Quantum error correction via purification using a single auxiliary

TL;DR

This work introduces a single auxiliary qudit (or qubit in reduced-resource variants) purification framework for quantum error correction, where errors moving a system out of its ground-state subspace are corrected via a joint system–auxiliary evolution and a single energy-basis measurement on the auxiliary. The method yields unit fidelity in the purified state with a finite success probability determined by the measurement outcome, and it is demonstrated on low-distance QECCs (3-, 4-, 5-qubit codes) as well as one-dimensional Heisenberg-encoded qubits. It further investigates reduced-resource schemes, two prescriptions for auxiliary-qubit implementations, and scenarios with resilient noise during correction, highlighting when purification extends the class of correctable errors beyond conventional QEC. The results suggest a thermodynamic and hardware-relevant perspective on QEC, offering a versatile alternative to syndrome-based protocols and motivating future scalable designs and robustness analyses.

Abstract

We propose a single auxiliary-assisted purification-based framework for quantum error correction, capable of correcting errors that drive a system from its ground-state subspace into excited-state sectors. The protocol consists of a joint time evolution of the system-auxiliary duo under a specially engineered interaction Hamiltonian, followed by a single measurement of the auxiliary in its energy eigenbasis and a subsequent post-selection of one of the measurement outcomes. We show that the resulting purified state always achieves unit fidelity, while the probability of obtaining any energy of the auxiliary other than its ground state energy yields the success rate of the protocol. We demonstrate the power of this proposed method for several low-distance quantum codes, including the three-, four-, and five-qubit codes, and for the one-dimensional isotropic Heisenberg model, subjected to bit-flip, phase-flip, and amplitude-damping noises acting on all qubits. Notably, the protocol expands the class of correctable errors for a given code, particularly in the presence of amplitude-damping noise. We further analyze the impact of replacing the auxiliary qudit with a single auxiliary qubit, and the changes in the performance of the protocol under the realistic scenario where noise remains active during the correction cycle.

Figures (8)

• Figure 1: (a) Splitting of $\mathcal{H}$ into the orthogonal degenerate energy subspaces $\mathcal{H}_i$ having energy $E_i$, such that $\mathcal{H}=\oplus_{i=0}^D\mathcal{H}_i$. An error corresponds to taking the system out of $\mathcal{H}_0$, while a recovery takes it back to $\mathcal{H}_0$. (b) There may exist errors that take the system out of $\mathcal{H}_0$ so that non-zero overlap can be found with more than one $\mathcal{H}_i$, eg. $\mathcal{H}_1$ and $\mathcal{H}_2$. An archetype state $\ket{\Psi_{12}}$ constructed in this case has non-zero overlap with both $\mathcal{H}_1$ and $\mathcal{H}_2$. (c) An uncorrectable error in our framework corresponds to the error that keeps the system within the ground state subspace $\mathcal{H}_0$.
• Figure 2: Three-qubit code. Variations of $P$ (ordinate) as a function of $\theta$ (abscissa) for different error rates $q$ when an arbitrary state in the logical subspace is subjected to (a)-(c) bit-flip, and (d)-(f) amplitude-damping errors on all qubits. The grey lines correspond to randomly sampled $\ket{\Psi_i}\in\mathcal{H}_i$, while the dark continuous line corresponds to the choice of equal superposition of contributing $\ket{E_{1,\alpha}}\in\mathcal{H}_1$ in $P$. The horizontal axes in all figures are in radian, and $t=\pi/2g$ for all plots.
• Figure 3: Four-qubit code.$P$ (ordinate) against $\theta$ (abscissa) for different $q$ values when an arbitrary state in the logical subspace is subjected to (a)-(c) bit-flip, (d)-(f) phase-flip, and (g)-(i) amplitude-damping errors on all qubits, and $t$ is fixed at $t=\pi/2g$. The gray lines correspond to the $H_{SA}$ constituted with randomly sampled $\ket{\Psi_i}\in\mathcal{H}_i$, the details of which are given in Sec. \ref{['subsec:four_qubit']}, while the dark continuous lines correspond to the interaction Hamiltonian constituted with the choice of equal superposition of contributing energy eigenstates from $\mathcal{H}_i$ as $\ket{\Psi_i}$ in the cases of (a)-(f), and equal superposition of all energy eigenstates from $\mathcal{H}_i$ as $\ket{\Psi_i}$ in the cases of (g)-(i). The horizontal axes in all figures are in radian.
• Figure 4: Five-qubit code.$P$ (ordinate) vs $\theta$ (abscissa) for different error rates $q$. (a)-(c) Bit-flip, and (d)-(f) phase-flip errors on all qubits. All other specifications are the same as in Fig. \ref{['fig:4qubit']}. The horizontal axes in all figures are in radian, and $t=\pi/2g$ for all figures.
• Figure 5: 1D isotropic Heisenberg model. Map plot of $P$ with $\theta$ (abscissa) and $q$ (ordinate) in the cases of (a)-(b) two-qubit isotropic Heisenberg encoding under (a) bit-flip and (b) phase-flip noise, and (c)-(d) two-qubit isotropic Heisenberg encoding under (c) bit-flip and (d) phase-flip noise, with $t=\pi/2g$ for all figures. The dashed lines mark boundaries of the regions where $P>0.01$, $0.1$, and $0.5$. The horizontal axis is in radian.