Digital-analog quantum computation

Digital quantum computing paradigm offers highly desirable features such as universality, scalability, and quantum error correction. However, physical resource requirements to implement useful error-corrected quantum algorithms are prohibitive in the current era of NISQ devices. As an alternative path to performing universal quantum computation, within the NISQ era limitations, we propose to merge digital single-qubit operations with analog multiqubit entangling blocks in an approach we call digital-analog quantum computing (DAQC). Along these lines, although the techniques may be extended to any resource, we propose to use unitaries generated by the ubiquitous Ising Hamiltonian for the analog entangling block and we prove its universal character. We construct explicit DAQC protocols for efficient simulations of arbitrary inhomogeneous Ising, two-body, and $M$-body spin Hamiltonian dynamics by means of single-qubit gates and a fixed homogeneous Ising Hamiltonian. Additionally, we compare a sequential approach where the interactions are switched on and off (stepwise DAQC) with an always-on multiqubit interaction interspersed by fast single-qubit pulses (banged DAQC). Finally, we perform numerical tests comparing purely digital schemes with DAQC protocols, showing a remarkably better performance of the latter. The proposed DAQC approach combines the robustness of analog quantum computing with the flexibility of digital methods.

Figure 1

Algorithm to simulate the general inhomogeneous Ising model from a fixed one. Each time step evolution $t_{\alpha }$ is sandwiched by a pair of single qubit gates ($X^i\equiv {\sigma }_x^i$) applied to qubits $(n,m)$, with $\alpha (n,m)$. Optimal sequences of SQRs can be used to simplify the number of pulses.

Figure 2

Algorithm to simulate the inhomogeneous $\mathit{XZ}$ model from an Ising model. The combination of four different general Ising evolutions and rotations handles the minimum degrees of freedom for each term $g_{jk}^{\mu \nu }$. $R_{{\theta }^{\left(s\right)}}$ rotations can be combined with inner ${\sigma }_x^n{\sigma }_x^m$ rotations required to implement the inhomogeneous ZZ Ising model into unique blocks of SQRs.

Figure 3

Algorithm to implement the $M$-body evolution. Generalized rotations $R^{\left(l\right)}={\otimes }_j^N{r}_j^{\left(l\right)}{\sigma }_x^j+s_j^{\left(l\right)}{\sigma }_y^j+t_j^{\left(l\right)}{\sigma }_z^j$ sandwich evolutions of $H_0^{\left(l\right)}={\sum }_{k=1}^4{\left(H_1^{\left(k\right)}\right)}^{\left(l\right)}+{\left(H_2^{\left(k\right)}\right)}^{\left(l\right)}$; see Eqs. (9) and (10) and Appendix pp4 for an example with $M=4$.

Figure 4

Comparison between a sDAQC and bDAQC protocols. (a) sDAQC digital evolutions with Hamiltonian $H_{R_k}$, and analog block ones evolving with Hamiltonians $H_I$ are well separated in time under the sudden approximation. (b) bDAQC: the adiabatic evolution $H_I$ is on during the whole time and fast rotations are added ($H_{R_k}+H_I$), also under the sudden approximation.

Figure 5

Five-qubit spin system. We have simulated the $\mathit{XZ}$ model with coupling parameters of the original system Hamiltonian with polynomial ${\overline{g}}_{jk}^a=J/{|j-k|}^{5/2}$ (squares), and exponential ${\overline{g}}_{jk}^b=J{e}^{-{\left(\right|j-k|-1)}^2}$ (triangles) decay, corresponding to the simulations in Fig. 6. On the other hand, the couplings of the simulated one are $g_{jk}^{\mu \nu }=J/{|j-k|}^{1/2}$ (circles); $\forall j<k$ and $j,k\in \{1-5\}$ and $\forall \mu ,\nu \in \{x,z\}$.

Figure 6

Exact Trotter evolution: sDAQC vs bDAQC vs DQC. Fidelity $F=|\langle {\mathrm{\Psi }}_e|{\mathrm{\Psi }}_c\rangle {|}^2$ between exact and ideal DQC, sDAQC, and bDAQC protocols for an evolution of a five-qubit $\mathit{XZ}$ model as a function of Trotter steps for a final time $t_F=1/J=2$. The couplings of the original system Hamiltonian are (a) polynomially and (b) exponentially decaying; see Fig. 5. The initial state is $\left|\mathrm{\Psi }\left(0\right)\right.〉=\left|\downarrow \downarrow \uparrow \downarrow \downarrow \right.〉$. The bDAQC protocol considers different finite times $\mathrm{\Delta }t$ for performing the SQRs blocks where the entangling Hamiltonian $H_I={\overline{H}}_{\mathrm{ZZ}}$ is still on.

Figure 7

Times in a Trotter step to perform the DQC and DAQC protocols in terms of the simulated two-body interaction term index $\beta$ and the analog block index $\alpha$, respectively, for the simulation in Fig. 6. (a) (Black) ATA DQC protocol improves considerably time resources when it is decomposed into NN SWAP gates (green). (b) However, both of them still perform worse than the DAQC protocol, implemented through four ZZ blocks of times $t_{\alpha }^{\left(s\right)}$ (blue points), in which the most negative times $|t_{\min }^{\left(s\right)}|$ have been highlighted (red points).

Figure 8

Total times (sums of points in Fig. 7) for each Trotter step as a function of number of qubits. The (blue) DAQC protocol requires (polynomially with $N$) less time than the (green) DQC protocol as the number of qubits increases. This systematic protocol used is not valid for $N=4$ qubits.

Figure 9

Trotter evolution with errors: sDAQC vs bDAQC vs DQC. The DQC protocol (blue) is simulated with a Gaussian phase noise with deviation ${\xi }_D=\mathcal{N}(0,{\sigma }_D)$ added to the $\pi /4$ phases, where ${\sigma }_D=0.009$. sDAQC (black) and bDAQC (red) have Gaussian time noise in the analog blocks ${\delta }_s=\mathcal{N}(0,r_s\mathrm{\Delta }t)$ and ${\delta }_b=\mathcal{N}(0,r_b\mathrm{\Delta }t),$ respectively, and $r_b=0.9=r_s/2$. All simulations include random axis magnetic field noise $\mathrm{\Delta }{B}_{\gamma }=\mathcal{U}(-r_U\mathrm{\Delta }t/2,r_U\mathrm{\Delta }t/2)$, with $r_U=0.002$. The rest of the parameters are equal to Fig. 6 where $\mathrm{\Delta }t=1/500J$ (orange line for bDAQC).

Figure 10

Sets of engineered interactions $O_{\mathrm{XX}}^s$ for creating all-support interactions. (Green) Two sets of generalized Hamiltonians required to create interactions in all supports for $M\le 4$. (Red) Three sets of generalized Hamiltonians required to create all interactions with support $M\le 6$.