1 Introduction

It is known that the calculation of the exact dynamics of a quantum many-body system is in general a challenging task. When the system is complex enough, analytical and numerical solutions are not possible. However, quantum simulation (QS) allows us to overcome this difficulty by using controllable and manipulable quantum systems, known as quantum simulators, to study another non-controllable one [1, 2]. QS can be classified into three different groups: analog quantum simulations (AQS), digital quantum simulations (DQS), and digital-analog quantum simulations (DAQS). Formally, DQS and Digital-Quantum Computing (DQC) are equivalent, and we will claim the same for DAQS and Digital-Analog Quantum Computing (DAQC).

In AQS, we may reproduce a given target Hamiltonian for some parameter regimes of the simulated model and of the quantum simulator, which is not universal [3, 4]. In DQS, we can perform a sequence of quantum gates in the quantum simulator, which is similar to what happens in DQC. In this case, we can approximate any unitary evolution, specifically, the unitary evolution of the target Hamiltonian model or the given quantum algorithm [5]. Even if DQC is universal, it is less accurate than AQS and Analog Quantum Computing, requiring quantum error correction to scale up and being impractical for current platforms. On the other hand, the recently proposed DAQS aims at getting the best of analog and digital paradigms, with more versatility in the target models or algorithms, higher accuracy, efficient administration of coherence time, and more suitable for current noise intermediate scale quantum (NISQ) architectures [6]. In DAQS, we use a continuous set of complex many-body interactions as a resource (analog blocks), offered naturally by the architecture possibilities of the quantum platform used as quantum computer. We complement it with accessible continuous sets of single-qubit operations (digital steps), providing versatility of target Hamiltonian models or algorithms [710].

Several physical platforms have been used as quantum simulators, such as optical lattices [11], trapped ions [12] and superconducting circuits [1315]. The latter has particular features allowing for current scalability and design flexibility, and produced the recent claim of quantum supremacy in quantum computing [16, 17]. At the same time, diverse target models have been proposed and implemented, principally for DQS and AQS, such as quantum chemistry systems [1821], high-energy physics [2225], and condensed matter physics [2628]. In the long run to reach fault-tolerant quantum computers in the future, proposals for DAQS appear promising for this and next generation of co-design quantum computers [2932]. To achieve that, it would be desirable to widely enhance the DAQC algorithmic mappings as well as the variety of accessible quantum computer geometries and topologies.

In this work, we propose a superconducting circuit design for DAQC. It is composed of a chain of charge qubits coupled through grounded superconducting quantum interference devices (SQUIDs), where the nearest-neighbor qubits are off resonance (different energy gap), similar ideas have been proposed using variable electrostatic fields [33, 34]. In this proposal, the SQUIDs modify the resonant condition among nearest-neighbor qubits, which allows us to produce different and independent interactions like an exchange or double excitation/de-excitation term. Also, as the SQUIDs are physically distant, we could manipulate them individually, activating/deactivating several interactions for obtaining parametrized multiqubit gates, also proving a large family of analog multibody Hamiltonians, being suitable for efficient implementations of DAQC protocols. Finally, we test our architecture performing the simulation of the Fermi-Hubbard model, where we need only \(2(2\ell + 1)^{2} + 24\) analog blocks for the simulation of a \(\ell \times h\) (\(\ell \le h\)) Fermi-Hubbard lattice.

2 The model

2.1 Two-qubit model

First, let us consider the two-qubit system showed in Fig. 1. It consists of two charge qubits coupled through a grounded symmetric SQUID. The Lagrangian of the circuit is given by

$$\begin{aligned} \mathcal{L} ={}& \sum_{j=1}^{2} \biggl[\frac{C_{g_{j}}}{2}\bigl(\Phi ^{ \prime }_{j}-V_{g_{j}} \bigr)^{2}+\frac{C_{J_{j}}}{2}\Phi ^{\prime 2}_{j}+E_{J_{j}} \cos {(\varphi _{j})} \biggr] \\ & {}+ \frac{C_{s}}{2}{\Phi _{s}}^{\prime 2}+E_{J_{s}}^{\textrm{eff}} \cos {( \varphi _{s})}+\frac{C_{c}}{2}\bigl(\Phi ^{\prime }_{1}-\Phi ^{\prime }_{s} \bigr)^{2}+ \frac{C_{c}}{2}\bigl(\Phi ^{\prime }_{s}- \Phi ^{\prime }_{2}\bigr)^{2}, \end{aligned}$$

where \(f^{\prime }=(d/dt)f(t)\), represent the time derivate of a function f, \(E_{J_{s}}^{\textrm{eff}}=2E_{J_{s}}\cos {(\varphi _{\mathrm{ext}})}\) is the effective Josephson energy of the SQUID, and \(\varphi _{j}=2\pi \Phi _{j}/\Phi _{0}\) is the superconducting phase, with the superconducting flux quantum \(\Phi _{0}=h/2e\), and 2e is the electrical charge of a Cooper pair. Moreover \(\Phi _{1}\), \(\Phi _{2}\), and \(\Phi _{s}\) are node fluxes defined in Fig. 1. The effective inductance of the SQUID \(L_{J_{s}}({\varphi _{\mathrm{ext}}})=({\Phi _{0}}/{2\pi })^{2}/E_{J_{s}}^{ \textrm{eff}}\) can be tuned by the external magnetic flux \(\varphi _{\mathrm{ext}}(t)\), providing a tunable boundary condition [35, 36].

Figure 1
figure 1

Circuit diagram of two charged qubits (green) coupled through a grounded SQUID (blue). \(E_{J_{1(2)}}\), \(C_{J_{1(2)}}\), \(C_{g_{1(2)}}\), and \(V_{g_{1(2)}}\) are the Josephson energy, Josephson capacitance, gate capacitance, and gate voltage of the qubit 1(2) respectively. \(E_{J_{s}}\) and \(C_{s}\) are the Josephson energy and effective capacitance of the SQUID. Moreover, \(C_{c}\) is the coupling capacitance, and \(\Phi _{1}\), \(\Phi _{2}\), and \(\Phi _{s}\) are node fluxes that define the degrees of freedom of the circuit

By applying the Legendre transformation, we obtain the Hamiltonian (see Appendix A)

$$\begin{aligned} \mathcal{H}&=\sum_{j=1}^{2} \mathcal{H}_{\textrm{qubit}}^{j}+ \mathcal{H}_{\textrm{SQUID}}+ \mathcal{H}_{\textrm{coupling}}, \end{aligned}$$


$$\begin{aligned} &\mathcal{H}_{\textrm{qubit}}^{j}=\frac{1}{2\tilde{C}_{J_{j}}}(Q_{j}-2e \tilde{n}_{g_{j}})^{2}-E_{J_{j}}\cos {\varphi _{j}}, \\ &\mathcal{H}_{\textrm{SQUID}}=\frac{1}{2\tilde{C}_{J_{s}}}(Q_{s}-2e \tilde{n}_{g_{s}})^{2}- E_{J_{s}}^{\textrm{eff}} \cos {(\varphi _{s})}, \\ &\mathcal{H}_{\textrm{coupling}}=g_{12}Q_{1} Q_{2}+g_{1s}Q_{1} Q_{s}+g_{2s}Q_{2} Q_{s}, \end{aligned}$$

where \(Q_{j}={\partial L}/{\partial \Phi ^{\prime }_{j}}\) is the charge (conjugate momenta) of the jth node given by

$$\begin{aligned} Q_{1(2)}=(C_{1(2)}+C_{c})\Phi ^{\prime }_{1(2)}-C_{g_{1(2)}}V_{g_{1(2)}}-C_{c} \Phi ^{\prime }_{s},~ Q_{s}=(C_{s}+2C_{c}) \Phi ^{\prime }_{s}-C_{c}\bigl( \Phi ^{\prime }_{1}+\Phi ^{\prime }_{2}\bigr), \end{aligned}$$

and the effective Josephson capacitances are defined as

$$\begin{aligned} \tilde{C}_{J_{1(2)}}= \frac{C_{\star }^{3}}{C_{2(1)}(2C_{c}+C_{s})+C_{c}(C_{c}+C_{s} )}, \qquad \tilde{C}_{J_{s}}= \frac{C_{c}C_{\star }^{3}}{(C_{c}+C_{1})(C_{c}+C_{2})}, \end{aligned}$$

with \(C_{\star }^{3}=C_{c}(C_{1}+C_{2})(C_{s}+C_{c})+C_{c}^{2}C_{s}+C_{1}C_{2}(2C_{c}+C_{s})\), and \(C_{j}=C_{g_{j}}+C_{J_{j}}\) (\(j=\{1,2\}\)). Moreover, the gate-charge numbers read

$$\begin{aligned} &\tilde{n}_{g_{1(2)}}=-\frac{C_{g_{1(2)}}}{2e}V_{g_{1(2)}}- \frac{\tilde{C}_{J_{1(2)}}C_{c}^{2}C_{g_{2(1)}}}{2eC_{\star }^{3}}V_{g_{2(1)}}, \\ &\tilde{n}_{g_{s}}=-\frac{\tilde{C}_{J_{s}}C_{c}}{2eC_{\star }^{3}} \bigl(C_{g_{1}}(C_{2} + C_{c} )V_{g_{1}}+C_{g_{2}}(C_{1} + C_{c} )V_{g_{2}} \bigr) , \end{aligned}$$

and couplings strengths are given by

$$\begin{aligned}& g_{12}=\frac{C^{2}_{c}}{C^{3}_{\star }}, \qquad g_{1s(2s)}= \frac{C_{c}(C_{2(1)} + C_{c} )}{C_{\star }^{3}} . \end{aligned}$$
$$\begin{aligned}& Q_{s}=-C_{c} \biggl(\frac{Q_{1}+C_{g_{1}}V_{g_{1}}}{C_{1}+C_{c}}+ \frac{Q_{2}+C_{g_{2}}V_{g_{2}}}{C_{2}+C_{c}} \biggr). \end{aligned}$$

Here, we consider the regime of high plasma frequency for the SQUID, where the charge energy is small compared to the Josephson energy, and the plasma frequency of the SQUID is far exceeding the frequency of the qubits (see Fig. 2), then we can consider \(\Phi ^{\prime }_{s}\ll \Phi ^{\prime }_{1}(\Phi ^{\prime }_{2})\) and \(\Phi ^{\prime \prime }_{s}\ll \Phi ^{\prime \prime }_{1}(\Phi ^{ \prime \prime }_{2})\) [35]. In addition, we also consider the low impedance for the SQUID (see Fig. 3), which allow us consider \(\Phi _{s}\ll \Phi _{1}(\Phi _{2})\). Based on the above conditions, we obtain the next relation for \(Q_{s}\) (see Appendix A)

Figure 2
figure 2

Transitions \(E_{n}-E_{0}\) (n = 1, 2, 3) of the qubits and SQUID Hamiltonians given in Eq. (3). (a) Qubit 1: \(E_{J_{1}} /E_{C_{1}}=0.303\). (b) Qubit 2: \(E_{J_{2}} /E_{C_{2}}=0.058\). (c) SQUID: \(E_{J_{s}}= 50\) (GHz) and \(C_{s}=12\) [fF]

Figure 3
figure 3

Ratio between the SQUID impedance \(Z_{s}\) and the qubit 1(2) impedance \(Z_{1(2)}\) as a function of the external magnetic flux \(\varphi _{\mathrm{ext}}\)

Now, using the Euler-Lagrange equations we obtain (see Appendix A)

$$\begin{aligned} & (C_{1(2)}+C_{c} )\Phi ^{\prime \prime }_{1(2)}-C_{c} \Phi ^{ \prime \prime }_{s}+\frac{2\pi }{\Phi _{0}}E_{J_{1(2)}} \sin {(\varphi _{1(2)})}=0, \\ &-C_{c}\Phi ^{\prime \prime }_{1}-C_{c} \Phi ^{\prime \prime }_{2}+2C_{s} \Phi ^{\prime \prime }_{s}+ \frac{2\pi E_{J_{s}}^{\textrm{eff}}}{\Phi _{0}}\sin {(\varphi _{s})}=0, \end{aligned}$$

using the same above conditions we get the relation for \({\varphi _{s}}\) as

$$\begin{aligned} &{\varphi _{s}}=\frac{-C_{c}}{E_{J_{s}}^{\textrm{eff}}} \biggl( \frac{E_{J_{1}}\sin {(\varphi _{1})}}{C_{1}+C_{c}}+ \frac{E_{J_{2}}\sin {(\varphi _{2})}}{C_{2}+C_{c}} \biggr), \end{aligned}$$

where we approximate \(\sin {(\varphi _{s})}\approx \varphi _{s}\).

We note that, we can write the charge in the node j as \(Q_{j}=2en_{j}\). Promoting the classical variables \(\{n_{j},\varphi _{j}\}\) to quantum operators \(\{\hat{n}_{j},\hat{\varphi }_{j}\}\) with the commutation relation \([e^{i\hat{\varphi }_{j}},\hat{n}_{j}]=e^{i\hat{\varphi }_{j}}\) [37], and applying Eqs. (8) and (10) to Eq. (2), we obtain the quantum mechanical Hamiltonian describing our circuit as

$$\begin{aligned} \hat{\mathcal{H}}=\sum_{j=1}^{2}\hat{ \mathcal{H}}_{\mathrm{sub}}^{j}+\gamma _{12}( \varphi _{\mathrm{ext}})\sin {(\hat{\varphi }_{1})}\sin {(\hat{\varphi }_{2})} , \end{aligned}$$

where the effective coupling strength

$$\begin{aligned} \gamma _{12}(\varphi _{\mathrm{ext}})= \frac{C_{c}^{2}E_{J_{1}}E_{J_{2}}}{(C_{1}+C_{c})(C_{2}+C_{c})E_{J_{s}}^{\textrm{eff}}} , \end{aligned}$$

and \(\hat{\mathcal{H}}_{\mathrm{sub}}^{j}\) is the Hamiltonian of the jth subsystem, given by

$$\begin{aligned} \hat{\mathcal{H}}_{\mathrm{sub}}^{j}=4E_{C_{j}}( \hat{n}_{j}-\bar{n}_{g_{j}})^{2}-E_{J_{j}} \cos {(\hat{\varphi }_{j})}+\gamma _{j}(\varphi _{\mathrm{ext}})\sin {( \hat{\varphi }_{j})}^{2} , \end{aligned}$$

with \(E_{C_{j}}=e^{2}/2(C_{j}+C_{c})\), \(\bar{n}_{g_{j}}=-C_{g_{j}}V_{g_{j}}/2e\), and

$$\begin{aligned} \gamma _{j}(\varphi _{\mathrm{ext}})= \frac{C_{c}^{2}E_{J_{j}}^{2}}{2E_{J_{s}}^{\textrm{eff}}(C_{j}+C_{c})^{2}} . \end{aligned}$$

In the following discussion, we consider \(\bar{n}_{g_{1}}=\bar{n}_{g_{2}}=0.5\), and \(\hbar =1\). It is convenient to write the circuit Hamiltonian in the charge basis, it means \(\hat{n}_{i}=\sum_{n_{j}}n_{j}|n_{j}\rangle \langle n_{j}|\) and \(\cos {(\hat{\varphi }_{j})}=1/2(\sum_{n_{j}}|n_{j}\rangle \langle n_{j}+1|+ \sum_{n_{j}}|n_{j}+1\rangle \langle n_{j}|)\) [37]. Due to the anharmonicity of \(\hat{\mathcal{H}}_{\mathrm{sub}}^{j}\) (see Appendix A), we can perform the two-level approximation in order to obtain the effective Hamiltonian

$$\begin{aligned} \hat{\mathcal{H}}=\frac{\omega _{1}}{2}\sigma _{1}^{z}+ \frac{\omega _{2}}{2}\sigma _{2}^{z}+ \frac{\gamma _{12}(\varphi _{\mathrm{ext}})}{4}\sigma _{1}^{y}\sigma _{2}^{y} , \end{aligned}$$

where \(\omega _{1}=E_{J_{1}}\), \(\omega _{2}=E_{J_{2}}\) and \(\sigma ^{\alpha }_{j}\) is Pauli matrix element of the jth charge qubit and \(\mathbb{I}\) is the identity operator.

Now, we will consider the external flux \(\varphi _{\mathrm{ext}}\) to be composed by a DC signal and a small AC signal as \(\varphi _{\mathrm{ext}}=\varphi _{\mathrm{ext}}(t)=\varphi _{DC}+\varphi _{AC}(t)\), where

$$\begin{aligned} \varphi _{AC}(t)=A_{1}\cos {(\nu _{1} t + \tilde{\varphi }_{1})}+A_{2} \cos {(\nu _{2} t +\tilde{\varphi }_{2})} , \end{aligned}$$

with \(|A_{1}|, |A_{2}|\ll |\varphi _{DC}|\), with which we can approximate

$$\begin{aligned} \frac{1}{E_{J_{s}}^{\textrm{eff}}}\approx \frac{1}{\bar{E}_{J_{s}}} \biggl[1+ \frac{\sin {({\varphi }_{DC})}}{\cos {({\varphi }_{DC}})}{ \varphi }_{AC}(t) \biggr] , \end{aligned}$$

where \(\bar{E}_{J_{s}}=2E_{J_{s}}\cos {({\varphi }_{DC})}\). Then, we can rewrite the Hamiltonian in Eq. (15) as

$$\begin{aligned} \hat{\mathcal{H}}&=\frac{\omega _{1}}{2}\sigma _{1}^{z}+ \frac{\omega _{2}}{2}\sigma _{2}^{z}+ \bigl[g_{0}+g_{1}\varphi _{AC}(t) \bigr] \sigma _{1}^{y}\sigma _{2}^{y} , \end{aligned}$$


$$\begin{aligned} g_{0}&= \frac{C_{c}^{2}E_{J_{1}}E_{J_{2}}}{4(C_{1}+C_{c})(C_{2}+C_{c})\bar{E}_{J_{s}}}, \qquad g_{1}= \frac{C_{c}^{2}E_{J_{1}}E_{J_{2}}}{4(C_{1}+C_{c})(C_{2}+C_{c})\bar{E}_{J_{s}}} \frac{\sin {({\varphi }_{DC})}}{\cos {({\varphi }_{DC})}} . \end{aligned}$$

Now, we write the Hamiltonian of Eq. (18) in the interaction picture with respect to \(\hat{\mathcal{H}}_{0}=\sum^{2}_{i=j}\omega _{j}\sigma _{j}^{z}/2\) and perform the rotating wave approximation (RWA), obtaining

$$\begin{aligned} \hat{\mathcal{H}}_{I}\approx{}& {-}\frac{g_{1}}{2} \sigma _{1}^{-}\sigma _{2}^{-} \bigl(A_{1}e^{i\tilde{\varphi }_{1}}e^{i(\nu _{1}-\mu _{12})t}+A_{2}e^{i \tilde{\varphi }_{2}}e^{i(\nu _{2}-\mu _{12})t} \bigr) \\ &{}+\frac{g_{1}}{2}\sigma _{1}^{-}\sigma _{2}^{+} \bigl(A_{1}e^{i \tilde{\varphi }_{1}}e^{i(\nu _{1}-\Delta _{12})t}+A_{2}e^{i \tilde{\varphi }_{2}}e^{i(\nu _{2}-\Delta _{12})t} \bigr) \\ &{}+\frac{g_{1}}{2}\sigma _{1}^{+}\sigma _{2}^{-} \bigl(A_{1}e^{-i \tilde{\varphi }_{1}}e^{-i(\nu _{1}-\Delta _{12})t}+A_{2}e^{-i \tilde{\varphi }_{2}}e^{-i(\nu _{2}-\Delta _{12})t} \bigr) \\ &{}-\frac{g_{1}}{2}\sigma _{1}^{+}\sigma _{2}^{+} \bigl(A_{1}e^{-i \tilde{\varphi }_{1}}e^{-i(\nu _{1}-\mu _{12})t}+A_{2}e^{-i \tilde{\varphi }_{2}}e^{-i(\nu _{2}-\mu _{12})t} \bigr). \end{aligned}$$

Here, we make use of \(\Delta _{12}=\omega _{1}-\omega _{2}\), \(\mu _{12}=\omega _{1}+\omega _{2}\) and we neglect the fast oscillating terms proportional to \(\operatorname{exp} ({\pm i(\Delta _{12}+\nu _{1(2)})t})\), \(\operatorname{exp} ({ \pm i(\mu _{12}+\nu _{1(2)})t})\), \(\operatorname{exp} ({\pm i\Delta _{12}t})\), and \(\operatorname{exp} ({ \pm i\mu _{12}t})\). As the qubits are far from resonance and considering \(\{g_{0}, A_{1} g_{1}/2, A_{2} g_{1}/2\}\ll \{\Delta _{12}, \mu _{12}, \nu _{1}, \nu _{2}\}\), the RWA is justified (for more details see Appendix A). Considering \(\nu _{1}=\Delta _{12}\) and \(\nu _{2}=\mu _{12}\), the Hamiltonian in Eq. (20) turns

$$\begin{aligned} \hat{\mathcal{H}}_{I}={}&\frac{g_{1}}{4} \bigl((A_{1}\cos { \tilde{\varphi }_{1}}-A_{2} \cos {\tilde{\varphi }_{2}})\sigma _{1}^{x} \sigma _{2}^{x}-(A_{1}\sin {\tilde{\varphi }_{1}}+A_{2}\sin { \tilde{\varphi }_{2}}) \sigma _{1}^{x}\sigma _{2}^{y} \\ &{}+(A_{1}\sin {\tilde{\varphi }_{1}}-A_{2} \sin {\tilde{\varphi }_{2}}) \sigma _{1}^{y} \sigma _{2}^{x}+(A_{1}\cos {\tilde{\varphi }_{1}}+A_{2} \cos {\tilde{\varphi }_{2}}) \sigma _{1}^{y}\sigma _{2}^{y} \bigr) , \end{aligned}$$

where we neglect the fast oscillating terms proportional to \(\operatorname{exp}({ \pm i(\Delta _{12}-\nu _{2})t})\) and \(\operatorname{exp}({\pm i(\mu _{12}-\nu _{1})t})\). We recall that, for a proper choice of the phases \(\tilde{\varphi }_{1}\) and \(\tilde{\varphi }_{2}\) in Eq. (21), we can engineer different interactions as those in Table 1.

Table 1 Interactions produced by different choosing of the phases \(\tilde{\varphi }_{1}\) and \(\tilde{\varphi }_{1}\) in Eq. (21)

2.2 Three-qubit model

In the three-qubit model, we consider the circuit given by Fig. 4. It is composed of a chain of three charge qubits coupled through grounded SQUIDs. As in the previous case, we consider far off-resonance nearest-neighbor qubits. Following the same procedure of the two-qubits model, we get the next effective Hamiltonian (see Appendix B)

$$\begin{aligned} \hat{\mathcal{H}}=\sum_{\ell =1}^{3} \frac{\omega _{\ell }}{2}\sigma _{\ell }^{z}+\sum _{j=1}^{2} \bigl[g^{(j)}_{0}+g^{(j)}_{1} \varphi ^{(j)}_{AC}(t) \bigr]\sigma _{j}^{y} \sigma _{j+1}^{y}, \end{aligned}$$

where \(\omega _{3}=\omega _{1}=E_{J_{1}}\), \(\omega _{2}=E_{J_{2}}\), and the time-dependent signal reads

$$\begin{aligned} \varphi ^{(j)}_{AC}(t)=A^{(j)}_{1} \cos {\bigl(\nu ^{(j)}_{1} t + \tilde{\varphi }^{(j)}_{1}\bigr)}+A^{(j)}_{2}\cos {\bigl(\nu ^{(j)}_{2} t + \tilde{\varphi }^{(j)}_{2}\bigr)}. \end{aligned}$$

Moreover the coupling strength \(g^{(j)}_{0}\) and \(g^{(j)}_{1}\) \((j=\{1,2\})\) are given by

$$\begin{aligned} \begin{gathered} g^{(j)}_{0}= \frac{C_{c}^{2}E_{J_{1}}E_{J_{2}}}{4(C_{1}+C_{c})(C_{2}+2C_{c})\bar{E}^{(j)}_{J_{s}}},\\ g^{(j)}_{1}= \frac{C_{c}^{2}E_{J_{1}}E_{J_{2}}}{4(C_{1}+C_{c})(C_{2}+2C_{c})\bar{E}^{(j)}_{J_{s}}} \frac{\sin { ({\varphi }^{(j)}_{DC} )}}{\cos { ({\varphi }^{(j)}_{DC} )}}, \end{gathered} \end{aligned}$$

with \(\bar{E}^{(j)}_{J_{s}}=2E_{J_{s}}\cos { ({\varphi }^{(j)}_{DC} )}\). To visualize the dynamics of the system, we write the Hamiltonian in the interaction picture. After we consider the resonant conditions \(\nu ^{(1)}_{1}=\nu ^{(2)}_{1}=\Delta _{12}\) and \(\nu ^{(1)}_{2}=\nu ^{(2)}_{2}=\mu _{12}\) and neglect the fast oscillating terms, the Hamiltonian in the interaction picture reads

$$\begin{aligned} \hat{\mathcal{H}}_{I}&=\hat{\mathcal{H}}^{1,2}_{I}+ \hat{\mathcal{H}}^{2,3}_{I}, \end{aligned}$$


$$\begin{aligned} \hat{\mathcal{H}}^{j,j+1}_{I}={}& \frac{g^{(j)}_{1}}{4} \bigl[ \bigl(A^{(j)}_{1} \cos { \tilde{\varphi }^{(j)}_{1}}-A^{(j)}_{2} \cos {\tilde{\varphi }^{(j)}_{2}} \bigr)\sigma _{j}^{x}\sigma _{j+1}^{x} \\ &{}+ \bigl((-1)^{j}A^{(j)}_{1}\sin { \tilde{\varphi }^{(j)}_{1}}-A_{2}\sin { \tilde{\varphi }^{(j)}_{2}} \bigr)\sigma _{j}^{x}\sigma _{j+1}^{y} \\ &{}+ \bigl((-1)^{j+1}A^{(1)}_{1}\sin {\tilde{\varphi }^{(j)}_{1}}-A^{(j)}_{2} \sin {\tilde{\varphi }^{(j)}_{2}} \bigr)\sigma _{j}^{y}\sigma _{j+1}^{x} \\ &{}+ \bigl(A^{(j)}_{1}\cos {\tilde{\varphi }^{(j)}_{1}}+A^{(j)}_{2}\cos { \tilde{\varphi }^{(j)}_{2}} \bigr)\sigma _{j}^{y}\sigma _{j+1}^{y} \bigr], \end{aligned}$$

is the interaction Hamiltonian between jth and \((j+1)\)th qubit. By choosing proper phase parameters, we can engineer different interaction operators between adjacent qubits, like in the previous case.

Figure 4
figure 4

Effective circuit diagram of three charge qubits (green) couple through grounded SQUIDs (blue) with Josephson energy \(E_{J_{s}}\) and effective capacitor \(C_{s}/2\). Moreover \(C_{c}\) is the coupling capacitance, and \(\Phi _{1}\), \(\Phi _{2}\), \(\Phi _{3}\), \(\Phi _{s_{1}}\) and \(\Phi _{s_{2}}\) are node fluxes that define the degrees of freedom of the qubits and SQUIDs

It is possible to generalize this expression for a chain of qubits coupled through grounded SQUIDs (see Fig. 5), where we define the qubits in odd positions as qubit 1 with frequency \(\omega _{1}\) and the qubits in even positions as qubit 2 with frequency \(\omega _{2}\). In the following discussion, we consider the amplitude of the two harmonic signals to be the same, that is \(A^{(j)}_{1}=A^{(j)}_{2}=A\) and the coupling strength \(g^{(j)}_{0}=g_{0}\) and \(g^{(j)}_{1}=g_{1}\).

Figure 5
figure 5

General circuit design of a chain of charge qubits (green part) coupling through grounded SQUIDs (blue part) with Josephson energy \(E_{J_{s}}\) and effective capacitor \(C_{s}\). \(\{\Phi _{1}, \Phi _{s_{1}},\dots ,\Phi _{s_{N-1}},\Phi _{N}\}\) are node fluxes that define the degrees of freedom of the circuit

By considering the resonant conditions \(\nu ^{(j)}_{1}=\Delta _{12}\), \(\nu ^{(j)}_{2}=\mu _{12}\) and choosing proper phase parameters \(\tilde{\varphi }^{(j)}_{1}\) and \(\tilde{\varphi }^{(j)}_{2}\), we can engineer again a family of interactions between nearest-neighbor qubits as is shown in Table 2. Note that the phase \(\tilde{\varphi }^{j}_{1}\) required to achieve \(\pm \sigma _{j}^{x}\sigma _{j+1}^{y}\) and \(\pm \sigma _{j}^{y}\sigma _{j+1}^{x}\) are different for odd and even j.

Table 2 Interactions between jth qubit and \((j+1)\)th qubit produced by different choice of the phases \(\tilde{\varphi }^{(j)}_{1}\) and \(\tilde{\varphi }^{(j)}_{1}\), where we take odd j and even j both into account

The controllability and flexibility of the interactions that our proposal offers, give us the possibility to the implement of a large variety of Hamiltonians in an analog way, such as Dzyaloshinskii-Moriya, XY, homogeneous and inhomogeneous spin chains. Such analog Hamiltonians could be very useful for DAQS and DAQC, where we can use such analog Hamiltonians like a resource (complex multibody gate) for simulating more complex systems, like quantum chemistry physics, condensed matter phenomena in spin lattices [38, 39], and shortcuts to adiabaticity in digitezed adiabatic quantum computing [40].

The approach we presented in this work is intimately linked to the nature of the Jordan-Wigner mapping that requires the quantum simulation algorithm follows a linear sorting of the lattice sites to reproduce the fermionic anti-commutation relation, being a qubit-chain a natural simulator of the fermion models. Naturally, the search for an experimentally feasible fermion to qubit mapping approaching a two-dimensional lattice beyond the Jordan-Wigner transformation is an open question that deserves further investigation. There is a recent work that proposed a novel mapping in this direction, but the experimental realization is still an open question [41]. Also, the use of qubit lattices for the Jordan-Wigner transformation has been proposed; nevertheless, the gates number scaling is the same as that in the qubit-chain case [8].

The current proposal could be extended to a two-dimensional (2D) array of charged qubits coupled through grounded SQUIDs generating a more complex family of Hamiltonians, opening the door to more efficient simulations. Nevertheless, for 2D structures, we can have the non-trivial problem of cross-talk between the different SQUID and loops in the circuit, requiring a deep feasibility study which is not the scope of this article. In the next section, we show a particular example about the efficient DAQC of a complex system, the Fermi-Hubbard model. This example will illustrate the versatility of our design.

3 Digital-analog quantum computation

Hubbard model represents the interaction between the neighboring sites, which is defined by hopping element and Coulombic interaction on the same site, called on-site interaction [42]. In this section, we are interested in the simulation of the hopping terms of a \(\ell \times h\) fermion-lattice (with \(\ell \leq h\)). The Hamiltonian of a \(\ell \times h\) fermion-lattice (see Fig. 6(a)) reads

$$\begin{aligned} \mathcal{H}_{\textrm{Hubb}}=\mathcal{A}\sum_{\alpha =\{\uparrow , \downarrow \}} \sum_{\langle j,k\rangle } \bigl(c^{\dagger }_{j,\alpha }c_{k, \alpha }+c^{\dagger }_{k,\alpha }c_{j,\alpha } \bigr)+\mathcal{B}\sum_{j}n_{j, \uparrow }n_{j,\downarrow } , \end{aligned}$$

where \(c_{j,\alpha }^{\dagger }\) (\(c_{j,\alpha }\)) are the creation (annihilation) operators of the jth site, with the number operator \(n_{j,\uparrow (\downarrow )}=c^{\dagger }_{j,\uparrow (\downarrow )}c_{j, \uparrow (\downarrow )}\), and spin-α, with \(\alpha =\{\uparrow ,\downarrow \}\). To suppress the index α, we map the \(\ell \times h\) lattice to a equivalent \(2\ell \times h\) spin-less lattice by

$$\begin{aligned} c_{j,\uparrow }^{\dagger }\rightarrow b_{2j-1}^{\dagger }, \qquad c_{j, \downarrow }^{\dagger }\rightarrow b_{2j}^{\dagger } , \end{aligned}$$

where \(b_{k}^{\dagger }\) are the creation operation over the site k for the lattice given by Fig. 6(b). Using these operators the Hamiltonian Eq. (27) can be rewrite as

$$\begin{aligned} \mathcal{H}_{\textrm{Hubb}}={}&\mathcal{A} \Biggl(\sum _{k=0}^{h-1} \Biggl[ \sum _{j=1}^{\ell -1} \bigl(b_{2k\ell +2j-1}^{\dagger }b_{2k\ell +2j+1}+b_{2k \ell +2j+1}^{\dagger }b_{2k\ell +2j-1} \bigr) \\ &{}+\sum_{j=1}^{\ell -1} \bigl(b_{2k\ell +2j}^{\dagger }b_{2k\ell +2(j+1)}+b_{2k \ell +2(j+1)}^{\dagger }b_{2k\ell +2j} \bigr) \Biggr] \\ &{}+\sum_{k=0}^{h-2}\sum _{j=1}^{2\ell } \bigl[b_{2k\ell +j}^{\dagger }b_{2(k+1) \ell +j}+b_{2(k+1)\ell +j}^{\dagger }b_{2k\ell +j} \bigr] \Biggr) \\ &{}+\mathcal{B} \Biggl(\sum_{j=1}^{h\ell }b^{\dagger }_{2j-1}b_{2j-1}b^{ \dagger }_{2j}b_{2j} \Biggr) . \end{aligned}$$

Finally, we map the \(2\ell \times h\) fermion lattice to a \(\mathrm{spin} -1/2\) chain using the Wigner-Jordan transformation (see Fig. 6(c)), where we represent \(b_{j}\) and \(b_{j}^{\dagger }\) as a combination of Pauli matrices

$$\begin{aligned} \begin{gathered} b_{j}= \Biggl[\prod_{l=1}^{j-1} \bigl(-\sigma _{l}^{z}\bigr) \Biggr]\sigma _{j}=(-1)^{j-1} \frac{1}{2} \Biggl[\prod _{l=1}^{j-1}\sigma _{l}^{z} \Biggr]\bigl(\sigma _{j}^{x}-i \sigma _{j}^{y}\bigr), \\ b_{j}^{\dagger }= \Biggl[\prod_{l=1}^{j-1} \bigl(-\sigma _{l}^{z}\bigr) \Biggr] \sigma _{j}^{\dagger }=(-1)^{j-1}\frac{1}{2} \Biggl[\prod_{l=1}^{j-1} \sigma _{\ell }^{z} \Biggr]\bigl(\sigma _{j}^{x}+i \sigma _{j}^{y}\bigr) . \end{gathered} \end{aligned}$$

Before to write the equivalent spin chain Hamiltonian, we define the operator

$$\begin{aligned} U_{(j,k)}^{\alpha ,\beta }=e^{-i\frac{\pi }{4}(\sigma _{j-1}^{\alpha } \sigma _{j}^{\alpha }+\sigma _{k}^{\beta }\sigma _{k+1}^{\beta })}={U}_{j-1}^{ \alpha }{U}_{k}^{\beta }, \end{aligned}$$

where \(j\ne k\), and \({U}_{j}^{\alpha }=e^{-i\frac{\pi }{4}\sigma _{j}^{\alpha }\sigma _{j+1}^{ \alpha }}\). After some algebraic manipulation, we obtain (for details see Appendix C)

$$\begin{aligned} \mathcal{H}_{\textrm{Hubb}}=\mathcal{H}_{\textrm{hori}}+ \mathcal{H}_{ \textrm{verti}}+\mathcal{H}_{\textrm{coul}}, \end{aligned}$$

where \(\mathcal{H}_{\textrm{hori}}\), \(\mathcal{H}_{\textrm{verti}}\) and \(\mathcal{H}_{\textrm{coul}}\) are given by

$$\begin{aligned} \begin{gathered} \begin{aligned} \mathcal{H}_{\textrm{hori}}={}&\frac{\mathcal{A}}{2} \bigl( \bigl[U_{(1,2)}^{x^{\dagger }} H_{1,2}^{(x,y)}U_{(1,2)}^{x} \bigr]+ \bigl[U_{(1,2)}^{y} H_{1,2}^{(y,x)}U_{(1,2)}^{y^{\dagger }} \bigr]+ \bigl[U_{(1,3)}^{x^{\dagger }} H_{1,3}^{(x,y)}U_{(1,3)}^{x} \bigr] \\ &{}+ \bigl[U_{(1,3)}^{y} H_{1,3}^{(y,x)}U_{(1,3)}^{y^{\dagger }} \bigr]+ \bigl[U_{(2,4)}^{x^{\dagger }} H_{2,4}^{(x,y)}U_{(2,4)}^{x} \bigr]+ \bigl[U_{(2,4)}^{y} H_{2,4}^{(y,x)}U_{(2,4)}^{y^{\dagger }} \bigr] \\ &{}+ \bigl[U_{(2,5)}^{x^{\dagger }} H_{2,5}^{(x,y)}U_{(2,5)}^{x} \bigr]+ \bigl[U_{(2,5)}^{y} H_{2,5}^{(y,x)}U_{(2,5)}^{y^{\dagger }} \bigr] \bigr), \end{aligned} \\ \begin{aligned} \mathcal{H}_{\textrm{verti}}={}&\frac{\mathcal{A}}{2}\sum _{j=1}^{2\ell } \bigl[ \bigl(\tilde{U}_{j,1}^{(x,x)} \tilde{U}_{j,2}^{(y,y)}\ldots \tilde{U}_{j,\ell -2}^{(x,x)} \tilde{U}_{j,\ell -1}^{(y,y)}\tilde{U}_{j}^{x^{\dagger }} \bigr) \\ &{}\cdot \Theta _{j}^{x,y} \bigl(\tilde{U}_{j}^{x}U_{j, \ell -1}^{(y,y)^{\dagger }} U_{j,\ell -2}^{(x,x)^{\dagger }}\ldots U_{j,2}^{(y,y)^{\dagger }}U_{j,1}^{(x,x)^{\dagger }} \bigr) \\ &{}+ \bigl(U_{j,1}^{(y,y)}U_{j,2}^{(x,x)}\ldots U_{j,\ell -2}^{(y,y)}U_{j, \ell -1}^{(x,x)}U_{j}^{y} \bigr)\\ &{}\cdot \Theta _{j}^{y,x} \bigl( \tilde{U}_{j}^{y^{\dagger }} \tilde{U}_{j,\ell -1}^{(x,x)^{\dagger }} \tilde{U}_{j,\ell -2}^{(y,y)^{\dagger }}\ldots U_{j,2}^{(x,x)^{\dagger }}U_{j,1}^{(y,y)^{\dagger }} \bigr) \bigr], \end{aligned} \\ \mathcal{H}_{\textrm{coul}}=\frac{\mathcal{B}}{4}\sum _{j=1}^{h\ell }\bigl( \sigma _{2j-1}^{z}+ \mathbb{I}\bigr) \bigl(\sigma _{2j}^{z}+\mathbb{I} \bigr), \end{gathered} \end{aligned}$$

which correspond to the Hamiltonian of the horizontal hopping, vertical hopping and coulomb interaction respectively, with

$$\begin{aligned} &U_{(n,i)}^{a}=\prod _{k=0}^{h-1} \prod_{j=1}^{m_{n}} U^{a}_{2k\ell +4(j-1)+i},\qquad H_{n,i}^{(a,b)}= \sum _{k=0}^{h-1}\sum _{j=1}^{m_{n}}\sigma _{2k\ell +4j-5+i}^{a} \sigma _{2k\ell +4(j-1)+i}^{b}, \\ &\Theta _{j}^{a,b}=\sum _{k=0}^{h-2}\sigma _{2k\ell +j+\ell -1}^{a} \sigma _{2k\ell +j+\ell }^{b},\qquad \tilde{U}_{j,i}^{(a,a)}= \prod_{k=0}^{h-2}U_{2k \ell +j+i,2k\ell +j+2\ell -i}^{(a,a)}, \\ &\tilde{U}_{j}^{a}=\prod_{k=0}^{h-2}U_{2k\ell +j+\ell }^{a} . \end{aligned}$$


$$\begin{aligned} m_{1}=\frac{2\ell -1-(-1)^{\ell +1}}{4},\qquad m_{2}= \frac{2\ell -3+(-1)^{\ell +1}}{4} . \end{aligned}$$
Figure 6
figure 6

Mapping of the fermionic lattice into a spin chain. (a) \(h\times \ell \) lattice where we have and spin-up or spin-down fermion in each site. (b) \(h\times 2\ell \) lattice where we in each odd site we have and spin-up fermion and in each even site we have a spin-down fermion, it represents the first mapping. (c) Spin chain with \(2\ell h\) sites resulting after the Wigner-Jordan Mapping

We notice that each term of \(\mathcal{H}_{\textrm{hori}}\) in Eq. (33) correspond to the set of horizontal hopping interactions which do not share sites in the lattice, these eight interactions are represented by arrows with different colors in Fig. 7 (blue, red, green and brown), and different textures (solid and dashed). Also, we highlight that each interactions given by Eq. (34) can be performed in an analog way since the sub-gates involved are applied in different qubits and can be done together without interfering with each other (see Appendix C), giving us the analog resource for the digital-analog simulation. The number \(m_{1}\) is the number of the hopping terms corresponding to the blue(red) solid/dash arrows, and \(m_{2}\) is the number of the hopping terms corresponding to the green(brown) solid/dash arrows (see Fig. 7), where \(m_{1}+m_{2}=\ell -1\).

Figure 7
figure 7

Diagram for the different horizontal hopping interactions. Different colors show the set of interactions that can be performed at the same time in an analog way. Solid arrows are for forward hopping and dashed arrows for backward hopping

Now, we approximate the time evolution of our system using the Trotter expansion [43] as follows

$$\begin{aligned} e^{-i\mathcal{H}_{\textrm{Hubb}}t}&\approx \bigl[e^{-i\mathcal{H}_{ \textrm{hori}}t/n}e^{-i\mathcal{H}_{\textrm{verti}}t/n}e^{-i\mathcal{H}_{ \textrm{coul}}t/n} \bigr]^{n} \\ &= \bigl[U_{\textrm{hori}}(t/n)U_{\textrm{verti}}(t/n)U_{\textrm{coul}}(t/n) \bigr]^{n}, \end{aligned}$$

where \(U_{\textrm{word}}(t)=e^{-i\mathcal{H}_{\textrm{word}}t}\). Using Eq. (33) we have

$$\begin{aligned} \begin{gathered} \begin{aligned} U_{\textrm{hori}}(t/n)\approx{}& U_{(1,2)}^{x^{\dagger }}e^{-i \frac{\mathcal{A}t}{2n}H_{1,2}^{(x,y)}}U_{(1,2)}^{x} \cdot U_{(1,2)}^{y} e^{-i\frac{\mathcal{A}t}{2n}H_{1,2}^{(y,x)}}U_{(1,2)}^{y^{\dagger }} \\ &{}\cdot U_{(1,3)}^{x^{\dagger }}e^{-i\frac{\mathcal{A}t}{2n}H_{1,3}^{(x,y)}}U_{(1,3)}^{x} \cdot U_{(1,3)}^{y} e^{-i\frac{\mathcal{A}t}{2n}H_{1,3}^{(y,x)}}U_{(1,3)}^{y^{\dagger }} \\ &{}\cdot U_{(2,4)}^{x^{\dagger }}e^{-i\frac{\mathcal{A}t}{2n}H_{2,4}^{(x,y)}}U_{(2,4)}^{x} \cdot U_{(2,4)}^{y} e^{-i\frac{\mathcal{A}t}{2n}H_{2,4}^{(y,x)}}U_{(2,4)}^{y^{\dagger }} \\ &{}\cdot U_{(2,5)}^{x^{\dagger }}e^{-i\frac{\mathcal{A}t}{2n}H_{2,5}^{(x,y)}}U_{(2,5)}^{x} \cdot U_{(2,5)}^{y} e^{-i\frac{\mathcal{A}t}{2n}H_{2,5}^{(y,x)}}U_{(2,5)}^{y^{\dagger }}, \end{aligned} \\ \begin{aligned} U_{\textrm{verti}}(t/n)\approx{} &\prod_{j=1}^{2\ell } \bigl[ \bigl( \tilde{U}_{j,1}^{(x,x)}\tilde{U}_{j,2}^{(y,y)}\ldots \tilde{U}_{j,\ell -2}^{(x,x)} \tilde{U}_{j,\ell -1}^{(y,y)} \tilde{U}_{j}^{x^{\dagger }} \bigr) \\ &{}\cdot e^{-i\frac{\mathcal{A}t}{2n}\Theta _{j}^{x,y}} \bigl(\tilde{U}_{j}^{x}U_{j, \ell -1}^{(y,y)^{\dagger }} U_{j,\ell -2}^{(x,x)^{\dagger }}\ldots U_{j,2}^{(y,y)^{\dagger }}U_{j,1}^{(x,x)^{\dagger }} \bigr) \\ &{}\cdot \bigl(U_{j,1}^{(y,y)}U_{j,2}^{(x,x)}\ldots U_{j,\ell -2}^{(y,y)}U_{j, \ell -1}^{(x,x)}U_{j}^{y} \bigr) \\ &{}\cdot e^{-i\frac{\mathcal{A}t}{2n}\Theta _{j}^{y,x}} \bigl(\tilde{U}_{j}^{y^{\dagger }} \tilde{U}_{j,\ell -1}^{(x,x)^{\dagger }}\tilde{U}_{j,\ell -2}^{(y,y)^{\dagger }}\ldots U_{j,2}^{(x,x)^{\dagger }}U_{j,1}^{(y,y)^{\dagger }} \bigr) \bigr], \end{aligned} \\ \begin{aligned} U_{\textrm{coul}}(t/n)&\approx \prod_{j=1}^{h\ell }e^{-i \frac{\mathcal{B}t}{4n}(\sigma _{2j-1}^{z}+\mathbb{I})(\sigma _{2j}^{z}+ \mathbb{I})} \\ &=\prod_{j=1}^{h\ell }e^{-i\frac{\mathcal{B}t}{4n}(\sigma _{2j-1}^{z} \sigma _{2j}^{z})}e^{-i\frac{\mathcal{B}t}{4n}(\sigma _{2j-1}^{z}+ \sigma _{2j}^{z})} . \end{aligned} \end{gathered} \end{aligned}$$

First, for \(U_{\textrm{hori}}\) we simulate eight types of interactions (see Fig. 7 and Eq. (33)), each of these interactions need three gates, as we mention above each of this gate can be simulated in an analog way. Therefore, to simulate \(U_{\textrm{hori}}\) (the horizontal hopping) we need 24 gates, as we can see in Eq. (34).

Second, for \(U_{\textrm{verti}}\), i.e. all the vertical hopping terms, we need \(2(2\ell +1)\) types of interactions, they are shown in Fig. 8 with different colors and different textures. We note that all the interactions of the same type can be performed at the same time, due to they do not share sites during its implementation, i.e. all the interactions with the same color and same texture can be simulated in parallel. Now, to simulate each type of interaction we need \(2\ell +1\) analog gates (see Eqs. (33) and (34)). Then, to simulate all the vertical hopping terms, we need \(2(2\ell +1)^{2}\) analog gates. Therefore to simulate all the hopping terms in a \(\ell \times h\) fermion lattice with \(\ell \le h\) we need only \(2(2\ell +1)^{2}+24=8\ell ^{2}+8\ell +26\) gates.

Figure 8
figure 8

Diagram for the different vertical hopping interactions. Different colors show the set of interactions that can be performed at the same time in an analog way. Solid arrows are for forward hopping and dashed arrows for backward hopping

It is necessary to highlight that to our knowledge, the more efficient proposal for the quantum simulation of the Hubbard model is a trapped ion one [44], which uses a multi-body entangling gate, needing \(8(2h\ell -\ell -h)+20\) gates for a \(\ell \times h\) fermion lattice. It means that for a square lattice \((\ell =h)\), the trapped ion proposal needs \(2(8\ell ^{2}-8\ell +10)\) gates, which for large need almost two times more gates than our proposal. It means that even if superconducting circuits cannot use multi-body entangling gates like trapped ions, it still is useful by a suitable design as in this work.

On the other hand, \(U_{\textrm{coul}}(t/n)\), correspond to the free energy of the original Hubbard model (Eq. (27)) and we will not consider for the Hopping dynamics simulation. Nevertheless, it can be simulated using three gates, i.e. the analog interaction \(\sum_{j}\sigma _{2j-1}^{x}\sigma _{2j}^{x}\) plus two rotations in the y-axis (the local terms \(\sigma _{z}^{j}\) can be mapped correspond to the free energy of our simulator).

The simulation time can be easily derivate as follow. Each type of interaction involve unitary gate of the form \(\mathcal{U}_{a}=\operatorname{exp}(-\frac{i\mathcal{A}}{2}\frac{t}{n} \hat{O})\) and of the form \(\mathcal{U}_{b}=\operatorname{exp}(-i\frac{\pi }{4}\hat{O})\). From Eq. (37) we obtain that the time for each kind of gates (a and b) is

$$\begin{aligned} \tau _{a}=\frac{\mathcal{A}t}{Ag_{1}n},\qquad \tau _{b}= \frac{\pi }{2Ag_{1}} , \end{aligned}$$

respectively. From the simulation of \(U_{\textrm{hori}}\) we have 8 gates of the class a, and from \(U_{\textrm{verti}}\) we have \(2(2\ell +1)\) gates of the class a. Then the time necessary to perform all these gates is \((4\ell +10)\tau _{a}\). As we have a total of \(2(2\ell +1)^{2}+24\) gates, the number of type b gates is \(8\ell ^{2}+4\ell +16\). Therefore, the total time for the simulation is

$$\begin{aligned} \tau _{\textrm{sim}}&=(4\ell +10)\tau _{a} + \bigl(8\ell ^{2}+4\ell +16\bigr) \tau _{b}=(4\ell +10) \frac{\mathcal{A}t}{Ag_{1}n}+\bigl(4\ell ^{2}+2\ell +8\bigr) \frac{\pi }{Ag_{1}} . \end{aligned}$$

We note that for the case \(\ell =2\), we need fewer gates, in particular, to simulate \(U_{\textrm{hori}}\) we need half of the gates, it means 12, for this case, the simulation time also decreases and is given by

$$\begin{aligned} \tau _{\textrm{sim}}^{*}=(4\ell +6)\tau _{a} + \bigl(8\ell ^{2}+4\ell +8\bigr) \tau _{b}=14 \frac{\mathcal{A}t}{Ag_{1}n}+24\frac{\pi }{Ag_{1}} . \end{aligned}$$

Finally, the character digital-analog of our simulation is given by the use of analog gates in each digital step, it means gates that act over several qubits simultaneously. In the next section, we present the numerical results of a quantum simulation of the hopping interaction of a \(2\times 3\) fermion-lattice.

4 Numerical results: \(2\times 3\) fermion lattice

As we mention above, for the case of \(\ell =2\) we only need 12 gates to simulate \(U_{\textrm{hori}}\), then for \(2\times h\) lattice, we need 62 gates per Trotter step

Figure 9 shows the types of hopping interactions to simulate for a \(2\times 3\) fermion-lattice. In Fig. 9 (a), we can see the four interactions to describe the horizontal hopping, where the solid arrows correspond to the forward hopping, and the dashed arrows correspond to the backward hopping. For vertical hopping, it requires ten types of interactions, as is shown in Fig. 9 (b). The sequence of the gates for different hopping interactions is shown in Appendix D and Appendix E.

Figure 9
figure 9

Diagram for the different hopping interactions in a \(2\times 3\) fermion lattice, with the solid arrows to be forward hopping and dashed arrows to be backward hopping (a) Horizontal hopping. (b) Vertical hopping

For the simulation, we map the \(2\times 3\) fermion lattice into a 12 qubit chain described by the Hamiltonian Eq. (26). The parameters that we consider for the simulation are summarized in the Table 3, where t represent the simulated time (evolution time of the system to be simulated). Figure 10 show the fidelity \(|\langle \psi _{\textrm{sim}}(t)|\psi (t)\rangle |^{2}\) of our simulation for different initial states and 10, 20 and 30 Trotter steps, where \(\vert \psi \rangle (t)\) is the state at time t of the real model, and \(\vert \psi _{\textrm{sim}}(t) \rangle \) is the state given by the simulation, which simulate the evolution at a time t. If we think the fermion lattice as a \(3\times 2\) matrix, where each element can be ↑, ↓ or vacuum, the initials states are: Fig. 10(a), ↑ for the sites \((1,1)\), \((2,2)\) and \((3,1)\), and the rest in vacuum; Fig. 10(b), ↑ for the sites \((1,1)\) and the rest in vacuum; Fig. 10(c), quantum superposition between the state ↑↓ for all sites and vacuum for all sites. Finally Fig. 10(d), shows the mean fidelity over 1000 random states.

Figure 10
figure 10

(a, b, c) Fidelity between the perfect evolution and DAQC for different numbers of Trotter steps and different initial states\(|\psi _{0} \rangle \) (shown in each subfigure). (d) Mean fidelity for 1000 random initial states for different Trotter steps. The physical parameters of the fermion model are shown Table 3

Table 3 Parameters for Fermion model of Eq. (15)

Table 4 summarize the time \(\tau _{a}\), \(\tau _{b}\) and \(\tau _{\textrm{sim}}^{*}\) defining in Eqs. (34) and (36), for the different number of Trotter steps. We can note that all the simulation times are below the 0.2 [μs], which means that the simulation can be implemented with the current technology, where the coherent times for superconducting qubits are in the order of 100 [μs] [45].

Table 4 Times involved in the fermion lattice simulations for different number of Trotter steps. The corresponding parameters of the fermion model are shown in Table 3

5 Conclusion

We have designed a superconducting circuit architecture suitable for DAQC, in the sense of providing a wide family of analog Hamiltonians as source of analog blocks and more flexibility for this pragmatic quantum computing paradigm. We test our design by the numerical calculation of the quantum simulation of a \(3\times 2\) fermion lattice described by the Hubbard model. We find that for a \(\ell \times h\) lattice (\(\ell \le h\)), we need \(2(2\ell +1)^{2}+24\) analog blocks, which depend only on one of the dimensions of the lattice and improves. To our knowledge, this result would improve previous achievements for simulating the Hubbard model by a factor 2 for large square lattice (\(\ell = h\)) [44]. Moreover, the total simulation time for 30 Trotter steps is less than 0.2 μs with an ideal fidelity around 0.97 (only digital error), which makes our proposal experimentally feasible. Finally, we consider this work provides an important boost to the DAQC paradigm, paving the way for computing and simulating complex systems in quantum platforms, while approaching us to useful quantum advantage with fewer algorithmic and hardware resources.