Sparse quantum states with long-range entanglement learned by single-qubit measurements
Learning sparse quantum states from single-qubit measurements
Data Structures and AlgorithmsInformation Theory
Summary
The problem is how to understand complex quantum states that have many hidden connections, called long-range entanglement. Such states are usually hard to learn without complicated measurements involving many qubits at once. The authors show that if the quantum state is sparse—meaning it has only a few significant parts in a certain way—it can still be learned efficiently using only simple single-qubit measurements. This makes it easier to study these quantum states on current quantum devices.
What this means in practice
- •For quantum computing engineers: Use single-qubit measurements to efficiently reconstruct sparse quantum states without complex entangling operations, enabling easier state diagnostics on current quantum devices.
- •For quantum hardware developers: Improve calibration and verification methods by learning sparse quantum states with only single-qubit measurements, reducing hardware demands during testing.
Authors
Su-un Lee, Liang Jiang, Kunal Sharma
Abstract
We study the problem of learning a sparse quantum state, an $n$-qubit quantum state whose density matrix has at most $s$ nonzero matrix entries in an unknown product basis. While such states admit compact classical descriptions, they can carry long-range entanglement that prevents reconstruction from local reduced density matrices alone. Therefore, previous learning approaches addressed such long-range-entangled states using many entangling gates to extract the necessary information. In this work, we show that sparse states can nevertheless be efficiently learned using only single-qubit measurements. Specifically, when the sparsity $s$ is constant, our algorithm can learn sparse states from single-qubit measurements with polynomial sample complexity and classical computational complexity. When $s$ grows polynomially with $n$, sparse states can still be learned from single-qubit measurements with polynomial sample complexity, although efficient classical computation is not guaranteed in general. In this regime, however, the classical computational complexity becomes quasipolynomial when the state is sparse in an unknown basis that is a product of a known fixed finite set of single-qubit bases (e.g., eigenbases of Pauli operators). These results establish efficient learning of sparse states with long-range entanglement without entangling gates, and the single-qubit measurement requirements make our algorithms compatible with current quantum devices.