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Quantum Sparse Recovery and Quantum Orthogonal Matching Pursuit

Armando Bellante, Stefano Vanerio, Stefano Zanero

TL;DR

This work introduces Quantum Orthogonal Matching Pursuit (QOMP), the first quantum analogue of the classical OMP greedy algorithm, and gives the first framework for sparse quantum tomography with non-orthogonal dictionaries in $\ell_2$ norm, achieving query complexity $\widetilde{O}(N/\epsilon)$ in favorable regimes and reducing tomography to estimating only $K$ coefficients instead of $N$ amplitudes.

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.

Quantum Sparse Recovery and Quantum Orthogonal Matching Pursuit

TL;DR

This work introduces Quantum Orthogonal Matching Pursuit (QOMP), the first quantum analogue of the classical OMP greedy algorithm, and gives the first framework for sparse quantum tomography with non-orthogonal dictionaries in norm, achieving query complexity in favorable regimes and reducing tomography to estimating only coefficients instead of amplitudes.

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 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 -sparse state in polynomial time. As an application, we give the first framework for sparse quantum tomography with non-orthogonal dictionaries in norm, achieving query complexity in favorable regimes and reducing tomography to estimating only coefficients instead of amplitudes. In particular, for pure-state tomography with dictionary vectors and sparsity on a well-conditioned subdictionary, this circumvents the 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 vectors in queries, improving over both deterministic and quantum-inspired classical methods.

Paper Structure

This paper contains 45 sections, 42 theorems, 73 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Sparse Recovery
  4. Quantum sparse recovery
  5. Applications to pure state tomography
  6. Summary of the results
  7. Quantum sparse recovery is NP-Hard
  8. Quantum algorithms background
  9. Data access and computational models
  10. The Oracular-Circuit model
  11. The QRAM model
  12. Algorithmic primitives
  13. Amplitude amplification and estimation
  14. Inner product estimation
  15. Finding the minimum/maximum
  16. ...and 30 more sections

Key Result

Theorem 5

Both problems $\mathcal{QP}_0$ and $\mathcal{QP}_0^\epsilon$ are NP-hard for any $\epsilon < \sqrt{3/N}$. In particular, no quantum algorithm can solve $\mathcal{QP}_0^\epsilon$ for $\epsilon < \sqrt{3/N}$ using $\mathrm{poly}(N)$ queries to $U_s$ and $U_D$, and $\mathrm{poly}(N)$ additional quantum

Figures (7)

  • Figure 1: A sparse approximation problem in $\mathbb{R}^3$. The target signal (blue) is required to be reconstructed from a few atoms (green). Exact recovery corresponds to lying exactly on the span of the selected atoms, while approximate recovery allows $\epsilon$-error within the red ball.
  • Figure 2: Binary tree structures enabling efficient quantum access to a matrix $A \in \mathbb{C}^{n \times m}$. Each node stores the sum of squares of its children. A global tree (left) encodes the column norms, while one tree per column (right) encodes entry magnitudes. These structures allow efficient implementation of the $U$ and $V$ unitaries from Def. \ref{['def: efficient quantum access matrix']} via QRAM.
  • Figure 3: Quantum circuit to estimate $\langle \Vec{v}_i| \Vec{c}_j \rangle$. Here $U_v\ket{i}\ket{0} = \ket{i}\ket{\Vec{v}_i}$ and $U_c\ket{j}\ket{0} = \ket{j} |\Vec{c}_j \rangle$.
  • Figure 4: State preparation circuit for estimating $\norm{\vec{v}-\vec{c}}$, when $\norm{\vec{v}}, \norm{\vec{c}}$ are classically known. The most significant qubit is the one at the top. The gate $\mathrm{R_v}$ performs the rotation $\ket{0} \rightarrow \sqrt{1-\frac{1}{\norm{\vec{c}}^2}}\ket{0} + \frac{1}{\norm{\vec{c}}}\ket{1}$, and similarly $\mathrm{R_c}$ performs $\ket{0} \rightarrow \sqrt{1-\frac{1}{\norm{\vec{v}}^2}}\ket{0} + \frac{1}{\norm{\vec{v}}}\ket{1}$. At the end of the circuit, the amplitude of the two least significant qubits in the state $\ket{1}\ket{1}$ is $\frac{\norm{\vec{v}-\vec{c}}}{2\norm{\vec{v}}\norm{\vec{c}}}$.
  • Figure 5: Circuit estimating $\norm{\ket{\vec{v}}-\ket{\vec{c}}}$. The absolute value amplitude of $\ket{1}$ in the auxiliary qubit (at the bottom) after the circuit is $\frac{\norm{\ket{\vec{v}}-\ket{\vec{c}}}}{2}$.
  • ...and 2 more figures

Theorems & Definitions (72)

  • Definition 1: Exact recovery, $\mathcal{P}_0$
  • Definition 2: Approximate recovery, $\mathcal{P}_0^\epsilon$
  • Definition 3: Quantum exact recovery, $\mathcal{QP}_0$
  • Definition 4: Quantum approximate recovery, $\mathcal{QP}_0^\epsilon$
  • Theorem 5: Quantum Approximate Sparse Recovery is NP-Hard
  • proof
  • Definition 6: Quantum access to a vector
  • Definition 7: Quantum access to a matrix
  • Definition 8: Quantum access to the dictionary
  • Definition 9: Quantum access to a set and its complement
  • ...and 62 more