[Paper Review] Simple digital quantum algorithm for symmetric first order linear hyperbolic systems

François Fillion‐Gourdeau, Emmanuel Lorin|arXiv (Cornell University)|May 25, 2017

Quantum Computing Algorithms and Architecture16 references3 citations

TL;DR

This paper presents a digital quantum algorithm for solving symmetric first-order linear hyperbolic systems using the reservoir method combined with alternate direction operator splitting. By reducing the scheme to simple unitary transformations, it achieves exponential speedup over classical counterparts for time evolution, though measurement requires post-processing due to amplitude encoding.

ABSTRACT

This paper is devoted to the derivation of a digital quantum algorithm for the Cauchy problem for symmetric first order linear hyperbolic systems, thanks to the reservoir technique. The reservoir technique is a method designed to avoid artificial diffusion generated by first order finite volume methods approximating hyperbolic systems of conservation laws. For some class of hyperbolic systems, namely those with constant matrices in several dimensions, we show that the combination of i) the reservoir method and ii) the alternate direction iteration operator splitting approximation, allows for the derivation of algorithms only based on simple unitary transformations, thus perfectly suitable for an implementation on a quantum computer. The same approach can also be adapted to scalar one-dimensional systems with non-constant velocity by combining with a non-uniform mesh. The asymptotic computational complexity for the time evolution is determined and it is demonstrated that the quantum algorithm is more efficient than the classical version. However, in the quantum case, the solution is encoded in probability amplitudes of the quantum register. As a consequence, as with other similar quantum algorithms, a post-processing mechanism has to be used to obtain general properties of the solution because a direct reading cannot be performed as efficiently as the time evolution.

Motivation & Objective

• To develop a quantum algorithm for symmetric first-order linear hyperbolic systems that leverages quantum speedup.
• To eliminate artificial diffusion in numerical schemes using the reservoir technique, ensuring stability and accuracy.
• To enable efficient quantum implementation by reducing the scheme to unitary operations via operator splitting.
• To demonstrate computational advantage over classical methods in time evolution complexity.
• To identify conditions under which quantum advantage is preserved despite amplitude encoding limitations.

Proposed method

• The reservoir method is applied to eliminate artificial diffusion in finite volume schemes for hyperbolic systems.
• Alternate direction iteration operator splitting is used to decompose multi-dimensional problems into sequential one-dimensional updates.
• The resulting scheme is reformulated as a sequence of streaming operations, which are implemented as unitary transformations on a quantum register.
• Amplitude encoding maps the solution wave function onto quantum state amplitudes using a discrete basis.
• The time evolution is approximated via Trotter-like decomposition of the evolution operator into unitary gates.
• For non-constant velocity in 1D, a non-uniform mesh is combined with the reservoir method to maintain accuracy and unitarity.

Experimental results

Research questions

• RQ1 Can the reservoir method be combined with operator splitting to produce a unitary quantum algorithm for symmetric hyperbolic systems?
• RQ2 What is the computational complexity of the quantum time evolution compared to classical finite volume schemes?
• RQ3 How does the algorithm handle non-constant velocity in one-dimensional scalar systems?
• RQ4 What are the limitations of extending this approach to non-symmetric or space-dependent matrices?
• RQ5 Can the quantum algorithm achieve exponential speedup despite the need for post-processing to extract solution properties?

Key findings

• The algorithm reduces the solution of symmetric hyperbolic systems to a sequence of unitary operations, making it suitable for quantum implementation.
• For constant symmetric matrices in multiple dimensions, the method achieves exponential speedup in time evolution complexity over classical finite volume schemes.
• The asymptotic computational complexity of the quantum algorithm scales favorably compared to classical versions, though measurement remains costly at O(N) operations.
• The method is generalized to 1D scalar systems with non-constant velocity using a non-uniform mesh, preserving unitarity and accuracy.
• For non-constant or non-symmetric matrices, the method requires non-unitary operations, which are more challenging to implement on quantum hardware.
• The approach highlights the importance of translation operators and efficient implementation of spatial shifts for broader applicability to linear PDEs.