Generic detection-based error mitigation using quantum autoencoders

Efficient error-mitigation techniques demanding minimal resources is key to quantum information processing. We propose a generic protocol to mitigate quantum errors using detection-based quantum autoencoders. In our protocol, the quantum data are compressed into a latent subspace while leaving errors outside, the latter of which is then removed by a measurement and postselection. Compared to previously developed methods, our protocol on the one hand requires no extra qubits, and on the other hand it has a near-optimal denoising power, in which under reasonable requirements all errors detected outside of the latent subspace can be removed, while those inside the subspace cannot be removed by any means. Our detection-based quantum autoencoders are therefore particularly useful for near-term quantum devices in which controllable qubits are limited while noise reduction is important.

Figure 1

(a) Sketch of the error detection and mitigation process. The large oval indicates the full Hilbert space $\mathcal{H}$, the green ellipse inside represents the latent subspace $\mathcal{L}$, while the remainder is the junk subspace $\mathcal{J}$. The noise effect is described by a quantum channel $\mathcal{E}(·)$ as shown in Eq. (3). We use dots to represent the error-free term $(1-\varepsilon )\rho$, and crosses denote the error term $\varepsilon {\rho }^{\text{err}}$. $U_{\text{e}}$ transfers the error-free term to $\mathcal{L}$, while most of the errors remain in $\mathcal{J}$. A measurement projects the state to the latent subspace, which detects and removes the errors in $\mathcal{J}$. Finally, $U_{\text{e}}^{†}$ is applied to recover the quantum data with error mitigated. (b) Relations between different types of errors. Errors removable by our method include but are not limited to decay-type errors outside $\mathcal{S}$, and a large portion of local and global depolarization (denoted as “dep”) errors. All errors recoverable by quantum operations are also removable by our method. (c) The detection-based quantum autoencoder in the circuit form.

Figure 2

Performance of a detection-based quantum autoencoder for W class states with forms of input states known, under (a) local depolarization noise ${\tilde{\mathcal{E}}}_{\text{lc}}(·)$ and (b) global depolarization noise ${\tilde{\mathcal{E}}}_{\text{gl}}(·)$. We set $\varepsilon =0.05$.

Figure 3

(a) An example of programmable quantum autoencoders with two groups of $N_{\text{ly}}$-layer networks, which compress the four-qubit states to two-qubit states in two stages. (b) The circuit structure of each layer.

Figure 4

Performance of a detection-based quantum autoencoder for W class states with forms of input states unknown, under (a) local depolarization noise ${\tilde{\mathcal{E}}}_{\text{lc}}(·)$ and (b) global depolarization noise ${\tilde{\mathcal{E}}}_{\text{gl}}(·)$. We set $n=4$.