Effects of cosine tapering window on quantum phase estimation

We provide a modification to the quantum phase estimation algorithm (QPEA) [Abrams and Lloyd, Phys. Rev. Lett. 83, 5162 (1999); Cleve et al., Proc. R. Soc. A 454, 339 (1998); Nielsen and Chuang, Quantum computation and quantum information, 2002.] inspired by classical windowing methods for spectral density estimation. From this modification we obtain an upper bound in the cost that implies a cubic improvement with respect to the algorithm’s error rate. Numerical evaluation of the costs also demonstrates an improvement. Moreover, with similar techniques, we detail an iterative projective measurement method for ground state preparation that gives an exponential improvement over previous bounds using QPEA. Numerical tests that confirm the expected scaling behavior are also obtained. For these numerical tests we have used a lattice Thirring model as testing ground. Using well-known perturbation theory results, we also show how to more appropriately estimate the cost scaling with respect to state error instead of evolution operator error.

Figure 1

Circuit to implement an $m$-qubit phase estimation algorithm. $U$ is $e^{2\pi i\lambda H}$. The ${\mathrm{QFT}}^{-1}$ applied on $|0{⟩}^{\otimes m}$ can be simplified to $H^{\otimes m}$ acting on $|0{⟩}^{\otimes m}$ as is typically shown. The red dashed line marks the samples of the filter function stored in the ancillary register and the black dashed line the samples of the corresponding window function.

Figure 2

Here we showcase the two different window functions and their corresponding filter functions, connected by DFT (QFT) and its inverse (${\mathrm{QFT}}^{-1}$). On the left, the black dots represent the discrete node points at which the window function is sampled. For example, these black dots represent the distribution of the ancillary register at the black dashed line on Fig. 1 for the rectangular window and on Fig. 3 for the cosine window. On the right, the red marks represent the distributions that we use to generate the windows through the inverse DFT (QFT) (these ancillary register distributions are marked with a red dashed line in Figs. 1 and 3 for the cosine and rectangular windows, respectively).

Figure 3

Circuit to implement improved $m$-qubit phase estimation algorithm. $U$ is $e^{2\pi i\lambda H}$. The red dashed line marks the samples of the filter function stored in the ancillary register and the black dashed line the samples of the corresponding window function.

Figure 4

Comparison of the filter functions in Fig. 2. We have shifted the cosine filter function in order for it to be centered at $q=0$. The filter corresponding to the cosine window has a wider main lobe, but it has a faster decaying behavior outside of that. As a consequence, when binning the probabilities, we get a higher peak for the cosine filter.

Figure 5

Here we plot the error rate $e$ defined in Eq. (27) against the extra number of qubits $p$ in Eq. (28). We have chosen $t=10$ and $2^m\delta \in \{0.0,-0.1,-0.2,-0.3,-0.4,-0.5\}$. It is clear that, except for the exceptional cases $2^m\delta =0$, the filter function coming from the cosine window outperforms the one from the rectangular window as we increase $p$.

Figure 6

Circuit to implement projection with peak centered at the estimate of the ground state energy. The resulting filter is the DFT of a rectangular window or a constant signal.

Figure 7

Circuit to implement projection with peak centered at the estimate of the ground state energy. The resulting filter is the DFT of a cosine tapering window. The last Hadamard gate is there to create a coherent binning of odd and even $x$ states.

Figure 8

Here we show comparisons of the excited state contamination, ${\sigma }_{\chi }$, in the chiral condensate for the values $d\in 1,2,3$ and $N\in 4,6,8$. The excited state contamination is plotted against the number of iterations, $r$, showing that the cosine-window filter achieves an improvement over the rectangular-window one.

Figure 9

Here, we have taken the circuit up to the black dashed line in Fig. 1 and simplified it given the initial state; same goes for Fig. 6.

Figure 10

Here, we have taken the circuit up to the black dashed line in Fig. 3 and simplified it given the initial state; same goes for Fig. 7.

Figure 11

Our convention for ${\mathrm{QFT}}^{-1}$. The corresponding QFT can be obtained by simply inverting the circuit. This circuit is most widely labeled as QFT due to conventions in quantum computing literature [14, 15].