Introduction Quantum Information Processing Superconducting Qubits and Circuit QED Circuit QED: Quantum Information Processing with a Photon Bus Experimental Setup and Details Initialization and Benchmarking of Single-Qubit Gates Two-Qubit Circuit QED: Riding the Quantum Bus Entanglement On-Demand and Joint Readout Two-Qubit Algorithms Conclusions and Future Work Bibliography ©byJerryMoyChow All rights reserved. Quantum Information Processing with Superconducting Qubits A Dissertation PresentedtotheFacultyoftheGraduateSchool of Yale University in Candidacy for the Degree of Doctor of Philosophy by Jerry Moy Chow Dissertation Director: Professor Robert J. Schoelkopf May 2010 Abstract Quantum Information Processing with Superconducting Qubits Jerry Moy Chow 2010 This thesis describes the theoretical framework, implementation, and measurements of a quantum processor comprised of superconducting qubits coupled in the circuit quantum electrodynamics (QED) architecture. In the realization of circuit QED, two superconducting ‘transmon’ charge qubits are capacitively coupled to a one-dimensional microwave transmis- sion line resonator which serves as a quantum bus. Single-qubit rotations can be applied through the resonator and their operation is characterized using various benchmarking techniques. Through a virtual photon interaction via the quantum bus, the two qubits can coherently swap a single excitation. A separate two-qubit conditional phase interaction is also observed which is attributable to an interaction in the two-excitation manifold of the transmons. Furthermore, the same quantum bus which couples the qubits can be used as a joint detector of the full two-qubit quantum state. Entanglement witnesses and a violation of a Bell-type inequality are found using this joint detector on highly entangled states. Fi- nally, combining the single-qubit rotations, conditional phase interaction, and joint readout, allows the realization and characterization of simple quantum algorithms, specifically the Deutsch-Jozsa and Grover’s search algorithms. Contents Contents v List of Figures xi List of Tables xv Acknowledgements xvii Publication list xix Nomenclature xxi Introduction . Computing with quantum mechanics . ...................... . Experimentalimplementationsofquantumprocessors............. . Overviewofthesis................................... Quantum Information Processing . Universalquantumcomputing........................... . Single-qubit gates ................................... . Two-qubitentanglementgates............................ .. cNOTgate.................................. .. c-Phase gate√ (cUij).............................. .. iSWAP and iSWAPgates......................... . Quantumalgorithms................................. .. Quantum parallelism in an algorithm .................. .. Deutsch-Jozsa algorithm .......................... v vi contents .. Grover’s search algorithm . ....................... .. Shor’s and other quantum algorithms ................... . Quantummeasurement............................... .. Density matrix ................................ .. State tomography .............................. . Entanglementmetrics................................ .. Concurrence ................................. .. Entanglement witnesses . ....................... . Belltests........................................ .. Clauser-Horne-Shimony-Holt inequality ................ .. CHSH entanglement witness . ....................... . Chaptersummary................................... Superconducting Qubits and Circuit QED . Superconductingqubits............................... .. Josephson junction as a non-linear inductor ............... .. The Cooper-pair box qubit . ....................... .. The transmon qubit ............................. . Couplingsuperconductingqubits.......................... .. Fixed capacitive coupling . ....................... .. Tunable inductive coupling . ....................... .. Quantum bus coupling ........................... . Cavity quantum electrodynamics .......................... .. Strong coupling regime ........................... .. Dispersive coupling regime . ....................... . CircuitQED...................................... .. Coupling a transmon to a coplanar waveguide resonator ....... .. Dispersive regime of circuit QED ..................... .. Strong dispersive regime . ....................... . Qubitdecoherence.................................. .. Relaxation and the Purcell effect ...................... .. Dephasing .................................. . Chaptersummary................................... Circuit QED: Quantum Information Processing with a Photon Bus . Initialization . ..................................... . Single-qubit gates in circuit QED .......................... .. Introducing a drive ............................. .. X-Y gates for a qubit ............................. .. X-Y gates for a transmon multi-level atom ................ .. Z (phase) gates ................................ . Two-qubitgatesincircuitQED........................... contents vii .. Two-qubits in the dispersive regime ................... .. Virtual qubit-qubit interaction . ...................... .. σz ⊗ σz higher level transmon interaction . ............. . Muliplexedjointqubitreadout........................... .. Deriving the measurement operator ................... .. State tomography in circuit QED ..................... .. Entanglement by joint measurement ................... Chaptersummary................................... Experimental Setup and Details . Experimentaltestsamples.............................. ResonatorFabrication................................ .. Resonator parameters ............................ .. Optical lithography ............................. Transmonfabrication................................. .. ‘Traditional’ transmon ........................... .. Balanced design ............................... .. Flux-bias transmon design . ...................... Sample boards and holders .............................. .. Coffin design ................................. .. Octobox design ............................... Cryogenic setup .................................... Roomtemperaturecontrol.............................. Pulsecontrolandmodulation............................ Chaptersummary................................... Initialization and Benchmarking of Single-Qubit Gates . Initializing pure states ................................ .. Nonlinear vacuum Rabi ........................... .. Cavity temperature ............................. .. Vacuum Rabi summary ........................... Characterizing single-qubit gates .......................... Single-qubit gate error experiments . ...................... .. Microwave pulse shaping .......................... .. Calibration of single-qubit gates ...................... .. Single qubit readout calibration ...................... .. Double π metric............................... .. Quantum process tomography . ...................... .. Randomized benchmarking . ...................... .. Summary of error metrics . ...................... Derivative-basedpulseshaping........................... .. Experimental details ............................. viii contents .. Results with standard pulse shaping ................... .. Experimentally implementing derivative pulse shaping ........ .. Summary of DPS .............................. Chaptersummary................................... Two-Qubit Circuit QED: Riding the Quantum Bus . Experimentaldetails................................. Two-qubitspectroscopy............................... .. Qubit-qubit avoided crossing ....................... .. The dark state ................................. Multiplexedjointqubitreadout........................... Coherentstatetransfer:Starkswap......................... Chaptersummary................................... Entanglement On-Demand and Joint Readout . Experimentalsetup.................................. Virtual swap interaction via flux bias . ...................... Higher-level transmon interaction . ...................... .. Tuning up a c-Phase gate . ....................... .. Generating Bell states ............................ Jointreadoutoftwoqubits.............................. Calibratingthemeasurementmodel........................ QuantumstatetomographyandthePauliset................... .. The density matrix representation ..................... .. Biasing of metrics by maximum-likelihood estimation ........ .. The Pauli set representation . ....................... Characterizing the quantum states . ...................... .. Fidelity to targeted states . ....................... .. Entanglement witnesses . ....................... .. Clauser-Horne-Shimony-Holt inequality violation . ........ Chaptersummary................................... Two-Qubit Algorithms . Experimentaldetails................................. Deutsch-Jozsaalgorithm............................... .. Breaking down the algorithm ....................... .. Deutsch–Jozsa results ............................ Groversearchalgorithm............................... .. The oracle ................................... .. Breaking down the algorithm ....................... .. Grover results and debugger . ....................... Chaptersummary................................... contents ix Conclusions and Future Work . Improving one and two-qubit operations ..................... .. Longer coherence