# Fermionic models with superconducting circuits

- Urtzi Las Heras
^{1}Email author, - Laura García-Álvarez
^{1}, - Antonio Mezzacapo
^{1}, - Enrique Solano
^{1, 2}and - Lucas Lamata
^{1}

**2**:8

https://doi.org/10.1140/epjqt/s40507-015-0021-5

© Las Heras et al.; licensee Springer. 2015

**Received: **24 November 2014

**Accepted: **5 March 2015

**Published: **24 March 2015

## Abstract

We propose a method for the efficient quantum simulation of fermionic systems with superconducting circuits. It consists in the suitable use of Jordan-Wigner mapping, Trotter decomposition, and multiqubit gates, be with the use of a quantum bus or direct capacitive couplings. We apply our method to the paradigmatic cases of 1D and 2D Fermi-Hubbard models, involving couplings with nearest and next-nearest neighbours. Furthermore, we propose an optimal architecture for this model and discuss the benchmarking of the simulations in realistic circuit quantum electrodynamics setups.

## Keywords

## PACS Codes

## 1 Background

Quantum simulations are one of the most promising research fields in quantum information, allowing the possibility of solving problems exponentially faster than classical computers [1]. In those cases in which analog quantum simulation is hard or impossible, one may decompose the simulated quantum dynamics in terms of discrete quantum gates through a technique known as digital quantum simulation [2–4]. Problems involving interacting fermions are frequently intractable for classical computers due to, among other features, the exponential growth of the Hilbert space dimension with the size of the system. Moreover, standard numerical methods such as quantum Monte Carlo algorithms, do not converge for fermionic systems. Indeed, neither fermionic models in more than one dimension nor systems with the well-known sign problem [5] can be efficiently simulated employing a classical computer. Quantum simulations allow us the reproduction and study of complex systems by means of the use of minimal experimental resources and going beyond mean field approximations in numerical calculations.

Circuit quantum electrodynamics (cQED) [6] is one of the most advanced quantum technologies in terms of coherent control and scalability aspects. Several analog quantum simulations have been proposed in this quantum platform, e.g., spin models [7], quantum phase transitions [8], spin glasses [9], disordered systems [10], metamaterials [11], time symmetry breaking [12], topological order [13], atomic physics [14], open systems [15], dynamical gauge theories [16], and fermionic models in one dimension [17], among others. Furthermore, digital quantum simulations have been recently proposed for superconducting circuits [18, 19] and two pioneering experiments have been performed [20, 21].

In this article, we present a method for encoding the simulation of fermionic systems for arbitrary spatial dimensions, long range or short range couplings, and highly nonlinear interactions, in superconducting circuits. For this purpose, we differentiate two kinds of cQED setups, those employing pairwise capacitive qubit interactions [22], and the ones employing microwave resonators as quantum buses [23]. Our method can be summarized in three steps. The first step consists in mapping a set of *N* fermionic modes to *N* spin operators via the Jordan-Wigner transformation [24]. Then, we make use of the Trotter expansion [2–4] to decompose the unitary evolution of the simulated system in a sequence of quantum gates. Finally, many-body interactions [25, 26] are implemented either with a sequence of capacitive two-qubit gates or by fast multiqubit gates mediated by resonators [27]. Our method allows to implement highly nonlinear and long-range interactions employing only polynomial resources, which makes it suitable for simulating complex physical problems intractable for classical computers. To this extent, we analyze the simulation of the Fermi-Hubbard model with different cQED architectures, considering couplings with nearest neighbours and next-nearest neighbours in two-dimensional fermionic lattices. The structure of the article is the following. In Section 2, we explain the method for decomposing a fermionic dynamics via digital techniques. In Section 3, we describe the proposal for implementing the Fermi-Hubbard model in two distinct situations, either with pairwise capacitive couplings or via resonators. Finally, in Section 4 we give our conclusions.

## 2 Jordan-Wigner mapping and Trotter expansion

The Jordan-Wigner (JW) transformation allows one to map fermionic creation and annihilation operators onto spin operators. When the fermionic lattice is two or three-dimensional, it is possible that local fermionic interactions are mapped onto nonlocal spin ones. The JW mapping reads \(b_{k}^{\dagger}=I_{N}\otimes I_{N-1}\otimes\cdots\otimes\sigma _{k}^{+}\otimes\sigma_{k-1}^{z}\otimes\cdots \otimes\sigma_{1}^{z}\), and \(b_{k}=(b_{k}^{\dagger})^{\dagger}\), where \(b_{k}(b_{k}^{\dagger})\) are the fermionic annihilation and creation operators and \(\sigma_{i}^{\alpha}\) are the spin operators of the *i*th site, being \(\sigma^{\alpha}\) for \(\alpha =x,y,z\) the Pauli matrices and \(\sigma^{+}=(\sigma^{x}+i\sigma^{y})/2\).

*H*is the simulated Hamiltonian, consisting of

*M*quantum gates \(e^{-iH_{i}t}\) that fulfill the condition \(H=\sum_{i}^{M} H_{i}\), being \(H_{i}\) the natural interaction terms of the controllable system. The Trotter expansion can be written as (\(\hbar=1\))

*l*is the total number of Trotter steps. By shortening the execution times of the gates and applying the protocol repeatedly, the digitized unitary evolution becomes more accurate. As can be seen in Eq. (1), the error estimate in this approximation scales with \(t^{2}/l\), such that the longer the simulated time is, the more digital steps we need to apply in order to get good fidelities.

## 3 Circuit QED implementation

### 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.

*m*.

*j*th axis of the Bloch sphere acting on both qubits.

*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.

*X*and

*Y*are

*π*pulses.

#### 3.1.1 Numerical analysis of the errors

*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.

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.

*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*.

*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.

*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.

*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.

*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.

## 4 Conclusions

We have presented a method for the digital quantum simulation of many-body fermionic systems in superconducting circuits with polynomial resources. Moreover, we have analyzed the efficiency of this method for the simulation of the Fermi-Hubbard model in 1D and 2D with different superconducting platforms. Finally, we have proposed an optimized circuit QED architecture where our ideas may be implemented. This work paves the way for the quantum simulation of complex fermionic dynamics in superconducting circuits.

## Declarations

### Acknowledgements

The authors acknowledge useful discussions with John Martinis group at Google/UCSB. We also acknowledge funding from Basque Government IT472-10 Grant, Spanish MINECO FIS2012-36673-C03-02, Ramón y Cajal Grant RYC-2012-11391, UPV/EHU PhD grant, UPV/EHU Project No. EHUA14/04, UPV/EHU UFI 11/55, CCQED, PROMISCE and SCALEQIT European projects.

**Open Access** This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

## Authors’ Affiliations

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