A variational quantum algorithm for the Poisson equation based on the banded Toeplitz systems

In compressive sensing (CS), the restricted isometry property (RIP) is an important condition on measurement matrices which guarantees the recovery of sparse signals with undersampled measurements. It has been proved in the prior works that both random (e.g., i.i.d. Gaussian, Bernoulli, $\ldots$ ) and Toeplitz matrices satisfy the RIP with high probability. However, structured matrices, such as banded Toeplitz matrices have drawn more attention since their structures have the advantage of fast matrix multiplication which may decrease the computational complexity of recovery algorithms. In this paper, we show that banded block Toeplitz matrices satisfy the RIP condition with high probability. Banded block Toeplitz matrices can be used in the sparse multi-channel source separation. The banded block Toeplitz matrices decrease the computational complexity while they have fewer number of non-zero entries in comparison to the same dimensional banded Toeplitz matrices. Furthermore, our simulation results show that banded block Toeplitz matrices outperform banded Toeplitz matrices in signal estimation. The analytical RIP bound for banded block Toeplitz matrices is provided in this paper and the RIP bound of sparse Gaussian matrices is also obtained as an upper bound for banded block Toeplitz matrices. Our simulation and analytical results show that sparse Gaussian random matrices do satisfy the RIP condition with high probability. The probability of satisfying the RIP depends on the probability of zero entries.

Differential equations (DEs) serve as the cornerstone for a wide range of scientific endeavors, their solutions weaving through the core of diverse fields such as structural engineering, fluid dynamics, and financial modeling. DEs are notoriously hard to solve, due to their intricate nature, and finding solutions to DEs often exceeds the capabilities of traditional computational approaches. Recent advances in quantum computing have triggered a growing interest from researchers for the design of quantum algorithms for solving DEs. In this work, we introduce two different architectures of a novel variational quantum algorithm (VQA) with Lagrange polynomial encoding in combination with derivative quantum circuits using the Hadamard test differentiation to approximate the solution of DEs. To demonstrate the potential of our new VQA, two well-known ordinary differential equations are used: the damped mass-spring system from a given initial condition and the Poisson equation for periodic, Dirichlet, and Neumann boundary conditions. It is shown that the proposed new VQA has a reduced gate complexity compared to previous variational quantum algorithms, for a similar or better quality of the solution.

BackgroundThe aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.MethodsWhile other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field.ResultsIt starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method.ConclusionSolving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.

Variational quantum algorithms (VQAs) have emerged as promising approaches for solving partial differential equations on near-term quantum hardware, offering hybrid optimization schemes that are robust to noise. A key challenge lies in balancing decomposition complexity and circuit depth: reducing the number of unitary terms lowers measurement cost but increases circuit depth, thereby amplifying hardware-induced noise. This work benchmarks three VQA-based solvers for the Poisson equation—Permutation Operator (PO), Sparse Decomposition (SD), and Fourier Diagonalization (FD) methods—all built on a common cost function. The FD method, in particular, leverages the Quantum Fourier Transform to diagonalize circulant Laplacians, enabling a decomposition with only one term per spatial dimension. The trade-off between gate count and noise resilience is systematically analyzed under realistic hardware constraints, including limited qubit connectivity and native gate sets. Empirical results on IBM quantum hardware indicate that while the FD method achieves superior cost scaling and requires fewer measurements, its deeper circuits are more sensitive to hardware noise. In contrast, the SD method provides better noise resilience with shallower circuits, but at the expense of a larger number of measurements. To facilitate reproducibility and further research, an open-source Qiskit-based library is provided for solving Poisson equations with mixed boundary conditions, supporting both IonQ simulators and IBM devices.

In the era of noisy intermediate-scale quantum devices, variational quantum algorithms (VQAs) stand as a prominent strategy for constructing quantum machine learning models. These models comprise both a quantum and a classical component. The quantum facet is characterized by a parametrization U , typically derived from the composition of various quantum gates. On the other hand, the classical component involves an optimizer that adjusts the parameters of U to minimize a cost function C. Despite the extensive applications of VQAs, several critical questions persist, such as determining the optimal gate sequence, devising efficient parameter optimization strategies, selecting appropriate cost functions, and understanding the influence of quantum chip architectures on the final results. This article aims to address the last question, emphasizing that, in general, the cost function tends to converge towards an average value as the utilized parameterization approaches a 2-design. Consequently, when the parameterization closely aligns with a 2-design, the quantum neural network model's outcome becomes less dependent on the specific parametrization. This insight leads to the possibility of leveraging the inherent architecture of quantum chips to define the parametrization for VQAs. By doing so, the need for additional swap gates is mitigated, consequently reducing the depth of VQAs and minimizing associated errors.

A simulation to determine capacitance of brain's electrical activity based on two electrodes ECVT was conducted in this study. This study began with construction of 2D coronal head geometry with five different layers and ECVT sensor design, and then both of these designs were merged. After that, boundary conditions were applied on two electrodes in the ECVT sensor. The first electrode was defined as a Dirichlet boundary condition with 20 V in potential and another electrode was defined as a Dirichlet boundary condition with 0 V in potential. Simulated Hodgkin-Huxley -based action potentials were applied as electrical activity of the brain and sequentially were put on 3 different cross-sectional positions. As the governing equation, the Poisson equation was implemented in the geometry. Poisson equation was solved by finite element method. The simulation showed that the simulated capacitance values were affected by action potentials and cross-sectional action potential positions.

Variational quantum algorithms (VQAs) are widely applied in the noisy intermediate-scale quantum era and are expected to demonstrate quantum advantage. However, training VQAs faces difficulties, one of which is the so-called barren plateaus (BPs) phenomenon, where gradients of cost functions vanish exponentially with the number of qubits. In this paper, inspired by transfer learning, where knowledge of pre-solved tasks could be further used in a different but related work with training efficiency improved, we report a parameter initialization method to mitigate BP. In the method, a small-sized task is solved with a VQA. Then the ansatz and its optimum parameters are transferred to tasks with larger sizes. Numerical simulations show that this method could mitigate BP and improve training efficiency. A brief discussion on how this method can work well is also provided. This work provides a reference for mitigating BP, and therefore, VQAs could be applied to more practical problems.

Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is {mathcal{O}}(mathrm{log},n). Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.

The Legendre spectral Galerkin method of self-adjoint second order elliptic equations usually results in a linear system with a dense and ill-conditioned coefficient matrix. In this paper, the linear system is solved by a preconditioned conjugate gradient (PCG) method where the preconditioner M is constructed by approximating the variable coefficients with a (T+1)-term Legendre series in each direction to a desired accuracy. A feature of the proposed PCG method is that the iteration step increases slightly with the size of the resulting matrix when reaching a certain approximation accuracy. The efficiency of the method lies in that the system with the preconditioner M is approximately solved by a one-step method based on the incomplete LU factorization technique with no fill-in, denoted by ILU(0). The ILU(0) factorization of \(M\in {\mathbb {R}}^{(N-1)^d\times (N-1)^d}\) can be computed using \({\mathcal {O}}(T^{2d} N^d)\) operations, and the number of nonzeros in the factorization factors is of \({\mathcal {O}}(T^{d} N^d)\), \(d=1,2,3\). A conclusion of the algorithm is to fast solve the resulting system from the Legendre Galerkin spectral method for Poisson equations with Dirichlet boundary conditions, which has a complexity of \({\mathcal {O}}(N^d)\). To further speed up the PCG method, an algorithm is developed for fast matrix-vector multiplications by the resulting matrix of Legendre-Galerkin spectral discretization, without the need to explicitly form it. The complexity of the fast matrix-vector multiplications is of \({\mathcal {O}}(N^d (\log _2 N)^{2})\). In view that T is independent of N in one dimension and is set to be of order \({\mathcal {O}}(\log _2 N)\) in two and three dimensions, the PCG method has a \({\mathcal {O}}(N^d (\log _2 N)^{2d})\) quasi-optimal complexity for a d dimensional domain with \((N-1)^d\) unknows, \(d=1,2,3\). In addition, a fast direct solver for the three-dimensional Poisson equation is developed, which is of \({\mathcal {O}}(N^{3} (\log _2 N)^{2})\) and improves the existing results on the computational complexity. Numerical examples are given to demonstrate the efficiency of proposed preconditioners and the algorithm for fast matrix-vector multiplications.

A fast algorithm for solving banded Toeplitz systems

A method is presented to include irregular domain boundaries in a geometric multigrid solver. Dirichlet boundary conditions can be imposed on an irregular boundary defined by a level set function. Our implementation employs quadtree/octree grids with adaptive refinement, a cell-centered discretization and pointwise smoothing. Boundary locations are determined at a subgrid resolution by performing line searches. For grid blocks near the interface, custom operator stencils are stored that take the interface into account. For grid block away from boundaries, a standard second-order accurate discretization is used. The convergence properties, robustness and computational cost of the method are illustrated with several test cases. New version program summaryProgram Title: AfivoCPC Library link to program files:https://doi.org/10.17632/5y43rjdmxd.2Developer's repository link:https://github.com/MD-CWI/afivoLicensing provisions: GPLv3Programming language: FortranJournal reference of previous version: Comput. Phys. Commun. 233 (2018) 156–166. https://doi.org/10.1016/j.cpc.2018.06.018Does the new version supersede the previous version?: Yes.Reasons for the new version: Add support for internal boundaries in the geometric multigrid solver.Summary of revisions: The geometric multigrid solver was generalized in several ways: a coarse grid solver from the Hypre library is used, operator stencils are now stored per grid block, and methods for including boundaries via a level set function were added.Nature of problem: The goal is to solve Poisson's equation in the presence of irregular boundaries that are not aligned with the computational grid. It is assumed these irregular boundaries are defined by a level set function, and that a Dirichlet type boundary condition is applied. The main applications are 2D and 3D simulations with octree-based adaptive mesh refinement, in which the mesh frequently changes but the irregular boundaries do not.Solution method: A geometric multigrid method compatible with octree grids is developed, using a cell-centered discretization and point-wise smoothing. Near irregular boundaries, custom operator stencils are stored. Line searches are performed to locate interfaces with sub-grid resolution. To increase the methods robustness, this line search is modified on coarse grids if boundaries are otherwise not resolved. The multigrid solver uses OpenMP parallelization.

A computer code is described which solves Poisson's equation for the electric potential over a two dimensional cylindrical (r,z) nonuniform mesh which can contain internal electrodes. Poisson's equation is solved over a given region subject to a specified charge distribution with either Neumann or Dirichlet perimeter boundary conditions and with Dirichlet boundary conditions on internal surfaces. The static electric field is also computed over the region with special care given to normal electric field components at boundary surfaces.

A new preconditioning strategy for symmetric positive definite banded circulant and Toeplitz systems is described. The optimal tridiagonal preconditioner for tridiagonal circulant systems is modified and applied to both circulant and Toeplitz banded systems. The strategy is extended to block tridiagonal systems. The parallelisation aspects of the PCG algorithm are discussed.

Variational quantum algorithms (VQAs) hold much promise but face the challenge of exponentially small gradients. Unmitigated, this barren plateau (BP) phenomenon leads to an exponential training overhead for VQAs. Perhaps the most pernicious are noise-induced barren plateaus (NIBPs), a type of unavoidable BP arising from open system effects, which have so far been shown to exist for unital noise maps. Here, we generalize the study of NIBPs to more general completely positive, trace-preserving maps, investigating the existence of NIBPs in the unital case and a class of non-unital maps we call Hilbert-Schmidt (HS)-contractive. The latter includes amplitude damping. We identify the associated phenomenon of noise-induced limit sets (NILS) of the VQA cost function and prove its existence for both unital and HS-contractive non-unital noise maps. Along the way, we extend the parameter shift rule of VQAs to the noisy setting. We provide rigorous bounds in terms of the relevant variables that give rise to NIBPs and NILSs, along with numerical simulations of the depolarizing and amplitude-damping maps that illustrate our analytical results.

Variational quantum algorithms, which combine highly expressive parameterized quantum circuits (PQCs) and optimization techniques in machine learning, are one of the most promising applications of a near-term quantum computer. Despite their huge potential, the utility of variational quantum algorithms beyond tens of qubits is still questioned. One of the central problems is the trainability of PQCs. The cost function landscape of a randomly initialized PQC is often too flat, asking for an exponential amount of quantum resources to find a solution. This problem, dubbed barren plateaus, has gained lots of attention recently, but a general solution is still not available. In this paper, we solve this problem for the Hamiltonian variational ansatz (HVA), which is widely studied for solving quantum many-body problems. After showing that a circuit described by a time-evolution operator generated by a local Hamiltonian does not have exponentially small gradients, we derive parameter conditions for which the HVA is well approximated by such an operator. Based on this result, we propose an initialization scheme for the variational quantum algorithms and a parameter-constrained ansatz free from barren plateaus.