Protected hybrid superconducting qubit in an array of gate-tunable Josephson interferometers

Abstract

We propose a protected qubit based on a modular array of superconducting islands connected by semiconductor Josephson interferometers. The individual interferometers realize effective cos 2φ elements that exchange 'pairs of Cooper pairs' between the superconducting islands when gate-tuned into balance and frustrated by a half flux quantum. If a large capacitor shunts the ends of the array, the circuit forms a protected qubit because its degenerate ground states are robust to offset charge and magnetic field fluctuations for a sizable window around zero offset charge and half flux quantum. This protection window broadens upon increasing the number of interferometers if the individual elements are balanced. We use an effective spin model to describe the system and show that a quantum phase transition point sets the critical flux value at which protection is destroyed.

Entangling transmons with low-frequency protected superconducting qubits

“…We initially consider an error term, H y = −ε y sin(φ p ), with an amplitude ε y that is small compared to the remaining energy scales of the setup. This sin(φ p ) error arises in a PPQ realized by a nanowire Josephson interferometer if the two interferometers junctions are not in balance [32]. Due to the error term, we find that the low-energy Hamiltonian of Eq.…”

“…For example, if the PPQ is realized in a nanowire Josephson interferometer, then sinusoidal term arises if the interferometer junctions are tuned out of balance by local gate electrodes. In contrast, the cosinusoidal term arise when the interferometer magnetic flux is biased away from half flux quantum [32,33].…”

“…It is now interesting to compare our results for sin(φ p ) errors with cos(φ p ) errors that are described by an error term H x = −ε x cos(φ p ) in the Hamiltonian. Such an error term can arise in an implementation of the PPQ with a nanowire Josephson interferometer if the external flux that threading the interferometer loop is detuned from half flux quantum [32]. In this situation, the lowenergy Hamiltonian of Eq.…”

“…While the sin(φ p ) errors lead primarily to additional twoqubit interactions that become less relevant in the limit when g xx ω t and g yy ω t , the cos(φ p ) errors lead primarily to additional single-qubit terms. Finding strategies of mitigating such flux errors, for example by concatenating multiple imperfect PPQs [32,35], is an important open challenge of the field.…”

“…Motivated by the challenge of exploring complementary qubit modalities, several alternative qubit encodings have been proposed [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] and experimentally studied [33][34][35][36][37][38][39][40][41][42][43][44][45]. A particular class of such novel qubit encodings are Cooper-pair parity-protected qubits (PPQ) [30][31][32][33][34], which rely on a special Josephson element that only permits tunneling of pairs of Cooper-pairs. Similar to the transmon qubit, the two nearly-degenerate ground states of the PPQ have a nearly flat charge dispersion, which makes them insensitive to charge-noise induced dephasing.…”