### 3.1 Fermi-Hubbard model: small lattices with pairwise interactions

In this section, we present a cQED encoding of the Fermi-Hubbard model, as an example of a fermionic model with nearest-neighbour pairwise interactions, which hence employs only pairwise capacitive spin-spin interactions. Although we focus on a model with three fermionic modes, for the sake of clarity, these techniques are straightforwardly extendable to arbitrary number of fermionic modes in two and three spatial dimensions. These cases are in general mapped into multi-qubit gates that can be always polynomially decomposed into sets of two-qubit gates, as shown below in Eq. (7). In the last part of Section 3, we focus on another cQED platform that uses resonators instead of direct qubit couplings to mediate the interactions.

The Fermi-Hubbard dynamics is a condensed matter model describing traveling electrons in a lattice. The model captures the competition between the kinetic energy of the electrons, discretized in a lattice and encoded in a hopping term, with their Coulomb interaction that is expressed by a nonlinear term. We begin by considering a small lattice realizable with current technology. We consider the Fermi-Hubbard-like model for three spinless fermions with open boundary conditions,

$$ H = -h \bigl(b^{\dagger}_{1} b_{2} + b^{\dagger}_{2} b_{1} + b^{\dagger}_{2} b_{3} + b^{\dagger }_{3} b_{2} \bigr) +U \bigl(b^{\dagger}_{1} b_{1} b^{\dagger}_{2} b_{2} + b^{\dagger}_{2} b_{2} b^{\dagger}_{3} b_{3} \bigr). $$

(2)

Here, \(b_{m}^{\dagger}\) and \(b_{m}\) are fermionic creation and annihilation operators for the site *m*.

We will use the Jordan-Wigner transformation in our derivation to encode the fermionic operators into tensor products of Pauli matrices. We will show below that the latter may be efficiently implemented in superconducting circuits. The Jordan-Wigner mapping reads,

$$\begin{aligned} b^{\dagger}_{1} =& \mathbb{I} \otimes\mathbb{I} \otimes \sigma^{+} , \\ b^{\dagger}_{2} =& \mathbb{I} \otimes \sigma^{+} \otimes\sigma^{z} , \\ b^{\dagger}_{3} =& \sigma^{+} \otimes \sigma^{z} \otimes\sigma^{z}. \end{aligned}$$

(3)

Afterwards, we rewrite the Hamiltonian in Eq. (2) in terms of spin-\(1/2\) operators,

$$\begin{aligned} H = & \frac{h}{2} \bigl(\mathbb{I} \otimes\sigma^{x} \otimes \sigma ^{x} + \mathbb{I} \otimes\sigma^{y} \otimes \sigma^{y} + \sigma^{x} \otimes\sigma^{x} \otimes \mathbb{I} + \sigma^{y} \otimes\sigma^{y} \otimes\mathbb{I} \bigr) \\ &{} + \frac{U}{4} \bigl(\mathbb{I} \otimes\sigma^{z} \otimes \sigma^{z} + \mathbb{I} \otimes\sigma^{z} \otimes\mathbb{I} + \mathbb{I} \otimes \mathbb{I} \otimes\sigma^{z} + \sigma^{z} \otimes\sigma^{z} \otimes \mathbb{I} \\ &{} + \sigma^{z} \otimes\mathbb{I} \otimes\mathbb{I} + \mathbb{I} \otimes\sigma^{z} \otimes\mathbb{I} \bigr). \end{aligned}$$

(4)

Here, the different interactions can be simulated via digital techniques using a specific sequence of gates. We will first consider the associated Hamiltonian evolution in terms of \(\exp( -i \phi\sigma ^{z} \otimes\sigma^{z} )\) interactions. These can be implemented in small steps of \(CZ_{\phi}\) gates, where an average single-qubit gate and two-qubit gate fidelities of 99.92% and up to 99.4%, respectively, have been recently achieved [22]. One can then use the following relations,

$$ \begin{aligned} \sigma^{x} \otimes\sigma^{x} &= R_{y} (\pi/2) \sigma^{z} \otimes\sigma ^{z} R_{y} (-\pi/2) , \\ \sigma^{y} \otimes\sigma^{y} &= R_{x} (-\pi/2) \sigma^{z} \otimes \sigma^{z} R_{x} (\pi/2) , \end{aligned} $$

(5)

where \(R_{j} (\theta) = \exp(-i\frac{\theta}{2} \sigma^{j})\) denote local rotations along the *j*th axis of the Bloch sphere acting on both qubits.

The evolution operator associated with the Hamiltonian in Eq. (4) can be expressed in terms of \(\exp( -i \phi\sigma^{z} \otimes\sigma^{z} )\) interactions. Moreover, the operators may be rearranged in a more suitable way in order to optimise the number of gates and eliminate global phases,

$$\begin{aligned} \exp(-iH t) \approx& \biggl[ R^{\prime}_{y} (\pi/2) \exp \biggl(-i \frac{h}{2} \mathbb{I} \otimes\sigma^{z} \otimes \sigma^{z} \frac{t}{n} \biggr) R^{\prime}_{y} (- \pi/2) R_{y} (\pi/2) \\ &{} \times\exp \biggl(-i \frac{h}{2} \sigma^{z} \otimes \sigma^{z} \otimes\mathbb{I} \frac{t}{n} \biggr) R_{y} (-\pi/2) R^{\prime}_{x} (-\pi/2) \\ &{} \times \exp \biggl(-i \frac{h}{2} \mathbb{I} \otimes \sigma^{z} \otimes\sigma^{z} \frac{t}{n} \biggr) R^{\prime}_{x} (\pi/2) R_{x} (-\pi/2) \\ & {}\times\exp \biggl(-i \frac{h}{2} \sigma^{z} \otimes \sigma^{z} \otimes\mathbb{I} \frac{t}{n} \biggr) R_{x} (\pi/2) \exp \biggl(-i \frac {U}{4} \mathbb{I} \otimes\sigma^{z} \otimes\sigma^{z} \frac {t}{n} \biggr) \\ & {}\times\exp \biggl(-i \frac{U}{2} \mathbb{I} \otimes \sigma^{z} \otimes\mathbb{I} \frac{t}{n} \biggr) \exp \biggl(-i \frac{U}{4} \mathbb {I} \otimes\mathbb{I} \otimes\sigma^{z} \frac{t}{n} \biggr) \\ &{} \times\exp \biggl(-i \frac{U}{4} \sigma^{z} \otimes \sigma^{z} \otimes\mathbb{I} \frac{t}{n} \biggr) \exp \biggl(-i \frac{U}{4} \sigma ^{z} \otimes\mathbb{I} \otimes\mathbb{I} \frac{t}{n} \biggr) \biggr]^{n}, \end{aligned}$$

(6)

where we use the prime notation in the rotation to distinguish between gates applied on different qubits, since \(R_{i}\) acts on the first and second qubits, while \(R^{\prime}_{i}\) acts on the second and the third. If we consider that \(R_{j} (\alpha) R_{j} (\beta) = R_{j} (\alpha+ \beta )\), the sequence of gates for one Trotter step in the digital simulation of the Hubbard model with three qubits is shown in Figure 1. There, gates *A* and *B* are two-qubit gates written in terms of \(\exp( -i \phi\sigma^{z} \otimes\sigma^{z} )\) interactions, \(A = \exp(-i\frac {{h}}{2}\sigma^{z} \otimes\sigma^{z}\frac{t}{n})\) and \(B = \exp(-i\frac {U}{4}\sigma^{z} \otimes\sigma^{z}\frac{t}{n})\). \(Z_{1}\) and \(Z_{2}\) are single-qubit phases, \(Z_{1} = \exp(-i\frac{U}{4}\sigma^{z}\frac{t}{n})\) and \(Z_{2} = \exp(-i\frac{U}{2}\sigma^{z}\frac{t}{n})\), while \(X_{\alpha }\) and \(Y_{\alpha}\) are rotations along the *x* and *y* axis, respectively.

The \(\exp( -i \phi\sigma^{z} \otimes\sigma^{z} )\) interaction can be implemented in small steps with optimized \(CZ_{\phi}\) gates,

$$\exp \biggl(-i\frac{\phi}{2} \sigma^{z} \otimes \sigma^{z} \biggr) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & e^{i\phi} & 0 & 0 \\ 0 & 0 & e^{i\phi} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$

Quantum circuits for simulating these gates are shown in Figures 2 and 3. In Figures 1-3, *X* and *Y* are *π* pulses.

#### 3.1.1 Numerical analysis of the errors

In this section, we present numerical simulations for specific values of model parameters, that is, the time *t*, the hopping coefficient *h*, and nonlinear coupling *U*. We compute numerically the results for the proposed model with three fermionic modes, as well as the equivalent one with two fermionic modes, for the sake of completeness. In Figures 4 and 5, we show the results of the Fermi-Hubbard model with two and three fermionic modes, respectively, for \(n=4\) and \(n=10\) Trotter steps. As shown below, the achieved fidelities can be large at the end of each digital protocol.

In Figure 6, we plot the fidelities of the digitally-evolved state with respect to the ideal dynamics associated with Eq. (2), where \(\theta\equiv Ut\), for \(n=4\) Trotter steps. The fidelities are defined as \(F=|\langle\Psi_{T}|\Psi\rangle|^{2}\), being \(|\Psi\rangle\) and \(|\Psi_{T}\rangle\) the states evolved with the exact unitary evolution and the digital one, respectively. Fidelities well above 90% can be achieved for a large fraction of the considered period.

Summarizing, we have analized the digital quantum simulation of the Fermi-Hubbard model with three fermionic modes in terms of simulatable spin operators with nearest neighbour interactions. Furthermore, we have considered the digital steps involving optimized gates (\(CZ_{\phi}\)).

### 3.2 Large lattices and collective gates mediated via quantum bus

Quantum simulations of fermionic and bosonic models, as well as quantum chemistry problems, have been recently proposed in trapped ions [28–31]. In these proposals, the use of nonlocal interactions via a quantum bus, together with digital expansion techniques, which have been implemented in recent ion-trap experimental setups [32, 33], allows for the retrieval of arbitrary fermionic dynamics. Most current superconducting circuit setups are composed of superconducting qubits and transmission line resonators [6]. A resonator is a useful tool with several applications such as single-qubit rotations, two-qubit gates between distant spins, and dispersive qubit readout [34, 35]. In this section, we analyze how a resonator permits the efficient reproduction of the dynamics of 2D and 3D fermionic systems.

Recently, engineering of fast multiqubit interactions with tunable transmon-resonator couplings has been proposed [27]. These many-body interactions allow for the realization of multipartite entanglement [36], topological codes [37], and as we show below, simulation of fermionic systems. Employing two multiqubit gates and a single-qubit rotation, the unitary evolution associated with a tensor product of spin operators can be constructed,

$$ U=U_{S_{z}^{2}}U_{\sigma_{y}}(\phi)U^{\dagger}_{S_{z}^{2}}=\exp \bigl[i\phi\sigma _{1}^{y}\otimes\sigma_{2}^{z} \otimes\sigma_{3}^{z}\otimes\cdots\otimes\sigma _{k}^{z}\bigr], $$

(7)

where \(U_{S_{z}^{2}}=\exp[-i\pi/4\sum_{i< j}\sigma_{i}^{z}\sigma_{j}^{z}]\) and \(U_{\sigma_{y}}(\phi)=\exp[-i\phi'\sigma_{1}^{y(x)}]\) for odd (even) *k*. The phase \(\phi'\) also depends on the number of qubits, i.e., \(\phi '=\phi\) for \(k=4n+1\), \(\phi'=-\phi\) for \(k=4n-1\), \(\phi'=-\phi\) for \(k=4n-2\), and \(\phi'=\phi\) for \(k=4n\), where *n* is a positive integer. Making use of this unitary evolution and introducing single qubit rotations, it is possible to generate any tensor product of Pauli matrices during a controlled phase that is given in terms of *ϕ*.

In Figure 7, we show how to implement the *i*th-site hopping terms of a system made of *N* fermionic sites onto *N* superconducting qubits coupled to a quantum bus. Notice that local interactions between nearest and next-nearest neighbours in the square lattice involve several qubits in the experimental setup.

In order to benchmark our protocol with a specific example, we consider the Hamiltonian of the Fermi-Hubbard model with both nearest and next-nearest neighbour couplings,

$$\begin{aligned} H = & \sum_{\langle i,j \rangle} \biggl[-h\bigl(b_{i}^{\dagger}b_{j}+{\mathrm{H.c.}}\bigr)+U \biggl(n_{i}-\frac{1}{2} \biggr) \biggl(n_{j}-\frac{1}{2} \biggr) \biggr] \\ &{}+\sum_{\langle\langle i,j \rangle\rangle} \biggl[-h' \bigl(b_{i}^{\dagger}b_{j}+{\mathrm{H.c.}} \bigr)+U' \biggl(n_{i}-\frac{1}{2} \biggr) \biggl(n_{j}-\frac{1}{2} \biggr) \biggr], \end{aligned}$$

(8)

where \(\langle i, j \rangle\) (resp., \(\langle\!\langle i, j \rangle\!\rangle \)) denote sum extended to nearest (next-nearest) neighbours, *h* (\(h'\)) is the hopping parameter and *U* (\(U'\)) is the interaction for nearest (next-nearest) neighbours. Here, \(b_{i}\) (\(b_{i}^{\dagger}\)) is the fermionic annihilation (creation) operator for site *i*, that satisfies the anticommutation relation \(\{b_{i},b_{j}^{\dagger}\}=\delta_{i,j}\), and \(n_{i}=b_{i}^{\dagger}b_{i}\) is the fermionic number operator.

Employing the method introduced before, it is possible to simulate any fermionic dynamics. Let us analyze the interactions we need to simulate in a superconducting qubit platform considering a two-dimensional lattice of \(4\times4\) sites. Taking as an example the 6th site in Figure 8, the simulation of hopping terms with sites 5 and 7 requires only two-qubit gates, since they are nearest neighbours in the order chosen for the mapping. However, the simulation of hopping terms between sites 2 and 6 involves 5 superconducting qubits, \(b_{6}^{\dagger}b_{2} + b_{2}^{\dagger}b_{6}=-(\sigma_{2}^{x}\otimes\sigma _{3}^{z}\otimes\sigma_{4}^{z}\otimes\sigma_{5}^{z}\otimes\sigma_{6}^{x}+\sigma _{2}^{y}\otimes\sigma_{3}^{z}\otimes\sigma_{4}^{z}\otimes\sigma_{5}^{z}\otimes\sigma _{6}^{y})/2\). The same thing happens for next-nearest neighbour interactions, which are simulated employing multiqubit gates made of either 4 or 6 spin operators. On the other hand, interaction terms between qubits *i* and *j* can be implemented by evolving the system with a global interaction involving all the qubits with labels between *i* and *j*, decoupling the rest of the qubits from the resonator.

The number of gates needed for realizing this simulation depends linearly on the number of qubits. Assuming that *N* is the number of fermionic sites in a 2D square lattice, the number of hopping and interaction terms that we need to simulate is \(2\sqrt{N}(\sqrt{N}-1)\) for nearest neighbours and \(2(\sqrt{N}-1)^{2}\) for next-nearest neighbours. As can be seen in Figure 7, every hopping term involving qubits with distant labels is made of 8 single-qubit rotations and 4 multiqubit gates. On the other hand, interaction terms can be simulated applying just one multiqubit gate.

The superconducting setup that we are considering for this quantum simulation is composed of *N* transmon qubits coupled to a single resonator. In order to perform highly nonlocal interactions between two distant qubits, every qubit with label inside the interval spanned by them should interact with the same resonator. Coupling several qubits to just one resonator can be a difficult task wherever the number of simulated sites is large enough. Therefore, we propose an optimized architecture for the simulation of Fermi-Hubbard model with up to next-nearest neighbours in 2D. As it is shown in Figure 8, we propose a setup with *N* superconducting circuits distributed in a square lattice [38]. Sequentially coupling two rows by a single transmission line resonator, one can reduce the number of qubits coupled to a single resonator. Nevertheless, all the interactions needed for satisfying the Jordan-Wigner mapping can be simulated with this architecture. Furthermore, one can achieve a speedup of the protocol by performing interactions that involve different qubits in a parallel way, e.g. the interaction between qubits 2 and 3 and the one between qubits 5 and 9 can be performed simultaneously using resonators 1 and 2, respectively.

In order to benchmark our protocol, we study its efficiency by computing the error associated with a digital decomposition. To this extent, we analyze the occupation of the fermionic sites in a \(3\times 3\) lattice. In Figure 9, we show a plot of these populations considering a perfect unitary evolution of the Fermi-Hubbard model versus the evolution associated with the digital decomposition, where *l* is the number of Trotter steps. As *l* increases, the fidelity \(F=|\langle\Psi_{T}|\Psi\rangle|^{2}\) improves, being \(|\Psi\rangle\) and \(|\Psi_{T}\rangle\) the states evolved with the exact unitary evolution and the digital one, respectively.