Correcting quantum errors using a classical code and one additional qubit

TL;DR

Hadamard-based Virtual Error Correction (H-VEC), a protocol that empowers any classical bit-flip code to correct arbitrary Pauli noise with the addition of only a single ancilla qubit and two layers of controlled-Hadamard gates, is introduced.

Abstract

Classical error-correcting codes are powerful but incompatible with quantum noise, which includes both bit-flips and phase-flips. We introduce Hadamard-based Virtual Error Correction (H-VEC), a protocol that empowers any classical bit-flip code to correct arbitrary Pauli noise with the addition of only a single ancilla qubit and two layers of controlled-Hadamard gates. Through classical post-processing, H-VEC virtually filters the error channel, projecting the noise into pure Y-type errors that are subsequently corrected using the classical code's native decoding algorithm. We demonstrate this by applying H-VEC to the classical repetition code. Under a code-capacity noise model, the resulting protocol not only provides full quantum protection but also achieves an exponentially stronger error suppression (in distance) than the original classical code. The improvements over the surface code are even more pronounced, while using far fewer qubits, simpler checks, and straightforward decoding. There are some limitations to the technique, most notably that H-VEC introduces a sampling overhead due to its post-processing nature. Nonetheless, it represents a fundamentally novel hybrid quantum error correction and mitigation framework that redefines the trade-offs between physical hardware requirements and classical processing for error suppression.

Figures (11)

• Figure 1: A diagram showing the H-VEC scheme presented in this work. The objective is to obtain an observable expectation value Tr$(O\rho_Z)$, where $\rho_Z$ is a bit-flip classical code state encoded in an $n$-physical-qubit logical register that suffers from an error channel $\mathcal{E}$, which comprises of both bit-flip and phase-flip errors. By initialising an ancilla qubit in $\ket{+}=(1/\sqrt{2})(\ket{0}+\ket{1})$ and sandwiching the error channel between two layers of C-H gates, the noise on the logical register, after the $X$ measurement on the control qubit and the bit-flip checks $S_{Z}$, is effectively projected into one that consists purely of $Y$-type errors. The projected errors can then be corrected using $Y$ gates via the measured syndrome $\vec{k}$, along with phase adjustments performed in classical software. The desired error-corrected expectation value can then be retrieved via the ratio of the two expectation values shown.
• Figure 2: Logical error rate $p_{\mathrm{L}}$ as a function of physical error rate $p$ simulated under the code-capacity error model with local depolarising errors, comparing the bit-flip repetition code (left), virtual quantum repetition code (middle), and unrotated surface code (right) for code distances $d \in \{1,3,5,7\}$. In contrast to the bit-flip repetition code, the virtual quantum repetition code is able to suppress both bit-flip and phase-flip errors, and it does so more effectively than the surface code. See \ref{['app: numerics']} for simulation details.
• Figure 3: The circuit in which C-H gates are applied on the whole register using one control qubit can be replaced by a circuit consisting of one control qubit per physical qubit, without altering the performance of H-VEC. The original control qubit measurement outcome simply corresponds to the product between the outcomes of all $n$ control qubits.
• Figure 4: EPPs viewed in terms of stabiliser projections performed in the (a) conventional and (b) H-VEC approach. Each vertical wavy line represents a noisy Bell pair $\mathcal{E}{[}\ketbra{\Psi^+}{]}$ established between two qubits, and "PS" denotes post-selection. (a) The basic 2-to-1 stabiliser EPP shi2025stabilizer sequentially measures the $Z\otimes Z$ and $X \otimes X$ stabilisers of the Bell state by interleaving Hadamard gates between two rounds. (b) When applying H-VEC to $\ket{\Psi^+}$, the first C-H layer can be removed so it can be applied directly to the noisy state rather than the noise channel. Only a single round is required in this case, leading also to a reduction in the number of noisy input Bell states. We refer the reader to \ref{['app: vepp_comments']} for the corresponding fanned-out circuits and details of their performance.
• Figure 5: Quantum circuit involved in the general VEC framework. The input state $\rho$ is some code state within the code space of $\Pi_E$. By choosing $U$ appropriately given the noise channel $\mathcal{N}$ that we wish to correct, the stabiliser measurement $S_E$, again of $\Pi_E$, effectively projects the noise in the logical register into one that is correctable using the obtained syndrome, upon post-processing via the $X$-basis measurement of the ancilla qubit.