This paper gives a quantum algorithm that implements the overlap fermion Hamiltonian while keeping lattice chiral symmetry nearly exact. In lattice particle physics, “chiral symmetry” is a property of massless fermions that is hard to keep when space is replaced by a discrete lattice. The overlap formulation and the Ginsparg–Wilson relation are known ways to restore that symmetry. The authors show how to realize the overlap Hamiltonian on a quantum computer using quantum signal processing (QSP), while controlling the small symmetry-breaking error ε_e.

The researchers build a QSP-based procedure to approximate the nonlocal “sign function” that appears in the overlap operator. QSP is a way to turn a desired function of an operator into a sequence of controlled quantum gates. To apply the fermion Hamiltonian they block-encode the single-particle Hamiltonian into a larger unitary and then use QSP to synthesize the overlap operator. They describe the technical steps needed for the lattice degrees of freedom, how to include gauge links, and how the required state-preparation and select operations scale with lattice size and dimension.

Why this matters: exact chiral symmetry matters for many calculations in lattice quantum chromodynamics and related field theories. The authors find that using QSP the cost of applying the overlap Hamiltonian is only larger than the simpler Wilson–Dirac Hamiltonian by a factor “logarithmic in ε_e.” In plain terms, getting a very small symmetry error does not dramatically raise the computational cost. Compared to domain-wall fermions, which add an extra spatial direction to restore chirality, the QSP overlap approach has a mild increase in gate complexity but reduces the number of qubits needed. The paper also explains that QSP effectively builds the extra dimension that domain-wall fermions use, tying the algorithmic cost to the underlying physics.