Constructing the Walk Operator in the Ladder Operator Block Encoding Framework

Estimating energy eigenvalues of a model Hamiltonian is an important goal of quantum simulation. Using quantum phase estimation on a block-encoded unitary which encodes a Hamiltonian does not work in general, since there is no clear map from eigenvalues of a block-encoded unitary to the spectrum of the Hamiltonian. Qubitization is the technique that defines a construction of the walk operator, built from the same unitaries that block-encode the Hamiltonian, with the key feature that the spectrum of the walk operator is a function of the spectrum of the Hamiltonian. In this work, we show how to obtain the walk operator via the Ladder Operator Block Encoding (LOBE) framework. This allows for efficient energy eigenvalue estimation of quantum field theory Hamiltonians with quantum phase estimation.