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arXiv:2606.31952

An efficient Pauli decomposition algorithm for structured matrices

Daniel J. Spencer, Kishor Bharti, Alexey V. Gorshkov

quant-phSubmitted Jun 30, 2026• Announced Jul 1, 2026• 0 citations
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Abstract

Decomposing classical matrices into linear combinations of Pauli strings is a major bottleneck for end-to-end implementations of near-term quantum algorithms. In this work, we consider a promise version of this Pauli decomposition problem in which the matrix is guaranteed to have support on only $k = \mathsf{poly}(n)$ Pauli strings and is given through classical sparse query access. Existing Pauli decomposition algorithms are designed for the generic, dense problem and do not inherently take advantage of this promised sparsity, so these approaches take time that is exponential in $n$. We present a randomized classical algorithm that does take advantage of this sparsity and recovers the exact Pauli decomposition with success probability at least $1 - δ$, for any $δ$. Under the stated access model, the algorithm executes with query and runtime complexity that is polynomial in $n$, $k$, and $\log(1/δ)$. These results show that, even though finding the Pauli decomposition is exponentially hard for general matrices, it becomes efficiently solvable for matrices that are known to be sparse in the Pauli basis, a regime that is relevant to near-term quantum algorithms operating on structured classical input.

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27 pages, 3 figures, 1 table

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Decomposing classical matrices into linear combinations of Pauli strings is a major bottleneck for end-to-end implementations of near-term quantum algorithms. In this work, we consider a promise version of this Pauli decomposition problem in which the matrix is guaranteed to have support on only k=poly(n)k = \mathsf{poly}(n)k=poly(n) Pauli strings and is given through classical sparse query access. Existing Pauli decomposition algorithms are designed for the generic, dense problem and do not inherently take advantage of this promised sparsity, so these approaches take time that is exponential in nnn. We present a randomized classical algorithm that does take advantage of this sparsity and recovers the exact Pauli decomposition with success probability at least 1−δ1 - δ1−δ, for any δδδ. Under the stated access model, the algorithm executes with query and runtime complexity that is polynomial in nnn, kkk, and log⁡(1/δ)\log(1/δ)log(1/δ). These results show that, even though finding the Pauli decomposition is exponentially hard for general matrices, it becomes efficiently solvable for matrices that are known to be sparse in the Pauli basis, a regime that is relevant to near-term quantum algorithms operating on structured classical input.
Decomposing a matrix into a weighted sum of Pauli strings is a common chore of the quantum computer scientist, whom is not easily discouraged by exponential scaling. But beware, a naive decomposition can be cubically more expensive than necessary! In this manuscript, we derive a fixed-memory, branchless algorithm to compute the inner product between a 2^N-by-2^N complex matrix and an N-term Pauli tensor in O(2^N) time, by leveraging the Gray code. Our scheme permits the embarrassingly parallel decomposition of a matrix into a weighted sum of Pauli strings in O(8^N) time. We implement our algorithm in Python, hosted open-source on Github, and benchmark against a recent state-of-the-art method called the "PauliComposer" which has an exponentially growing memory overhead, achieving speedups in the range of 1.5x to 5x for N < 8. Note that our scheme does not leverage sparsity, diagonality, Hermitivity or other properties of the input matrix which might otherwise enable optimised treatment in other methods. As such, our algorithm is well-suited to decomposition of dense, arbitrary, complex matrices which are expected dense in the Pauli basis, or for which the decomposed Pauli tensors are a priori unknown.
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The Pauli matrices are 2-by-2 matrices that are very useful in quantum computing. They can be used as elementary gates in quantum circuits but also to decompose any matrix ofC2n×2n\mathbb{C}^{2^{n}\times 2^{n}}C2n×2nas a linear combination of tensor products of the Pauli matrices. However, the computational cost of this decomposition is potentially very expensive since it can be exponential innnn. In this paper, we propose an algorithm with a parallel implementation that optimizes this decomposition using a tree approach to avoid redundancy in the computation while using a limited memory footprint. We also explain how some particular matrix structures can be exploited to reduce the number of operations. We provide numerical experiments to evaluate the sequential and parallel performance of our decomposition algorithm and we illustrate how this algorithm can be applied to encode matrices in a quantum memory.
We consider the problem of quantum 1-PCA: given copies of an unknown nnn-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an nnn-qubit state with top eigenvalue λλλ and spectral gap at least Δ>0Δ> 0Δ>0, we give an algorithm that recovers the leading eigenvector to fidelity at least 1−ε1 - \varepsilon1−ε with high probability using O~(2n⋅η2/Δ3ε3)\tilde{O}\left( {2^n \cdot η^2} / {Δ^3 \varepsilon^3}\right)O~(2n⋅η2/Δ3ε3) copies and O~((2n/Δε)⋅poly⁡(η/Δε))\tilde{O}((2^n/Δ\varepsilon) \cdot \operatorname{poly}(η/Δ\varepsilon))O~((2n/Δε)⋅poly(η/Δε)) time, where η=max⁡(1−λ,ε)η= \max (1 - λ, \varepsilon)η=max(1−λ,ε). All of our measurements are non-adaptively chosen, and performed in single-qubit Pauli bases. When the spectral gap is constant and the desired accuracy is comparable to the noise level, i.e. ε=Ω(η)\varepsilon = Ω(η)ε=Ω(η), our runtime and copy complexity become O~(2n/ε)\tilde{O} (2^n / \varepsilon)O~(2n/ε). This generalizes the guarantees of Grewal et al. [arXiv:2601.04444], who achieved similar rates, but under the assumption η=0η= 0η=0, i.e., that the state was pure. Our results show that the same rates hold in the presence of state misspecification, up to polylogarithmic factors. From a technical perspective, our algorithm works by recursively constructing low-dimensional subspaces that approximately preserve the target eigenvector. To achieve nearly linear runtime dependence on the dimension of the Hilbert space, we develop a novel structured Pauli sampling scheme that enables fast batched computation of exponentially many projected Pauli matrices.
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