A Unified Error Correction Code for Universal Quantum Computing with Identical Particles

TL;DR

It is proposed that the simplest quantum error correction code can be realized directly within the physical qubit, provided that conventional correction and restoration are generalized beyond unitary operations to employ physically implementable reversal operations -- naturally placing logical and physical qubits on equal footing.

Abstract

We present a universal fault-tolerant quantum computing architecture based on identical particle qubits (IPQs), where we find that the first-order IPQ - bath interaction fundamentally differs from the conventional first-order qubit-bath interaction. This key distinction necessitates a redesign of existing strategies to fight decoherence. We propose that the simplest quantum error correction code can be realized directly within the physical qubit, provided that conventional correction and restoration are generalized beyond unitary operations to employ physically implementable reversal operations -- naturally placing logical and physical qubits on equal footing. We further demonstrate that dynamical decoupling (DD) remains effective within this unified framework, and that a decoherence-free subspace (DFS) -- like structure emerges. Unlike previous approximate treatments, our analytically solvable IPQ-Bath model enables rigorous testing of these strategies, with numerical simulations validating their effectiveness.

Figures (3)

• Figure 1: (a) The evolution of the average excitation number for a single identical particle. (b-d) The infidelity of the effective density matrices during (b) information storage, (c) Z-gate operation, and (d) X-gate operation. The dashed purple lines represent the system without applying LEO pulses, while the blue solid lines show the results with LEO pulses applied. The temperature parameters are set as $\beta = \omega_{0}^{-1}$ and $\Omega = \omega_{0}$. The dotted green lines and the yellow dashed-dotted lines correspond to the decoherence-free subspace (DFS) state for ${a}_{0}(t)$ with $\beta = \omega_{0}^{-1}$ and $\beta = 10\omega_{0}^{-1}$, respectively. System parameters are chosen as $\Gamma = 5G_{k}$, $\gamma = 0.5G_{k}$, and $\omega_{0} = 100G_{k}$. The LEO pulse parameters are configured with pulse strength $c_{0} = 50G_{k}$, pulse width $\Delta\tau = 0.02\pi G_{k}^{-1}$, and pulse spacing $\delta\tau = 0.005\pi G_{k}^{-1}$. The unit $G_{k} = 1$, where $k = 0, x, z$, is used for other parameters, with $G_{0} = 1$ for $H_{\text{S}}^{0} = G_{0} {I}_{2}$.
• Figure 2: (a) The average excitation number as a function of dimensionless time $\Gamma t$. (b) The steady state average excitation number as a function of $1/\beta$. The system parameters are set to $\gamma = 2.5\Gamma$, and $\omega_{0} = 100\Gamma$. No gate operation is used. All other parameters use $\Gamma= 1$.
• Figure 3: The infidelity as a function of dimensionless time $G_{k}t$. (a) Infidelity for different gate operations with $\beta_{1} = \beta_{2} = 0.1\omega_{0}^{-1}$. (b) Infidelity for varying reservoir temperature differences without applying LEO pulses. The system parameters are set to $\Gamma = 5G_{k}$, $\gamma = 0.5G_{k}$, and $\omega_{0} = 100G_{k}$. The LEO pulse parameters include pulse strength $c_{0} = 80G_{k}$, pulse width $\Delta\tau = \pi / 50G_{k}$, and pulse spacing $\delta\tau = \pi / 200G_{k}$. All other parameters use $G_{k} = 1$, with $k = 0, x, z$.