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To measure the state of a transmon qubit, we make use of the capacitive coupling between the qubit and a resonator captured by the Hamiltonian H=g[σ+a+σ−a†]{“version”:”1.1″,”math”:”\(H = g [\sigma_+ a + \sigma_- a^\dagger]\)”}. Given that the qubit frequency ωq=(2π)5{“version”:”1.1″,”math”:”\(\omega_q = (2\pi)\, 5 \)”} GHz, the resonator frequency ωr=(2π)4{“version”:”1.1″,”math”:”\(\omega_r = (2\pi)\,4 \)”} GHz, and the capacitive coupling g=(2π)10{“version”:”1.1″,”math”:”\(g = (2\pi)\, 10\)”} MHz: Considering the qubit to be in the state |0⟩{“version”:”1.1″,”math”:”\(\vert 0 \rangle\)”}, what is the magnitude of the shift in the resonator’s frequency (assuming ℏ=1{“version”:”1.1″,”math”:”\(\hbar = 1\)”})?

Suppose we are tasked to design a quantum harmonic oscillator having well-defined flux states, which of the following energy scale relation regimes will you choose?

What is the average flux carried by the first excited state of the quantum LC oscillator with resonant frequency ω0=(2π)×2GHz{“version”:”1.1″,”math”:”\(\omega_0 = (2\pi) \times 2 \, \text{GHz}\)”} ? Note that Φ0{“version”:”1.1″,”math”:”\(\Phi_0\) “} denotes the flux quantum.

Consider a quantum LC oscillator with capacitor energy EC/h=100{“version”:”1.1″,”math”:”\(E_C/h = 100\)”} MHz and inductor energy EL/h=5{“version”:”1.1″,”math”:” \(E_L/h = 5\)”} GHz in its ground state. What is the fluctuation in flux Φzpf{“version”:”1.1″,”math”:”\(\Phi_\text{zpf}\)”} (expressed in terms of the magnetic flux quantum Φ0{“version”:”1.1″,”math”:”\(\Phi_0\)”})?

We have seen that to design a two-qubit gate, two transmon qubits are capacitively coupled giving rise to a transverse coupling between the two transmon qubits. When on-resonance, the Hamiltonian of this interaction in the rotating wave approximation is given by H=g[σ−σ++σ+σ−]{“version”:”1.1″,”math”:”\(H = g [\sigma_- \sigma_+ + \sigma_+ \sigma_-]\)”}. What is the final state of the two qubits [up to an overall phase] starting from the initial state is |10⟩{“version”:”1.1″,”math”:”\(\vert 1 0 \rangle\)”} when the time duration for which the transmons are in-resonance is t=π/(4g){“version”:”1.1″,”math”:”\(t=\pi/(4g)\)”}?

An important characteristic of a quantum LC oscillator is that its energy eigenstates are equally spaced.  In contrast, a Transmon (i.e., quantum anharmonic oscillator with capacitive energy EC{“version”:”1.1″,”math”:”\(E_C\)”} and inductive energy EJ{“version”:”1.1″,”math”:”\(E_J\)”}) due to Josephson junction introduces shifts in the energy levels of the oscillator.  To first order in perturbation theory, what is the energy spacing between the ground (i.e., |0⟩{“version”:”1.1″,”math”:”\(\vert 0 \rangle\)”}) and the first (i.e., |1⟩{“version”:”1.1″,”math”:”\(\vert 1 \rangle\)”}) excited state?

An endorsement in which the payee specifies the person to whom, or to whose order, it is to be paid.  

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