Hardware of a Quantum Computer #7 Superconducting Qubit

• Superconducting qubits vs natural qubits
  • Multi-level vs Two-level
    • Possible to create superconducting qubits by restricting all dynamics to (generally, the lowest) 2 levels of the multiple levels of the system
  • Can fabricate vs Cannot fabricate
    • +) freedom to design
    • -) cannot make same qubits
• Consists of
  • Superconducting electrodes or islands
    • interconnected by Josephson junctions (pairs of superconducting electrons could “tunnel” right through the non-superconducting barrier from one superconductor to another)
• Transmon Qubit
  • an example of superconducting charge qubit
  • 2 islands interconnected by one junction and a large capacitor
  • inductance provided by a Josephson junction and not by a typical coil inductor
    • disrupts the harmonic energy spectrum and helps confine the system of two levels
  • inductive energy not quadratic, but a cosine function of the generalized flux through it
• Circuit Quantum Electrodynamics (QED)
  • combining of quantum hardware with resonators
  • second-generation processors for superconducting qubits
    • dedicated resonators for readout of each qubit
    • 1 common ‘bus’ resonator connecting to all the qubits
  • frequency of a readout resonator depends on the qubit state
    • this dependence is the key ingredient that facilitates the measurement of qubits
  • mediating the inter-qubit interaction on the common ‘bus’ resonator → possible to perform two-qubit opperations
• Assembling a quantum processor from circuit QED hardware
  • Surface Code (fault-tolerant quantum computing in QuTech)
    • 2D square lattice of qubits with only nearest-neighbour interactions
    • individually addressable for single-qubit gating and measurement
    1. Add a coupling bus resonator to interconnect them & add a dedicated flux-bias control line to each qubit → to perform two-qubit conditional-phase gates between nearest neighbours
    2. Add a dedicated microwave drive line to each qubit → to perform single-qubit gates
    3. Add a dedicated readout resonators to each qubit → to measure each qubit individually
    4. Readout resonators coupled to diagonally running feedlines and probed independently using frequency division multiplexing
    • 4 frequencies are sufficient to control a Surface Code of any size. This leads to an arbitrary-sized Surface Code being composed of repetitive 8-qubit unit cells.

Operations on Superconducting qubits

• Measurements
  • readout resonator
  • resonance frequency of the resonator quite far away from the qubit transition frequency
    • BUT due to the coupling, shift in the resonator’s frequency depending on the qubit state
      • evidence : measured resonator transmission dips of the qubit in the ground state and the excited state
        • observe this shift → qubit state
        • by injecting the resonator with a pulse near the resonator frequency → reflected by the resonator
        • output voltage as a function of time for the qubit in the ground state and excited state looks different
  • quantify performance of qubit measurement in the presence of noise
    • record the integrated voltage of thousands of individual traces
    • plot the individual shots in the histograms → extract the fidelity of the measurement
      • highest achievable fidelity aided by the world’s lowest noise, superconducting amplifiers = 99%
  • simultaneous readout for multiple qubits
    • each qubit coupled to its own resonator with unique lengths and thus readout frequency
• Single-qubit gates
  • singel-qubit gates → rotations of the Bloch sphere
  • Rabi oscillation
    • apply an external oscillating electric field at the qubit frequency = energy difference between the ground and excited state
      • usually in the microwave range, between 3 and 10 GHz
    • drives the qubit from ground → first excited state → back to ground
      • Bloch sphere - roation with constant speed, axis of rotation lies in the x-y plane
        • sine wave - rotation around the x-axis
        • cosine wave - rotation around the y-axis
  • any single-qubit gate can be performed using no more than 3 microwave pulses in sequence
  • shorter pulse for quicker computation
    • BUT shorter the pulse, more frequency components it has
      → broader frequency spectrum
      → also drives oscillations between states 1 and 2
      → not a qubit anymore
      - so, use Gaussian shaped envelope pulse instead of square pulse
      - Derivative Removal by Adiabatic Gate (DRAG) technique : superimposing a fine-tuned out-of-phase component with an envelope proportional to the derivative of the original pulse → push the limit further
      - pulse length reduced to below 5 ns
• Two-qubit gates
  • Conditional phase gae (Cphase-gate / CZ-gate)
    • implemented by tuning in and out of resonance with an interaction in the two excitation manifolds
  • Traversal qubit-qubit coupling
    • mediated through a coupling resonator
    • transition frequency of a transplant qubit must be controlled → able to tune in and out of resonance with the interaction
    • apply a current to the flux bias line
      → flux through the SQUID loop of the transmon
      → control qubit’s frequency
    • apply a pulse to the flux bias line
      → qubit detuned from its operating frequency for the duration of the pulse
    • To avoid energy transfer to undesired states, fast-adiabatic pulse that minimizes leakage is used
    • Causes of distortions
      • AWG response
      • Impedance mismatch
      • Cables (skin effect)
      • Filters
      • On-chip response
    • Key challenges
      • Measuring distortions (in the fridge)
      • Correcting distortions
        • caused by electrical components in the signal path between the wave generator and the qubits
      • Mitigating the effect of distortions
• The pulses for operations and measurements have different frequencies
  • Coupling the qubits to resonators of different length
    → alter the readout qubits’ frequencies separately
    → measurements on multiple qubits at the same time