Symmetric channel verification for purifying noisy quantum channels

TL;DR

This work introduces symmetric channel verification (SCV), a single-copy protocol that purifies quantum channels by exploiting symmetry in the channel structure. It formalizes SCV with a phase gadget across symmetry eigenspaces and a quantum phase-estimation-like circuit, and provides a hardware-friendly variant (virtual SCV) that uses one ancilla and Clifford operations to estimate purified expectation values. The authors develop a Pauli-symmetry specialization showing SCV is optimal under Clifford restrictions and derive a resource-theoretic bound on worst-case fidelity, demonstrating that SCV under Pauli symmetry can saturate this bound in relevant regimes. They also extend the framework to particle-number symmetry, analyze error-correction via SCV with feedback, and discuss limitations when restricted to Clifford unitaries. Collectively, SCV and virtual SCV offer robust, low-overhead strategies for mitigating symmetry-breaking noise in Hamiltonian simulation, phase estimation, and other quantum algorithms, with clear paths toward fault-tolerant channel-level error correction and integration with existing symmetry-verification methods.

Abstract

Symmetry inherent in quantum states has been widely used to reduce the effect of noise in quantum error correction and a quantum error mitigation technique known as symmetry verification. However, these symmetry-based techniques exploit symmetry in quantum states rather than quantum channels, limiting their application to cases where the entire circuit shares the same symmetry. In this work, we propose symmetric channel verification (SCV), a channel purification protocol that leverages the symmetry inherent in quantum channels. By introducing different phases to each symmetric subspace and employing a quantum phase estimation-like circuit, SCV can detect and correct symmetry-breaking noise in quantum channels. We further propose a hardware-efficient implementation of SCV at the virtual level, which requires only a single-qubit ancilla and is robust against the noise in the ancilla qubit. Our protocol is applied to various Hamiltonian simulation circuits and phase estimation circuits, resulting in a significant reduction of errors. Furthermore, in setups where only Clifford unitaries can be used for noise purification, which is relevant in the early fault-tolerant regime, we show that SCV under Pauli symmetry represents the optimal purification method.

Figures (15)

• Figure 1: Schematic of channel purification. (a) We aim to reduce the effect of the noise $\mathcal{N}$ by preparing $m$-qubit ancilla initialized in $\ket{0^m}$, applying $(n+m)$-qubit unitary operations $U_{\mathrm{E}}$ and $U_{\mathrm{D}}$ before and after the noisy channel, and measuring the ancilla qubits in the computational basis either to (i) post-select, (ii) average, or (iii) discard the measurement results. These operations can be regarded as applying a quantum supermap $\Theta$ to the noisy channel. (b) Applying feedback control to the system can be interpreted as applying control operation and discarding the measurement results.
• Figure 2: Circuit structure of symmetric channel verification (SCV). We prepare $m = \lceil \log_2M \rceil$-qubit ancilla, apply Hadamard gate, apply controlled-$V_S^{2^{k-1}}$ for $k=1,\ldots,m$ where the $k$-th ancilla is the control qubit and $V_S = \sum_j \exp[\frac{2\pi i}{2^m}j]\Pi_j$, apply the noisy channel $\mathcal{U}_{\mathcal{N}} = \mathcal{N}\circ\mathcal{U}$, apply controlled-$V_S^{\dag2^{k-1}}$ for $k=1,\ldots,m$, perform the inverse quantum Fourier transform, measure the ancilla qubits in the computational basis, and post-select the measurement result $\ket{0^m}$. This operation transforms the noisy channel $\mathcal{U}_\mathcal{N}(\cdot)$ into $(\Theta^{\mathrm{det}}_S(\mathcal{U}_\mathcal{N}))(\cdot)=\sum_{ij} \Pi_i \mathcal{U}_{\mathcal{N}}(\Pi_i\cdot \Pi_j)\Pi_j$.
• Figure 3: Circuit structure of SCV for the set of Pauli operators $\mathcal{Q}_U^{\mathrm{com}}$. We concatenate the SCV gadget for the generator $Q_1,\ldots,Q_r\in\mathcal{Q}_U^{\mathrm{com}}$ as $\Theta^{\mathrm{det}}_{\mathcal{Q}^{\mathrm{com}}_{U}} = \Theta^{\mathrm{det}}_{Q_1} * \cdots * \Theta^{\mathrm{det}}_{Q_r}$.
• Figure 4: Performance of SCV in detecting errors on Hamiltonian simulation circuits for (a) 1D Heisenberg model and (b) Floquet dynamics. The "Raw" line represents the trace distance between the noisy quantum state and the ideal quantum state, while the "SV" line represents the trace distance for the quantum state purified by symmetry verification. The "SCV" line represents the trace distance where the SCV is applied to the entire noisy circuit $\bigcirc_{l=1}^L \mathcal{N}_l \circ \mathcal{U}_l$, and the "SCV (LW)" line represents the layer-wise application of SCV, meaning that SCV is applied to every noisy layer [$\mathcal{N}_l \circ \mathcal{U}_l$ for (a) Heisenberg model and $\mathcal{N}_l \circ \mathcal{U}_{X,l}$ and $\mathcal{N}_l \circ \mathcal{U}_{Z,l}$ for (b) Floquet dynamics].
• Figure 5: Performance of SCV in detecting errors on a Hamiltonian simulation circuit for the H$_2$ molecule. The "Raw" line represents the trace distance between the noisy quantum state and the ideal quantum state. The "SCV (parity, noisy)" line represents the result using SCV for the parity operator $\prod_i Z_i$, where we assume that the SCV gadget is affected by noise. The "SCV (U(1), noiseless)" and "SCV (U(1), noisy)" lines represent the trace distance for the quantum state with the noise detected by SCV for the particle number operator $\sum_i Z_i$, where "noiseless" and "noisy" represent the results using noiseless and noisy SCV gadgets.

Theorems & Definitions (17)

• Theorem 1
• Corollary 1
• Theorem 2
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• proof
• proof
• proof