Speaker
Description
Neural-Shadow Quantum State Tomography (NSQST) [Wei24] combines classical-shadow measurements [HKP20] with Neural Quantum States (NQS) [CT17] in order to reconstruct unknown quantum states from classical-shadow data. The learned state is represented as
$$ |\psi_\theta\rangle = \sum_{s\in\{0,1\}^N} \psi_\theta(s)|s\rangle = \sum_{s\in\{0,1\}^N} \sqrt{p_\theta(s)}\,e^{i\phi_\theta(s)}|s\rangle , $$ where $p_\theta(s)$ and $\phi_\theta(s)$ define the probability and phase sectors of the wavefunction. In the original NSQST formulation, these sectors are modeled using transformer networks [Wei24, Vas17]. In this work, we study a convolutional variant in which the wavefunction is modeled by an autoregressive one-dimensional causal Convolutional Neural Network (CNN), inspired by causal sequence models such as WaveNet [Oor16], with separate output heads for the conditional probabilities and phase contributions. This preserves the autoregressive structure of the NQS while introducing a computationally lighter inductive bias, particularly suited to states with local or structured correlations. The training objective is based on the infidelity between the unknown target state $|\psi\rangle$ and the learned state $|\psi_\theta\rangle$, $$ F(|\psi\rangle,|\psi_\theta\rangle) = |\langle\psi|\psi_\theta\rangle|^2, \qquad \mathcal{L}(\theta)=1-\widehat{F}(\theta). $$ In a tomography setting, the exact fidelity cannot be computed directly because the target state is not known explicitly. Classical-shadow data provide an experimentally accessible estimator of this fidelity. In the noise-free global Clifford setting, the estimator used in NSQST is $$ \widehat{F}(\theta) = \frac{1}{M} \sum_{i=1}^{M} \left[ (d+1) |\langle b_i|U_i|\psi_\theta\rangle|^2 -1 \right], \qquad d=2^N . $$ Here, each pair $(U_i,b_i)$ comes from the classical-shadow measurement procedure: a random Clifford unitary $U_i$ is applied to the target state, followed by a computational-basis measurement producing the bitstring outcome $b_i$. The inverse measurement channel for global Clifford shadows is what leads to the prefactor $(d+1)$ and the subtraction of $1$ in the fidelity estimator. Via Monte Carlo, the overlaps appearing in this estimator are approximated from samples $x_k\sim p_\theta$ drawn from the neural distribution [Wei24], $$ \langle b|U|\psi_\theta\rangle \approx \frac{1}{K} \sum_{k=1}^{K} \frac{\langle b|U|x_k\rangle\psi_\theta(x_k)} {p_\theta(x_k)}, \qquad p_\theta(x_k)=|\psi_\theta(x_k)|^2 . $$ We benchmark the CNN-based implementation on representative six-qubit target states: minus product, GHZ, phase-shifted GHZ, W, one-dimensional cluster, and MPS-like circuit-generated states. In parallel, we have developed an improved implementation of the NSQST training pipeline with respect to the original code [Wei24]. The improved version includes fixed shadow-pool reuse, multiple shots per Clifford, histogram-weighted shadow outcomes, and a more vectorized Monte Carlo estimator which significantly improves computation times. To make a controlled comparison, we also run the original transformer ansatz inside this modified implementation. This allows us to assess separately the effect of the neural architecture and the effect of the updated shadow-training protocol. The proposed talk will discuss these preliminary results and possible extensions toward structured many-qubit systems beyond the exact-enumeration regime. In particular, we aim to investigate pretrainable target families such as GHZ states, one-dimensional cluster states, and Dicke states with different excitation numbers, potentially in the range of approximately $20$--$40$ qubits. Following the strategy used in the original NSQST work [Wei24], pretraining can first bias the NQS toward the relevant computational-basis probability distribution before the shadow-based optimization stage. This is especially natural for sparse target families such as GHZ and low-excitation Dicke states. For cluster states, whose computational-basis distribution is uniform, the useful prior is instead expected to come from the local structure of the phase.
Finally, we outline a more exploratory direction based on QAOA-inspired pretraining [FGG14]. For state families whose relevant computational-basis support is related to low-energy configurations of a classical cost Hamiltonian, QAOA may provide a physically motivated initialization for the NQS. This would not replace the shadow-based reconstruction step, but could provide a useful prior for structured states whose amplitudes are concentrated on selected bitstrings. Overall, the approach combines classical shadows, autoregressive NQS, convolutional architectures, improved shadow-pool training, and pretraining as a possible route toward tomography of structured many-qubit quantum states.
References:
[Wei24] Y. Wei, W. A. Coish, P. Ronagh, and C. A. Muschik, “Neural-shadow quantum state tomography,” Physical Review Research 6, 023250 (2024).
[HKP20] H.-Y. Huang, R. Kueng, and J. Preskill, “Predicting many properties of a quantum system from very few measurements,” Nature Physics 16, 1050--1057 (2020).
[CT17] G. Carleo and M. Troyer, “Solving the quantum many-body problem with artificial neural networks,” Science 355, 602--606 (2017).
[Vas17] A. Vaswani et al., “Attention is all you need,” Advances in Neural Information Processing Systems 30 (2017).
[Oor16] A. van den Oord et al., “WaveNet: A generative model for raw audio,” arXiv:1609.03499 (2016).
[FGG14] E. Farhi, J. Goldstone, and S. Gutmann, “A quantum approximate optimization algorithm,” arXiv:1411.4028 (2014).