Abstract:
Solving partial differential equations (PDEs), which is pervasive in science and engineering, is emerging as a promising application area for quantum computing because it can be reduced to Hamiltonian simulation. Unlike quantum chemistry, Pauli-term expansion is not useful for a PDE Hamiltonian, since it ends up with an exponential number of Pauli terms. Recently, it has been shown that, by diagonalizing tensor products of ladder operators, such a Hamiltonian can be mapped into a scalable circuit with a polynomial number of multi-controlled gates. When optimized by an industry-grade circuit compiler, the resulting circuit is realized with a quadratic number of CX gates with respect to the number of qubits. To further optimize PDE circuits, we propose Multilevel Gate Set Optimization (MGSO), an approach for identifying effective decomposition methods for high-level gates. In MGSO, we define multiple levels of gate sets, from higher to lower. At each level, we optimize the circuit and then lower it using decomposition methods carefully selected based on circuit structure to maximize optimization opportunities at subsequent levels. Applying MGSO to optimize the PDE circuits, we achieved a quadratic reduction in the number of CX gates with a linear increase in the number of qubits.