# QCJC: Barends 2020

There is always a tradeoff between gate speed and control of qubits. At the most fundamental limit, there is Heisenberg uncertainty, $\Delta E \Delta t \geq \frac{\hbar}{2}$. However, before we even reach that limit, we tend to be bound by other types of leakages. When we try to move a qubit state too rapidly, as it passes a resonance of a higher, unwanted level, there is an increased likelihood that there would be leakage into those levels. Since that higher level is not controlled for, this would likely show up as loss in fidelity of the qubit, and even worse, having some part of the quantum state in those higher level can lead to cascading errors down the line. This is the motivation for many common features in superconducting qubits, such as the implementation of resonator cavities and slow(er), adiabatic[1] gates of the past.

Here, Google’s team uses the advantages of having a frequency-tunable transmon qubit to execute faster, dibatic gates through some clever manipulation of the resonant energies of the qubit. They argue that their capacitatively coupled two-qubit system typically exhibits 6 states: four computational states (|00>, |01>, |10>, |11>) as well as two noncomputational (ie, garbage) states: |02> and |20>. They don’t seem to offer a full energy diagram or two-tone spectroscopy here, so it’s hard to tell if there are even higher states that they are just ignoring, or why there are not states like |12>, |21>, |22> that also might come into effect. However, my assumption will be that those remaining states are very different in energy, and therefore are much less likely to come in resonance with the transition of the swap gates.

The gate that this team tries to implement is the SWAP gate, where the population in |01> will transfer with the population in |10>. However, during this gate, any population that is in the |11> state would have the chance to interact with the state $|\psi_b\rangle = \frac{ |20\rangle + |02\rangle}{\sqrt{2}}$. The complimentary Bell-like state, $|\psi_d\rangle = \frac{ |02\rangle - |20\rangle }{\sqrt{2}}$ is decoupled and does not interact. In equation 2, the authors express the probability that there would be a transfer of population from the |11> state to the $|\psi_b\rangle$ state, finding that there exists a null point at which there would be no transfer. That null point then implies a certain relationship between the interqubit coupling and the nonlinearity of the qubits, as shown in equation 3. Both of those parameters are able to be controlled – the coupling strength by the size of the capacitor, and the nonlinearity/anharmonicity of the qubit through the ratio of $\frac{E_J}{E_C}$[2] of the qubit.

However, there remains the question if such a relationship will hold for non-ideal SWAP gates, where instead of a perfect square pulse, there is some ramp-up and ramp-down slopes. In addition, I think there is also a concern here that about frequency dependent couplings that I do not fully follow. In particular, the authors identify in equation 3 an integer $n$ that needs to be satisfied, and in Fig 1(c), plot curves in frequency/hold space of that integer. I’m not sure what it actually represents, and what the choice of n=4 means physically.

There is an interesting discussion here about how this SWAP gate is actually implemented. First, they drive flux pulses to the two qubits so that they meet at the interaction frequency, allowing the two states to interact. However, in order to drive that swap transition, it would be similar to rotating the state vector of the system by angle $\pi$ about the initial Bloch vector, m(0). However, this initial Bloch vector is not exactly aligned with the x-axis of the experiment, due to the large z component of the magnetic field that controls frequency detuning. Therefore, a single pulse is insufficient to fully execute the SWAP gate, and would require some “overshoot” of the control pulse, on the order of a few MHz. Several of the remaining plots – especially those in Fig. 3, focus on using the parameter of this overshoot $\Delta$ to tune this gate. This overshoot parameter, as well as a hold time parameter, can be identified from 1D microwave scans.

There was one line that seemed a bit confusing – the authors note that, in Fig 2(c), “the minima of the leakage channel dip down to different values – a consequence of the qubits having dissimilar nonlinearities”. It’s unclear to me what the effect of the different minima is, but the “dissimilar nonlinearities” are very close already – less than 17 MHz difference. In fact, in the earlier paragraph, the authors even note these “nearly constant nonlinearities $\eta / 2\pi$ of 223 and 240 MHz. Therefore, if they really need constant dips, then is that able to be engineered?

Much of the remaining paper focuses on the benchmarking of their qubits and gates, especially using their own cross-entropy benchmarking scheme. I think I will pass on discussing that for now. Overall, it seems like this is an interesting gate scheme, even if it requires the frequency dependent qubit in order to properly operate.

[1] that is, gates that move sufficiently slowly that there is no energy exchange with the environment. I typically think of the picture presented in Griffiths QM – if you transport a moving pendulum sufficiently slowly, it will keep the same oscillation, but if you jerk it around, it will be affected. Similarly, if you have a particle in the ground state of the infinite square well and very slowly expand the walls, the particle will remain in the ground state.
[2] See D. I. Schuster’s thesis, Circuit Quantum Electrodynamics, 4.3.2