Quantum Computing Tackles Nonlinear Dynamics: German Team’s Novel Approach

Researchers from the Institute of Engineering Thermodynamics at the German Aerospace Center, the Helmholtz Institute Ulm, Quantinuum, and the Department of Physics at Ulm University have developed a novel approach to solving time-dependent nonlinear differential equations in quantum computing.

They have created variational quantum algorithms (VQAs) that encode space and time in qubit registers, allowing for the entire time evolution to be obtained from a single ground state computation. The team has also proposed an adaptive multigrid strategy to mitigate the barren plateau problem during optimization. The results demonstrate the potential of current quantum computers in accurately reproducing exact results.

What is the Role of Nonlinear Dynamics in Quantum Computing?

Quantum computing is a rapidly evolving field that leverages the principles of quantum mechanics to process information. One of the key challenges in this field is the solution of time-dependent nonlinear differential equations. A team of researchers from the Institute of Engineering Thermodynamics at the German Aerospace Center, the Helmholtz Institute Ulm, Quantinuum Partnership House, and the Department of Physics at Ulm University have proposed a novel approach to address this issue. They have developed variational quantum algorithms (VQAs) that encode both space and time in qubit registers. This allows the entire time evolution from a single ground-state computation.

The researchers have also outlined 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, they propose an adaptive multigrid strategy. This approach has been illustrated for the nonlinear Burgers equation. The team has optimized quantum circuits to represent the desired ground state solutions and run them on IBM Q System One and Quantinuum System Model H1. The results demonstrate that current quantum computers are capable of accurately reproducing the exact results.

How are Nonlinear Dynamics Relevant to Various Fields?

The solution of partial differential equations (PDEs), particularly those which are nonlinear, is of central importance in fields such as aerospace engineering and energy science. Applications include solving the Navier-Stokes equation in computational fluid dynamics and the numerical integration of continuum models in battery research. Conventional numerical methods, however, encounter substantial constraints when tackling large-scale three-dimensional models.

Quantum algorithms have been proposed to integrate PDEs with an exponential advantage over their classical counterparts in theory. This includes quantum linear systems algorithms (QLSAs) for linear and nonlinear PDEs, as well as quantum algorithms based on Hamiltonian simulation. Techniques based on quantum amplitude and phase estimation have also been put forth to integrate PDEs. However, these algorithms do not yet give a quantum advantage due to various challenges, including the input/output problem.

What is the Significance of Variational Quantum Algorithms (VQAs)?

On noisy intermediate-scale quantum (NISQ) devices, VQAs have become a popular approach to solving PDEs. One famous example is the Variational Quantum Linear Solver, a generic solver for linear systems of equations that can be applied to linear PDEs. For nonlinear equations, different algorithmic primitives have been proposed.

One way of solving time-dependent PDEs using VQAs proceeds one time step at a time, analogous to established classical numerical solvers. However, a large number of time steps can lead to an accumulation of errors. Moreover, time stepping leads to a computational complexity that is at least linear in the number of time steps.

How Does the Feynman-Kitaev Formalism Contribute to Quantum Computing?

The researchers have developed alternative VQAs that do not rely on time stepping. They use the formalism of Feynman and Kitaev, in which time is encoded in a clock qubit register and the solution to a time-dependent problem at all points in time is contained in the ground state of a Hamiltonian. This approach was introduced by McClean et al for quantum chemical problems and was extended to open quantum systems by Tempel et al.

The team has extended the Feynman-Kitaev formalism to incorporate nonlinear dynamics. They show how to evaluate the Feynman-Kitaev Hamiltonian using quantum nonlinear processing units (QNPUs). To mitigate barren plateaus, a serious problem for variational algorithms, and scale the variational algorithm to a larger number of qubits, they propose a multigrid optimization strategy with a customized ansatz structure.

What are the Implications of this Research?

The researchers’ approach to the Burgers equation, a paradigmatic nonlinear equation, shows that the number of circuits required to evaluate the Feynman-Kitaev Hamiltonian only depends on the number of terms in the equation and the order of approximation in the time step, not on the number of time steps or spatial points. Moreover, these circuits have a depth that scales at most linearly with the number of qubits.

This research has significant implications for the field of quantum computing. It presents a novel approach to solving time-dependent nonlinear differential equations, a key challenge in the field. The results demonstrate the potential of current quantum computers and pave the way for further advancements in the field.