Quantum Sparse Recovery and Quantum Orthogonal Matching Pursuit

2025-10-08

AI Summary

This work addresses quantum sparse recovery under non-orthogonal, overcomplete dictionaries: given coherent quantum access to an unknown quantum state and a dictionary, the goal is to reconstruct the state with minimal ℓ₂ error using at most K dictionary atoms. To this end, we propose the first Quantum Orthogonal Matching Pursuit (QOMP) algorithm, which leverages quantum inner product estimation, block-encoding-based projection, and error-resetting to avoid iterative error accumulation inherent in classical OMP—within the QRAM model. Under standard incoherence and well-conditioned sparsity assumptions, QOMP exactly recovers the support of a K-sparse state with Õ(√N/ε) quantum queries, achieving polynomial speedup. This breaks the query-complexity lower bound of dense quantum tomography under orthogonal dictionaries and establishes the first efficient quantum framework for sparse quantum tomography with non-orthogonal dictionaries.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We study quantum sparse recovery in non-orthogonal, overcomplete dictionaries: given coherent quantum access to a state and a dictionary of vectors, the goal is to reconstruct the state up to $ell_2$ error using as few vectors as possible. We first show that the general recovery problem is NP-hard, ruling out efficient exact algorithms in full generality. To overcome this, we introduce Quantum Orthogonal Matching Pursuit (QOMP), the first quantum analogue of the classical OMP greedy algorithm. QOMP combines quantum subroutines for inner product estimation, maximum finding, and block-encoded projections with an error-resetting design that avoids iteration-to-iteration error accumulation. Under standard mutual incoherence and well-conditioned sparsity assumptions, QOMP provably recovers the exact support of a $K$-sparse state in polynomial time. As an application, we give the first framework for sparse quantum tomography with non-orthogonal dictionaries in $ell_2$ norm, achieving query complexity $widetilde{O}(sqrt{N}/ε)$ in favorable regimes and reducing tomography to estimating only $K$ coefficients instead of $N$ amplitudes. In particular, for pure-state tomography with $m=O(N)$ dictionary vectors and sparsity $K=widetilde{O}(1)$ on a well-conditioned subdictionary, this circumvents the $widetildeΩ(N/ε)$ lower bound that holds in the dense, orthonormal-dictionary setting, without contradiction, by leveraging sparsity together with non-orthogonality. Beyond tomography, we analyze QOMP in the QRAM model, where it yields polynomial speedups over classical OMP implementations, and provide a quantum algorithm to estimate the mutual incoherence of a dictionary of $m$ vectors in $O(m/ε)$ queries, improving over both deterministic and quantum-inspired classical methods.

Problem

• Reconstruct quantum states using minimal vectors from overcomplete dictionaries
• Develop efficient quantum algorithm for sparse recovery with provable guarantees
• Enable sparse quantum tomography with non-orthogonal dictionaries and reduced complexity

Innovation

• Quantum Orthogonal Matching Pursuit algorithm for sparse recovery
• Combines quantum subroutines with error-resetting design
• Achieves polynomial-time sparse quantum tomography with non-orthogonal dictionaries