Control and coherence time enhancement of the 0–π qubit

Abstract

Kitaev's 0-π qubit encodes quantum information in two protected, near-degenerate states of a superconducting quantum circuit. In a recent work, we have shown that the coherence times of a realistic 0-π device can surpass that of today's best superconducting qubits (Groszkowski et al 2018 New J. Phys. 20 043053). Here we address controllability of the 0-π qubit. Specifically, we investigate the potential for dispersive control and readout, and introduce a new, fast and high-fidelity singlequbit gate that can interpolate smoothly between logical X and Z. We characterize the action of this gate using a multi-level treatment of the device, and analyze the impact of circuit-element disorder and deviations in control and circuit parameters from their optimal values. Furthermore, we propose a cooling scheme to decrease the photon shot-noise dephasing rate, which we previously found to limit the coherence times of 0-π devices within reach of current experiments. Using this approach, we predict coherence time enhancements between one and three orders of magnitude, depending on parameter regime. ζ-mode) [22]. Nevertheless, we found that the 0-π qubit still has the potential to outperform state-of-the-art superconducting devices. Below, we propose a method to further enhance the coherence time by orders of magnitude by cooling the ζ-mode.However, as can be expected, the price of intrinsic noise protection in the 0-π qubit is that it is difficult to perform logical operations on this device. In particular, protection from noise comes in part from the exponentially small overlap of its logical wave functions. As a result, matrix elements of local operators between the two logical states will also be small, thus resulting in extremely slow gates. In [20], Brooks et al proposed a universal set of protected logical operations based on coupling to an ultra-high impedance LC oscillator. However, these operations were based on an idealized model of the qubit and with parameters that are difficult to realize in practice. Further work is required to determine the potential of this approach in a more realistic setting.Motivated by the prospect of realizing 0-π qubits in the near term, we investigate alternative approaches to measurement and control with lower experimental complexity. The operations we propose are not protected in the same sense as those proposed in [20], because they rely either on operating the device in a regime where the qubit is not fully isolated from the environment, or they make use of excited states outside the qubit manifold. In particular, we develop a single-qubit gate based on a multi-level excursion through higher energy levels. Nevertheless, we hope that these schemes will be useful for both characterization and control of 0-π qubits in near-to-medium-term experiments.This work is organized as follows. In section 2, we introduce the 0-π qubit and provide a simplified effective model for the 0-π circuit with only a single degree of freedom. In section 3, we discuss general coupling strategies for qubit co...

Universal gates for protected superconducting qubits using optimal control

“…The 0-π circuit [27,[32][33][34] further extends the protection afforded by the heavy-fluxonium qubit by combining exponential suppression of both relaxation and dephasing due to disjoint wave-function support and robust ground-state degeneracy. We briefly review the physics of the 0-π circuit and the parameters required for its protected regime.…”

“…where n g is the offset charge, and q θ = 2en θ , q φ = 2en φ are the charge operators canonically conjugate to the two degrees of freedom θ and φ. The effective capacitances associated with these two variables are C θ = 2(C + C J ) + C g and C φ = 2C J + C g , where C g is a small capacitance due to coupling to ground and external voltage lines [34]. The external magnetic flux threading the circuit loop is denoted Φ ext .…”

“…Similar to the situation with the heavy-fluxonium qubit, disjoint support of the computational basis states in the 0-π qubit provides intrinsic protection from decoherence, but inevitably also prevents one from driving direct transitions between the two qubit states. Di Paolo et al [34] achieved gate operations between states indirectly via a square voltage pulse that drives transitions via intermediate higher excited levels. Depending on device parameters, this strategy results either in an X gate or a Hadamard gate, but does not readily yield a gate set universal for single-qubit operations.…”

“…With careful tuning of cost-function weights, this process converges to an average solution balancing all possible values of n g . A second challenge concerns the inevitable presence of disorder in circuit components which can lead to spurious coupling to a harmonic, low-energy degree of freedom, the ζ-mode [32][33][34]. To avoid the overhead of a significant increase in Hilbert space dimension, we apply the optimal-control formalism to the ideal 0-π system, and verify subsequently that weak coupling to the ζ-mode does not significantly reduce gate fidelities.…”

“…We next investigate the performance of the optimized pulses in the presence of circuit disorder -in particular, in C and L. Such disorder leads to spurious coupling to the harmonic ζ-mode with mode frequency Ω ζ = 8E L E C ζ / , where E C ζ = e 2 /2C ζ and C ζ = 2C + C g ≈ 2C [33,34]. The resulting Hamiltonian reads…”