Fig. 1. Diagram of HHL quantum circuit. (a) Quantum phase estimation module (QPE); (b) Quantum state controlled rotation module; (c) Time-reversal of the QPE module (QPE†) ^[15]

Fig. 2. (a) Quantum circuit framework and (b) diagram of quantum phase estimation algorithm

Fig. 3. Some quantum circuits in decomposition of unitary matrix U. (a) Single qubit operation ei2πaX/m; (b) Two qubits quantum operation ei2πaX⊗X/m; (c) Multi-qubit quantum operation ei2πaX⊗X⊗X/m; (d) Quantum operation ei2πaZ/m;

Fig. 4. Equivalent quantum circuits of controlled ei2πa(I⊗I⊗⋯⊗I)/m before (a) and after (b) the Phase Kickback

Fig. 5. Circuit of quantum Fourier transform

Fig. 6. Quantum adder of two binary strings dn-1dn-2⋯d1d0 and cn-1cn-2⋯c1c0

Fig. 7. Quantum multiplier of two binary strings dn-1dn-2⋯d1d0 and cn-1cn-2⋯c1c0 where Σ corresponds to the quantum phase addition module in Fig. 6

Fig. 8. Quantum circuit of the controlled rotation operation

Fig. 9. An example of HHL quantum circuit designed by IBM qiskit and its simulation results. (a) Quantum circuit diagram for solving linear equations; (b) Probability histogram of the quantum measurement; (c) QPE module encapsulated in diagram (a)