On exploring the potential of quantum auto-encoder for learning quantum systems

TL;DR

This work devise three effective QAE-based learning protocols to address three classically computational hard learning problems when learning quantum systems, which are low-rank state fidelity estimation, quantum Fisher information (QFI) estimation, and Gibbs state preparation.

Abstract

The frequent interactions between quantum computing and machine learning revolutionize both fields. One prototypical achievement is the quantum auto-encoder (QAE), as the leading strategy to relieve the curse of dimensionality ubiquitous in the quantum world. Despite its attractive capabilities, practical applications of QAE have yet largely unexplored. To narrow this knowledge gap, here we devise three effective QAE-based learning protocols to address three classically computational hard learning problems when learning quantum systems, which are low-rank state fidelity estimation, quantum Fisher information estimation, and Gibbs state preparation. Attributed to the versatility of QAE, our proposals can be readily executed on near-term quantum machines. Besides, we analyze the error bounds of the trained protocols and showcase the necessary conditions to provide practical utility from the perspective of complexity theory. We conduct numerical simulations to confirm the effectiveness of the proposed three protocols. Our work sheds new light on developing advanced quantum learning algorithms to accomplish hard quantum physics and quantum information processing tasks.

Figures (8)

• Figure 1: The paradigm of AE and QAE. The upper left panel illustrates the classical auto-encoder (AE). The upper right panel shows the quantum AE (QAE). In the training process, all quantum operations in the grey box are removed. The state $\sigma$ corresponds to the compressed state of $\rho$ (see Section \ref{['subsec:QAE']} for explanations). The lower panel depicts the data type suited for AE and QAE. Two typical datasets explored by QAE are collected from chemistry and material science.
• Figure 2: The paradigm of the QAE-based fidelity estimator. The implementation of the QAE-based fidelity estimator is composed of four steps. (i) QAE in Eqn. (\ref{['eqn:loss_G1']}) is applied to optimize $\bm{\theta}$ with $T$ iterations. (ii) After optimization, the compressed state $\sigma^{(T)}$ in Eqn. (\ref{['eqn:sigma_T_1']}) is extracted into the classical register via quantum state tomography. Once the classical form of $\sigma^{(T)}$ is accessible, the classical Eigen-solver is employed to acquire its spectral information, i.e., $\{\lambda_i, \ket{\varpi_i}\}$. (iii) When the spectral information of $\rho$ is available and the state $\kappa$ is accessible, we can calculate $W$ in Eqn. (\ref{['eqn:QAE-W']}) based on Ref. cerezo2020variational. (iv) Given access to $W$, the fidelity $F(\rho,\kappa)$ can be effectively obtained.
• Figure 3: Simulation results for low-rank states fidelity estimation with the number of qubits $N=8$. (a) The implementation of variational circuit $U(\bm{\theta})$ used in the quantum encoder. The trainable parameters are contained in the rotational single-qubit gates RZ and RY. The entangled layer is composed of CNOT gates. (b) The average training loss of the QAE-based fidelity estimator in the first stage. The label '$K = a$' refers to the employed number of latent qubits of the QAE-based fidelity estimator is $a$. The shadow region refers to the variance. (c) The top-1 training loss achieved by QAE-based fidelity estimator. All notations follow the same meaning as those in (b). (d) The average fidelity estimation results returned by QAE-based fidelity estimator. The labels QAEFL and QAEFU refer to the bound in Theorem 2, respectively. The label 'SSFBU' ('SSFBL') is super-fidelity (sub-fidelity) bounds. (e) The top-1 fidelity estimation results achieved by QAE-based fidelity estimator. All notations follow the same meaning as those in (d).
• Figure 4: The simulation results for full-rank states fidelity estimation. The left panel demonstrates the training loss of the QAE-based fidelity estimator in the first stage when it applies to estimate $\mathop{\mathrm{F}}\nolimits(\rho, \rho)$. The right panel illustrates the estimated fidelity bounds and SSFB. The meaning of labels is identical to those in Figure \ref{['fig:QAE']} and Figure \ref{['fig:QAE-fide-low-append']}.
• Figure 5: The paradigm of the QAE-based QFI estimator. The QAE-based QFI estimator is implemented on a quantum-classical hybrid system. In the quantum part, a variational quantum circuit $U(\bm{\gamma})$ is employed to prepare the tunable probe state $\rho(\bm{\gamma})$. This probe state is separately interacted with the source described by the parameter $\vartheta$ and $\vartheta+\tau$ to prepare the state $\rho_{\vartheta}(\bm{\gamma})$ and $\rho_{\vartheta+\tau}(\bm{\gamma})$. The two generated states are fed into the QAE-based fidelity estimator to estimate $\mathop{\mathrm{F}}\nolimits(\rho_{\vartheta}(\bm{\gamma}), \rho_{\vartheta+\tau}(\bm{\gamma}))$, as highlighted by the red box. The calculated fidelity is used to compute $\mathcal{I}_{\tau}(\vartheta, \rho_{\vartheta}(\bm{\gamma}))$ in Eqn. (\ref{['eqn:q_metro_def']}) and then update parameters $\bm{\gamma}$ to maximize this quantity. Repeating the above procedures with $T$ times, the classical processor outputs $\gamma^{(T)}$, which can be used to prepare the estimated probe state $\rho(\bm{\gamma}^{(T)})$.

Theorems & Definitions (13)

• Theorem 1
• Lemma 1
• Lemma 2: Proposition 5, cerezo2020variational
• Theorem 2
• Lemma 3
• Lemma 4
• proof : Proof of Theorem \ref{['thm_QAE_noiseless']}
• proof : Proof of Lemma \ref{['lem:sigma_eigen']}
• Lemma 5: Lemma 1, cerezo2020variational
• proof : Proof of Theorem \ref{['thm:fide_bound']}