# Learning shallow quantum circuits

arXiv: https://arxiv.org/abs/2401.10095 · HTML: https://mlsanity.com/paper/2401.10095/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 95 references that are in the map.
- First posted on arXiv 18 Jan 2024.

## Builds on (most cited first, 26 of 26)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Recurrent Quantum Neural Networks](https://mlsanity.com/paper/2006.14619/) (2020)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [Trainability of Dissipative Perceptron-Based Quantum Neural Networks](https://mlsanity.com/paper/2005.12458/) (2020)
- [Out-of-distribution generalization for learning quantum dynamics](https://mlsanity.com/paper/2204.10268/) (2022)
- [Enhancing Generative Models via Quantum Correlations](https://mlsanity.com/paper/2101.08354/) (2021)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning many-body Hamiltonians with Heisenberg-limited scaling](https://mlsanity.com/paper/2210.03030/) (2022)
- [Practical Black Box Hamiltonian Learning](https://mlsanity.com/paper/2206.15464/) (2022)
- [Dynamical simulation via quantum machine learning with provable generalization](https://mlsanity.com/paper/2204.10269/) (2022)
- [The Complexity of NISQ](https://mlsanity.com/paper/2210.07234/) (2022)
- [The power and limitations of learning quantum dynamics incoherently](https://mlsanity.com/paper/2303.12834/) (2023)
- [Improved Stabilizer Estimation via Bell Difference Sampling](https://mlsanity.com/paper/2304.13915/) (2023)
- [Scalably learning quantum many-body Hamiltonians from dynamical data](https://mlsanity.com/paper/2209.14328/) (2022)
- [Low-Stabilizer-Complexity Quantum States Are Not Pseudorandom](https://mlsanity.com/paper/2209.14530/) (2022)

## Built on by (most cited first, 20 of 20)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Generative quantum advantage for classical and quantum problems](https://mlsanity.com/paper/2509.09033/) (2025)
- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [Certifying almost all quantum states with few single-qubit measurements](https://mlsanity.com/paper/2404.07281/) (2024)
- [Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits](https://mlsanity.com/paper/2408.12199/) (2024)
- [Hamiltonian Property Testing](https://mlsanity.com/paper/2403.02968/) (2024)
- [Learning quantum states prepared by shallow circuits in polynomial time](https://mlsanity.com/paper/2410.23618/) (2024)
- [Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation](https://mlsanity.com/paper/2408.06967/) (2024)
- [Agnostic Tomography of Stabilizer Product States](https://mlsanity.com/paper/2404.03813/) (2024)
- [Predicting quantum channels over general product distributions](https://mlsanity.com/paper/2409.03684/) (2024)
- [Learning and Generating Mixed States Prepared by Shallow Channel Circuits](https://mlsanity.com/paper/2604.01197/) (2026)
- [Separable Power of Classical and Quantum Learning Protocols Through the Lens of No-Free-Lunch Theorem](https://mlsanity.com/paper/2405.07226/) (2024)
- [Provably scalable learning from quantum many-body data via locality](https://mlsanity.com/paper/2509.13705/) (2025)
- [Efficient learning of bosonic Gaussian unitaries](https://mlsanity.com/paper/2510.05531/) (2025)
- [Noise-tolerant learnability of shallow quantum circuits from statistics and the cost of quantum pseudorandomness](https://mlsanity.com/paper/2405.12085/) (2024)
- [Learning to erase quantum states: thermodynamic implications of quantum learning theory](https://mlsanity.com/paper/2504.07341/) (2025)
- [Fragmentation is Efficiently Learnable by Quantum Neural Networks](https://mlsanity.com/paper/2512.00751/) (2025)
- [Quantum Algorithm for Distributed Reduction of Entanglements (QADR): A Trainable and Simulation-Efficient QML Framework](https://mlsanity.com/paper/2606.01291/) (2026)

## Abstract (arXiv)

Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown n-qubit shallow quantum circuit U (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of U. We also provide a polynomial-time classical algorithm for learning the description of any unknown n-qubit state lvert psi rangle = U lvert 0n rangle prepared by a shallow quantum circuit U (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of lvert psi rangle. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.