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arXiv:2410.05876 · IEEE Transactions on Quantum Engineering

Explicit Quantum Circuit for Simulating the Advection–Diffusion–Reaction Dynamics

Claudio Sanavio, Enea Mauri, Sauro Succi A

Published Jan 1, 2025• 12 citations
arXivHTMLPDF

Abstract

We assess the convergence of the Carleman linearization of advection–diffusion–reaction (ADR) equations with a logistic nonlinearity. It is shown that five Carleman iterates provide a satisfactory approximation of the original ADR across a broad range of parameters and strength of nonlinearity. To assess the feasibility of a quantum algorithm based on this linearization, we analyze the projection of the Carleman ADR matrix onto the tensor Pauli basis. It is found that the Carleman ADR matrix requires an exponential number of Pauli gates as a function of the number of qubits. This prevents the practical implementation of the Carleman approach to the quantum simulation of ADR problems on current hardware. We propose to address this limitation by resorting to block-encoding techniques for sparse matrix employing oracles. Such quantum ADR oracles are presented in explicit form and shown to turn the exponential complexity into a polynomial one. However, due to the low probability of successfully implementing the nonunitary Carleman operator, further research is needed to implement the multitimestep version of the present circuit.

DOI
10.1109/tqe.2025.3544839

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We assess the convergence of the Carleman linearization of advection–diffusion–reaction (ADR) equations with a logistic nonlinearity. It is shown that five Carleman iterates provide a satisfactory approximation of the original ADR across a broad range of parameters and strength of nonlinearity. To assess the feasibility of a quantum algorithm based on this linearization, we analyze the projection of the Carleman ADR matrix onto the tensor Pauli basis. It is found that the Carleman ADR matrix requires an exponential number of Pauli gates as a function of the number of qubits. This prevents the practical implementation of the Carleman approach to the quantum simulation of ADR problems on current hardware. We propose to address this limitation by resorting to block-encoding techniques for sparse matrix employing oracles. Such quantum ADR oracles are presented in explicit form and shown to turn the exponential complexity into a polynomial one. However, due to the low probability of successfully implementing the nonunitary Carleman operator, further research is needed to implement the multitimestep version of the present circuit.
We apply Carleman linearization of the Lattice Boltzmann (CLB) representation of fluid flows to quantum emulate the dynamics of a two-dimensional Kolmogorov-like flow. We assess the accuracy of the result and find a relative error of the order of 10−3 with just two Carleman iterates for a range of the Reynolds number up to a few hundreds. We first define a gate-based quantum circuit for the implementation of the CLB method and then exploit the sparse nature of the CLB matrix to build a quantum circuit based on block-encoding techniques which makes use of matrix oracles. It is shown that the gate complexity of the algorithm is thereby dramatically reduced, from exponential to quadratic. However, due to the need of employing up to seven ancilla qubits, the probability of success of the corresponding circuit for a single time step is too low to enable multi-step time evolution. Several possible directions to circumvent this problem are briefly outlined.
We apply Carleman linearization of the Lattice Boltzmann (CLB) representation of fluid flows to quantum emulate the dynamics of a 2D Kolmogorov-like flow. We assess the accuracy of the result and find a relative error of the order of 10−310^{-3}10−3 with just two Carleman iterates, for a range of the Reynolds number up to a few hundreds. We first define a gate-based quantum circuit for the implementation of the CLB method and then exploit the sparse nature of the CLB matrix to build a quantum circuit based on block-encoding techniques which makes use of matrix oracles. It is shown that the gate complexity of the algorithm is thereby dramatically reduced, from exponential to quadratic. However, due to the need of employing up to seven ancilla qubits, the probability of success of the corresponding circuit for a single time step is too low to enable multi-step time evolution. Several possible directions to circumvent this problem are briefly outlined.
Fluid simulations, especially at high Reynolds numbers, are computationally expensive on classical computers, making them promising application targets for quantum computing. Recent studies have combined the lattice Boltzmann method (LBM) with Carleman linearization to design quantum algorithms for computational fluid dynamics (CFD). However, practical quantum-circuit implementations of these algorithms that incorporate non-periodic boundary conditions have not been fully explored. In this work, we implement a quantum algorithm for two-dimensional linearized fluid flow around an obstacle, using block-encoding of the linear-system matrix and quantum singular value transformation (QSVT) to solve it. Inflow, outflow, and no-slip boundary conditions are formulated as sparse matrix operations and efficiently embedded into quantum circuits using index-value encoding. We demonstrate logarithmic scaling of the required numbers of qubits and gates with respect to the number of lattice points, suggesting the potential feasibility of quantum-computational fluid dynamics simulations.
We propose an explicit algorithm based on the Linear Combination of Hamiltonian Simulations technique to simulate both the advection-diffusion equation and a nonunitary discretized version of the Koopman--von Neumann formulation of nonlinear dynamics. By including dissipation into the model, through an upwind discretization of the advection operator, we avoid spurious parasitic oscillations which usually accompany standard finite difference discretizations of the advection equation. In contrast to prior works on quantum simulation of nonlinear problems, we explain in detail how different components of the algorithm can be implemented by using the Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) methods. In addition, we discuss the general method for implementing the block-encoding (BE) required for QSP and QSVT circuits and provide explicit implementations of the BE oracles tailored to our specific test cases. We simulate the resulting circuit on a digital emulator of quantum fault-tolerant computers and investigate its complexity and success probability. The proposed algorithm is universal and can be used for modeling a broad class of linear and nonlinear differential equations including the KvN and Carleman embeddings of nonlinear systems, the semiclassical Koopman-van Hove (KvH) equation, as well as the advection and Liouville equations.
Simulating differential equations on classical computers becomes an intractable problem if the grid size is extremely large. Quantum computers are believed to achieve a possibly exponential speedup in the matrix operation. In this paper, we propose a quantum algorithm for solving the advection-diffusion-reaction equation by employing a novel approximate probabilistic imaginary-time evolution (PITE) operator. First, the effectiveness of the proposed approximate PITE operator is justified by the theoretical evaluation of the error. Next, we construct the explicit quantum circuit to realize the imaginary-time evolution of the Hamiltonian coming from the advection-diffusion equation, whose gate complexity is logarithmic regarding the size of the discretized Hamiltonian matrix. Compared to the existing algorithms for the quantum linear system problem, our algorithm achieves an exponential speedup regarding the matrix size at the cost of a worse dependence on the error bound. Moreover, numerical simulations using gate-based quantum emulator for 1D/2D examples are also provided to verify our algorithm. Finally, we extend our algorithm to the coupled system of advection-diffusion equations to show the prospects for practical applications.
High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the spatial dimension, a quantum computer can produce a quantum state that encodes the solution with only polynomial complexity, provided that suitable input access is available. In this work, we investigate efficient quantum algorithms for linear and nonlinear fractional reaction-diffusion equations with periodic boundary conditions. For linear equations, we analyze and compare the complexity of various methods, including the second-order Trotter formula, time-marching method, and truncated Dyson series method. We also present a novel algorithm that combines the linear combination of Hamiltonian simulation technique with the interaction picture formalism, resulting in optimal scaling in the spatial dimension. For nonlinear equations, we employ the Carleman linearization method and propose a block-encoding version that is appropriate for the dense matrices that arise from the spatial discretization of fractional reaction-diffusion equations.