTION 37 2 MEASUREMENT SCHEMES 37 3 EXPERIMENTAL DEMONSTRATION 37 4 CONCLUSIONS 38 ENTANGLEMENT IN QUANTUM-CRITICAL SPIN SYSTEMS 38 1 INTRODUCTION 38 2 THE MODEL 38 3 LINEAR CHAIN: CRITICAL POINT AND FACTORIZED STATE 38 4 ENTANGLEMENT ESTIMATORS 38 5 RESULTS 38 6 TWO-LEG LADDER 38 7 CONCLUSIONS 39 CONTROL OF NUCLEAR SPINS BY QUANTUM HALL EDGE CHANNELS 39 1 INTRODUCTION 39 2 SAMPLES 39 3 DYNAMICAL NUCLEAR POLARIZATION 39 4 LOCAL AND COHERENT CONTROL 39 5 DISCUSSION 39 6 SUMMARY 40 CLONING OF SINGLE PHOTON BY HIGH GAIN AMPLI?ER IN THE NEXT SECTION, THE EXPERIMENTAL SETUP AND ITS NOISE SOURCES ARE DESCRIBED, AND THE RELE- VANT DECOHERENCE RATES IS INTRODUCED IN SECTION 3, CHARACTERIZATION OF DECOHERENCE AT DIFFERENT OPERATING POINTS ARE EXPLAINED, AND THE EXPERIMENTAL RESULTS WITHIN A SIMPLE MODEL FOR THE NOISE SOURCES ARE DISCUSSED ?N V(T) S FIGURE 1 1 TOP: CIRCUIT DIAGRAM OF THE QUANTRONIUM QUBIT THE CONTROL PARAMETERS ARE NG = CGU/2E AND THE PHASE ?, WHICH IS DETERMINED BY THE ?UX ? IMPOSED THROUGH THE LOOP AND BY THE BIAS-CURRENT IB THE SWITCHING TO ?NITE VOLTAGE OF THE LARGER READOUT JUNCTION IN RESPONSE TO AN IB(T) PULSE ENABLES DISCRIMINATION OF THE QUBIT STATES BOTTOM: BLOCH SPHERE REPRESENTATION IN THE ROTATING FRAME GATE MICROWAVE PULSES INDUCE ROTATIONS OF THE EFFECTIVE SPIN S AROUND THE R AXIS, WHEREAS ADIABATIC NG(T) OR IB(T) PULSES INDUCE ROTATIONS AROUND Z QUANTUM COMPUTATION IN SOLID STATE SYSTEMS DECOHERENCE SOURCES THE QUANTRONIUM IS SUBJECT TO DECOHERENCE DUE TO ITS INTERACTIONS WITH UNCONTROLLED DEGREES OF FREEDOM INDUCING NOISE IN ? = NG OR ? = ?/2 (WE NEGLECT HERE NOISE ON EJ, WHICH IS SMALLER) DECOHERENCE IS DESCRIBED HERE IN TERMS OF RELAXATION OF THE QUBIT ENERGY INTO A QUAN- TUM SYSTEM ON ONE HAND, AND IN TERMS OF RANDOM DEPHASING BETWEEN THE TWO QUBIT STATES DUE TO ADIABATIC VARIATIONS OF ?01, ON THE OTHER HAND EACH PROCESS IS CONVENIENTLY DESCRIBED BY A QUANTUM SPECTRAL DENSITY S?(?) = 1/(2?) DT???(0)??(T)?EXP(?I?T) THAT QUANTI?ES, AT POSI- TIVE ?, THE ABILITY OF THE SOURCE TO ABSORB ONE ENERGY QUANTUM ¯H? CLASSICAL SPECTRAL DENSITIES QUANTUM COMPUTATION IN SOLID STATE SYSTEMS /(RAD/S) (1 7) IN ADDITION, THE NOISE ?IB OF THE ARBITRARY WAVEFORM GENERATOR (AWG) USED FOR THE READOUT PULSE HAS BEEN MEASURED TO BE WHITE UP TO 200 MHZ, AND CORRESPONDS TO A SPECTRAL DENSITY SCAWG ?/2? (|?| < 2? 200MHZ) = (15 × 10?9/ COS ? )2/(RAD/S) (1 8) BESIDES, AS ANY OTHER COULOMB BLOCKADE DEVICE, THE QUANTRONIUM SUFFERS FROM A BACKGROUND CHARGE NOISE (BCN) DUE TO CHARGED TWO-LEVEL ?UCTUATORS THE CORRESPONDING SPECTRAL DENSITY IS OF THE 1/F TYPE AT LOW FREQUENCY, SCBCN NG (|?| < 2? 100KHZ) ?AG/|?|, (1 9) 80 AF (4 K) (0 6K) YR 75 ? LJ 50 ? 40AF NG (20MK) K? K? FMICRO D DIB FIGURE 1 2 EQUIVALENT CIRCUIT DIAGRAM FOR DECOHERENCE THE 50 AND 1 K RESISTANCES REPRESENT THE REAL PART OF THE SERIES GATE IMPEDANCE AT THE QUBIT FREQUENCY AND AT LOW FREQUENCY, RESPECTIVELY INDUCTANCE LJ REPRESENTS THE READOUT JUNCTION WHEREAS THE 75 AND 4 K RESISTANCES ARE PARALLEL ADMITTANCES AT THE QUBIT AND AT LOW FREQUENCY, RESPECTIVELY MICROSCOPIC CHARGED TWO-LEVEL ?UCTUATORS ARE DISPLAYED AS DOUBLE ARROWS AND ?UX ?UCTUATORS ARE ENCAPSULATED IN ?MICRO QUANTUM COMPUTATION IN SOLID STATE SYSTEMS ? 2? ?,ZS?(0) FROM THE SAMPLE PARAMETERS WE COMPUTE D?/2?,Z(? = 0 OR NG = 1/2) = ?850 ? 2? 109 RAD/S, (1 16) DNG,Z(? = 0 ORNG = 1/2) = +290(NG ?1/2)109 RAD/S (1 17) QUANTUM COMPUTATION IN SOLID STATE SYSTEMS FROM P0 TO A POINT P WHERE T2 IS TO BE MEASURED THE MEASURED SIGNAL OSCILLATES WITH T1 AT THE NEW DETUNING FREQUENCY V(P), WITH AMPLITUDE THAT DECAYS WITH A CHARACTERISTIC TIME T2(P) THE USEFULNESS OF THE METHOD IS THAT THE PARAMETERS OF THE RAMSEY PULSES CAN BE OPTIMIZED AND KEPT CONSTANT FOR ALL POINTS P THESE ?RST TWO METHODS ARE OF COURSE APPLICABLE ONLY TO POINTS WHERE T2 > 1/V(P) WHEN T2 IS TOO SHORT, IT IS MORE CONVENIENT TO OPERATE IN THE FREQUENCY DOMAIN, BY RECORDING THE LINE SHAPE OF THE RESONANCE LINE AT LOW MICROWAVE POWER (DESATURATED LINE) IN THIS CASE THE LINESHAPE IS THE FOURIER TRANSFORM OF THE RAMSEY SIGNAL ONE HAS T2 = ?/(?V?W) WHERE V?W IS THE FULL WIDTH AT HALF MAXIMUM AND ? IS A COEF?CIENT THAT DEPENDS ON THE EXACT SHAPE OF THE LINE (1 FOR LORENTZIAN, 0 8 FOR GAUSSIAN) IN PRACTICE, DESATURATING THE LINE LEADS TO A RATHER LOW SIGNAL TO NOISE RATIO AND IT IS DIF?CULT TO DISCRIMINATE BETWEEN THE TWO SHAPES NEAR P0, THE RAMSEY DECAYS LOOK CLOSE TO EXPONENTIAL AND WE TAKE ? = 1 ANYWAY, OUR ERROR BARS ON T2 ARE LARGER THAN THE DIFFERENCE BETWEEN THE TWO ?’S SPIN-ECHO-LIKE EXPERIMENTS WERE PERFORMED BY INSERTING A MICROWAVE ? PULSE IN THE MIDDLE OF A RAMSEY SEQUENCE 5,8 IT WAS POSSIBLE EXPERIMENTALLY TO VARY THE SEQUENCE DURATION T WHILE KEEPING THE ? PULSE PRECISELY IN QUANTUM COMPUTATION IN SOLID STATE SYSTEMS S/RAD LOG(W) LOG(W) | NG - 1/2 | -| D /2P | 0 9 8/|| 1 9 6/|W| 0 4 MH< D = DISCUSSION FIGURE 1 3 SHOWS ?RST THAT DECOHERENCE OF THIS QUANTRONIUM IS LIMITED AT ALL P BY PURE DEPHASING RATHER THAN BY RELAXATION NEVERTHELESS TE(P0) APPROACHES THE LIMIT 2T1 NEXT, WE OBSERVE THAT RAMSEY DECAYS “DETUNING PULSES,” AND LINESHAPES MEASUREMENTS ALL LEAD TO T2’S IN REASONABLE AGREEMENT WITH EACH OTHER THE CONCEPT OF OPTIMAL POINT P0, WHERE T2’S AND TE’S ARE MAXIMUM, QUANTUM COMPUTATION IN SOLID STATE SYSTEMS E ?1 MHZ ANOTHER IMPORTANT RESULT IS THAT THE T2’S ARE TOO SHORT FOR THE DECOHERENCE TO BE EXPLAINED BY THE DISSIPATIVE ELEMENTS OF THE CIRCUIT INJECTING SPECTRAL DENSITIES EQS (1 3), (1 6) AND (1 8) IN EQS (1 14)–(1 17) LEADS TO THEORETICAL T2(P0) VALUES OF A FEW TENTH OF SECOND FOR THE GATE LINE, OF 160 ?S FOR THE READOUT IMPEDANCE AND OF 7 ?S FOR THE WHITE NOISE FROM THE AWG, RESPECTIVELY THE MAIN SOURCES OF NOISE ARE THUS OF MICROSCOPIC NATURE GIVEN THAT THEIR SPECTRAL DENSITIES ARE KNOWN TO BE ESSENTIALLY OF THE 1/F TYPE, WE TRIED TO ?T THE DATA WITH PURE 1/F SPECTRA, USING EQS (1 14)–(1 17) THE ?TS WERE GOOD FOR THE T2’S HOWEVER WE COMPUTED TE/T2 ?4 EVERYWHERE (AS EXPECTED FROM REF [8]), WHICH IS TOO LARGE IN THE PHASE DIRECTION AND TOO SMALL IN THE CHARGE DIRECTION, WHEN COMPARED TO THE DATA CONSEQUENTLY, WE ARE LED TO INTRODUCE PURELY EMPIRICALLY A HIGH FREQUENCY CUT-OFF IN THE CHARGE NOISE AND A WHITE CONTRIBUTION TO THE PHASE NOISE, AS SHOWN IN FIG 1 3 THE CORRESPONDING BEST ?TS (SHOWN ON FIG 1 3) ARE NOW IN BETTER AGREEMENT WITH THE DATA NOTE THAT THE SECOND ORDER CONTRIBUTION OF THE NOISES, WHICH DOMINATE VERY CLOSE TO P0, WERE ROUGHLY ESTIMATED ASSUMING GAUSSIAN N IS CHARGE NOISE WE HAVE ANALYSED THE DATA USING A SIMPLE MODEL FOR THE INTERACTION OF THE QUBIT WITH ITS ENVIRONMENT, AND FOR THE NOISE SOURCES A MORE COMPLETE ANALYSIS INCLUDING DECOHERENCE DURING DRIVEN EVOLUTION WILL BE PUBLISHED LATER ACKNOWLEDGMENTS THE ESSENTIAL TECHNICAL CONTRIBUTIONS OF P F OR?LA, P SENAT AND J C TACK, AND THE ?NANCIAL SUPPORT FROM THE EUROPEAN COMMISSION (SQUBIT2) AND OF THE “DYNASTY FOUN- DATION” ARE GRATEFULLY ACKNOWLEDGED REFERENCES 1 Y NAKAMURA, YU A PASHKIN, J S TSAI, NATURE 398, 786 (1999) 2 D VION ET AL , SCIENCE 296, 886 (2002) 3 I CHIORESCU ET AL , SCIENCE 299, 1869 (2003) 4 M A NIELSEN AND I L CHUANG, QUANTUM COMPUTATION AND QUANTUM INFORMATION (CAMBRIDGE UNIVER- SITY PRESS, CAMBRIDGE, 2000) 5 E COLLIN ET AL , PHYS REV LETT 93, 157005 (2004) QUANTUM COMPUTATION IN SOLID STATE SYSTEMS 6 V BOUCHIAT ET AL , PHYS SCR T76, 165 (1998) 7 A COTTET, PHD THESIS, UNIVERSIT´PARIS VI, (2002); WWW-DRECAM CEA FR/ DRECAM/SPEC/PRES/QUANTRO/ 8 D VION, IN QUANTUM ENTANGLEMENT AND INFORMATION PROCESSING, D EST`EVE, J M RAIMOND, AND J DAL- IBARD (EDS), (ELSEVIER, 2004), AND REFS THEREIN 9 G ITHIER ET AL , PHYS REV LETT 10 F C WELLSTOOD, C URBINA AND J CLARKE, APPL PHYS LETT 50, (1987) 11 Y MAKHLIN AND A SHNIRMAN, PHYS REV LETT 92, 178301 (2004) 12 E PALADINO ET AL , COND-MAT/0407484 (2004) QUANTUM COMPUTATION IN SOLID STATE SYSTEMS BESIDES ALL THIS, COUPLING MULTIPLE QUBITS IS ANOTHER IMPORTANT ISSUE, BECAUSE TO REALIZE THE UNI- VERSAL GATE IN QUANTUM COMPUTATION, TWO-QUBIT CONDITIONAL GATE IS REQUIRED ON TOP OF THE SINGLE- QUBIT ROTATION GATE THE ?RST EXPERIMENT ON COUPLED SUPERCONDUCTING QUBITS HAS BEEN REPORTED BY OUR GROUP USING CAPACITIVELY-COUPLED COOPER-PAIR BOXES 16 BY APPLYING PULSE VOLTAGE ON THE GATE, TWO QUBITS ARE BROUGHT TO THE RESONANCE SIMULTANEOUSLY, WHICH INDUCES THE BEATING D C GATE PULSE GATE GROUND PROBE PROBE BOX CONTROL BOX TARGET RESERVOIR RESERVOIR COUPLING CAPACITOR DEVICE STRUCTURE FIGURE 2 1(A) SHOWS THE SCHEMATIC DEVICE STRUCTURE OF COUPLED SUPERCONDUCTING CHARGE QUBITS THE DEVICE WAS FABRICATED BY ELECTRON-BEAM LITHOGRAPHY AND THREE-ANGLE EVAPORATION OF AL ON A SINX INSULATING LAYER ABOVE A GOLD GROUND PLANE ON THE OXIDIZED SI SUBSTRATE TWO COOPER-PAIR BOXES ARE CAPACITIVELY COUPLED BY AN ON-CHIP CAPACITOR THE RIGHT QUBIT HAS SUPERCONDUCTING QUANTUM INTERFERENCE DEVICE (SQUID) GEOMETRY SO THAT WE CAN CONTROL THE JOSEPHSON COU- PLING OF THE BOX TO THE RESERVOIR BY AN EXTERNAL MAGNETIC ?ELD IN THE EXPERIMENT, WE USE THIS QUBIT AS THE CONTROL QUBIT AND THE LEFT ONE AS THE TARGET QUBIT, ALTHOUGH THIS STRUCTURAL ASYMME- TRY IS NOT ESSENTIAL FOR THE LOGIC OPERATION BOTH QUBITS HAVE INDEPENDENT PULSE GATES AND THEY ENABLE US TO ADDRESS EACH QUBIT INDIVIDUALLY WE MEASURE THE PULSE-INDUCED CURRENTS THROUGH PROBES 1 AND 2, WHICH ARE BIASED AT ?650 ?V TO ENABLE JOSEPHSON-QUASIPARTICLE (JQP) CYCLE THESE CURRENTS ARE PROPORTIONAL TO THE PROBABILITY OF THE RESPECTIVE QUBIT HAVING ONE EXTRA COOPER PAIR QUANTUM COMPUTATION IN SOLID STATE SYSTEMS THE CHARGE STATES SHOWN IN THE ?GURE DENOTE THE GROUND STATES IN EACH HEXAGONAL CELL IN THE ABSENCE OF JOSEPHSON COUPLING AT BOUNDARIES, CHARGING ENERGIES OF TWO (OR THREE AT TRIPLE POINT) NEIGHBOURING STATES DEGENERATE PULSED VOLTAGE APPLIED TO PULSE GATES 1 AND 2 MOVE THE SYSTEM NON-ADIABATICALLY ALONG NG1 AND NG2 AXES, RESPECTIVELY IN OUR PREVIOUS EXPERIMENT TO DEMONSTRATE QUANTUM BEATING,16 WE ?RST SET THE OPERATION POINT (POINT A) SUF?CIENTLY FAR AWAY FROM THE CO- RESONANT POINT (NG1, NG2)=(0 5,0 5) AND BROUGHT THE SYSTEM NON-ADIABATICALLY TO THE CO-RESONANT POINT, WHERE FOUR EIGENENERGIES BECOME CLOSE TO EACH OTHER AND LET THE SYSTEM EVOLVE FREELY FOR THE CONDITIONAL GATE, HOWEVER, WE ?X NG1 AS A CONSTANT VALUE AND OPERATE ALONG THE DASHED LINE IN THE ?GURE FIGURE 2 2(A) SHOWS THE ENERGY BANDS OF THE PRESENT SYSTEM ALONG THE DASHED LINE IN FIG 2 1(B) HERE, FOUR ENERGY BANDS CAN BE REGARDED AS TWO PAIRS OF NEARLY INDEPENDENT SINGLE-QUBIT ENERGY BANDS FOR THE LOWER TWO BANDS, THE ?RST QUBIT (CONTROL QUBIT) IS ALWAYS IN THE |0> STATE, WHILE FOR THE HIGHER TWO BANDS, THE ?RST QUBIT IS IN THE |1> STATE IMPORTANTLY, THE CHARGING ENERGIES OF EACH OF THE TWO-LEVEL SYSTEMS DEGENERATE AT DIFFERENT NG2 THIS DIFFERENCE ORIGINATES FROM THE ELECTROSTATIC COUPLING BETWEEN THE QUBITS AND CAN BE UTILIZED FOR THE CONDITIONAL GATE OPERATION FOR THE CONDITIONAL GATE OPERATION, WE APPLY THE VOLTAGE PULSE TO PULSE GATE 2 SO THAT IT BRINGS THE SYSTEM TO THE DEGENERACY POINT FOR LOWER TWO BANDS (FIG 2 2(A)) SUPPOSE WE START FROM THE |00 > STATE APPLICATION OF THIS PULSE INDUCES THE OSCILLATION BETWEEN |00> AND |01> STATES WITH MAXIMUM AMPLITUDE, AS SCHEMATICALLY SHOWN BY THE BLOCH SPHERE IN FIG 2 2(B) BY PROPERLY TUNING THE LENGTH OF THE PULSE, THE OSCILLATION CAN BE STOPPED WHEN THE SYSTEM IS IN THE |01> STATE BY THE SAME PULSE, WE OBTAIN |00> STATE FROM THE INPUT STATE OF |01> ON THE OTHER HAND, WHEN THE INITIAL STATE IS |10> OR |11>, THE PULSE INDUCES OSCILLATION BETWEEN |10> AND |11> STATES HOWEVER, BECAUSE THE SYSTEM IS NOT BROUGHT TO THE DEGENERACY POINT FOR |10> AND |11> STATES, THE OSCILLATION AMPLITUDE IS SUPPRESSED DUE TO THE ?NITE ?CTITIOUS MAGNETIC ?ELD ALONG Z AXIS (FIG 2 2(C)) THE MAGNITUDE OF THIS ?CTITIOUS ?ELD IS PROPORTIONAL TO THE COUPLING ENERGY IN THE IDEAL CASE, INPUT STATES REMAIN THE SAME THUS, THIS PULSE PERFORMS THE CONDITIONAL GATE OPERATION, I E , THE STATE OF THE TARGET QUBIT IS ?IPPED ONLY WHEN THE CONTROL QUBIT IS IN THE |0> STATE QUANTUM COMPUTATION IN SOLID STATE SYSTEMS EXPERIMENTAL SETUP THE SAMPLE CHIP IS MOUNTED IN A SHIELDED COPPER BOX AND IS COOLED DOWN TO THE BASE TEMPERATURE OF THE DILUTION REFRIGERATOR (?40 MK) THE OVERALL WIRING IS ALMOST THE SAME AS THAT REPORTED IN REF [22], EXCEPT THAT NOW WE HAVE TWO COAXIAL CABLES FOR THE TRANSMISSION OF FAST VOLTAGE PULSES AS THE HIGH-SPEED PULSE GENERATOR, WE USE AN ANRITSU MP1758A PULSE-PATTERN GENERATOR IT CAN CREATE AN ARBITRARY DIGITAL PATTERN WITH NON-RETURN ZERO (NRZ) PULSES FOR UP TO FOUR CHANNELS ALTHOUGH THE DIGITAL PATTERN AND THE OUTPUT LEVEL CAN BE SET INDEPENDENTLY FOR EACH CHANNEL, THE BASE FREQUENCY, WHICH IS TUNABLE UP TO 12 5 GHZ, IS COMMON FOR ALL CHANNELS THUS, ONCE THE BASE FREQUENCY IS ?XED, SAY, AT 10 GHZ, ALL THE DIGITAL PATTERNS ARE CREATED WITH THE UNIT LENGTH OF 100 PS IN THE EXPERIMENT REPORTED IN REF [20], TWO OF THESE CHANNELS ARE USED TO CONTROL THE TWO QUBITS FIGURE 2 3(A) SHOWS ONE OF THE PULSE SEQUENCES UTILIZED IN REF [20] IT IS THE REAL OUTPUT SIGNAL FROM THE PULSE GENERATOR MEASURED AT THE TOP OF THE CRYOSTAT THE OUTPUTS OF CHANNELS AND 2 ARE SENT TO PULSE GATE 1 (CONTROL QUBIT) AND PULSE GATE 2 (TARGET QUBIT), RESPECTIVELY THE INSET SHOWS THE CORRESPONDING BIT PATTERN TO OPERATE THE GENERATOR HERE, THE BASE FREQUENCY IS SET TO 11 8 GHZ, THUS ONE BIT CORRESPONDS TO 85 PS AND THE REPETITION TIME IS 85×1505 = 127925 PS ?128 NS QUANTUM COMPUTATION IN SOLID STATE SYSTEMS (A) (B) (C) FIGURE 2 3 (A) OUTPUT SIGNAL OF THE ANRITSU MP1758A PULSE-PATTERN GENERATOR THE INSET SHOWS THE CORRE- SPONDING BIT PATTERN TO GENERATE THIS PULSE PATTERN THE NOMINAL LENGTH OF ONE BIT IS SET AS 85 PS THIS PULSE SEQUENCE IS USED TO DEMONSTRATE CONDITIONAL GATE OPERATION, ALTHOUGH THE AMPLITUDES OF THE PULSES MUST BE PROPERLY ADJUSTED (B) SCHEMATIC DIAGRAM TO ILLUSTRATE HOW TO USE THE ANRITSU MP1707A PULSE CONTROLLER WITH PULSE-PATTERN GENERATOR THE LENGTH OF THE OUTPUT PULSE OF MP1707A CAN BE CHANGED BY THE INTERNAL PHASE SHIFTERS, WHILE KEEPING THE FREQUENCY OF MP1758A CONSTANT (C) AN EXAMPLE OF THE PULSE PATTERN PRODUCED BY THE SETUP IS DESCRIBED IN (B) HERE THE BASE FREQUENCY OF MP1758A IS ?XED AS 12 5 GHZ THE NOMINAL LENGTH OF THE PULSE FROM MP1758A (LOWER SIGNAL) IS ?XED AS 80 PS, WHILE THE LENGTH OF THE PULSE FROM MP1707A (UPPER SINGNAL) IS CHANGED FROM 100 TO 300 PS WITH 50 PS STEPS THE ?RST PULSE APPLIED TO PULSE GATE 1 CREATES A SUPERPOSITION STATE OF THE CONTROL QUBIT THERE- FORE, THE STATE ?|00 > +?|10 > IS PREPARED AS AN INPUT STATE THE SECOND PULSE IS APPLIED TO PULSE GATE 2 AND IT CORRESPONDS TO THE CONDITIONAL GATE OPERATION EXPLAINED IN THE PREVIOUS SEC- TION ACCORDINGLY, THIS PULSE IS EXPECTED TO TRANSFORM THE INPUT STATE TO ?|01 > +?|10 > IN REF [20], WE SWEPT A MAGNETIC ?ELD TO CHANGE THE COEF?CIENTS ? AND ? THE SQUID GEOMETRY OF THE CONTROL QUBIT ENABLES US TO CONTROL EJ1 BY A MAGNETIC ?ELD SINCE EJ1 GIVES THE OSCILLATION FREQUENCY DURING THE ?RST PULSE, ? AND ? CAN BE CHANGED BY A MAGNETIC ?ELD, WHILE KEEPING THE PULSE LENGTH CONSTANT AN ALTERNATIVE (AND PROBABLY MORE NATURAL) WAY TO CHANGE ? AND ? IS TO CHANGE THE LENGTH OF THE ?RST PULSE FOR THAT PURPOSE, WE USE AN ANRITSU MP1707A PULSE CONTROLLER IT IS A LOGIC DEVICE WITH TUNABLE DELAY LINES THE SETUP OF THIS DEVICE COMBINED WITH MP1758A IS SHOWN IN FIG 2 3 (B) TWO OUTPUTS OF MP1758A ARE USED AS INPUTS OF MP1707A THEY ARE LOGICALLY OPERATED (LOGICAL OR) AND ONE OUTPUT IS PRODUCED BY MP1758A AS SHOWN IN THE ?GURE SINCE THE DELAY FOR THE INPUTS CAN BE SET WITH A STEP OF INCREMENT LESS THAN 1 PS, THE LENGTH OF THE OUTPUT PULSE FROM MP1707A CAN ALSO BE CHANGED BY THE SAME STEP FIGURE 2 3(C) SHOWS AN EXAMPLE OF THE PULSE PATTERN PRODUCED BY THIS SETUP WE CAN CHANGE THE LENGTH OF THE ?RST PULSE, WHILE KEEPING THAT OF THE SECOND ?XED IF WE SET THE LENGTH OF THE SECOND PULSE PROPERLY, WE CAN DEMONSTRATE THE CONDITIONAL GATE OPERATION BY THIS PULSE SEQUENCE QUANTUM COMPUTATION IN SOLID STATE SYSTEMS ?T (PSEC) AND VG2, AT WHICH POINT THE SYSTEM IS BROUGHT TO THE RESONANCE FOR |00> AND |10> BY VP1 OR THE RESONANCE FOR |00> AND |01> BY VP2 FIGURE 2 4(C) SHOWS THE DEPENDENCE OF I2 ON THE LENGTH OF THE PULSE APPLIED TO PULSE GATE 2 AT PROPERLY CHOSEN VG1 AND VG2 THE COHERENT OSCILLATION WITH THE FREQUENCY OF EJ2/¯H IS OBSERVED THE LENGTH OF THE PULSE FOR CONDITIONAL GATE OPERATION SHOULD BE ?XED AT THE PEAK OF THIS OSCILLATION, 255 PS, FOR EXAMPLE QUANTUM COMPUTATION IN SOLID STATE SYSTEMS CONTROL QUBIT TARGET QUBIT CONTROL QUBIT TARGET QUBIT PULSE-INDUCED CURRENT (PA) CONCLUSIONS THE CONCEPT AND EXPERIMENTAL DETAILS CONCERNING THE CONDITIONAL GATE OPERATION USING CAPACITIVELY-COUPLED SUPERCONDUCTING CHARGE QUBITS HAVE BEEN DESCRIBED THE TWO-QUBIT QUANTUM COMPUTATION IN SOLID STATE SYSTEMS SOLID-STATE CIRCUIT WAS CONTROLLED BY APPLYING A SEQUENCE OF PULSES AND THE CONDITIONAL GATE OPERATION AS FOR THE AMPLITUDE OF THE QUANTUM STATE DEMONSTRATED FOR THE REALIZATION OF THE QUANTUM C-NOT GATE, IT IS ALSO NEEDED TO EXAMINE THE PHASE EVOLUTION DURING THE GATE OPERA- TION IN ADDITION, IT IS DESIRABLE THAT THE STRENGTH OF THE COUPLING CAN BE EXTERNALLY CONTROLLED SEVERAL SCHEMES OF THE TUNABLE COUPLING, WHICH ARE APPLICABLE TO THE PRESENT SYSTEM, HAVE BEEN THEORETICALLY PROPOSED23–25 AND THE EXPERIMENTS ARE NOW UNDERWAY ACKNOWLEDGMENTS WE THANK B L ALTSHULER, D V AVERIN, S ISHIZAKA, J LANTZ, S LLOYD, F NORI, T TILMA, C URBINA, G WENDIN AND J Q YOU FOR MANY FRUITFUL DISCUSSIONS REFERENCES 1 Y MAKHLIN, G SCH¨ON, AND A SHNIRMAN, QUANRUM-STATE ENGINEERING WITH JOSEPHSON-JUNCTION DEVICES, REV MOD PHYS 73, 357 (2001) 2 Y NAKAMURA, YU A PASHKIN, AND J S TSAI, COHERENT CONTROL OF MACROSCOPIC QUANTUM STATES IN A SINGLE-COOPER-PAIR BOX, NATURE 398, 786–788 (1999) 3 D VION ET AL , MANIPULATING THE QUANTUM STATE OF AN ELECTRICAL CIRCUIT, SCIENCE 296, 886–889 (2002) 4 Y YU, S HAN, X CHU, S I CHU, AND Z WANG, COHERENT TEMPORAL OSCILLATIONS OF MACROSCOPIC QUANTUM STATES IN A JOSEPHSON JUNCTION, SCIENCE 296, 889–892 (2002) 5 J M MARTINIS, S NAM, J AUMENTADO, AND C URBINA, RABI OSCILLATIONS IN A LARGE JOSEPHSON-JUNCTION QUBIT, PHYS REV LETT 89, 117901 (2002) 6 I CHIORESCU, Y NAKAMURA, C J P M HARMANS, AND J E MOOIJ, COHERENT QUANTUM DYNAMICS OF A SUPERCONDUCTING ?UX QUBIT, SCIENCE 299, 1869–1871 (2003) 7 O ASTA?EV, YU A PASHKIN, T YAMAMOTO, Y NAKAMURA, AND J S TSAI, SINGLE-SHOT MEASUREMENT OF THE JOSEPHSON CHARGE QUBIT, PHYS REV B 69, 180507(R) (2004) 8 I SIDDIQI ET AL , AN RF-DRIVEN JOSEPHSON BIFURCATION AMPLI?ER FOR QUANTUM MEASUREMENTS, COND- MAT/0312623 9 A LUPACU, C J M VERWIJS, R N SCHOUTEN, C J P M HARMANS, AND J E MOOIJ, NONDESTRUCTIVE READOUT FOR A SUPERCONDUCTING ?UX QUBIT, COND-MAT/0311510 10 E COLLIN, G ITHIER, A AASSIME, P JOYEZ, D VION, AND D ESTEVE, NMR-LIKE CONTROL OF A QUANTUM BIT SUPERCONDUCTING CIRCUIT, COND-MAT/0404503 11 Y NAKAMURA, YU A PASHKIN, T YAMAMOTO, AND J S TSAI, CHARGE ECHO IN A COOPER-PAIR BOX, PHYS REV LETT 88, 047901 (2002) 12 K W LEHNERT, K BLADH, L F SPIETZ, D GUNNARSSON, D I SCHUSTER, P DELSING, AND R J SCHOELKOPF, PHYS REV LETT 90, 027002 (2003) 13 T DUTY, D GUNNARSSON, K BLADH, AND P DELSING, COHERENT DYNAMICS OF A JOSEPHSON CHARGE QUBIT, PHYS REV B 69, 140503(R) (2004) 14 R W SIMMONDS, K M LANG, D A HITE, D P PAPPAS, AND J M MARTINIS, DECOHERENCE IN JOSEPHSON QUBITS FROM JUNCTION RESONANCES, COND-MAT/0402470 15 O ASTA?EV, YU A PASHKIN, Y NAKAMURA, T YAMAMOTO, AND J S TSAI, PRIVATE COMMUNICATION 16 YU A PASHKIN ET AL , QUANTUM OSCILLATIONS IN TWO COUPLED CHARGE QUBITS, NATURE 421, 823–826 (2003) 17 A J BERKLEY ET AL , ENTANGLED MACROSCOPIC QUANTUM STATES IN TWO SUPERCONDUCTING QUBITS, SCIENCE 300, 1548–1550 (2003) 18 J B MAJER, F G PAAUW, A C J TER HAAR, C J P M HARMANS, AND J E MOOIJ, SPECTROSCOPY ON TWO COUPLED ?UX QUBITS, COND-MAT/0308192 19 A IZMALKOV ET AL , EXPERIMENTAL EVIDENCE FOR ENTANGLED STATES FORMATION IN A SYSTEM OF TWO COUPLED ?UX QUBITS, COND-MAT/0312332 20 T YAMAMOTO, YU A PASHKIN, O ASTA?EV, Y NAKAMURA, AND J S TSAI, DEMONSTRATION OF CONDITIONAL GATE OPERATION USING SUPERCONDUCTING CHARGE QUBITS, NATURE 425, 941–944 (2003) QUANTUM COMPUTATION IN SOLID STATE SYSTEMS 21 H POTHIER, P LAFARGE, C URBINA, D ESTEVE, AND M H DEVORET, SINGLE-ELECTRON PUMP BASED ON CHARGING EFFECTS, EUROPHYS LETT 17, 249–254 (1992) 22 Y NAKAMURA AND J S TSAI, QUANTUM-STATE CONTROL WITH A SINGLE-COOPER-PAIR BOX, J LOW TEMP PHYS 118, 765–779 (2000) 23 J Q YOU, J S TSAI, AND F NORI, CONTROLLABLE MANIPULATION AND ENTANGLEMENT OF MACROSCOPIC QUANTUM STATES IN COUPLED CHARGE QUBITS, PHYS REV B 68, 024510 (2003) 24 D V AVERIN AND C BRUDER, VARIABLE ELECTROSTATIC TRANSFORMER: CONTROLLABLE COUPLING OF TWO CHARGE QUBITS, PHYS REV LETT 91, 057003 (2003) 25 J LANTZ, M WALLQUIST, V S SHUMEIKO, AND G WENDIN, JOSEPHSON JUNCTION QUBIT NETWORK WITH CURRENT- CONTROLLED INTERACTION, COND-MAT/0403285 QUANTUM COMPUTATION IN SOLID STATE SYSTEMS IN THIS PAPER, WE CONSIDER THE CHARGE-PHASE QUBIT1 COMPRISING A MACROSCOPIC SUPERCONDUCT- ING RING INCLUDING TWO SMALL JOSEPHSON JUNCTIONS WITH A SMALL ISLAND IN BETWEEN (SEE FIG 3 1), I E , A COOPER PAIR BOX6 OF SQUID CON?GURATION THIS SETUP IS, IN PRINCIPLE, SIMILAR TO THAT OF QUBIT PARAMETERS AND THE MODEL FOR THE VANISHINGLY SMALL VALUE OF INDUCTANCE L THE BEHAVIOR OF THE CHARGE-PHASE QUBIT IS PRI- MARILY DETERMINED BY THE RELATIONSHIP BETWEEN THE ENERGIES ASSOCIATED WITH THE ISLAND AND THE JUNCTIONS THESE ARE THE COULOMB ENERGY OF THE ISLAND, EC = E2/2C, AND THE JOSEPHSON COU- PLING ENERGIES OF THE JUNCTIONS, EJ1,J2 = ¯HIC1,C2/2E, WHERE C = C1 + C2 + CG IS THE ISL