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  • Goal: quantum computer
  • Paper in 2000 in which author proposed cirtiera for a qunatum computer. has been guide in the fieldscaleable: can double number of qubits with polynomial increasein resources -Most of the criteria have been met in different qubits-focus on 2 qubit gates
  • -Why use photons: highly coherent and can mediate interactions between distance objects-In a cavity QED system, we have enhanced qubit-photon interactions-If we work in the strong coupling limit, we have a coherent interaction between TLS-photon, permitting information transfer.
  • Start with JCM in the RWA
  • One of the regimes well consider is the strongly dispersive
  • No TLS-cavity interaction: no exchange of excitationsIt will be useful to turn off the qubit-qubit interaction in order to apply 1-qubit gates
  • Top: schematic of cavity. -2 transmon qubits, located at opposite cavity ends -Coplanar waveguide interrupted by 2 coupling capacitors (mirrors)
  • Why work in the strongly dispersive limit?----- Meeting Notes (12/5/12 12:45) -----In this limit we can use the cavity as a medium connecting the qubits and obtain a spin-spin Hamiltonian
  • First qubit absorbs a nonresonant photon: non-energy conserving process, so the probability of happening is small. This information is transmitted through the cavity to the other qubit that goes to the ground state such that the whole process is energy conserving. That means that the probability of happening is small.
  • Satisfy criteria 4 and 3 (sort of), in that we can manipulate the qubits before they decohere
  • Solid line is theory, dots are actual dataDecay is decoherence: <a+a^\\dagger> decays exponentially
  • Black line: homodyne voltage due to Stark pulse without pi-pulse being applied to either qubitThin lines: without stark pulseAverage over 3E6 tracesiSWAP- Universal quantum gate.How do we know this is really a manifestation of the qubit-oscillation term in the Hamiltonian?
  • Plot of the obersved qubit oscillation frequency and the calculated frequency splittingWhy should we observe a dark stateQubit1: 6.469, Qubit2: 6.546
  • Virtual particle: a particle that exists for a limited time and space. Obeys energy-time uncertainty.Virtual photons have mass (from borrowed energy) b/c they exist for limited time, giving them limited “range”
  • Presentation

    2. 2. DiVincenzo’s Criteria for Quantum Computing1. Scalability and well-defined qubits2. Initialization of qubits3. Small decoherence4. 1 and 2 qubit gates5. Measurement arXiv:cond-mat/96121265/12/2012 Majer et al. Nature (2007) 2
    3. 3. Motivationo Previous studies have shown that two nearby qubits can be coupled with local interactionso Highly desirable to perform gate operations between two distant qubits – How to accomplish? – Use a quantum bus (cavity photons) to transfer information Strong coupling limit o Why use photons?5/12/2012 Majer et al. Nature (2007) 3
    4. 4. Experiment Goalso Demonstrate a coherent, nonlocal coupling between two qubits in a transmission line cavityo Cavity mediates the qubit-qubit interaction via photons Blais et al. Rhy. Rev. A (2007)5/12/2012 Majer et al. Nature (2007) 4
    5. 5. THEORYo Begin with the Jaynes Cummings (JC) Hamiltoniano Eigenstates and eigenenergies readily obtainedo Vacuum Rabi splitting can be observed by moving to a rotating frame and solving the equations of motion5/12/2012 Majer et al. Nature (2007) 5
    6. 6. THEORYo The solution is , which produces 2 peaks ato This allows for a measurement of the coupling constant5/12/2012 Majer et al. Nature (2007) 6
    7. 7. Theoryo Consider the strongly dispersive limit,o Using the canonical Schrieffer-Wolf Transformation, eliminate interaction term to 1st order5/12/2012 Majer et al. Nature (2007) 7
    8. 8. THEORYo That was for 1 qubit. How about 2?o Natural generalization:o Can easily show that5/12/2012 Majer et al. Nature (2007) 8
    9. 9. THEORYo Salient features: – No TLS-cavity interaction (no energy is exchanged) – Cavity frequency shifted by qubit state – Qubit-qubit interaction can be effectively turned off by making the qubits strongly detuned from one another:5/12/2012 Majer et al. Nature (2007) 9
    10. 10. THEORYo A.C. Stark Shift: rearrange the Hamiltoniano By applying a strongly detuned drive, we can adjust the number of photons in the cavity, thereby adjusting the qubit transition frequency.5/12/2012 Majer et al. Nature (2007) 10
    11. 11. Experiment Setup5/12/2012 Majer et al. Nature (2007) 11
    12. 12. Strong Qubit-Cavity Couplingo Demonstrate that each-qubit can be strongly coupled to the cavityo Use vacuum Rabi splitting to determine the coupling constantso Ensures that we can go into strongly dispersive limit and that qubit-qubit coupling is big5/12/2012 Majer et al. Nature (2007) 12
    13. 13. Strongly Dispersive Limito Cavity shields qubits from the environment.o Can further isolate qubit by strongly detuning it from the cavity5/12/2012 Majer et al. Nature (2007) 13
    14. 14. Qubit-Qubit Interactiono Qubits interact by exchanging their excitations through virtual photons in the cavity:5/12/2012 Majer et al. Nature (2007) 14
    15. 15. Single Qubit Controlo Demonstrate fast control of each qubit individually in order to satisfy the 1st part of criteria 4o Detune the qubits from one another:o Apply a pulse at , then apply a measurement pulse at to monitor transmissiono From transmission, infer :5/12/2012 Majer et al. Nature (2007) 15
    16. 16. Single Qubit Control o Response consistent with that of single qubit Rabi oscillation  coupling does not affect single qubit operation o Determine decoherence time to be 78 and 120 ns, which is larger than the coherent manipulation time5/12/2012 Majer et al. Nature (2007) 16
    17. 17. Multiplex Measuremento Use π pulses to put your qubits into desired states:o Use probing field resonant with cavity and compare theoretical (via master equation) with actual value5/12/2012 Majer et al. Nature (2007) 17
    18. 18. Coherent State Transfer Between Qubitso Can transfer the state of one qubit to the other by turning qubit-qubit coupling on and offo Use off-resonant Stark drive to quickly push qubits into resonance5/12/2012 Majer et al. Nature (2007) 18
    19. 19. Coherent State Transfer 1. Initially qubits are 80 MHz detuned, and are allowed to relax to 2. Apply π pulse to create or 3. Apply Stark pulse to bring qubits into resonance for some variable time Because not eigenstates of system, we’ll see oscillation5/12/2012 Majer et al. Nature (2007) 19
    20. 20. Coherent State Transfer o Quarter period of oscillation between qubits is o This is the second part of DiVincenzo’s criterion 45/12/2012 Majer et al. Nature (2007) 20
    21. 21. Coherent State Transfero Observed qubit-qubit oscillation frequency agrees very well with value of J measured from CW spectroscopy5/12/2012 Majer et al. Nature (2007) 21
    22. 22. Summaryo Demonstrated non-local coupling of qubitso Qubit-qubit interaction is due to the exchange of virtual photons, protecting against cavity induced losseso Qubits may be manipulated individually and a universal 2 qubit gate can be performed5/12/2012 Majer et al. Nature (2007) 22
    23. 23. Summary1. ✓ Scalability and well-defined qubits2. Initialization of qubits ✓3. Small decoherence ✓4. 1 and 2 qubit gates ✓5. Measurement ✓5/12/2012 Majer et al. Nature (2007) 23
    24. 24. Thank you for your time and attention.5/12/2012 Majer et al. Nature (2007) 24
    25. 25. 5/12/2012 Majer et al. Nature (2007) 25
    26. 26. 5/12/2012 Majer et al. Nature (2007) 26