Control, readout, and realistic-decoherence certification for the protected current-mirror superconducting qubit

The current-mirror qubit — a Mobius-strip array of $2N$ Josephson junctions bridged by large capacitors, proposed by Kitaev — derives intrinsic protection from logical states with nearly disjoint support in configuration space: all matrix elements of local noise operators between the two lowest eigenstates are exponentially suppressed with array size, and a 2019 full circuit analysis (Weiss, Li, Ferguson, Koch) predicted coherence times exceeding $1\,\mathrm{ms}$ . That analysis, however, left the device non-operational and only partially certified: no practical scheme for single-qubit gates or readout exists for this circuit, quasiparticle tunneling has not been analyzed, the effect of disorder among the $2N$ nominally identical junctions and capacitors is unknown, and coherence was computed only at the charge sweet spot, leaving robustness to offset-charge jumps larger than $0.1e$ uncharacterized. Later literature validated the strategy of gating protected qubits through higher-lying states and improved full-circuit simulation methods, but only on the successor $0$ - $\pi$ qubit and other protected circuits — the current-mirror circuit itself remains without control, readout, or realistic-decoherence analysis. The problem is to make the current-mirror qubit's built-in protection compatible with operability, and to certify whether its predicted millisecond coherence survives quasiparticles, fabrication disorder, and offset-charge fluctuations under operating conditions.

Background

Superconducting qubits store quantum information in the collective states of electrical circuits of Josephson junctions (tunnel elements joining two superconductors, each characterized by a Josephson energy $E_J$ and a junction capacitance) and capacitors; the transmon, the current workhorse, uses a single degree of freedom and reaches relaxation and dephasing times of order $100\,\mu\mathrm{s}$ . Fault-tolerance roadmaps motivate qubits with intrinsic protection, in which the logical wavefunctions themselves decouple from local noise. One of the earliest such proposals is Kitaev's current-mirror circuit (2006): two chains of Josephson junctions bridged at $N$ rungs by large capacitors (capacitance $C_B$ ), twisted and closed into a Mobius strip, with each node also shunted by a small capacitance $C_g$ to ground. In the regime where the rung capacitance dominates ( $E_{C_B}$ smallest charging energy, $E_J < E_{C_J}$ ), the low-energy charge excitations are Cooper-pair excitons — a Cooper pair and a Cooper-pair hole on the two sides of one rung, which hop between rungs while moving together; higher-energy single-charge excitations are called agitons. For zero offset charge (the residual gate charge $n_g$ on each node) and large $N$ , the two lowest eigenstates $|0\rangle$ and $|\pi\rangle$ localize in opposite wells of an $N$ -dimensional double-well potential and have nearly disjoint support, so matrix elements $\langle 0|M|\pi\rangle$ of any local operator $M$ are exponentially small in $N$ : relaxation is suppressed, and the near-degeneracy (split only by an $N$ -th-order cotunneling process with coefficient $K$ ) suppresses dephasing. Kitaev's original note also sketched a conceptually universal gate set in an idealized limit. The successor $0$ - $\pi$ qubit (Brooks, Preskill, Kitaev 2013) implements the same disjoint-support idea in a compact four-node circuit and has since been realized experimentally with gates that temporarily route population through higher-lying states. Weiss, Li, Ferguson, and Koch (Phys. Rev. B 100, 224507 (2019)) performed the definitive circuit-level analysis of the current-mirror device itself: an effective low-energy Hamiltonian (treating Josephson tunneling perturbatively, halving the degrees of freedom) diagonalized by DMRG — the density-matrix renormalization group, a variational method for one-dimensional many-body systems — up to $N=12$ , yielding coherence-time predictions above $1\,\mathrm{ms}$ with an optimal size near $N=11$ (below that, dephasing from $1/f$ charge noise dominates; above it, the $1/N$ -scaled excitation energies invite thermal depolarization). Three limitations frame the open problem. First, at points of maximal charge frustration ( $n_g^+ = 1/2$ on a rung) the effective model breaks down because the nature of the low-energy excitations changes from excitons to Cooper-pair and void excitons, and full-model simulation is hindered by long-range interactions mediated by agiton charging energies. Second, the analysis assumed all $2N$ junctions, rung capacitors, and ground capacitances identical. Third, coherence was computed at the charge sweet spot ( $n_g = 0$ ), whereas real devices exhibit discrete offset-charge jumps larger than $0.1e$ . Operating at half a flux quantum ( $\phi_{\mathrm{ext}}=\pi$ ) symmetrizes the double well but destroys the disjoint support, so it does not rescue operability.

Problem Statement

Make Kitaev's current-mirror qubit — as concretely analyzed by Weiss, Li, Ferguson, and Koch (Phys. Rev. B 100, 224507 (2019)) — an operable protected qubit, by solving the interlinked open issues that the source itself poses. The common objective is to establish whether the circuit's predicted millisecond-scale intrinsic protection survives the conditions of actual operation; the requested results jointly serve this objective, because the residual noise channels determine the error budget and design constraints under which any control and readout scheme must operate. Specifically: (i) develop concrete modes of control and readout for the realistic circuit — single-qubit gates must overcome the exponentially suppressed matrix elements between the disjoint-support logical states (e.g., via higher-lying eigenstates or temporary, controlled lifting of protection) while quantifying the incurred relaxation/dephasing cost, and a measurement of the logical states must be specified; (ii) analyze quasiparticle tunneling as a relaxation and dephasing mechanism for this circuit, which requires going beyond the low-energy effective model at points of maximal charge frustration (where excitons give way to Cooper-pair and void excitons) and hence a full-model treatment of the multi-node circuit; (iii) determine the effect of disorder in the circuit elements — junctions, rung capacitors, and ground capacitances — which breaks the $2N$ -fold symmetry underlying the protective near-degeneracy; and (iv) characterize coherence away from the charge sweet spot, including robustness to offset-charge jumps larger than $0.1e$ . A complete answer must address all four jointly: resolving any single issue, or any proper subset, does not resolve the task, because the top-level objective requires both an operable control/readout scheme and a certification of the predicted coherence under each residual noise channel. Depth is part of the standard: (ii)-(iv) must each be reported as a quantitative contribution to $T_1$ and $T_\phi$ — or a bound establishing that channel is negligible against the predicted millisecond scale — evaluated in the regime each part invokes (the full model at charge frustration; the $2N$ element variations; static and jumping offset charges beyond $0.1e$ ) and at circuit sizes that are stated and justified against the small- versus large- $N$ trade-off the source identifies, while the scheme of (i) is correspondingly held to its stated relaxation/dephasing cost. The answer must reach a definite verdict on whether the millisecond-scale protection is preserved, degraded, or destroyed. Answers may be theoretical designs certified by simulation of the published circuit model, or experimental demonstrations on a fabricated current-mirror device.

References

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