Variational optimization of PREPARE circuit for block-encoding Hamiltonians without ancillary qubits

In the state-of-the-art quantum phase estimation algorithms, a quantum state, whose amplitudes correspond to the coefficients of the Hamiltonian, needs to be prepared to block-encode the Hamiltonian. While this preparation process can be done with T gates scaling linearly to the number of coefficients by using quantum read-only memory(QROM), it requires a large number of ancilla qubits. Here we propose a method to construct a quantum circuit to prepare an arbitrary quantum state without using any additional ancilla qubit. Specifically, we construct a PREPARE circuit with a small number of T gates by using automatic quantum circuit encoding (AQCE)[arXiv:2112.14524], which is a method to encode a given arbitrarily quantum state with a finite number of two-qubit quantum gates in an optimal way. To evaluate the performance of the proposed method, we decompose the generated PREPARE-circuit into Clifford+T gate set and estimated the number of T gates required to achieve chemical accuracy. As a result, the number of T gates is significantly reduced compared with a method that exactly prepares the quantum state without using ancilla qubits. The number of logical qubits is reduced to less than half compared to the preparation process by using QROM. Since the number of available logical qubits is expected to be limited in an early stage of FTQC, the proposed method without ancilla qubits and with smaller T gates is promising to construct a hardware-efficient quantum phase estimation algorithm.