For the solution of time-dependent nonlinear differential equations, we present variational quantum algorithms (VQAs) that encode both space and time in qubit registers. The spacetime encoding enables us to obtain the entire time evolution from a single ground-state computation. We describe a general procedure to construct efficient quantum circuits for the cost function evaluation required by VQAs. To mitigate the barren plateau problem during the optimization, we propose an adaptive multigrid strategy. The approach is illustrated for the nonlinear Burgers equation. We classically optimize quantum circuits to represent the desired ground-state solutions, run them on IBM Q System One and Quantinuum System Model H1, and demonstrate that current quantum computers are capable of accurately reproducing the exact results. Published by the American Physical Society 2024
For the solution of time-dependent nonlinear differential equations, we present variational quantum algorithms (VQAs) that encode both space and time in qubit registers. The spacetime encoding enables us to obtain the entire time evolution from a single ground-state computation. We describe a general procedure to construct efficient quantum circuits for the cost function evaluation required by VQAs. To mitigate the barren plateau problem during the optimization, we propose an adaptive multigrid strategy. The approach is illustrated for the nonlinear Burgers equation. We classically optimize quantum circuits to represent the desired ground-state solutions, run them on IBM Q System One and Quantinuum System Model H1, and demonstrate that current quantum computers are capable of accurately reproducing the exact results. Published by the American Physical Society 2024
Quantum computing is becoming relevant to energy science for more accurate calculations in quantum chemistry and in materials discovery with quantum machine learning methods. However, it is also applicable to cell models thanks to recent works solving nonlinear partial differential equations (PDEs). Various methods have been recently proposed, including fault-tolerant quantum linear systems algorithms (QLSAs) for linear and nonlinear PDEs. In practice, however, these fault-tolerant algorithms do not yet give a quantum advantage. On current noisy intermediate-scale quantum (NISQ) devices, these algorithms are affected by strong limitations on the depth of the quantum circuits that can be coherently executed. On NISQ devices, variational quantum algorithms (VQAs) have become a popular approach to solving PDEs. Many algorithms have been proposed that solve one time step at a time, which inherently presents a limitation to the scalability and accuracy. In our paper, we combine quantum nonlinear processing units (QNPUs) with the Feynman-Kitaev Hamiltonian to achieve a spacetime solution instead, and apply it to the nonlinear Burgers equation. We find the ground state of this Hamiltonian using an adaptive optimization procedure to avoid barren plateaus. Here, we also apply this method to an adapted version of the single particle model with electrolyte (SPMe).
From fluid flow and transport to collective dynamics, numerical simulation of nonlinear partial differential equations underpins modern scientific computing. Extending this capability to quantum computers remains a longstanding challenge because nonlinear and non-Hermitian evolution is fundamentally incompatible with conventional Hamiltonian-based quantum simulation. Here we experimentally realize the time evolution of nonlinear fluid dynamics on a quantum processor using a hybrid variational framework for the viscous and inviscid Burgers equations. Our approach directly encodes the nonlinear dynamics into a variational optimization procedure, avoiding the enlarged linear embeddings and truncation overhead associated with Carleman linearization-based quantum algorithms. We further demonstrate convection-dominated dynamics corresponding to Reynolds numbers of order 102. We encode the governing evolution into parametrized quantum circuits and iteratively reconstruct the time-dependent field through quantum-classical optimization. By introducing a zero-noise extrapolation method without additional circuit-folding overhead, we accurately execute deep error-circuits with entangling-gate counts beyond those typical of Hadamard test circuits. We accurately reconstruct the time evolution across multiple timesteps despite hardware noise and finite device coherence. Our results constitute, to our knowledge, the first experimental realization of nonlinear time propagation on a quantum processor, extending quantum simulation beyond predominantly linear settings and establishing a route toward quantum computation for nonlinear continuum dynamics.
Real- and imaginary-time quantum state evolutions are crucial in physics and chemistry for exploring quantum dynamics, preparing ground states, and computing thermodynamic observables. On near-term devices, variational quantum time evolution is a promising candidate for these tasks, as the required circuit model can be tailored to the available devices' capabilities. Due to the evaluation of the quantum geometric tensor (QGT), however, this approach quickly becomes infeasible for relevant system sizes. Here, we propose a dual formulation for variational time evolution, which replaces the calculation of the QGT by solving a fidelity-based optimization to compute updates to the dynamics in each time step. We demonstrate our algorithm for the time evolution of the Heisenberg Hamiltonian and show that it accurately reproduces the system dynamics at a fraction of the cost of standard variational quantum time evolution algorithms. Published by the American Physical Society 2024
Efficiently solving nonlinear ordinary and partial differential equations using a quantum computer is a major challenge due its inherent linearity. To circumvent this challenge, the Carleman linearization method has been proposed to transform a nonlinear ordinary differential equation into a linear system of equations, the primary advantage being that existing quantum linear systems algorithms may then be applied to obtain a solution. However, this methodology also brings forth several major challenges that must be addressed to attain a quantum advantage. Herein, we address several of these challenges enabling us to solve the Carleman linearized one-dimensional Burgers' equation on real and simulated quantum hardware. All simulations were performed on BlueQubit's platform allowing for quantum circuits to be run on GPU or QPU's seamlessly. We first demonstrate that the Carleman linearized Burgers' equation can be efficiently loaded onto a quantum computer using the linear combination of non-unitaries method, an alternative to the linear combintaiton of unitaries approach. Once loaded, the linear system is then solved using the variational quantum linear solver. Since a naive implementation of this solver is hindered by the barren plateau phenomenon, we introduce a multigridding method to solve the problem in a series of stages with the solution of the previous stage acting as a warm start for the next stage. This approach is found to significantly improve the accuracy of the solution compared with a naive cold start. Finally, circuits with a combined number of spatial and temporal discretization points totaling up to 280≈1024 are transpiled onto real quantum hardware demonstrating that the proposed methodology could feasibly produce a quantum advantage on future hardware.
Differential equations underpin a vast range of scientific and engineering domains, including structural mechanics, fluid dynamics, and financial modelling. Their intricate nature often renders classical numerical methods computationally demanding, requiring fine discretization or large basis expansions, which can lead to prohibitive costs. Quantum computing offers a fundamentally different paradigm, leveraging superposition and entanglement with the promise of computational speedup. However, in the current Noisy Intermediate-Scale Quantum (NISQ) era, fully fault-tolerant quantum computers remain decades away, motivating the development of hybrid strategies such as Variational Quantum Algorithms (VQAs), which train parametrised quantum circuits using classical optimizers. This work investigates spectral VQAs for solving differential equations, extending the Chebyshev-encoded VQA of Kyriienko et al. and introducing a novel quantum spectral approach: the Lagrange-encoded VQA. This approach shifts the paradigm by incorporating entanglement directly within the encoding block while maintaining a shallow variational ansatz. Inspired by the Hadamard test structure, the design enables an efficient representation of the complete set of Lagrange polynomial basis functions using a limited number of gates and facilitates the use of the Hadamard test differentiation method, thereby reducing the computational load. VQA performance is analysed through the accuracy of the result in combination with the gate count and circuit count metrics, which serve as proxies for runtime and noise accumulation on NISQ devices. The proposed VQA is benchmarked on multiple problems: (i) a first-order linear ODEs and (ii) the second-order ODEs damped mass-spring system, both comparing Lagrange- and Chebyshev-encoded VQAs, (iii) the Poisson equation, comparing the Lagrange-encoded VQA with the discretized approach of Sato et al., and (iv) a first-order ODE with nonlinear coefficients addressed with the Lagrange-encoded VQA. Across these cases, the Lagrange-encoded VQA consistently achieves comparable or improved accuracy while significantly reducing circuit depth and computational overload, making it a promising candidate for NISQ devices.
We propose a distinct approach to solving linear and nonlinear differential equations (DEs) on quantum computers by encoding the problem into ground states of effective Hamiltonian operators. Our algorithm relies on constructing such operators in the Chebyshev space, where an effective Hamiltonian is a sum of global differential and data constraints. Once the effective Hamiltonian is formed, solutions of differential equations can be obtained using the ground state preparation techniques (e.g. imaginary-time evolution and quantum singular value transformation), bypassing variational search. Unlike approaches based on discrete grids, the algorithm enables evaluation of solutions beyond fixed grid points and implements constraints in the physics-informed way. Our proposal inherits the best traits from quantum machine learning-based DE solving (compact basis representation, automatic differentiation, nonlinearity) and quantum linear algebra-based approaches (fine-grid encoding, provable speed-up for state preparation), offering a robust strategy for quantum scientific computing in the early fault-tolerant era.