Learning shallow quantum circuits (2401.10095v1)

Abstract: 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 $\lvert \psi \rangle = U \lvert 0^n \rangle$ 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.

• The paper demonstrates that shallow quantum circuits can be efficiently learned in polynomial time using classical algorithms with sample complexity O(n² log(n)/ε²).
• It shows that geometrically local circuits on k-dimensional lattices retain their locality when learned, enhancing their practical applicability in constrained quantum systems.
• The authors propose a verification method to ensure that the learned circuit approximates the original in an average-case scenario, effectively addressing realistic noise concerns.

An Analysis of "Learning shallow quantum circuits"

The paper "Learning shallow quantum circuits" addresses a fundamental question in quantum computing: can shallow quantum circuits, which are theoretically more powerful than classical circuits of similar depth, be learned efficiently by classical means? This paper presents several algorithms that answer this question affirmatively under specific conditions.

Main Contributions

The authors introduce a classical algorithm that can learn the description of an unknown $n$-qubit shallow quantum circuit $U$. This algorithm operates in polynomial time and uses single-qubit measurement data on the output states of $U$. Additionally, they provide a method for learning the description of any unknown $n$-qubit state prepared by a shallow quantum circuit with a 2D lattice structure.

Technical Achievements

1. Learning General Shallow Circuits:
   • The proposed algorithm can learn any constant-depth $n$-qubit quantum circuit to an $\varepsilon$ diamond distance with high probability. The learning is efficient, with sample complexity scaling as $$\mathcal{O}\left(\frac{n^2 \log(n)}{\varepsilon^2}\right)$$.
   • The algorithm only requires classical computation and can also handle circuits constructed from finite gate sets, achieving exact recovery in these cases.
2. Learning Geometrically-Local Circuits:
   • The algorithm provides optimized methods for circuits laid out on a $k$-dimensional lattice. For such geometries, the learned circuit maintains geometric locality, enhancing the practical applicability of the results in physically constrained quantum systems.
3. Quantum State Learning:
   • The paper includes a procedure for learning pure quantum states prepared by shallow circuits on 2D lattices. This involves a comprehensive approach to disentangle and learn correlations within the quantum state, leveraging properties of finite correlation length in quantum systems.
4. Verification:
   • The authors propose a verification algorithm to ensure that the learned circuit closely approximates the original circuit with respect to average-case distance, rather than the worst-case diamond distance. This aspect is critical when dealing with circuits that may include noise.

Implications and Future Directions

The results in this paper have significant implications for both quantum learning theory and practical quantum computing. By demonstrating that shallow circuits can be efficiently learned, it opens potential pathways for hybrid classical-quantum algorithms where the classical part efficiently handles the learning task. Additionally, understanding how to effectively verify quantum devices and their states remains an important challenge; the provided verification algorithm addresses this within the specified constraints.

Future research could explore extending these results to deeper quantum circuits or reducing the computational overhead for specific classes of quantum circuits. Another promising direction is studying similar learnability questions for other classes of quantum states, such as those appearing in fault-tolerant quantum computing.

In conclusion, this paper represents a considerable step forward in understanding the computational learnability of quantum systems, particularly shallow circuits. It combines theoretical innovation with practical relevance, showing convincingly that such models are amenable to efficient classical learning under the conditions explored.