Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows

Extracting information efficiently from quantum systems is a major component of quantum information processing tasks. Randomized measurements, or classical shadows, enable predicting many properties of arbitrary quantum states using few measurements. While random single qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Prepending a shallow random quantum circuit before measurements maintains this experimental friendliness, but also has favorable sample complexities for observables beyond low-weight Paulis, including high-weight Paulis and global low-rank properties such as fidelity. However, in realistic scenarios, quantum noise accumulated with each additional layer of the shallow circuit biases the results. To address these challenges, we propose the robust shallow shadows protocol. Our protocol uses Bayesian inference to learn the experimentally relevant noise model and mitigate it in postprocessing. This mitigation introduces a bias-variance trade-off: correcting for noise-induced bias comes at the cost of a larger estimator variance. Despite this increased variance, as we demonstrate on a superconducting quantum processor, our protocol correctly recovers state properties such as expectation values, fidelity, and entanglement entropy, while maintaining a lower sample complexity compared to the random single qubit measurement scheme. We also theoretically analyze the effects of noise on sample complexity and show how the optimal choice of the shallow shadow depth varies with noise strength. This combined theoretical and experimental analysis positions the robust shallow shadow protocol as a scalable, robust, and sample-efficient protocol for characterizing quantum states on current quantum computing platforms.

Digital-analog quantum learning on Rydberg atom arrays

We propose hybrid digital-analog learning algorithms on Rydberg atom arrays, combining the potentially practical utility and near-term realizability of quantum learning with the rapidly scaling architectures of neutral atoms. Our construction requires only single-qubit operations in the digital setting and global driving according to the Rydberg Hamiltonian in the analog setting. We perform a comprehensive numerical study of our algorithm on both classical and quantum data, given respectively by handwritten digit classification and unsupervised quantum phase boundary learning. We show in the two representative problems that digital-analog learning is not only feasible in the near term, but also requires shorter circuit depths and is more robust to realistic error models as compared to digital learning schemes. Our results suggest that digital-analog learning opens a promising path towards improved variational quantum learning experiments in the near term.

Discovery of Optimal Quantum Error Correcting Codes via Reinforcement Learning

The recently introduced Quantum Lego framework provides a powerful method for generating complex quantum error correcting codes (QECCs) out of simple ones. We gamify this process and unlock a new avenue for code design and discovery using reinforcement learning (RL). One benefit of RL is that we can specify \textit{arbitrary} properties of the code to be optimized. We train on two such properties, maximizing the code distance, and minimizing the probability of logical error under biased Pauli noise. For the first, we show that the trained agent identifies ways to increase code distance beyond naive concatenation, saturating the linear programming bound for CSS codes on 13 qubits. With a learning objective to minimize the logical error probability under biased Pauli noise, we find the best known CSS code at this task for $\lesssim 20$ qubits. Compared to other (locally deformed) CSS codes, including Surface, XZZX, and 2D Color codes, our $[[17,1,3]]$ code construction actually has \textit{lower} adversarial distance, yet better protects the logical information, highlighting the importance of QECC desiderata. Lastly, we comment on how this RL framework can be used in conjunction with physical quantum devices to tailor a code without explicit characterization of the noise model.

Differentiable Programming of Isometric Tensor Networks

Differentiable programming is a new programming paradigm which enables large scale optimization through automatic calculation of gradients also known as auto-differentiation. This concept emerges from deep learning, and has also been generalized to tensor network optimizations. Here, we extend the differentiable programming to tensor networks with isometric constraints with applications to multiscale entanglement renormalization ansatz (MERA) and tensor network renormalization (TNR). By introducing several gradient-based optimization methods for the isometric tensor network and comparing with Evenbly-Vidal method, we show that auto-differentiation has a better performance for both stability and accuracy. We numerically tested our methods on 1D critical quantum Ising spin chain and 2D classical Ising model. We calculate the ground state energy for the 1D quantum model and internal energy for the classical model, and scaling dimensions of scaling operators and find they all agree with the theory well.

Hamiltonian-Driven Shadow Tomography of Quantum States

Classical shadow tomography provides an efficient method for predicting functions of an unknown quantum state from a few measurements of the state. It relies on a unitary channel that efficiently scrambles the quantum information of the state to the measurement basis. Facing the challenge of realizing deep unitary circuits on near-term quantum devices, we explore the scenario in which the unitary channel can be shallow and is generated by a quantum chaotic Hamiltonian via time evolution. We provide an unbiased estimator of the density matrix for all ranges of the evolution time. We analyze the sample complexity of the Hamiltonian-driven shadow tomography. We find that it can be more efficient than the unitary-2-design-based shadow tomography in a sequence of intermediate time windows that range from an order-1 scrambling time to a time scale of $D^{1/6}$, given the Hilbert space dimension $D$. In particular, the efficiency of predicting diagonal observables is improved by a factor of $D$ without sacrificing the efficiency of predicting off-diagonal observables.

RG-Flow: A hierarchical and explainable flow model based on renormalization group and sparse prior

Flow-based generative models have become an important class of unsupervised learning approaches. In this work, we incorporate the key idea of renormalization group (RG) and sparse prior distribution to design a hierarchical flow-based generative model, called RG-Flow, which can separate different scale information of images with disentangle representations at each scale. We demonstrate our method mainly on the CelebA dataset and show that the disentangled representation at different scales enables semantic manipulation and style mixing of the images. To visualize the latent representation, we introduce the receptive fields for flow-based models and find receptive fields learned by RG-Flow are similar to convolutional neural networks. In addition, we replace the widely adopted Gaussian prior distribution by sparse prior distributions to further enhance the disentanglement of representations. From a theoretical perspective, the proposed method has $O(\log L)$ complexity for image inpainting compared to previous flow-based models with $O(L^2)$ complexity.