Simulations of Topological Phases in Superconducting Circuits

Project Information

TOPOCIRCUS

Grant agreement ID: 841894

DOI

10.3030/841894

Project closed

EC signature date 29 March 2019

Start date 16 September 2019

End date 15 September 2022

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 251 002,56

EU contribution

€ 251 002,56

Coordinated by

CONSIGLIO NAZIONALE DELLE RICERCHE
Italy

Periodic Reporting for period 2 - TOPOCIRCUS (Simulations of Topological Phases in Superconducting Circuits)

Reporting period: 2021-09-16 to 2022-09-15

Summary of the context and overall objectives of the project

The goals of the TOPOCIRCUS project is to view a superconducting circuit as a playground where novel topological phases of matter can be engineered and simulated. A topological phase is characterized by a Hamiltonian whose typically low energy eigenstates has a non-local character, so that they are insensitive to local perturbations.

Objective i) Simulation of topological phases in two-terminal Josephson circuits
Objective ii) Search for new fermions in multi-terminal Josephson junctions

One of the main ideas of the present Marie Curie action proposal is to simulate topological states of matter with superconducting circuits. In the proposed Objectives i) and ii) of the present TOPOCIRCUS project specific mention is made of the possibility to engineer the spectrum of a superconducting circuits in such a way that it contains bulk topologically non-trivial features.

The achievement of such a goal had to go through singling out what are the possible players that can be exploited for engineering the Josephson potential. Any superconducting circuit is generically described by a charging energy, that is a quadratic form in the charges on the islands, and the Josephson potential, that is periodic on the phase. All properties of superconducting circuits depends on the details of the Josephson potential.

The main change of speed in the project occurred when realizing that some complicate circuits can effectively show a π-periodicity with respect to the typical 2π-periodicity in the phase difference, that opened up completely novel possibilities.
We managed to accomplish the goals of Objective i) and ii) in an excellent way by showing how groups of flat bands can be engineered by tailoring the Josephson potential and exploiting the achievement to enhance coherence.

The TOPOCIRCUS project ended mid September 2022. The main result of the project is the identification of π-periodic Josephson junction as a novel element in superconducting circuits that allows augmented freedom of manipulation of the Josephson potentials and enhanced coherence of superconducting qubits. Coupling to Majorana qubit shows also full versatility of parity-protected superconducting qubits based on π-periodic Josephson junction. In addition the spectrum of a π-periodic qubit realizes a SSH topological model, that is one of the main goals of the present TOPOCIRCUS projects.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Enhanced coherence in superconducting circuits via band engineering (Phys Rev Lett 126, 187701 (2021)): we show that higher-harmonic Josephson elements described by a cos(2φ) energy-phase relation provide an increased freedom to tailor the shape of the Josephson potential and design flat bands providing noise-insensitive quantum states.

Impact of electrostatic fields in layered crystalline BCS superconductors (Phys Rev Research 3, 023135 (2021)): we have shown that when the lateral sizes of a superconducting slab is reduced and the average level spacing becomes comparable to the gap, the latter becomes extremely sensitive to the applied voltage.

Colossal orbital-Edelstein effect in non-centrosymmetric superconductors (Phys Rev Lett 128, 217703 (2022)): we generalize the concept of Edelstein response to systems characterized by an orbital-Rashba coupling. A much richer dependence on the effective masses and crystal field splitting appears, that is proper of the orbital-Rashba coupling and does not appear in the spin-Rashba case.

Anyons in Quantum Hall Interferometry (Nature Reviews Physics 3, 698 (2021)): review illustrating the advances in interferometry in the quantum Hall effect with emphasis on anyonic states. Two main interferometric setups are discussed, the Fabry-Perot interferometer and the Mach-Zehnder interferometer.

Frustration driven Josephson phase dynamics (Phys. Rev. B 105, 134503 (2022)): in this work we study a driven frustrated Josephson system and we demonstrate that a dynamical regime with an oscillating phase evolution is a general hallmark of driven Josephson systems exhibiting sign competition in the Josephson couplings.

SWAP gate between a Majorana qubit and a parity-protected superconducting qubit (Phys. Rev. Lett. 129, 177701 (2022)): in this work we couple a parity-protected superconducting qubit based on a π-periodic Josephson junction to a Majorana qubit and show that the two systems can interact in a way that allows the implementation of a SWAP gate and the use of the system as a processor.

Anomalous periodicity and parafermion hybridization in superconducting qubits (arXiv:2208.07408): in this work we study a parafermionic fluxonium circuit and show that the presence of topological states yields a striking periodicity in the qubit spectrum. In addition, peculiar and marked signatures of different parafermion coupling can be detected in the qubit microwave spectrum.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

In developing the project a main question has been asked by the PI. How can we exploit concepts of non-locality developed in the context of topological material for the benefit superconducting circuits? One of possible answers is that the coherence properties of a quantum device constructed via superconducting circuits can enormously benefit for a topological encoding of the wave function. In an original proposal that goes back to 2006 Kitaev proposed to use a cos(2φ) as a 0-π qubit that is topologically protected. The key player is a cos(2φ) Josephson junction, that captures the tunneling of pairs of Cooper pairs. The way the 0-π qubit has been studied in the literature is with the goal to encode the qubit degree of freedom in parity protected doublets. At the same time, the strong protection of these doublets prevents an efficient quantum computation in terms of gates.

We have thoroughly investigated the cos(2φ) qubit and the possibility to use its parity-protected degeneracy to store quantum information. This has been achieve by combining the qubit with a Majorana qubit. This achievement is perfectly in line with the general goals of the present TOPOCIRCUS project. The main idea is to exploit both sides of a parity-protected qubi based on pi-periodic Josephson junctionst: i) the ordinary level staircase associated to the principal quantum number, that couples via a dipole matrix element to other qubits and transmission line resonators, to perform quantum computation through gates, and ii) the parity-protected subspace to store the information. In order to achieve that a SWAP gate is necessary. By hybridizing a Majorana qubit through a 4π-Josephson junction we are able to coherently couple the otherwise orthogonal sectors of the parity-protected superconducting qubit and generate an effective term in the Hamiltonian. In turn, by allowing a sensitivity to charge in the regime we are able to generate a coupling term. We the aid of single qubit operations on the ordinary level staircase it is possible to perform a SWAP gate. The work is about to be finalized and submitted for publication.

The impact of the present project is fully and entirely scientific. The results obtained can be exploited by experimentalists to build devices with higher degree of coherence and simulate topological states of matter, and as a basis for other theorists to further develop novel results. Besides, the present project has an indirect socio-economic impact in the future of quantum computing technologies, a field that is actually booming nowadays. A wider societal implication is not clearly identifiable.

Poster of the main publications