Explicit Quantum Circuit for Simulating the Advection-Diffusion-Reaction Dynamics

Abstract

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 multi-timestep version of the present circuit.

24 pages, 12 figures

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Cited by in corpus (6)

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• Quantum Dynamics Simulation of the Advection-Diffusion Equation
• Quantum Algorithm for the Advection-Diffusion Equation by Direct Block Encoding of the Time-Marching Operator
• Ladder Operator Block-Encoding
• Performance evaluation of variational quantum eigensolver and quantum dynamics algorithms on the advection-diffusion equation
• An efficient explicit implementation of a near-optimal quantum algorithm for simulating linear dissipative differential equations