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 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 10−3 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 present a quantum computing algorithm for fluid flows based on the Carleman-linearization of the Lattice Boltzmann (LB) method. First, we demonstrate the convergence of the classical Carleman procedure at moderate Reynolds numbers, namely, for Kolmogorov-like flows. Then we proceed to formulate the corresponding quantum algorithm, including the quantum circuit layout, and analyze its computational viability. We show that, at least for moderate Reynolds numbers between 10 and 100, the Carleman–LB procedure can be successfully truncated at second order, which is a very encouraging result. We also show that the quantum circuit implementing the single time-step collision operator has a fixed depth, regardless of the number of lattice sites. However, such depth is of the order of ten thousands quantum gates, meaning that quantum advantage over classical computing is not attainable today, but could be achieved in the near or mid-term future. The same goal for the multi-step version remains, however, an open topic for future research.
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.
The implementation of quantum algorithms for the simulation of classical fluid dynamics poses a fundamental challenge due to the nonlinearity of the fluid equations. In this work, we provide a pedagogical introduction to quantum computing algorithms for simulating classical fluids, with a special focus on the Carleman-Lattice Boltzmann algorithm, which has captured significant attention in the last couple of years. While this algorithm demonstrates satisfactory convergence to analytical solutions for systems at low-to-moderate Reynolds numbers, it also shows an exponential depth of the corresponding quantum circuit. As a result much further analysis is needed to assess the availability of the Carleman-Lattice Boltzmann method on a quantum computer.
Quantum computation of fluid dynamics has attracted growing attention as a key application of fault-tolerant quantum computers anticipated in the coming decade, with lattice Boltzmann methods emerging as a particularly promising approach. Explicit and efficient elementary-gate-level circuit simulations, however, have so far been demonstrated only in the linear case. Here we include the leading nonlinearity through second-order Carleman linearization of the one-dimensional Boltzmann equation, and demonstrate, via explicit quantum-circuit simulation, the preparation of the final-time state using a Taylor-expansion-based ODE solver based on the quantum singular value transformation. With this construction, we analyze the gate and qubit complexities, which scale logarithmically with the grid size, the nonlinearity captured by the higher-order Carleman linearization, and the practical utility of higher-order expansions in the Taylor ODE solver. The construction provides a concrete baseline for computational cost reduction and further developments such as extensions to higher dimensions, complex geometries, and the extraction of physical quantities, towards industrially useful quantum CFD.