Digital-analog quantum algorithm for the quantum Fourier transform

Quantum computers will allow calculations beyond existing classical computers. However, current technology is still too noisy and imperfect to construct a universal digital quantum computer with quantum error correction. Inspired by the evolution of classical computation, an alternative paradigm merging the flexibility of digital quantum computation with the robustness of analog quantum simulation has emerged. This universal paradigm is known as digital-analog quantum computing. Here, we introduce an efficient digital-analog quantum algorithm to compute the quantum Fourier transform, a subroutine widely employed in several relevant quantum algorithms. We show that, under reasonable assumptions about noise models, the fidelity of the quantum Fourier transformation improves considerably using this approach when the number of qubits involved grows. This suggests that, in the noisy intermediate-scale quantum era, hybrid protocols combining digital and analog quantum computing could be a sensible approach to reach useful quantum supremacy.

Figure 1

Comparison between the sDAQC and the bDAQC protocols. The blue blocks $U_{\mathrm{int}}\left(t\right)$ represent the analog blocks, and the single-qubit gates $X$ refer to the Pauli matrix ${\sigma }_x$. In the sDAQC protocol, the digital and the analog blocks alternate with each other. The evolution of the interaction Hamiltonian is turned on and off several times. When applying the bDAQC protocol, the analog block is turned on during the whole simulation, and the digital blocks are performed on top of the analog evolution.

Figure 2

Digital implementation of the $\mathrm{Q}\mathcal{F}\mathrm{T}$ for a $n$-qubit system. The single-qubit gate $H$ corresponds to the Hadamard gate [see Eq. (9)]. The rest are the controlled rotations defined by $c{R}_k=|0\rangle \langle 0|\otimes \mathbb{1}+|1\rangle \langle 1|\otimes R_k$, where $R_k=|0\rangle \langle 0|+e^{2\pi i/2^k}|1\rangle \langle 1|$. The swap gates at the end of the circuit needed to correctly read the transformed state are not shown.

Figure 3

Implementation of the $\mathrm{Q}\mathcal{F}\mathrm{T}$ for a three-qubit system using three different protocols: DQC, sDAQC, and bDAQC. Digital implementation: We show the transformation between the usual DQC implementation of the $\mathrm{Q}\mathcal{F}\mathrm{T}$ (see Fig. 2) and the one that follows the Hamiltonian described by Eq. (14). Following Eq. (11), the controlled-rotation $c{R}_2$ and $c{R}_3$ correspond to the controlled-phase gate $cS$ and the controlled-$\pi /8$ gate $cT$, respectively. For the implementation that follows the Hamiltonian described by Eq. (14), each entangling two-qubit gate is applied according to the ATA DQC protocol using a fixed $\pi /4$ phase [see Eq. (18)]. DAQC implementation: The blue blocks $U_{\mathrm{int}}\left(t\right)$ represent the analog blocks, and each of them is applied during different times $t$. The single-qubit gates $X$ refer to the Pauli matrix ${\sigma }_x$ and act for a time $\mathrm{\Delta }t$. We apply the DAQC protocol for each block of controlled rotations of the DQC implementation, which is detailed by the red line over each of those blocks. The sDAQC switches on and off the analog evolution before applying the single-qubit rotations $X$. In contrast, in the bDAQC protocol, the single-qubit rotations are performed on top of the analog evolution. Since we are applying a Suzuki-Lie-Trotter decomposition to minimize the error, between the single-qubit rotations, each analog block acts for different times $t_i-\mathrm{\Delta }t$, except for the first and the last block, which act for times $t_i-\frac{3}{2}\mathrm{\Delta }t$.

Figure 4

Fidelity of the transformation of the family of states $|{\psi }_0\rangle =sin\beta |W_n\rangle +cos\beta |{\mathrm{GHZ}}_n\rangle$ using the three protocols. (a) Ideal implementation. Both the DQC and the sDAQC protocols perform the $\mathrm{Q}\mathcal{F}\mathrm{T}$ with fidelity $F_{\mathrm{DQC}}=F_{\mathrm{sDAQC}}=1$. The bDAQC has an intrinsic error due to the fact that the analog block is applied during the whole process. In this case, the fidelity is $F_{\mathrm{bDAQC}}\in (0.90,1)$. (b) Realistic noisy implementation. In this situation, the intrinsic error of the bDAQC is less significant than the experimental errors of the DQC and the sDAQC. The fidelity of the DQC decreases fast with the number of qubits. For a six-qubit system, the fidelity is around 50%, so the DQC protocol is no longer useful. The fidelity of both DAQC protocols behaves better than the one of the DQC with the number of qubits and remains above 70% for the sDAQC and over 80% for the bDAQC. For the case of a seven-qubit system, we obtain a similar fidelity for both the DQC and the sDAQC protocols, but the one obtained for the bDAQC remains remarkably higher. This shows that the bDAQC protocol is the best option if one wants to implement the $\mathrm{Q}\mathcal{F}\mathrm{T}$ on a system built up from several qubits. (c) Fidelity evolution with growing errors. We want to show how the fidelity behaves if the errors we have estimated are slightly different. We have computed the $\mathrm{Q}\mathcal{F}\mathrm{T}$ of state $|{\psi }_0\rangle =sin\frac{\pi }{4}|W_n\rangle +cos\frac{\pi }{4}|{\mathrm{GHZ}}_n\rangle$ for a system of three, five, six, and seven qubits. The fidelity of the two DAQC protocols is better than the fidelity obtained with the DQC, no matter what the errors are. Note that, in the DQC protocol, the error of the two-qubit gates dominates in the total error. Similarly, the fidelity in the DAQC is mainly affected by the errors in the analog blocks,