New Quantum Algorithm Efficiently Solves Linear Differential Equations, Revolutionising Science and Engineering

A team of researchers has developed a new quantum algorithm for solving linear differential equations (LDEs), which are prevalent in science and engineering. The algorithm uses a variational method with shallow circuits to encode the equations into a ground state problem of the Hamiltonian, which is then solved using hybrid quantum-classical methods. This approach requires fewer qubit resources than previous methods and can restore the entire evolutionary process. The algorithm has been demonstrated in simulating the evolution of harmonic oscillators and dynamics of non-Hermitian systems with PT-symmetry.

What is the New Quantum Algorithm for Linear Differential Equations?

A team of researchers, including Junxiang Xiao, Jingwei Wen, Zengrong Zhou, Ling Qian, Zhiguo Huang, Shijie Wei, and Guilu Long, have developed a new quantum algorithm for solving linear differential equations (LDEs). LDEs are common in almost all fields of science and engineering, and solving them can be a challenging task, especially for classical high-performance computers when the configuration space has a large dimension, such as in systems of quantum mechanics and hydrodynamics.

The researchers’ algorithm uses a variational method with shallow circuits to solve LDEs. The core idea of the algorithm is to encode the equations into a ground state problem of the Hamiltonian, which is then solved using hybrid quantum-classical methods with high fidelities. This approach requires the least qubit resources compared to previous works and can restore the entire evolutionary process. The researchers demonstrated its application in simulating the evolution of harmonic oscillators and dynamics of non-Hermitian systems with PT-symmetry.

How Does the Quantum Algorithm Work?

The researchers’ quantum algorithm for solving LDEs is inspired by quantum-classical hybrid variational methods. The original problem is first approximated using difference equations and then transformed into a linear system problem with the help of an auxiliary qubit. The complete time interval is then split into slices, and in each slice, there is a Hamiltonian whose unique ground state encodes the solution to the linear equations.

These Hamiltonian problems are solved with variational quantum algorithms, and their ground states are also the initial states for the next time slice. The output state at the end time point is post-processed by taking a measurement on the auxiliary qubit, and then the quantum state encoding the solution is obtained from the work qubits. This approach requires the least number of qubits compared to previous works, and the gate complexity of the trained circuit is logarithmic, keeping the same as the existing algorithms.

What is the Significance of the Quantum Algorithm?

Quantum computers have the potential to provide an exponential advantage over classical computers for certain problems. Various effective quantum computation protocols have emerged, such as circuit-based quantum computation, adiabatic quantum computation, and duality quantum computation. Efficient algorithms for quantum simulation, quantum search, and algebra problems have been devised and demonstrated on various quantum devices.

A scalable quantum computer is expected to be able to obtain the solution of linear systems of equations and even linear differential equations exponentially faster than the fastest high-performance classical computers. In the past few years, some efficient quantum algorithms for solving linear differential equations have been proposed, which are based on HHL algorithms or Taylor expansions with linear combination of unitaries. However, these require complex operations and a large number of qubits, making them far from current noisy intermediate-scale quantum (NISQ) devices.

How Does the Quantum Algorithm Compare to Previous Works?

Compared to previous works, the researchers’ quantum algorithm for solving LDEs requires the least qubit resources and can restore the entire evolutionary process. The gate complexity of the trained circuit is logarithmic, keeping the same as the existing algorithms. Therefore, the algorithm is compatible with shallow circuits, which are a key feature of NISQ devices.

In the NISQ era, quantum-classical hybrid algorithms utilize both the performance of classical computers and the intrinsic quantum properties of intermediate-scale quantum systems, leading to lower hardware requirements and robustness against certain noise. These variational algorithms have already provided efficient solutions for quantum chemistry, linear algebra, and quantum machine learning. Some of them can be applied to the problem of solving differential equations, although their performance may be highly reliant on the pre-design of ansatz or sufficient qubit resources.

What are the Applications of the Quantum Algorithm?

The researchers demonstrated the application of their quantum algorithm for solving LDEs in simulating the evolution of harmonic oscillators and dynamics of non-Hermitian systems with PT-symmetry. This shows that the algorithm can be used to solve many important problems whose essence is the solution of linear differential equations.

The algorithm provides a key technique for solving LDEs, which are common in almost all fields of science and engineering. It can be particularly useful in fields where LDEs play key roles, such as the Schrödinger equation in quantum mechanics, the Stokes equations in fluid dynamics, and the Maxwell equations in electromagnetism.