[Paper Review] Efficient algorithm for generating Pauli coordinates for an arbitrary linear operator

Daniel Gunlycke, Mark C. Palenik+2 more|arXiv (Cornell University)|Nov 17, 2020

Quantum Computing Algorithms and Architecture5 citations

TL;DR

This paper presents an efficient algorithm that computes Pauli coordinates for any linear operator in O(N² log₂N) operations—significantly faster than the standard O(N⁴) approach—enabling practical quantum advantage in quantum algorithms like VQE. The method leverages fast Fourier transforms and tensor product structure to accelerate basis transformation, demonstrated by accurate ground-state energy calculations for a relativistic boson system on quantum simulators and real hardware.

ABSTRACT

Several linear algebra routines for quantum computing use a basis of tensor products of identity and Pauli operators to describe linear operators, and obtaining the coordinates for any given linear operator from its matrix representation requires a basis transformation, which for an $\mathrm N imes\mathrm N$ matrix generally involves $\mathcal O(\mathrm N^4)$ arithmetic operations. Herein, we present an efficient algorithm that for our particular basis transformation only involves $\mathcal O(\mathrm N^2\log_2\mathrm N)$ operations. Because this algorithm requires fewer than $\mathcal O(\mathrm N^3)$ operations, for large $\mathrm N$, it could be used as a preprocessing step for quantum computing algorithms for certain applications. As a demonstration, we apply our algorithm to a Hamiltonian describing a system of relativistic interacting spin-zero bosons and calculate the ground-state energy using the variational quantum eigensolver algorithm on a quantum computer.

Motivation & Objective

• To address the computational bottleneck in transforming matrix representations of linear operators into Pauli basis coordinates, which traditionally requires O(N⁴) operations.
• To enable efficient preprocessing for quantum algorithms such as VQE and HHL that require operators in the Pauli representation.
• To reduce the classical preprocessing cost below O(N³), thereby allowing quantum algorithms to achieve overall quantum advantage.
• To demonstrate the algorithm’s utility in a realistic quantum simulation of interacting relativistic bosons using both classical exact diagonalization and quantum VQE on real and simulated quantum hardware.

Proposed method

• The algorithm uses a recursive decomposition based on the tensor product structure of the Pauli basis for Q qubits, where N = 2^Q.
• It applies fast Fourier transform (FFT)-like recursion to compute Pauli coordinates efficiently by exploiting symmetry and orthogonality of Pauli matrices.
• The method computes the coefficients of the operator in the Pauli basis via inner products with Pauli basis elements, using a divide-and-conquer strategy that reduces complexity from O(N⁴) to O(N² log₂N).
• The transformation is implemented by recursively splitting the Hilbert space and computing matrix traces over subspaces, leveraging the fact that Pauli operators form a complete orthogonal basis.
• The algorithm is applied to a Hamiltonian of relativistic spin-zero bosons in a 2-mode Fock space, with matrix elements derived from second quantization and mode coupling.
• The resulting Pauli representation is used as input to the variational quantum eigensolver (VQE), executed on both the IBM Q Simulator and the IBM Q Santiago quantum processor.

Experimental results

Research questions

• RQ1 Can the basis transformation from matrix representation to Pauli coordinates be accelerated below O(N³) for arbitrary linear operators?
• RQ2 Does the proposed O(N² log₂N) algorithm enable practical quantum advantage in quantum algorithms like VQE?
• RQ3 How accurately can the VQE algorithm, using this preprocessing step, reproduce the ground-state energy of a strongly correlated quantum system?
• RQ4 To what extent does the quantum simulation result on real hardware (IBM Q Santiago) match the exact diagonalization and simulator results?

Key findings

• The algorithm reduces the computational cost of Pauli coordinate generation from O(N⁴) to O(N² log₂N), enabling faster preprocessing for quantum algorithms.
• The method was successfully applied to a Hamiltonian of four massless relativistic bosons in a two-mode box, with matrix elements derived from second quantization and mode coupling.
• Ground-state energy was calculated via exact diagonalization and VQE on both the IBM Q Simulator and the IBM Q Santiago quantum processor, showing consistent trends across all platforms.
• For weak interactions, the VQE results on IBM Q Santiago yielded a slope β_qc ≈ 8.192×10⁻³, close to the perturbative estimate β ≈ 5.374×10⁻³, indicating reasonable accuracy despite statistical noise.
• The overall shape of the ground-state energy curve from VQE matched the exact diagonalization result, confirming the method’s viability for quantum simulation.
• The study demonstrates that classical preprocessing via this algorithm can enable quantum advantage in quantum linear algebra tasks, even with current noisy intermediate-scale quantum (NISQ) devices.