Efficient Quantum Error Correction of Dephasing Induced by a Common Fluctuator

Quantum error correction is expected to be essential in large-scale quantum technologies. However, the substantial overhead of qubits it requires is thought to greatly limit its utility in smaller, near-term devices. Here we introduce a new family of special-purpose quantum error-correcting codes that offer an exponential reduction in overhead compared to the usual repetition code. They are tailored for a common and important source of decoherence in current experiments, whereby a register of qubits is subject to phase noise through coupling to a common fluctuator, such as a resonator or a spin defect. The smallest instance encodes one logical qubit into two physical qubits, and corrects decoherence to leading-order using a constant number of one- and two-qubit operations. More generally, while the repetition code on $n$ qubits corrects errors to order $t^{O(n)}$, with $t$ the time between recoveries, our codes correct to order $t^{O(2^n)}$. Moreover, they are robust to model imperfections in small- and intermediate-scale devices, where they already provide substantial gains in error suppression. As a result, these hardware-efficient codes open a potential avenue for useful quantum error correction in near-term, pre-fault tolerant devices.

Figure 1

Comparison of QEC codes performance. We assume that the effect of the quantum fluctuator is to impart a random phase, $\theta$, which follows a Gaussian distribution $\theta \sim \mathcal{N}(0,\sigma )$ with standard deviation $\sigma$. By normalizing the $g_j\mathrm{s}$ to lie in $[0,1{]}^n$, $\sigma$ describes the noise strength. CFD followed by a QEC recovery (if applicable) results in an effective phase- or bit-flip channel $\rho \mapsto (1-p)\rho +pA\rho A$, where $A=Z$ for the physical qubits, $X_L$ for the repetition codes, and $Z_L$ for hardware-efficient codes. The average infidelity, average trace distance and diamond distance to $I$ are all $\propto p$. As the performance of all strategies shown depends on $\{g_j\}$, we plot the average of $p$ over $\{g_j\}\in [0,1{]}^n$. The error bands for the hardware-efficient codes denote the standard error of the mean from Monte Carlo integration. More details on the numerical implementation are given in [31].

Figure 2

A recovery procedure for $n=2$ qubits where $|{\psi }_L⟩=\alpha |0_L⟩+\beta |1_L⟩$ for arbitrary $\alpha$ and $\beta$, $H$ denotes a Hadamard gate, and $\theta$ is a random variable. The unitaries $U_x$ and $U_z$ are both $\pi$ rotations about orthogonal axes on the Bloch sphere which are determined by $g_1$, $g_2$, and $\vartheta$.