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Fewqubit quantumclassical simulation of strongly correlated lattice fermions
EPJ Quantum Technology volume 3, Article number: 11 (2016)
Abstract
We study a proofofprinciple example of the recently proposed hybrid quantumclassical simulation of strongly correlated fermion models in the thermodynamic limit. In a ‘twosite’ dynamical meanfield theory (DMFT) approach we reduce the Hubbard model to an effective impurity model subject to selfconsistency conditions. The resulting minimal twosite representation of the nonlinear hybrid setup involves four qubits implementing the impurity problem, plus an ancilla qubit on which all measurements are performed. We outline a possible implementation with superconducting circuits feasible with nearfuture technology.
Introduction
Using highly controllable quantum devices to study other quantum systems, i.e., quantum simulation [1–4], offers a means to tackle strongly correlated fermion models that are intractable on classical computers. This is vital for understanding complex quantum materials [5] with strong electronic correlations that exhibit a plethora of exciting physical phenomena of immediate technological interest. Examples of such effects include the Mott metalinsulator transition [6, 7], colossal magnetoresistance [8], and hightemperature superconductivity [9, 10].
Classical numerical methods have limited ability to study even significantly simplified toy models of strongly correlated fermions. For instance, exact diagonalization faces exponential scaling with the system size, while quantum Monte Carlo methods [11, 12] are often crippled by the infamous fermionic sign problem [13]. Tensor network methods [14–18] are powerful in one spatial dimension where they track strong correlations accurately. However, in higher dimensional systems, correlations tend to grow more quickly with system size, making these methods computationally challenging.
Another wellestablished approach to the study of strongly correlated fermionic lattice systems is dynamical meanfield theory (DMFT) [19]. It reduces the complexity of the original problem, e.g., the Hubbard model [20] in the thermodynamic limit, by mapping it onto a simpler impurity problem that is subject to a selfconsistency condition relating its properties to those of the original model. Since an impurity problem is local, the mapping corresponds to neglecting spatial fluctuations. In the limit of infinite spatial dimensions this mapping is exact, but for finite dimensions it is an approximation. Nonetheless for lattice geometries with a large coordination number, selfconsistently solving the impurity problem can yield an accurate approximate solution to the original Hubbard problem.
The ‘impurity’ itself consists of a single lattice site taken from the original problem, and so inherits onsite interactions from the Hubbard model. This impurity site is then immersed into a timedependent, selfconsistent meanfield with which it can dynamically exchange fermions. The meanfield thus attempts to model the rest of the lattice and by being dynamical can describe retardation phenomena. Overall the impurity problem can be represented by a Hamiltonian in which the interacting impurity site is coupled to a discrete set of noninteracting ‘bath’ sites. The bath sites represent the meanfield and if there is an infinite number of them then the selfconsistency condition is guaranteed to be satisfied. However, in practical implementations only a finite number of bath sites are used, which restricts the frequency resolution of the bath so selfconsistency condition can only be fulfilled approximately. Nevertheless, many stronglycorrelated features, e.g., the Mott transition, are still be captured correctly [19]. For a study of different bath discretisation strategies in DMFT, see [21].
Although DMFT maps a Hubbard model to an impurity model this is still a nontrivial quantum manybody problem to solve because of the interactions at the impurity site. It is usually solved by classical numerical methods, e.g., specialised versions of those used to tackle the original problem, which attempt to keep track of the quantum correlations between impurity and bath sites. Again this limits the number of bath sites that can be treated accurately.
Here, we consider an alternative approach where the impurity problem is solved with a quantum simulator, thus avoiding many issues that are inherent to the classical methods. Quantum simulation of fermionic models has so far been mostly restricted to the analogue paradigm, especially with ultracold atoms in optical lattices [22]. Digital simulation approaches, akin to universal quantum simulators [23], have started to emerge in recent years, for example based on superconducting circuits [24–27]. Different quantum simulation schemes for the Hubbard model have been proposed [28–31]. The number of qubits in these digital simulators is, however, presently rather small. A direct implementation of the Hubbard model would suffer from severe finite size effects. It is nevertheless still possible for a digital quantum simulator with a restricted number of qubits to describe fermionic models directly in the thermodynamic limit when the DMFT approach is adopted.
To demonstrate this method we focus on the minimal incarnation of DMFT, the socalled ‘twosite’ DMFT [32], where the impurity model consists of one impurity site and only one bath site, both with local Hilbert space dimension four, subjected to two specially chosen selfconsistency conditions. Since twosite DMFT considers only the smallest possible impurity model, the approach cannot match the accuracy of full DMFT, but it can still give a qualitatively correct description of the infinitedimensional Hubbard model, and its simplicity makes it a good starting point before advancing to more accurate schemes. For explicit details of twosite DMFT and its features compared to full DMFT we refer to Ref. [32].
The twosite system corresponds to four qubits, two for the impurity site and two for the bath site, while a fifth, ancillary qubit is used for measurements. This number of qubits is readily available in current digital quantum simulator platforms, with IBM having made a fivequbit quantum processor available to the public [33]. A ninequbit processor has already been demonstrated in superconducting circuits [26, 27, 34]. Trappedion technologies also allow for digital quantum simulations with up to six qubits [35, 36]. Being commensurate with current stateoftheart technology is a further justification for studying this minimal model. Our scheme is readily generalisable to a larger number of qubits allowing for more accurate simulations and potentially offering an exponential speedup over classical Hamiltonianbased DMFT methods [37]. For example, the number of multiqubit MølmerSørensen gates scales only linearly with the number of bath sites, enabling efficient simulations [28, 29, 37, 38].
The selfconsistency conditions are taken care of iteratively in a classical feedback loop, which thus completes the nonlinear, hybrid quantumclassical device we introduce. Dynamical meanfield simulations have already been proposed for such hybrid devices [37, 39]. Quantum gates similar to the ones needed in the twosite scheme have been used in demonstrating digital quantum simulation of fermionic models with superconducting circuits [26, 38]. We thus focus on superconducting circuits as a candidate platform, although, e.g., trapped ions [28, 35, 40, 41] could also be considered.
This paper is organised as follows. In Section 2, we further elucidate the framework of DMFT applied to the Hubbard model in infinite dimensions. Section 3 introduces the twosite DMFT scheme in detail. Section 4 discusses the implementation of this twosite scheme with special attention to superconducting circuits. In Section 5, we show the results of our analysis. We end with a summary in Section 6 and give an outline of the singlequbit interferometry measurement scheme in the Appendix.
Hubbard model in infinite dimensions and dynamical meanfield theory
A standard model to describe strongly correlated electron systems in thermodynamic equilibrium is the Hubbard Hamiltonian
In this model, electrons with spin projections \(\sigma=\downarrow, \uparrow\) ‘hop’ between adjacent lattice sites with tunnelling energy t. This process is described in the first term, where \(\langle j, k \rangle\) denotes the sum over all nearestneighbour sites j and k, and \(\hat{c}^{\dagger}_{j,\sigma}\) and \(\hat{c}_{k,\sigma}\) denote the fermionic creation and annihilation operators, respectively. The electrons interact with onsite Coulomb repulsion \(U>0\), described in the latter term by the product of the local number operators \(\hat {n}_{j,\downarrow}=\hat{c}^{\dagger}_{j,\downarrow}\hat {c}_{j,\downarrow}\) and \(\hat{n}_{j,\uparrow}=\hat{c}^{\dagger }_{j,\uparrow}\hat{c}_{j,\uparrow}\).
Here, we consider the paramagnetic Hubbard model in an infinitedimensional Bethe lattice in the thermodynamic limit at zero temperature. This setup has very simple selfconsistency relations, which makes it an ideal testbed for a proofofprinciple demonstration of a hybrid quantumclassical scheme.
The DMFT approach [19] to solving this model consists in neglecting spatial fluctuations around a single lattice site and replacing the rest of the manybody lattice in the thermodynamic limit by a timetranslationinvariant, selfconsistent meanfield \(\Delta(\tau\tau')\) (or \(\Delta(\omega)\) in the frequency domain), as illustrated in Figure 1(a). The isolated lattice site can dynamically exchange fermions with the meanfield at time instants \(\tau'\) and τ. This allows one to include retardation effects that are important in the presence of strong correlations. In short, the dynamical meanfield approach reduces the complexity of the full Hubbard model to an effective singlesite system which is a slightly more benign manybody problem to solve. In infinite dimensions, DMFT becomes exact as the irreducible selfenergy of the lattice model becomes strictly local in space, \(\Sigma_{{\mathrm{latt}},jk}(\omega)=\delta_{jk} \Sigma_{{\mathrm{latt}},jj}(\omega)\), and its skeleton diagrams agree with those of a singlesite, or impurity, model [19].
The solution of the effective singlesite, or impurity, problem also yields the solution of the infinitedimensional Hubbard model due to the selfconsistency condition. This leads to the retarded singleparticle impurity Green function in the frequency domain being given by
where μ is the chemical potential, and \(\Sigma_{\mathrm{imp}}(\omega )\) denotes the impurity selfenergy. We set \(\hbar=1\) throughout the paper. The impurity Green function describes the response of the manybody system after a localized removal or addition of a particle on the impurity site and is defined in the time domain and at zero temperature as
where i is the imaginary unit, τ is real time, \(\{\cdot,\cdot \}\) denotes the anticommutator, \(\theta(\tau)\) is the Heaviside step function, and the average is computed in the groundstate \(GS\rangle\) of the impurity model. The fermionic creation and annihilation operators are given in the Heisenberg picture. In the paramagnetic phase the Green function is spin symmetric and we therefore only need to work out \(G^{R}_{\mathrm{imp}}(\omega)\) for one spin configuration.
The initially unknown meanfield \(\Delta(\omega)\) has to be chosen such that \(G^{R}_{\mathrm{imp}}(\omega)\) matches the local part of the retarded lattice Green function \(G^{R}_{{\mathrm{latt}},jj}(\omega)\), i.e.,
where j is the (arbitrarily chosen) lattice site from which the removal or addition of a particle occurs in the translationally invariant lattice model. The DMFT selfconsistency condition Eq. (4) implies
i.e., the impurity selfenergy matches the local selfenergy of the Hubbard model in the infinitedimensional Bethe lattice.
In the general case, the DMFT selfconsistency loop is iterated as follows (see also Ref. [19]). (i) First, guess the local selfenergy \(\Sigma_{{\mathrm{latt}},jj}(\omega)\). (ii) The local lattice Green function can be computed as \(G^{R}_{{\mathrm{latt}},jj} (\omega)=\int_{\infty}^{\infty} d\epsilon \rho _{0}(\epsilon)/ [\omega+\mu\epsilon\Sigma_{{\mathrm{latt}},jj}(\omega) ]\), where \(\rho_{0}(\epsilon)=\sqrt {4{t^{*}}^{2}\epsilon^{2}}/2\pi{t^{*}}^{2}\) is the noninteracting density of states of a Bethe lattice. The constant \({t^{*}}\) emerges from the requirement that the Hubbard hopping needs to be scaled as \(t \sim t^{*}/\sqrt{z}\) to avoid a diverging kinetic energy per lattice site in the limit of infinite coordination, \(z \rightarrow\infty\) [19]. (iii) With Eqs. (4) and (5), we obtain \(\Delta(\omega)\) from Eq. (2) and the impurity model is then defined. (iv) Compute the impurity Green function and obtain the impurity selfenergy \(\Sigma_{\mathrm{imp}}(\omega )\). There are several means to do this [19]. (v) Set \(\Sigma^{\mathrm{new}}_{{\mathrm{latt}},jj}(\omega)=\Sigma_{\mathrm{imp}} (\omega)\). (vi) Check if the selfenergy has converged. If not, go to step (ii) and repeat.
Once selfconsistent, the solution of the impurity problem then gives access to local singleparticle properties of the original lattice model. For example, the local lattice spectral function is given by
where η is a positive infinitesimal.
In Hamiltonianbased impurity solvers, one parameterizes \(\Delta (\omega)\) by a set of bath sites (see Figure 1(b)). For any finite number of bath sites, the selfconsistency condition (4) can only be approximately satisfied and in the extreme ‘twosite’ DMFT it turns out to be more suitable to reformulate Eq. (4) in a manner specially focused on this minimal representation [32] (see Section 3). Note that twosite DMFT is only able to provide a qualitatively correct description of the Hubbard model even in infinite dimensions [32].
Quantum simulator based on twosite DMFT
In terms of the singleimpurity Anderson model (SIAM), the smallest impurity problem involves one fermionic site corresponding to the impurity and only one fermionic site corresponding to the entire meanfield as described in the previous section. Since two qubits are needed to encode the local Hilbert space of a fermionic site, we only require four physical qubits to implement this representation in the lab. The SIAM Hamiltonian for only one bath site reads
Here, U is the Hubbard interaction at the impurity site 1, and μ is the chemical potential that controls the electron filling in the grand canonical ensemble. Furthermore, \(\epsilon_{c}\) and V describe the onsite energy of the noninteracting bath site 2 and hybridization between the impurity and the bath site, respectively, and give the meanfield as
See Figure 1(c) for illustration of the twosite SIAM. The parameters \(\epsilon_{c}\) and V are initially unknown and they need to be determined iteratively such that two selfconsistency conditions are satisfied. For details of the derivation and motivation of these conditions we refer to Ref. [32].
The first condition is that the electron filling \(n_{\mathrm{imp}}\) of the impurity site and the filling \(n=\langle n_{j\downarrow} \rangle+ \langle n_{j\uparrow} \rangle\) of the lattice model match, i.e.,
The second selfconsistency condition is given by
where quasiparticle weight reads
In Eq. (9), \(M_{2}^{(0)}\) is the second moment of the noninteracting density of states, and the final equality follows from the semicircular density of states of the Bethe lattice.
Twosite DMFT protocol
The hybrid quantumclassical device implementing twosite DMFT consists of a fewqubit digital quantum simulator in which the impurity Green function is measured and of a classical feedback loop in which the parameters of the twosite SIAM are updated. The twosite DMFT protocol is summarized in Figure 2 and proceeds as follows (see also Ref. [32]).

1.
First fix U and μ to the desired values in the SIAM and set the unknown parameters \(\epsilon_{c}\) and V equal to an initial guess.

2.
Measure the interacting Green function \(iG^{R}_{\mathrm {imp}}(\tau)\). This can be done using, e.g., singlequbit interferometry (see details in the Appendix).

3.
After Fouriertransforming the impurity Green function, the impurity selfenergy is obtained classically from the Dyson equation
$$\begin{aligned} \Sigma_{\mathrm{imp}}(\omega) = {G}_{\mathrm{imp}}^{R(0)}( \omega )^{1}G^{R}_{\mathrm{imp}}(\omega)^{1}. \end{aligned}$$(11)Here, the noninteracting impurity Green function is given by
$$\begin{aligned} {G}_{\mathrm{imp}}^{R(0)}(\omega)^{1}=\omega+ \mu\Delta(\omega). \end{aligned}$$(12)From the derivative of the selfenergy one obtains the quasiparticle weight \(\mathcal{Z}\) which directly yields the updated hopping parameter V via Eq. (9). The update for \(\epsilon_{c}\) is found by minimizing the difference \(n_{\mathrm{imp}}n\) [32].

4.
Steps 2 and 3 need to be repeated until V and \(\epsilon _{c}\) are selfconsistent, and \(n_{\mathrm{imp}}=n\).
The selfconsistent Green function \(G^{R}_{\mathrm{imp}}(\omega)\) and selfenergy \(\Sigma_{\mathrm{imp}}(\omega)\) thus obtained are used to calculate approximations to local singleparticle properties of the Hubbard model. Note that for larger systems the twosite DMFT steps need to be replaced with the general DMFT selfconsistency loop outlined in Section 2.
Quantum algorithm for the singleimpurity Anderson model with superconducting circuits
Here, we consider the quantum gates of the digital quantum simulator part in Figure 2, with special focus on superconducting circuits as the platform of choice [26, 27, 38].
JordanWigner transformation of the SIAM
To implement the twosite SIAM with qubits, the fermionic creation and annihilation operators need to be mapped onto tensor products of spin operators which then act on the qubits via quantum gates. In order to obtain as simple quantum gates as possible in Section 4.3 and in the Appendix, we consider an ordering of the qubits where the first two qubits encode the spin ↓ for both fermionic sites while the last two correspond to spin ↑. This is achieved via the JordanWigner transformation given explicitly as
and \(\hat{c}_{j\sigma}= (\hat{c}_{j\sigma}^{\dagger} )^{\dagger}\). Here, \(\hat{\sigma}^{x}_{l}\), \(\hat{\sigma}^{y}_{l}\), and \(\hat{\sigma}^{z}_{l}\) are spin\(\frac{1}{2}\) Pauli operators for qubit l. For larger systems the use of the JordanWigner transformation, e.g., \(\hat{c}^{\dagger}_{j\downarrow}= ( \prod_{p<2j1}\hat{\sigma}^{z}_{p} ) \hat{\sigma}^{}_{2j1}\), \(\hat {c}^{\dagger}_{j\uparrow}= ( \prod_{p<2j}\hat{\sigma}^{z}_{p} ) \hat{\sigma}^{}_{2j}\), \(\hat{c}_{j\sigma}= (\hat {c}_{j\sigma}^{\dagger} )^{\dagger}\), becomes important, as considered in Refs. [29, 37]. The hybridization terms in the SIAM, which now involve many spins, can be implemented efficiently and scalably with multiqubit MølmerSørensen gates, the number of which scales only linearly with the number of bath sites [28, 29, 37, 38].
With the mappings in Eqs. (13)(16), the hybridization terms in the SIAM described in Eq. (6) transform into
and
The number operators become
and thus the interaction term can be written as
up to a constant. The total Hamiltonian then reads
where we have dropped constant terms.
Quantum gates in superconducting circuits
We now consider how the JordanWigner transformed SIAM in Eq. (24) can be implemented in an experimental arrangement based on superconducting circuits. We present two alternative approaches. The first one couples the qubits with a transmission line resonator, which leads to the socalled XY gate between the qubits. The second approach is the ControlledZ_{ ϕ } (CZ_{ ϕ }) gate, which can be obtained via a capacitive coupling of nearestneighbour transmon qubits without using a resonator. These CZ_{ ϕ } gates have been implemented with high fidelities of above 99% for a variant of transmon qubits called ‘Xmon’ qubits [42].
XY gates with resonators
The basic Hamiltonian coupling a set of qubits to the resonator has the form of a detuned JaynesCummings model. By adiabatically eliminating the resonator one obtains, when the resonator is in the vacuum state, the wellknown XY model for a pair of qubits l and m as
Here, Δ is the detuning between the qubit levelspacing and the resonator mode, and \(g_{l}\) is the coupling constant between qubit l and the resonator. The XY gate is universal for quantum computation and simulation in combination with single qubit gates, and is the natural interaction customarily employed in superconducting circuits.
CZ_{ ϕ } gates with capacitive couplings
To perform the CZ_{ ϕ } gate, one qubit is kept at a fixed frequency while the other carries out an adiabatic trajectory near an appropriate resonance of the twoqubit states. By varying the amplitude of this trajectory one can tune the conditional phase ϕ. The unitary for the CZ_{ ϕ } is given by
Quantum gate decomposition of the timeevolution operator
In order to use quantum gates for timeevolution, we utilize a Trotter decomposition of the timeevolution operator corresponding to \(\hat {H}_{\mathrm{SIAM}}\) in Eq. (24). The first order Trotter expansion is given by
Here, N is the number of Trotter (i.e., time) steps and \(\frac{\tau }{N}\) is the size of the time step. In what follows, we use the two alternative approaches for quantum gates outlined in Section 4.2 to implement Eq. (27).
XY gates
As shown in Section 4.2, the XY gate, given by the expression \(XY = {\exp [i\frac{V}{2}(\hat{\sigma}^{x}_{l} \hat {\sigma}^{x}_{m} + \hat{\sigma}^{y}_{l} \hat{\sigma}^{y}_{m}) \frac {\tau}{n} ]}\), naturally appears when considering the use of a resonator quantum bus [43]. The quantum circuit for a single Trotter step with these gates is shown in Figure 3(a).
CZ_{ ϕ } gates
To be able to utilize the CZ_{ ϕ } gates, we write the timeevolution operator in Eq. (27) in terms of \(\hat{\sigma}^{z}_{l} \hat{\sigma}^{z}_{m}\) (ZZ) interactions, taking into account that
and
where \(\mathcal{R}^{(l)}_{\alpha} (\theta) = \exp(i\frac{\theta }{2} \hat{\sigma}_{l}^{\alpha})\) is the rotation along the αaxis of qubit l. Note that in the computational basis, one can write, e.g.,
where we have neglected global phases. Thus, we have the decomposition
where the tunable \({\mathrm{CZ}}_{\phi}\)gate is given by Eq. (26).
The timeevolution operator in Eq. (27) in terms of ZZ interactions is given by
where \(\mathcal{R}^{(1234)}_{\alpha}(\phi)=\mathcal{R}^{(1)}_{\alpha } (\phi)\mathcal{R}^{(2)}_{\alpha} (\phi) \mathcal {R}^{(3)}_{\alpha} (\phi)\mathcal{R}^{(4)}_{\alpha}(\phi) \). The sequence of gates for one Trotter step is depicted in Figure 3(b).
A single Trotter step contains 5 ZZ twoqubit gates (corresponding to the A and B gates in Figure 3(b)) between nearestneighbour qubits, 2 SWAP gates (for the B gate which acts on qubits 1 and 3), and 20 singlequbit rotations. We note that a SWAPgate amounts to three CZ_{ ϕ } gates, and a ZZgate amounts to two CZ_{ ϕ } gates (see Eq. (31)). This number can be optimised further if we consider different orderings for odd and even Trotter steps as in Figure 4, such that subsequent gates may be suppressed. This reorganisation of interactions does not in principle affect the Trotter error. Hence, for a pair of Trotter steps, the number of gates is reduced, and we may only consider 10 ZZ twoqubit gates between nearestneighbour qubits, 4 SWAP gates, and 32 singlequbit rotations.
Results
We focus on the halffilled case, i.e., \(\mu=\frac{U}{2}\) and \(\epsilon_{c}=0\), which requires the least amount of quantum gates, since the C and D gates in Section 4 vanish. Note that since the value of \(\epsilon_{c}\) is fixed in this case, it need not be updated in the selfconsistency loop. We use \(t^{*}\), the Hubbard hopping in infinite dimensions, as our unit of energy, hence time τ is measured in units of \(1/t^{*}\). Note that τ refers here to the time in the evolution operator \(\hat{U}(\tau)\), not to the actual time to run the experiment.
We show in Figure 5 the state fidelities \(\mathcal{F}= \langle\Psi(\tau)  \Psi_{T}(\tau) \rangle^{2}\), where \(\Psi(\tau )\rangle\) denotes the state obtained with exact timeevolution using the full, nonTrotterized operator \(\hat{U}(\tau)=\exp(i\tau\hat {H}_{\mathrm{SIAM}})\) corresponding to the twosite SIAM in Eq. (6), and \( \Psi_{T}(\tau) \rangle\) is the state evolved using either the XY or CZ_{ ϕ } quantum gates, for various Trotter steps N up to time \(\tau=6/t^{*}\). Note that the number of qubits corresponding to the twosite SIAM is fixed, leaving only N as the parameter to be varied for increased accuracy. We use the initial state \(\Psi(\tau=0)\rangle=\hat{c}^{\dagger}_{1\downarrow}GS\rangle /\hat{c}^{\dagger}_{1\downarrow}GS\rangle\), where \(GS\rangle \) is the groundstate of the twosite SIAM in Eq. (6), which is a relevant state for obtaining the impurity Green function at zero temperature (see Eq. (3)). As expected, using XY gates displays superior fidelities, since CZ_{ ϕ } gates require an extra factorization of the hybridization term (see Section 4). For \(N=24\) steps, the state fidelity using XY gates remains over 99% throughout the evolution. In what follows, we use only XY gates for the timeevolution for concreteness.
As shown in Section 3, the main object of interest is the retarded impurity Green function. One possibility to measure \(iG^{R}_{\mathrm{imp}}(\tau)\) is singlequbit interferometry (see the Appendix for details), which raises the total number of qubits in the experimental arrangement to five. In Figure 6 we plot the impurity Green function obtained from evolving the state with XY gates compared to exact evolution of the twosite SIAM for different N. We see that the Green function from the XY approach starts to follow the curve of the exact Green function better for increasing N. In our subsequent analysis, we use \(N=24\) up to \(\tau=6/t^{*}\) to study what twosite DMFT physics can be captured with the digital approach.
To obtain the impurity Green function in the frequency domain, we first consider some known and general analytic properties of the retarded Green function in Eq. (3). This Green function can be written as a sum of the particle and hole contributions as
where \(j\rangle\) is an eigenstate of \(\hat{H}_{\mathrm{SIAM}}\) with eigenenergy \(E_{j}\), and \(\omega_{j}=E_{j}E_{GS}\). In twosite DMFT, the interacting Green function is a fourpole function [32], which limits the number of terms in the above summation to four. Moreover, in the presence of particlehole symmetry, we have \( \langle j  \hat{c}^{\dagger}_{1\sigma}  GS \rangle ^{2}=  \langle j  \hat{c}_{1\sigma}  GS \rangle ^{2}\), and Eq. (33) can be written as
where \(\alpha_{j}=\vert \langle j  \hat{c}^{\dagger}_{1\sigma}  GS \rangle \vert ^{2}\). Thus, to obtain the impurity Green function in the frequency domain as
we need to extract the unknown residues \(\alpha_{j}\) and poles \(\omega _{j}\) by fitting an expression of the form in Eq. (34) to the measurement data of \(iG^{R}_{\mathrm{imp}}(\tau)\), as shown in Figure 7(a). This method to determine \(\alpha_{j}\) and \(\omega_{j}\) is far more reliable and requires fewer time steps than numerically Fouriertransforming the \(iG^{R}_{\mathrm{imp}}(\tau)\) data. It can also be readily generalised to larger systems by including more terms in the sum in Eq. (33). Figure 7(b) shows the real part of the impurity Green function in the frequency domain, \({\operatorname{Re}} [ G^{R}_{\mathrm{imp}}(\omega+i\eta) ]\), with residues and poles obtained from the fit in Figure 7(a), while in Figure 7(c) we plot the real part of the impurity selfenergy, \({\operatorname{Re}} [ \Sigma_{\mathrm{imp}}(\omega+i\eta) ]\), obtained utilizing the Dyson equation (11). We clearly see the fourpole structure of the Green function, while the selfenergy has two poles. The results are in excellent agreement with the exact solution of the twosite SIAM, with the poles of the selfenergy using fitted \(\alpha_{j}\) and \(\omega_{j}\) differing from the exact solution by 2%.
Once we have obtained the impurity Green function, and thus the impurity selfenergy, we proceed according to the twosite DMFT protocol in Section 3 until selfconsistency has been reached. In DMFT we are interested in the local lattice spectral function \(A_{{\mathrm{latt}},jj}(\omega)\) which, at selfconsistency, is given by the impurity spectral function \(A_{\mathrm{imp}}(\omega)\). In the paramagnetic phase of the infinitedimensional Hubbard model, the spectral function has a three peak structure with an upper and a lower Hubbard band, corresponding to empty and doubly occupied sites, respectively, and a quasiparticle peak with integrated spectral weight \(\mathcal{Z}\) between the bands [19]. In twosite DMFT, since the selfenergy has two poles, this three peak structure can be qualitatively reproduced with the spectral function [32]
where \(\rho_{0}\) is the noninteracting density of states of the Bethe lattice. Figure 8 shows the spectral function in Eq. (36) where the impurity selfenergy has been obtained both from the XY method and from exact numerics of the twosite SIAM using the interactions \(U=5t^{*}\) and \(U=8t^{*}\). We notice that for \(U=5t^{*}\), the Hubbard bands from the XY method are slightly dislocated and the quasiparticle peak is slightly narrower compared with the exact solution of the twosite SIAM, but the agreement is still very good. The overall shape of the spectral function from the XY method is unchanged compared to the exact case. This underestimation of the width of the quasiparticle peak stems from the fact that the fitting procedure in Figure 7(a) causes the negative of the derivative of the selfenergy in the XY method to be a bit larger than the exact value from the twosite SIAM, i.e.,
which leads to \(\mathcal{Z}\) in Eq. (10) from the XY method to be slightly smaller than in the exact solution of the twosite SIAM, i.e., \(\mathcal{Z}^{XY} \lesssim\mathcal{Z}^{\mathrm{exact}}\). For \(U=8t^{*}\), the two spectral functions agree with maximum relative error of 10^{−8}, since in this case \(V=0\) is found to be the selfconsistent solution, whence the Trotterized evolution operator in Eq. (27) matches full evolution operator of the twosite SIAM, and thus there is no Trotter error. We observe that in Figure 8 the central quasiparticle peak vanishes, which is characteristic of insulating behaviour. See Ref. [32] for a discussion of the artifacts of the spectral functions in twosite DMFT compared to full DMFT.
To study the transition between the two types of spectral functions in Figure 8, we plot in Figure 9 the selfconsistent quasiparticle weight \(\mathcal{Z}\) obtained from the XY method as a function of the interaction U for different Trotter steps N. We also show \(\mathcal{Z}\) from the exact solution of the twosite SIAM for comparison. We see that the digital approach captures the correct trend of the curve, but in the metallic side underestimates to a small degree the values of \(\mathcal{Z}\) for interactions close to \(U=U_{c}=6t^{*}\), which is the critical interaction for Mott transition in twosite DMFT at halffilling [32]. These results are consistent with the spectral functions in Figure 8. The underestimation of \(\mathcal{Z}\) can be diminished by increasing N, as shown in Figure 9. It is noteworthy to mention that twosite DMFT overestimates the quasiparticle weight compared to full DMFT for interactions \(U< U_{c}\), as demonstrated in Ref. [32]. Above \(U_{c}\), we find \(\mathcal{Z}=0\) to be the selfconsistent solution, corresponding to the insulating phase.
Summary
We have proposed a quantum algorithm for twosite DMFT to be run on a small digital quantum simulator with a classical feedback loop, allowing the qualitative description of the infinitedimensional Hubbard model in the thermodynamic limit. We have considered two alternative quantum gate decompositions consistent with stateoftheart technology in superconducting circuits for the timeevolution operator. We found that an increasing number of Trotter steps improves the fidelity of our digital scheme to qualitatively describe the Mott transition. Our work therefore provides an interesting application for smallscale quantum devices. It also paves the way for more accurate quantum simulations of strongly correlated fermions in various lattice geometries, which are relevant to novel quantum materials, when the general selfconsistency condition and larger number of qubits are used.
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Acknowledgements
We acknowledge AnnaMaija Uimonen as well as Ian Walmsley and his group members for useful discussions. JMK acknowledges financial support from Christ Church, Oxford and the Osk Huttunen Foundation. LGÁ, LL and ES acknowledge support from a UPV/EHU PhD grant, Spanish MINECO/FEDER FIS201569983P, UPV/EHU UFI 11/55 and Project EHUA14/04, and Ramón y Cajal Grant RYC201211391. DJ was supported by the EPSRC National Quantum Technology Hub in Networked Quantum Information Processing (NQIT) EP/M013243/1.
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Authors’ contributions
JMK conceived the project and performed the numerical simulations under the supervision of SRC and DJ. LGÁ, LL and ES worked out the superconducting circuit implementation. All authors contributed to interpreting the results and writing of the manuscript.
Appendix: singlequbit interferometry for the impurity Green function
Appendix: singlequbit interferometry for the impurity Green function
Here, we present a measurement scheme for the retarded impurity Green function.
A.1 Definitions
The retarded zero temperature impurity Green function in the time domain can be written as
where the ‘greater’ and ‘lesser’ Green functions are given by
respectively. The average is computed in the groundstate \(GS\rangle\) of the twosite SIAM in Eq. (6). Here, σ can be either ↓ or ↑ since we are considering a spinsymmetric case (i.e., \(G^{R}_{\downarrow }=G^{R}_{\uparrow}\)), and the ĉoperators are given in the Heisenberg picture with respect to \(\hat{H}_{\mathrm{SIAM}}\), i.e.,
One possibility to measure the impurity Green function \(G^{R}_{\mathrm{imp}}(\tau)\) is to use a singlequbit Ramsey interferometer [44] which was used in Ref. [37] in the more general nonequilibrium case. To this end, we introduce an ancilla qubit in addition to the ‘system’ qubits, raising the total number of qubits needed to implement the twosite DMFT scheme to five.
A.2 JordanWigner transformation
The greater and lesser components, \(G^{>}_{\mathrm{imp}}(\tau)\) and \(G^{<}_{\mathrm{imp}}(\tau)\), must be written in terms of spin operators by again mapping the \(\hat{c}_{1\sigma}\) and \(\hat{c}^{\dagger }_{1\sigma}\) operators onto Pauli operators via the JordanWigner transformation. For concreteness, we focus on the case \(\sigma= \downarrow\). We obtain
and
A.3 Measurement protocol
Each of the terms of the form \(\langle \hat{U}^{\dagger}(\tau) \hat {\sigma}^{\alpha}_{1} \hat{U}(\tau) \hat{\sigma}^{\beta}_{1} \rangle \), where \(\alpha, \beta\in\{x, y\}\), can be measured in the interferometer. This can be seen as follows. We denote the state of the system qubits by \(\hat{\rho}_{\mathrm{sys}}=GS\rangle\langle GS\), where \(GS\rangle\) is the groundstate of the system. We initialize the ancilla qubit in the state \(0\rangle\), yielding the total density operator \(\hat{\rho}_{\mathrm{tot}}=0\rangle\langle0  \otimes\hat {\rho}_{\mathrm{sys}}\). The total system then undergoes the following evolution:

1.
At time \(t=0\), a Hadamard gate \(\hat{\sigma}_{H}=\frac {1}{\sqrt{2}} (\hat{\sigma}^{z}+\hat{\sigma}^{x} )\) is applied on the ancilla qubit, creating the superposition \(0\rangle _{\mathrm{ancilla}} \rightarrow (0\rangle_{\mathrm{ancilla}} + 1\rangle _{\mathrm{ancilla}} )/\sqrt{2}\).

2.
A ControlledPauli gate \(\hat{\sigma}^{\alpha}_{1}\) is applied on the impurity qubit 1 if the ancilla qubit has state \(0\rangle\).

3.
The system qubits undergo time evolution according to the unitary \(\hat{U}(\tau)\) which is decomposed into quantum gates.

4.
Another controlled Pauli gate \(\hat{\sigma}^{\beta}_{1}\) is applied on the impurity qubit 1 if the ancilla qubit has state \(1\rangle\).

5.
Another Hadamard gate is applied on the ancilla qubit.
Denoting the total unitary in steps 24 by T̂, the state of the ancilla qubit after this evolution is given by
where \(F(\tau)=\operatorname {Tr}_{\mathrm{sys}} [ \hat{T}_{1}^{\dagger }(\tau) \hat{T}_{0}(\tau) \hat{\rho}_{\mathrm{sys}} ]\). We have denoted the controlled unitaries as \(\hat{T}_{1}(\tau)=\hat {\sigma}^{\alpha}_{1} \hat{U}(\tau)\) and \(\hat{T}_{0}(\tau)= \hat {U}(\tau)\hat{\sigma}^{\beta}_{1}\). Note that since the same \(\hat {U}(\tau)\) appears in both unitaries, only the Pauli gates \(\hat {\sigma}^{\alpha/ \beta}_{1}\) need to be controlled, as described above. Note also that \(F(\tau)=\langle\hat{U}^{\dagger}(\tau) \hat {\sigma}^{\alpha}_{1} \hat{U}(\tau) \hat{\sigma}^{\beta}_{1} \rangle \). We can rewrite the state of the ancilla qubit as
whence \(\operatorname {Tr}_{\mathrm{ancilla}} [\hat{\rho}_{\mathrm{ancilla}} \hat {\sigma}^{z} ]=\operatorname{Re}[F(\tau)]\), and \(\operatorname {Tr}_{\mathrm{ancilla}} [\hat{\rho}_{\mathrm{ancilla}} \hat {\sigma}^{y} ]=\operatorname{Im}[F(\tau)]\). Thus, repeated measurements of the \(\hat{\sigma}^{z}\) and \(\hat{\sigma }^{y}\) components of the ancilla qubit yield the real and imaginary parts of the term \(\langle \hat{U}^{\dagger}(\tau) \hat{\sigma}^{\alpha }_{1} \hat{U}(\tau) \hat{\sigma}^{\beta}_{1} \rangle\). See Figure 10 for the quantum network of the scheme.
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Kreula, J.M., GarcíaÁlvarez, L., Lamata, L. et al. Fewqubit quantumclassical simulation of strongly correlated lattice fermions. EPJ Quantum Technol. 3, 11 (2016). https://doi.org/10.1140/epjqt/s4050701600491
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Keywords
 quantum simulation
 dynamical meanfield theory
 superconducting circuits