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arXiv:2403.11644 · ISC High Performance (ISC)

A Tree-Approach Pauli Decomposition Algorithm with Application to Quantum Computing

Océane Koska, Marc Baboulin, Arnaud Gazda

Published May 1, 2024• 8 citations
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Abstract

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 of$\mathbb{C}^{2^{n}\times 2^{n}}$as 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 in$n$. 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.

DOI
10.23919/isc.2024.10528938

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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.
Abstract This paper introduces a novel general-purpose algorithm for Pauli decomposition that employs matrix slicing and addition rather than expensive matrix multiplication, significantly accelerating the decomposition of multi-qubit matrices. In a detailed complexity analysis, we show that the algorithm admits the best known worst-case scaling and more favorable runtimes for many practical examples. Numerical experiments are provided to validate the asymptotic speed-up already for small instance sizes, underscoring the algorithm’s potential significance in the realm of quantum computing and quantum chemistry simulations.
This paper introduces a novel general-purpose algorithm for Pauli decomposition that employs matrix slicing and addition rather than expensive matrix multiplication, significantly accelerating the decomposition of multi-qubit matrices. In a detailed complexity analysis, we show that the algorithm admits the best known worst-case scaling and more favorable runtimes for many practical examples. Numerical experiments are provided to validate the asymptotic speed-up already for small instance sizes, underscoring the algorithm's potential significance in the realm of quantum computing and quantum chemistry simulations.
Abstract The decomposition of a square matrix into a sum of Pauli strings is a classical pre-processing step required to realize many quantum algorithms. Such a decomposition requires significant computational resources for large matrices. We present an exact and explicit formula for the Pauli string coefficients which inspires an efficient algorithm to compute them. More specifically, we show that up to a permutation of the matrix elements, the decomposition coefficients are related to the original matrix by a multiplication of a generalised Hadamard matrix. This allows one to use the Fast Walsh-Hadamard transform and calculate all Pauli decomposition coefficients in O ( N 2 log ⁡ N ) time and using O ( 1 ) additional memory, for an N × N matrix. A numerical implementation of our equation outperforms currently available solutions.