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Qubit—qubit coupling (tunable coupler)

This experiment measures the strength of qubit–qubit coupling in an iSWAP-like gate as a function of coupler flux. This is done by fitting oscillations in the qubits' population exchange.

Fixed coupler

For determining the coupling strength of a fixed coupler, see Qubit–qubit coupling (fixed coupler).

Description

In a standard circuit QED (quantum electrodynamics) setup, qubit–qubit coupling can result from either longitudinal (along the qubit \(z\)-axis) or transverse (along the \(x\)- or \(y\)-axes) interactions. For transversely coupled qubits, the effective coupling Hamiltonian can be reduced to the excitation–exchange form under the following conditions1:

  1. The qubits are on resonance with each other.
  2. The rotating wave approximation is applied at the qubit frequency.
  3. If the interaction is mediated by a resonator rather than direct capacitive coupling, the resonator must be far detuned from the qubits (dispersive regime).

The qubit–qubit interaction Hamiltonian becomes

\[ H_{qq} = g (\sigma^{+}_{A}\sigma^{-}_{B} + \sigma^{-}_{A}\sigma^{+}_{B}), \]

where \(g\) is the effective (coupler flux-dependent) qubit–qubit coupling strength, and \(\sigma_i^{+}\) and \(\sigma_i^{-}\) are the raising and lowering operators, respectively, for the \(i^{\text{th}}\) qubit (\(i=A,B\)).

In this experiment, the two qubits are capacitively coupled to a third, tunable qubit, known as the coupler. By adjusting the coupler’s frequency, the virtual exchange interaction between the two qubits can be tuned. At a specific, critical flux bias, the mediated coupling through the coupler cancels the direct qubit–qubit (next-nearest-neighbour) coupling, resulting in a coupling-free operating point. This essentially decouples the two qubits.

Qubit A is prepared in its ground state and qubit B in its excited state, with qubit A (the higher frequency qubit) flux-biased to bring it into resonance with qubit B. Simultaneously, a flux is applied to qubit C (the coupler), inducing Rabi oscillations between the \(|0_A1_B\rangle\) and \(|1_A0_B\rangle\) states. For each amplitude of the coupler flux pulse, the populations of the two states oscillate in time at a (angular) frequency given by \(2g\), with \(g\) the effective coupling between qubits A and B. By scanning the coupler flux, we can therefore determine how \(g\) varies with the coupler-flux amplitude2.

Experimental steps

  1. Qubits A and B are initialised in their ground states, \(|0\rangle\).

  2. A \(\pi\)-pulse (\(R_x(\pi)\)) is applied to qubit B to excite it to the \(|1\rangle\) state.

  3. A flux pulse is applied to qubit A to bring it into resonance with qubit B. Simultaneously, a flux pulse is applied to qubit C (the coupler). The amplitude of the coupler pulse and the duration of both pulses are swept over a range of values, while the flux amplitude on qubit A is kept constant to maintain resonance between qubits A and B.

  4. For each combination of applied flux amplitude and duration, qubits A and B are measured in their respective \(|0\rangle\) - \(|1\rangle\) bases.

Analysis steps

  1. The qubit populations for the \(|0_A1_B\rangle\) and \(|1_A0_B\rangle\) states are computed as a function of the amplitude and duration of the flux pulse applied to qubit C (see top figure below for the evolution of the \(|0_A1_B\rangle\) state). Here, the qubit states are identified from the \(I\)–\(Q\) data by applying the composite discriminator obtained in the Correlated readout error experiment.

  2. For each coupler flux amplitude, the Fourier transform of the population oscillations is computed, giving the population exchange frequency, \(2g\). This gives \(g\), the coupler-flux-dependent qubit–qubit coupling rate (see bottom plot below; note that \(g\) is an angular frequency). The flux scan reveals that the critical flux biases are at -0.036V and +0.028V, where the effective qubit–qubit coupling is zero.

image

  1. Alexandre Blais, Arne L. Grimsmo, S. M. Girvin, and Andreas Wallraff. Circuit quantum electrodynamics. Rev. Mod. Phys., 93:025005, May 2021. URL: https://link.aps.org/doi/10.1103/RevModPhys.93.025005, doi:10.1103/RevModPhys.93.025005. ↩

  2. Fei Yan, Philip Krantz, Youngkyu Sung, Morten Kjaergaard, Daniel L. Campbell, Terry P. Orlando, Simon Gustavsson, and William D. Oliver. Tunable coupling scheme for implementing high-fidelity two-qubit gates. Phys. Rev. Appl., 10:054062, Nov 2018. doi:10.1103/PhysRevApplied.10.054062. ↩