Exact chiral symmetry with quantum signal processing

Quantum simulations of lattice gauge theories often struggle to preserve chiral symmetry without introducing fermion doubling. While domain-wall fermions provide a path to chiral symmetry by adding an extra spatial dimension, overlap fermions offer a more direct, albeit mathematically complex, approach. This paper investigates how to efficiently implement the overlap fermion Hamiltonian on a quantum computer using Quantum Signal Processing (QSP).

The authors utilize QSP to approximate the sign function, which is the core component of the overlap operator. By promoting the single-particle Wilson-Dirac Hamiltonian to a block-encoded operator, they construct a quantum circuit that approximates the overlap Hamiltonian. This approach allows for the simulation of Dirac fermions with exact chiral symmetry, where the Ginsparg-Wilson relation is preserved up to a controllable error parameter, ε_e.

The study demonstrates that simulating overlap fermions is nearly as efficient as simulating Wilson-Dirac fermions. Specifically, the gate complexity scales as O(Qκ⁻¹ log(1/ε_e)), where Q is the number of fermionic degrees of freedom and κ is the spectral gap. The authors show that the QSP algorithm effectively constructs an extra dimension, providing a quantum-algorithmic realization of the known physical correspondence between overlap fermions and the boundary of domain-wall fermions. While overlap fermions require more gate depth than domain-wall fermions due to their nonlocal structure, they offer a reduction in the required number of qubits.

This work provides a rigorous framework for implementing overlap fermions on quantum hardware. By reducing the overhead associated with maintaining chiral symmetry, this algorithm enables more accurate simulations of nonperturbative phenomena in quantum chromodynamics (QCD). It bridges the gap between theoretical lattice formulations and practical quantum algorithmic implementations, offering a clear path for future high-precision studies of chiral fermions.