Parallelization techniques for quantum simulation of fermionic systems

Jacob Bringewatt1,2,3 and Zohreh Davoudi1,4,5

1Department of Physics, University of Maryland, College Park, Maryland 20742, USA
2Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
3Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
4Maryland Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, USA
5Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA

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Abstract

Mapping fermionic operators to qubit operators is an essential step for simulating fermionic systems on a quantum computer. We investigate how the choice of such a mapping interacts with the underlying qubit connectivity of the quantum processor to enable (or impede) parallelization of the resulting Hamiltonian-simulation algorithm. It is shown that this problem can be mapped to a path coloring problem on a graph constructed from the particular choice of encoding fermions onto qubits and the fermionic interactions onto paths. The basic version of this problem is called the weak coloring problem. Taking into account the fine-grained details of the mapping yields what is called the strong coloring problem, which leads to improved parallelization performance. A variety of illustrative analytical and numerical examples are presented to demonstrate the amount of improvement for both weak and strong coloring-based parallelizations. Our results are particularly important for implementation on near-term quantum processors where minimizing circuit depth is necessary for algorithmic feasibility.

While the basic element of a quantum computer is a qubit—a two-level quantum system—most matter is made of fermions, which have unique statistical properties, namely they obey the Pauli exclusion principle, that differ from the "native" qubit statistics of a quantum computer. Therefore, when simulating physical systems, such as electrons, quarks, nucleons, and certain nuclei and atoms, on a quantum computer, one must encode these fermionic statistics into the qubits, resulting in additional qubit and/or quantum-operation requirements. In this work, we consider a particular class of mappings from fermions to qubits, focusing on how one can optimize the mapping to the hardware in such a way that maximizes the amount of parallelization possible in quantum simulation algorithms for fermionic systems. We do this optimization by viewing the parallelization of fermion to qubit mappings as a graph coloring problem, where one seeks to color the vertices of a collection of vertices and edges so that no vertices that share an edge get the same color. Such optimizations are particularly important in the near-to-intermediate term when the runtime of quantum algorithms is sharply limited by noisy hardware.

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