Efficient state initialization by a quantum spectral filtering algorithm

An algorithm that initializes a quantum register to a state with a specified energy range is given, corresponding to a quantum implementation of the celebrated Feit-Fleck method. This is performed by introducing a nondeterministic quantum implementation of a standard spectral filtering procedure combined with an apodization technique, allowing for accurate state initialization. It is shown that the implementation requires only two ancilla qubits. A lower bound for the total probability of success of this algorithm is derived, showing that this scheme can be realized using a finite, relatively low number of trials. Assuming the time evolution can be performed efficiently and using a trial state polynomially close to the desired states, it is demonstrated that the number of operations required scales polynomially with the number of qubits. Tradeoffs between accuracy and performance are demonstrated in a simple example: the harmonic oscillator. This algorithm would be useful for the initialization phase of the simulation of quantum systems on digital quantum computers.

Figure 1

Circuit diagram for the quantum implementation of the spectral filtering method. The gate $U_i$ advances the trial solution by $\mathrm{\Delta }t$. The gate $B_i$ is a nonunitary operation. It is defined in Eq. (8) and its quantum circuit is displayed in Fig. 2. The gate $|{\mathrm{\Psi }}_{\mathrm{trial}}\rangle$ initializes the quantum register to an arbitrary state. The projective measurement operator implements the projective measurement $|1\rangle \langle 1|$ that collapses the register to the wanted wave function. Finally, the gate $\widehat{U}$ performs the quantum simulation by evolving in time the initial state constructed by the filtering algorithm.

Figure 2

Circuit diagram for the implementation of the nonunitary operation. The upper qubit is an ancilla qubit prepared in the state $|0\rangle$. The measurement operator implements the projective measurement $|0\rangle \langle 0|$. If the measurement yields the state $|1\rangle$, the calculation has to be redone from the beginning.

Figure 3

Power spectra for the ground state generated from the filtering method using the constant or Hann window function. The spectrum is also given for the trial state, prior to filtering.

Figure 4

Circuit diagram for the quantum implementation of the measurement of the correlation function. The gate $U_t$ advances the trial solution by $t$. The gate $H$ is a Hadamard gate. The gate $|{\mathrm{\Psi }}_{\mathrm{trial}}\rangle$ initializes the quantum register to an arbitrary state. Finally, the quantity $\langle {\sigma }_{x,y}\rangle$ are measured on the ancilla qubit.