Abstract
We propose a nonlinear, hybrid quantumclassical scheme for simulating nonequilibrium dynamics of strongly correlated fermions described by the Hubbard model in a Bethe lattice in the thermodynamic limit. Our scheme implements nonequilibrium dynamical mean field theory (DMFT) and uses a digital quantum simulator to solve a quantum impurity problem whose parameters are iterated to selfconsistency via a classically computed feedback loop where quantum gate errors can be partly accounted for. We analyse the performance of the scheme in an example case.
Introduction
Next generation scalable quantum devices^{1,2} promise a step change in our ability to do computations. Direct quantum simulation^{3,4,5} using highly controllable quantum systems^{6,7,8} has already led to numerous insights into manybody quantum physics, despite limitations in the size of the simulated system.
Recently, quantum computer simulations of strongly correlated fermion models have been proposed^{9,10}. We suggest a hybrid quantumclassical scheme to simulate nonequilibrium dynamics of the Hubbard model in a Bethe lattice directly in the thermodynamic limit. Our scheme implements the nonequilibrium extension of the wellestablished dynamical meanfield theory (DMFT) method (for extensive reviews of DMFT, see, e.g. refs 11 and 12). Instead of the traditional allclassical method, the proposed scheme uses a digital quantum simulator to efficiently solve the DMFT impurity problem, the parameters of which are iterated to selfconsistency via a classically computed feedback loop. This setup promises an exponential speedup over the best currentlyknown Hamiltonianbased classical algorithms. We show how quantum gate errors can be partly accounted for in the feedback loop, improving simulation results. The scheme also avoids the sign problem in classical quantum Monte Carlo methods and works for all interaction strengths, unlike classical methods based on perturbation theory. Presently, nonequilibrium DMFT is one of the most promising methods to study timedependent phenomena in highdimensional correlated lattice models, and could thus be of interest for current efforts to develop scalable quantum technologies^{1,6,13,14}. Examples of applications of nonequilibrium DMFT include the dielectric breakdown of Mott insulators^{15}, damping of Bloch oscillations^{16}, and thermalization after parameter quenches^{17,18}.
Further to this, driven strongly correlated quantum materials are now being extensively investigated experimentally. A large motivation for this is the possibility of manipulating correlated phases of matter with strong pulses of light, such as photodoping of Mott insulators^{19} or inducing superconductivity^{20}. The underlying physical mechanisms are, however, still poorly understood. Even the dynamical behaviour of conceptually simple and commonly used quantum lattice models is yet not fully grasped. Solving these model systems could elucidate physical phenomena underlying currently unexplained experimental results. A standard example of this kind of idealised model for nonequilibrium problems is the timedependent Hubbard Hamiltonian
In this model, electrons with spin projections σ =↓,↑ move only between adjacent lattice sites i and j with timedependent ‘hopping’ energy v(t), where t denotes time. This process is described in the first sum, which is over all nearestneighbour sites, with fermionic creation and annihilation operators and , respectively. The electrons interact with Coulomb repulsion U(t) only if they occupy the same lattice site i, given in the latter term by the product of the number operators and .
This and similar models are extremely challenging to study numerically due to the exponential growth of the Hilbert space with system size. One thus often resorts to mean field approximations which typically consider only a single lattice site and replace interactions with its neighbourhood by a mean field Λ. This turns a linear quantum problem in an exponentially large Hilbert space into a much smaller but nonlinear problem where Λ needs to be determined selfconsistently. Such mean field approximations become increasingly accurate with the number of nearest neighbours. A classic example of this approach is the Weiss theory of ferromagnetism^{21}. For mean field theory to be applicable to strongly correlated Fermi systems in thermal equilibrium, the mean field Λ_{σ}(t) has to be dynamical to account for correlations between interactions with the environment that are separated by t in time, as schematically shown in Fig. 1a,b.
This highly successful approach is called DMFT^{11}. DMFT can be extended to nonequilibrium systems^{12} by letting Λ_{σ}(t, t′), which is often called hybridization function, depend on two interaction times t and t′ explicitly. Note that nonlocal spatial fluctuations can be included in DMFT by going beyond the singlesite approximation and considering a cluster of isolated sites^{22,23}, but this is beyond the scope of this work.
In general, it is a complex task to determine Λ_{σ}(t, t′) and the related local singleparticle Green’s function (where is the timeordering operator), describing the response of the manybody system after a localized removal and addition of a particle at times t and t′. Commonly used numerical methods for solving the nonequilibrium DMFT problem include continuoustime quantum Monte Carlo, which suffers from a severe dynamical sign problem, and perturbation theory which can only address the weak and strong coupling regimes^{12}.
In infinite dimensions, the system can also be explicitly mapped onto a single impurity Anderson model (SIAM)^{24}
where the selected lattice site is represented by an impurity, with the creation (annihilation) operator () and number operator , whose interaction with Λ_{σ}(t, t′) is mimicked by a collection of N noninteracting bath sites with onsite energies ε_{pσ}(t), as shown in Fig. 1c. The timedependent hybridization energy V_{pσ}(t) describes the amplitude for exchange of fermions between the impurity site and bath site p. These must be determined selfconsistently: for given V_{pσ}(t) the quantum dynamics of the SIAM is solved and its Green’s function and corresponding hybridization function Λ_{σ}(t, t′) are determined. From Λ_{σ}(t, t′) a new set of V_{pσ}(t) is worked out which is then fed back into the SIAM. These steps are repeated until convergence is achieved^{24}. The dynamics of the SIAM is usually worked out with exact diagonalization (ED)^{24} for small systems or with tensor network theory (TNT) methods^{25}. However, the dynamical generation of entanglement in these problems has severely hampered the efficiency of TNT methods^{25,26}. Furthermore, the required number of bath sites increases with the maximum simulation time t_{max}. This makes solving the SIAM the exponentially difficult bottleneck^{24,25,27} in purely classical DMFT solvers.
Here, we propose and analyze a hybrid quantumclassical computing scheme for DMFT to efficiently solve the Hubbard model in a Bethe lattice. The Bethe lattice is chosen for the simplicity of its selfconsistency condition. It is conceptually straightforward to extend the scheme to other types of lattices. A small digital quantum coprocessor solves the SIAM evolution with the resulting G_{σ}(t, t′) being processed by a classical computer to complete the nonlinear feedback loop as shown in Fig. 1d. We consider a trapped ion coprocessor for concreteness, although any other platform for quantum computing could implement the coprocessor as well. Even for imperfectly implemented quantum gates with realistic errors of 1% we find accurate solutions to a simple model problem in small systems. In addition, our numerical evidence suggests that gate errors mainly lead to a smearing of the bath energies, which can be accounted for in the classical feedback loop to improve the solution.
Figure 2 shows an example coprocessor quantum network for computing a contribution to the Green’s function (see Methods for details). The real and imaginary contributions to the impurity Green’s function are encoded as 〈σ^{z}〉 and 〈σ^{y}〉 of a probe qubit by interacting it with the impurity state at times t′ and t via controlled quantum gates^{28}. We decompose the unitary dynamics of the SIAM into a network of quantum gates^{29,30} by discretising time as t_{n} = nΔt, where Δt is a small timestep. We then breakup the evolution from t = 0 to t = t_{n} into a product of Trotter steps . The Trotter steps can readily be implemented by single qubit rotations and multiqubit entangling MølmerSørensen (MS) gates^{30,31} that have recently been realized in ion traps with high fidelity^{13,14}. The total number of MS gates per Trotter step scales only linearly with the number of bath sites.
We analyze the performance of our simulation scheme by considering a simple example system^{24}. We study the infinitedimensional timedependent Hubbard model (1) with constant onsite interaction U and tunneling matrix element v(t). The simulation starts in the halffilled paramagnetic atomic limit with tunneling v(t = 0) = 0, which is then dynamically ramped up to its final value v_{0} after quench time 1/4v_{0} and is kept at v_{0} until the final simulation time t_{max} is reached^{24} (setting ħ = 1). Such a sudden quench is representative of experimental ultracold atom dynamics^{32,33} and also ultrafast dynamics probed in condensed matter systems^{19}. The initial state of the system has a singly occupied impurity site in the completely mixed state of spin ↑ and spin ↓, and one half of the bath sites are doubly occupied and the other half empty (for explicit details, see ref. 24). In practice, we prepare the system in two pure fermion occupational number states, where one has the impurity in state ↑〉 and the other in state ↓〉, along with the bath states^{24}. The results are then averaged over these two pure states. These initial number states are mapped onto product states of qubits via the JordanWigner transformation (see Methods). The initial qubit configuration is that shown in Fig. 2, where . We emulate the operation of the quantum coprocessor by classically evaluating the quantum networks, and the classical exponential scaling limits our simulations to small systems. The selfconsistency condition for the Bethe lattice calculated in the classical feedback loop is Λ_{σ}(t, t′) = v(t)G_{σ}(t, t′)v(t′), from which we obtain the SIAM coupling to bath p efficiently via a Cholesky decomposition , where * denotes complex conjugation (see Supplementary Material for details). The impurity site double occupancy obtained from the selfconsistent hybrid simulation is compared to the exact result in Fig. 3a and shows that Trotter errors do not noticeably affect our results.
Next we assume imperfect gates characterized by phase errors that are described by normally distributed random variables with zero mean^{34}. We choose their standard deviations consistent with current experimental capabilities^{1,13,35} setting the single qubit error to σ = 10^{−6} and allowing MS gate errors σ_{MS} to vary between 0.1% and 10%. We obtain accurate results for the dynamics of the double occupancy even in the presence of gate errors. As shown in Fig. 3a the double occupation differs from the exact result by only ≈3% for σ_{MS} = 1%. For a smaller gate error of σ_{MS} = 0.1% the difference is insignificant up to t = 1.5/v_{0}. In Fig. 3b we plot the error in the imaginary part of the lesser Green’s function induced by imperfect gates. The diagonal values , which determine timelocal singleparticle observables, are almost unaffected even for large MS gate errors. Gate errors in general make the Green’s function decay faster with t − t′ than in the ideal case and will thus affect unequal time correlation functions.
We further investigate the effect of imperfect gates by considering the impurity site coupled to two bath sites via constant V_{pσ}(t). We find that the imaginary part of the mean field differs from the exact solution by a factor of approximately exp(−ηt′ − t) as shown in Fig. 4a. The decay rate η increases with σ_{MS} as displayed in the inset of Fig. 4a. This numerical evidence suggests that gate errors have the same effect as smearing out the bath energies ε_{pσ}(t) to a similar width η. The impurity model including errors would then be equivalent to the bath sites possessing a finite coherence time 1/η. Since the number of gates is ∝N we expect η to only depend weakly on N.
A bath site with coherence time 1/η can be modelled by allowing an ideal bath to incoherently exchange particles with a reservoir at an ‘error’ rate Γ = η. This exchange of particles modifies the bath’s Green’s function from its ideal value of g_{pσ}(t,t′) = 1 and correspondingly modifies the relation between impurity bath couplings and mean field to ref. 24 . This relation does not necessarily allow for an exact solution for V_{pσ}(t) even for large N. The effect of noise therefore limits the mean fields Λ_{σ}(t, t′) that the bath sites can model.
We investigate if the noise induced by gate errors can be partly compensated by implementing selfconsistency via this modified relation. For the noninteracting impurity with bath sites coupled to a particle reservoir we solve numerically for the bath Green’s functions g_{pσ}(t, t′), exploiting the superfermion formalism^{36} (see Supplementary Material). We minimize using the Frobenius norm over the V_{pσ}(t) to obtain the hybridizations in the noisy system. This modification of the classical feedback loop significantly reduces the effect of gate errors as demonstrated in Fig. 4b, showing the reduction in average absolute error in the mean field Λ_{σ}(t, t′). In the hybrid simulation scheme a slight modification of the quantum network shown in Fig. 2 allows the probe qubit to measure the bath Green’s functions, thus providing the information required for this noisereduction scheme to be implemented.
Finally, we emphasize that our scheme works directly in the thermodynamic limit and, since it does not require a small expansion term, gives accurate results for all values of U, in particular for the challenging situation of intermediate interactions like the example U = 2v_{0} considered here. The number of available qubits only limits the number of bath sites that can be included in the simulation and hence the maximally reachable simulation time t_{max}. Purely classical simulations are currently limited to approximately 25 bath sites^{25} and, because of fast growing SIAM entanglement^{24,25}, scale exponentially with t_{max} despite efficiently implementing the feedback loop. Therefore, a quantum coprocessor with only about 50 qubits^{1} coupled to a classical feedback loop would be able to improve upon current purely classical algorithms. Our hybrid simulation scheme thus provides an interesting scientific application of next generation, possibly imperfect, quantum devices. While preparing this manuscript, we became aware of related work by B. Bauer et al.^{37}.
Methods
Implementing the singleimpurity Anderson model with the digital quantum simulator
To implement the SIAM in Eq. (2) in the main text with the digital quantum simulator, we first map the creation and annihilation operators in onto spin operators that act on the qubits in the coprocessor. This is achieved via the JordanWigner transformation , , and (we take p = 1 to be the impurity). Here, , and , , and are the Pauli spin operators. The transformation maps N fermionic sites onto a string of 2N qubits such that two adjacent qubits represent one lattice site. The correspondences between the qubit states and fermionic states are 0, 0〉 = vac〉, 1, 0〉 = ↓〉, 0, 1〉 = ↑〉, and 1, 1〉 = ↓↑〉.
To obtain the necessary quantum gates to approximate the unitary evolution operator we use a Trotter decomposition on the propagator between each time t_{n} and t_{n+1} as , where . Each term can be readily implemented using spin rotations where φ is the angle of rotation, and multiqubit MølmerSørensen (MS) gates^{30,31}, characterized by two phases θ and ϕ as , with (see Supplementary Material). Here, the MS gate acts on qubits l, l + 1, ⋯, m, and the phase θ controls the amount of entanglement, while varying ϕ allows a shift between a or a type gate.
Measuring the impurity Green’s function with singlequbit interferometry
Using the JordanWigner transformation, the lesser and greater impurity Green’s functions for each spin σ can be written as a sum of four expectation values of products of Pauli operators and evolution operators (see Supplementary Material). We use a singlequbit interferometry scheme^{28} to measure each of the expectation values F(t, t′) that constitute the Green’s function. We introduce a probe qubit which is coupled to the string of 2N system qubits. We assume that the probe qubit is prepared in the pure state 0〉, yielding the total systemprobe density operator . The combined system is then run through a Ramsey interferometer sequence, in which first a π/2 pulse (or Hadamard gate ) is applied to the probe qubit, the state of which will transform into the superposition . The two states in the superposition provide the necessary interference paths. Following the π/2 pulse, we apply the unitary evolution on the system of interest up to a certain time t′. The Pauli operators are then applied on the system as controlled quantum gates with either 0〉 or 1〉 as the control state. This is followed by evolution up to the final time t′, another controlled application of Pauli gates, and finally another π/2 pulse is applied on the probe qubit, bringing the interference paths together. The output state of the probe qubit at the end of the Ramsey sequence is given by
where Here, the unitary operators and , in which and are Pauli operators or tensor products of Pauli operators (see Supplementary Material), act only on the system and not on the probe qubit. Note that we can write so that we have and . Therefore repeated measurements (which can be done in parallel) of the and components of the probe qubit for all times t′ and t yields a contribution to the impurity Green’s function G_{σ}(t, t′). For a spinsymmetric system, on the order of 80,000 measurements per time step are required. See Supplementary Material for details.
Additional Information
How to cite this article: Kreula, J. M. et al. Nonlinear quantumclassical scheme to simulate nonequilibrium strongly correlated fermionic manybody dynamics. Sci. Rep. 6, 32940; doi: 10.1038/srep32940 (2016).
References
A bet on quantum. Nature Phys. 11, 89 (2015).
Barends, R. et al. Digital quantum simulation of fermionic models with a superconducting circuit. Nature Comm. 6, 7654 (2015).
Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).
Buluta, I. & Nori, F. Quantum simulators. Science 326, 108–111 (2009).
Johnson, T. H., Clark, S. R. & Jaksch, D. What is a quantum simulator? EPJ Quantum Technology 1, 1–12 (2014).
Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nature Phys. 8, 277–284 (2012).
Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nature Phys. 8, 267–276 (2012).
Houck, A. A., Türeci, H. E. & Koch, J. Onchip quantum simulation with superconducting circuits. Nature Phys. 8, 292–299 (2012).
Wecker, D. et al. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A 92, 062318 (2015).
DallaireDemers, P.L. & Wilhelm, F. K. Method to efficiently simulate the thermodynamic properties of the FermiHubbard model on a quantum computer. Phys. Rev. A 93, 032303 (2016).
Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical meanfield theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13 (1996).
Aoki, H. et al. Nonequilibrium dynamical meanfield theory and its applications. Rev. Mod. Phys. 86, 779–837 (2014).
Benhelm, J., Kirchmair, G., Roos, C. F. & Blatt, R. Towards faulttolerant quantum computing with trapped ions. Nature Phys. 4, 463–466 (2008).
Lanyon, B. P. et al. Universal digital quantum simulation with trapped ions. Science 334, 57–61 (2011).
Eckstein, M., Oka, T. & Werner, P. Dielectric breakdown of mott insulators in dynamical meanfield theory. Phys. Rev. Lett. 105, 146404 (2010).
Eckstein, M. & Werner, P. Damping of Bloch oscillations in the Hubbard model. Phys. Rev. Lett. 107, 186406 (2011).
Eckstein, M., Kollar, M. & Werner, P. Thermalization after an interaction quench in the Hubbard model. Phys. Rev. Lett. 103, 056403 (2009).
Eckstein, M., Kollar, M. & Werner, P. Interaction quench in the Hubbard model: Relaxation of the spectral function and the optical conductivity. Phys. Rev. B 81, 115131 (2010).
Wall, S. et al. Quantum interference between charge excitation paths in a solidstate Mott insulator. Nature Phys. 7, 114–118 (2011).
Fausti, D. et al. Lightinduced superconductivity in a stripeordered cuprate. Science 331, 189–191 (2011).
Cardy, J. Scaling and Renormalization in Statistical Physics, vol. 5 (Cambridge University Press, 1996).
Maier, T., Jarrell, M., Pruschke, T. & Hettler, M. H. Quantum cluster theories. Rev. Mod. Phys. 77, 1027–1080 (2005).
Tsuji, N., Barmettler, P., Aoki, H. & Werner, P. Nonequilibrium dynamical cluster theory. Phys. Rev. B 90, 075117 (2014).
Gramsch, C., Balzer, K., Eckstein, M. & Kollar, M. Hamiltonianbased impurity solver for nonequilibrium dynamical meanfield theory. Phys. Rev. B 88, 235106 (2013).
Wolf, F. A., McCulloch, I. P. & Schollwöck, U. Solving nonequilibrium dynamical meanfield theory using matrix product states. Phys. Rev. B 90, 235131 (2014).
Cirac, J. I. & Zoller, P. Goals and opportunities in quantum simulation. Nature Phys. 8, 264–266 (2012).
Balzer, K., Li, Z., Vendrell, O. & Eckstein, M. Multiconfiguration timedependent Hartree impurity solver for nonequilibrium dynamical meanfield theory. Phys. Rev. B 91, 045136 (2015).
Dorner, R. et al. Extracting quantum work statistics and fluctuation theorems by singlequbit interferometry. Phys. Rev. Lett. 110, 230601 (2013).
Casanova, J., Mezzacapo, A., Lamata, L. & Solano, E. Quantum simulation of interacting fermion lattice models in trapped ions. Phys. Rev. Lett. 108, 190502 (2012).
Müller, M., Hammerer, K., Zhou, Y. L., Roos, C. F. & Zoller, P. Simulating open quantum systems: from manybody interactions to stabilizer pumping. New J. Phys. 13, 085007 (2011).
Mølmer, K. & Sørensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835 (1999).
Esslinger, T. FermiHubbard physics with atoms in an optical lattice. Annu. Rev. Condens. Matter Phys. 1, 129 (2010).
Langen, T., Geiger, R. & Schmiedmayer, J. Ultracold atoms out of equilibrium. Annu. Rev. Condens. Matter Phys. 6, 201–217 (2015).
Fowler, A. G., Hill, C. D. & Hollenberg, L. C. L. Quantumerror correction on linearnearestneighbor qubit arrays. Phys. Rev. A 69, 042314 (2004).
Harty, T. P. et al. Highfidelity preparation, gates, memory, and readout of a trappedion quantum bit. Phys. Rev. Lett. 113, 220501 (2014).
Dzhioev, A. A. & Kosov, D. S. Superfermion representation of quantum kinetic equations for the electron transport problem. J. Chem. Phys. 134, 044121 (2011).
Bauer, B., Wecker, D., Millis, A. J., Hastings, M. B. & Troyer, M. Hybrid quantumclassical approach to correlated materials. arXiv:1510.03859 (2015).
Acknowledgements
The authors would like to thank Simon Benjamin as well as Ian Walmsey and his group members for useful discussions. The research leading to these results has received funding from the EPSRC National Quantum Technology Hub in Networked Quantum Information Processing (NQIT). J.M.K. acknowledges financial support from Christ Church, Oxford and the Osk. Huttunen Foundation. D.J. acknowledges financial support from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement no. 319286 QMAC and the EU Collaborative project QuProCS (Grant Agreement 641277). The data presented in this work is contained in the source files of the arXiv submission arXiv:1510.05703.
Author information
Affiliations
Contributions
J.M.K. and S.R.C. carried out the numerical calculations. J.M.K. decomposed the SIAM into quantum networks and carried out the analytical calculations. D.J., S.R.C. and J.M.K. wrote the manuscript. D.J. conceived and coordinated the project.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Electronic supplementary material
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Kreula, J., Clark, S. & Jaksch, D. Nonlinear quantumclassical scheme to simulate nonequilibrium strongly correlated fermionic manybody dynamics. Sci Rep 6, 32940 (2016) doi:10.1038/srep32940
Received
Accepted
Published
DOI
Further reading

Dynamical quantum phase transitions in Weyl semimetals
Physical Review B (2019)

Variational Quantum Simulation for Quantum Chemistry
Advanced Theory and Simulations (2019)

Onedimensional quantum computing with a ‘segmented chain’ is feasible with today’s gate fidelities
npj Quantum Information (2018)

Experimental demonstration of a measurementbased realisation of a quantum channel
New Journal of Physics (2018)

Quantum Chemistry Calculations on a TrappedIon Quantum Simulator
Physical Review X (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.