Highly robust logical qubit encoding in an ensemble of V-symmetrical qutrits

TL;DR

This work introduces a robust logical qubit encoded in an ensemble of V-configured qutrits, using even and odd Schrödinger cat states of SU(3) that are parameter-independent dark states of an engineered master equation. Through a detailed parity analysis and construction of parameter-insensitive stationary states, the authors show immunity to single-qutrit decay, two-qutrit correlated decay/driving, and certain dephasing channels, with parity-sensitive robustness and mixed-symmetry extensions to counteract symmetry-breaking noise. A concrete physical origin is provided via a tripartite cavity-QED system, and a multi-step adiabatic elimination plus averaging procedure yields the effective master equation underpinning the protected qubit and accessible quantum gates (NOT, Hadamard with phase, and Z-phase variants). The results extend to larger qutrit ensembles, propose practical preparation and gate schemes, and outline experimental routes using coplanar-waveguide resonators and alkali-atom ensembles, highlighting a promising route to decoherence-robust quantum information processing beyond standard qubit encodings.

Abstract

We propose using even and odd Schödinger cat states formed from coherent states of U(3) of an ensemble of qutrits with a symmetrical V-configuration (a qubit-disguised qutrit) to encode a logical qubit. These carefully engineered logical qubit states are parameter independent stationary states of the effective master equation governing the evolution of the ensemble and, consequently, constitute dark states and are invulnerable to dissipation and correlated collective dephasing. In particular, the logical qubit states are immune to single qutrit decay (the analogous of single photon loss process for qutrits) and simultaneous decay and driving of two qutrits (the analogous two-photon loss and driving processes for qutrits). In addition, we show how to implement the single-qubit quantum NOT gate and the Hadamard gate followed by either the phase gate or the phase and $Z$ gates. We study analytically the case of two qutrits and conclude that the logical qubit states exhibit parity-sensitive inhomogeneous broadening and local correlated dephasing: the even logical state is completely immune to these processes, while odd one is vulnerable. Nevertheless, in the presence of these interactions one can also define another odd state with mixed permutation symmetry that is immune to both inhomogeneous broadening and local correlated dephasing. We suggest that these results can be extrapolated to an arbitrary number of qutrits. The effective master equation is deduced from a physical system composed of two parametrically coupled cavities with one of them interacting dispersively with an ensemble of three-level atoms (the qutrits). In principle this physical system can be implemented by means of two coplanar waveguide resonators, a SQUID parametrically coupling them, and a cloud of alkali atoms close to one of the resonators.

Figures (3)

• Figure 1: Each qutrit is a quantum $3$-level system with a V-configuration where $\vert 1\rangle$ is the ground level and $\vert 2 \rangle$ and $\vert 3 \rangle$ are the excited levels. The angular frequency associated with the transition $\vert j \rangle \leftrightarrow \vert 1 \rangle$ is $\omega_{j} >0$ with $j=2,3$. In the context of the physical system of Sec. \ref{['Origen']}, the qutrits are coupled to a single-mode cavity quantum electromagnetic field called the signal field. The coupling strengths are $g_{2}$ and $g_{3}$.
• Figure 2: The figures show the fidelity $F_{\mathbb{j}}(t)$ of $\rho(t)$, evolving under $\mathcal{L}+\mathcal{L}_{\mathrm{ud}}$, with $\vert 0_{L} \rangle\langle 0_{L} \vert$ (red line) and $\vert 1_{L} \rangle\langle 1_{L} \vert$ (blue-dashed line) as a function of time $t$ when $\rho(0) = \vert 0_{L} \rangle\langle 0_{L} \vert$, Fig. \ref{['Figure3a']}, and when $\rho(0) = \vert 1_{L} \rangle\langle 1_{L} \vert$, Fig. \ref{['Figure3b']}. In both figures the steady state values of the fidelity are $F_{0}^{\infty} = 0.016$ and $F_{1}^{\infty} = 0.52$. The values of the parameters in both figures are $\delta_{1} = 0$, $\xi = 0.81$, $\alpha_{0} = 6.29 + i0.37$, $\delta = -3.15$, $\kappa_{1} = 7.99$, $\kappa_{2} = 0.65$, $\Gamma_{1} = 0.024$, $\Gamma_{2} = 0.032$, $\Gamma_{3} = 0.038$.
• Figure 3: The figure presents a schematic of the physical system under consideration. It is composed of a collection of $N\geq 2$ qutrits (represented by the black dots on the righthand side) and two single-mode cavity electromagnetic fields called the pump field and the signal field. The pump field has frequency $\omega_{p}$, is subject to dissipation at a rate $\kappa_{p}$, and is also driven by a classical field with frequency $\omega_{d}$ and driving strength $\Omega_{d}$. The signal field has frequency $\omega_{s}$, is subject to dissipation at a rate $\kappa_{s}$, and interacts with the pump field with coupling strength $J$. The qutrits have a V-configuration with transition frequencies $\omega_{j}$ and interact with the signal field with coupling strengths $g_{j}$$(j=2,3)$, see also Fig. \ref{['Figure1']}.