  Qubitization is a modern approach to estimate Hamiltonian eigenvalues without
simulating its time evolution. While in this way approximation errors are
avoided, its resource and gate requirements are more extensive: qubitization
requires additional qubits to store information about the Hamiltonian, and
Toffoli gates to probe them throughout the routine. Recently, it was shown that
storing the Hamiltonian in a unary representation can alleviate the need for
such gates in one of the qubitization subroutines. Building on that principle,
we develop an entirely new decomposition of the entire algorithm: without
Toffoli gates, we can encode the Hamiltonian into qubits within logarithmic
depth.
