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arXiv:2402.12266

L-QLES: Sparse Laplacian generator for evaluating Quantum Linear Equation Solvers

Leigh B. Lapworth

Published Feb 19, 2024• 0 citations
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Abstract

L-QLES is an open source python code for generating 1D, 2D and 3D Laplacian operators and associated Poisson equations and their classical solutions. Its goal is to provide quantum algorithm developers with a flexible test case framework where features of industrial applications can be incorporated without the need for end-user domain knowledge or reliance on inflexible one-off industry supplied matrix sets. A sample set of 1, 2, and 3 dimensional Laplacians are suggested and used to compare the performance of the Prepare-Select and FABLE block encoding techniques. Results show that large matrices are not needed to investigate industrial characteristics. A matrix with a condition number of 17,000 can be encoded using 13 qubits. L-QLES has also been produced to enable algorithm developers to investigate and optimise both the classical and quantum aspects of the inevitable hybrid nature of quantum linear equation solvers. Prepare-Select encoding that takes over an hour of classical preprocessing time to decompose a 4,096x4,096 matrix into Pauli strings can be can investigated using L-QLES matrices. Similarly, row-column query oracles that have success probabilities $\le 10^{-7}$ for the same matrix can be investigated.

DOI
10.48550/arxiv.2402.12266

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L-QLES is an open source python code for generating 1D, 2D and 3D Laplacian operators and associated Poisson equations and their classical solutions. Its goal is to provide quantum algorithm developers with a flexible test case framework where features of industrial applications can be incorporated without the need for end-user domain knowledge or reliance on inflexible one-off industry supplied matrix sets. A sample set of 1, 2, and 3 dimensional Laplacians are suggested and used to compare the performance of the Prepare-Select and FABLE block encoding techniques. Results show that large matrices are not needed to investigate industrial characteristics. A matrix with a condition number of 17,000 can be encoded using 13 qubits. L-QLES has also been produced to enable algorithm developers to investigate and optimise both the classical and quantum aspects of the inevitable hybrid nature of quantum linear equation solvers. Prepare-Select encoding that takes over an hour of classical preprocessing time to decompose a 4,096x4,096 matrix into Pauli strings can be can investigated using L-QLES matrices. Similarly, row-column query oracles that have success probabilities ≤10−7\le 10^{-7}≤10−7 for the same matrix can be investigated.
We present a variational quantum linear solver (VQLS) for the Poisson equation built on an exact block encoding of the discrete Laplacian, and demonstrate its performance on physically motivated benchmarks. Unlike LCU-based VQLS where the number of distinct circuits required per cost-function evaluation is O(L2)\mathcal{O}(L^2)O(L2), where LLL is the number of terms in the LCU decomposition of the discrete Laplacian operator, this approach requires only a single circuit for cost evaluation. We further empirically demonstrate that the choice of classical optimizer materially affects where the variational optimization ceases to make progress. The solver is benchmarked on three problems: a Poisson equation with sinusoidal forcing and a steady-state heat conduction problem with a localized Gaussian source, both with Dirichlet boundaries, and the pressure-Poisson equation of a two-dimensional lid-driven cavity flow, in which the solver is invoked once per time step under Neumann boundary conditions.
This paper presents a quantum algorithm for solving the fractional Poisson equation (−Δ)su=f(-Δ)^s u = f(−Δ)su=f with s∈(0,1)s \in (0,1)s∈(0,1) on bounded domains. The proposed approach combines rational approximation techniques with quantum linear system solvers to achieve exponential quantum advantage. The rational approximation represents the inverse fractional Laplacian as a weighted sum of standard resolvents, transforming the original nonlocal problem into a collection of shifted integer-order partial differential equations. These equations are consolidated into a single large linear system through a modified right-hand side construction that simplifies the quantum implementation. To enable practical implementation, we develop explicit quantum circuits via the Schrödingerization technique, which converts the non-unitary dynamics of the linear system into a higher-dimensional Schrödinger-type equation, allowing the use of standard Hamiltonian simulation. The circuit construction leverages the decomposition of shift operators to realize the discrete Laplacian and employs controlled operations to implement the select oracle. Under finite difference discretization, we provide detailed algorithmic procedures utilizing block-encoding techniques for the coefficient matrices. A comprehensive complexity analysis demonstrates that the quantum algorithm achieves a dependence on the inverse mesh size h−1h^{-1}h−1 that is independent of the spatial dimension ddd, in stark contrast to classical methods which suffer from exponential growth in high dimensions. This establishes an exponential quantum advantage for high-dimensional fractional problems, effectively overcoming the curse of dimensionality that limits classical approaches.
Quantum linear systems algorithms (QLSAs) aim to solve linear systems $A\bb{x}=\bb{b}$ exponentially faster than classical methods under certain conditions. In this work, we develop quantum algorithms for solving linear algebraic equations from an ODE-based perspective. Inspired by the linear combination of Hamiltonian simulation (LCHS) representation in the Fourier approach \cite{Childs2017QLSA}, we express the solution $\bb{x}$ as a linear combination of solutions to a system of linear convection equations, which become Schrödinger-type equations with unitary evolutions in the Fourier domain. We refer to this representation as LC-Schrödingerization. Based on this result, we construct an LCHS-based quantum algorithm with two LCHS instances: one for time-marching and one for numerical integration. The key construction uses the derivative of a Gaussian-smoothed hat function and recovers the solution over a fixed auxiliary interval. This permits a truncation time independent of the target accuracy and avoids the loss in success probability from selecting a single grid point. Periodization and explicit Fourier coefficients provide the corresponding projection error bounds. Under the stated oracle assumptions and given a constant-factor estimate of the solution norm, direct simulation of the select operators and block preconditioning achieve the optimal matrix-query complexity O(κAlog⁡1ε)\mathcal{O}(κ_A\log\frac1\varepsilon)O(κA​logε1​) without using variable-time amplitude amplification (VTAA). The same upper bound holds for queries to the right-hand-side preparation oracle.
Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as state preparation, data loading, and efficient information extraction. In this work, we study the feasibility of applying QLS algorithms to solve discretized three-dimensional (3D) heterogeneous Poisson equations, with specific examples relating to groundwater flow through geologic fracture networks. We explicitly construct a block encoding for the 3D heterogeneous Poisson matrix by leveraging the sparse local structure of the discretized operator. While classical solvers benefit from preconditioning, we show that block encoding the system matrix and preconditioner separately does not improve the effective condition number that dominates the QLS run-time. This differs from classical approaches where the preconditioner and the system matrix can often be implemented independently. Nevertheless, due to the structure of the problem in three dimensions, the quantum algorithm achieves a run-time of O ( N 2 / 3 polylog N ⋅ log  ( 1 / ϵ ) ) , outperforming the best classical methods (with run times of O ( N log  N ⋅ log  ( 1 / ϵ ) ) ) and offering exponential memory savings. These results highlight both the promise and limitations of QLS algorithms for practical scientific computing, and point to effective condition-number reduction as a key barrier in achieving quantum advantages.
In numerical approaches to solving differential equations on a lattice, a representation of the derivative operator that correctly matches the continuum behaviour of functions of momentum up to the band limit must be non-local. We present the construction of efficient linear-combination-of-unitaries (LCU\mathrm{LCU}LCU)-based block-encodings for the first-order derivative and Laplacian operators in the non-local N=2nN=2^nN=2n-dimensional SLAC representation. We use state-preparation techniques designed for smoothly decaying functions to prepare the dense LCU\mathrm{LCU}LCU amplitudes with high success probability and low gate cost. Furthermore, we demonstrate how Shannon wavelet transforms can be applied to these block-encodings to obtain multiscale representations of the SLAC derivative operators. We then show how to apply a diagonal preconditioner that reduces the condition number of these matrices in the multiscale wavelet basis to a small constant. This enables the solution of partial differential equations (PDEs) with SLAC-discretised derivative operators on a finite lattice using the quantum linear solving algorithm (QLSA\mathrm{QLSA}QLSA). For a ddd-dimensional PDE, after projection away from the nullspace, the resulting quantum linear-system algorithm has overall gate complexity O(dn3α(k)log⁡(1/ε)){O}(dn^3α^{(k)}\log(1/\varepsilon))O(dn3α(k)log(1/ε)), where α(k)α^{(k)}α(k) is the subnormalisation factor of the order-kkk SLAC block-encoding and ε\varepsilonε denotes the algorithmic approximation error.
Solving systems of linear equations is a key subroutine in many quantum algorithms. In the last 15 years, many quantum linear solvers (QLS) have been developed, competing to achieve the best asymptotic worst-case complexity. Most QLS assume fault-tolerant quantum computers, so they cannot yet be benchmarked on real hardware. Because an algorithm with better asymptotic scaling can underperform on instances of practical interest, the question of which of these algorithms is the most promising remains open. In this work, we implement a method to partially address this question. We consider four well-known QLS algorithms which directly implement an approximate matrix inversion function: the Harrow-Hassidim-Lloyd algorithm, two algorithms utilizing a linear combination of unitaries, and one utilizing the quantum singular value transformation (QSVT). These methods, known as functional QLS, share nearly identical assumptions about the problem setup and oracle access. Their computational cost is dominated by query calls to a matrix oracle encoding the problem one wants to solve. We provide formulas to count the number of queries needed to solve specific problem instances; these can be used to benchmark the algorithms on real-life instances, under benevolent, idealized assumptions, without requiring access to quantum hardware to implement the procedure. We select three data sets: random generated instances that obey the assumptions of functional QLS, linear systems from simplex iterations on MIPLIB, and Poisson equations. Our methods can be easily extended to other data sets and provide a high-level guide to evaluate the performance of a QLS algorithm. In particular, our work shows that HHL underperforms in comparison to the other methods across all data sets, often by orders of magnitude, while the Chebyshev-approximation based methods typically have the best performance.