Mmathematical description of a LCU qubisation embedded in a QPE algorithm.

1. A Hermetian operator $H$ is written as a linear combination of unitaries.

$$H =\ \sum_{i=0}^{L-1} ,\omega_i,U_i,$$

where Ua are unitary matrices with coefficents. The one-norm of the LCU,

$$\lambda =\sum_{i=0}|\omega_i|$$

For a one‐qubit toy model we take $$U_0 = I,\qquad U_1 = X,\qquad U_2 = Z, \quad H = 1.5,I + 0.5,X - 0.5,Z . $$

2. Normalisation Factor

$$\lambda =\sum_{i=0}^{L-1}|\omega_i| =\ 1.5 + 0.5 + 0.5 =\ 2.5 . $$

Hence

$$\frac{H}{\lambda} = \sum_{i=0}^{L-1} \frac{|\omega_i|}{\lambda}; s_i,U_i, \qquad s_i = {sgn}(\omega_i)\in{\pm1}. $$

3. Ancilla Preparation PREP

Let $m=\lceil\log_2 L\rceil$ (here $m=2$).

Prepare

$$|\chi\rangle= \sum_{i=0}^{L-1} \sqrt{\tfrac{|\omega_i|}{\lambda}} |i\rangle = \sqrt{0.6}|00\rangle +\sqrt{0.2}|01\rangle +\sqrt{0.2}|10\rangle, $$ via a unitary $$PREP: |0^{\otimes m}\rangle!\mapsto|\chi\rangle .$$

4. SELECT

$$SELECT = \sum_{i=0}^{L-1}|i\rangle\langle i|\otimes s_i,U_i = |00\rangle\langle00|\otimes I + |01\rangle!\langle01|\otimes X + |10\rangle!\langle10|\otimes(-Z). $$

5. Block‐Encoding Unitary

$$U = (PREP^\dagger\otimes I) SELECT (PREP\otimes I). $$

Projecting the ancilla onto

$|0^{\otimes m}\rangle$ returns $$\bigl(\langle0^{\otimes m}|\otimes I\bigr), U, \bigl(|0^{\otimes m}\rangle\otimes I\bigr) ;=; \frac{H}{\lambda}. $$

6. Walk Operator (Qubitisation)

Define the reflection $$R = 2|0^{\otimes m}\rangle!\langle0^{\otimes m}| - I_{2^{m}},$$ and the walk operator $$W= R,U .$$ For every eigenvector $|\psi_j\rangle$ of $H$ with eigenvalue $E_j$,

$$W\bigl(|0^{\otimes m}\rangle\otimes|\psi_j\rangle\bigr) = e^{\pm i\theta_j}, \bigl(|0^{\otimes m}\rangle\otimes|\psi_j\rangle\bigr), \qquad \cos\theta_j = \frac{E_j}{\lambda}.$$

7. Using QPE

Because $W$ is unitary and its phases $\theta_j$ directly encode the eigen-energies, standard Quantum Phase Estimation on $W$ yields $$E_j = \lambda,\cos\theta_j .$$

8. Walkoperator in QPE

The walk operator is the essential bridge between a block-encoded Hamiltonian and the phase spectrum required by Quantum Phase Estimation. Its theoretical justification follows directly from the qubitisation results in Low & Chuang, npj Quantum Information 3, 13 (2017).