The Unitary in the bottom register is squared because it is part of the quantum phase estimation algorithm, which is used to estimate the phase of a unitary operator.
In the quantum phase estimation algorithm, the goal is to estimate the phase of an eigenvalue of a unitary operator U. To do this, the algorithm applies powers of the unitary operator U to the input state, and then performs a quantum Fourier transform on the results.
The bottom register in the circuit diagram contains the powers of the unitary operator U, specifically U^(2^t-1), U^(2^t-2), ..., U^2, U^1, U^0. By squaring these unitary operators, the algorithm is effectively applying the unitary operator U to the input state multiple times, which is a key step in the phase estimation process.
The squaring of the unitary operators in the bottom register allows the algorithm to estimate the phase of the eigenvalue of U with increasing precision as the number of qubits in the bottom register increases.
The quantum phase estimation algorithm estimates the phase of the eigenvalue of the unitary operator U by applying powers of U to the input state and then performing a quantum Fourier transform on the results.
Specifically, the bottom register in the circuit diagram contains the powers of the unitary operator U, specifically U^(2^t-1), U^(2^t-2), ..., U^2, U^1, U^0. By applying these powers of U to the input state, the algorithm is effectively creating a superposition of states that encodes the phase information.
The quantum Fourier transform is then applied to this superposition, which allows the phase information to be extracted and measured. The more qubits in the bottom register, the more precise the phase estimation can be.
So in summary, the algorithm uses the repeated application of the unitary operator U and the quantum Fourier transform to estimate the phase of the eigenvalue of U with increasing precision as the number of qubits in the bottom register increases.