Abstract
One of the key applications of quantum information is simulating nature. Fermions are ubiquitous in nature, appearing in condensed matter systems, chemistry and high energy physics. However, universally simulating their interactions is arguably one of the largest challenges, because of the difficulties arising from anticommutativity. Here we use digital methods to construct the required arbitrary interactions, and perform quantum simulation of up to four fermionic modes with a superconducting quantum circuit. We employ in excess of 300 quantum logic gates, and reach fidelities that are consistent with a simple model of uncorrelated errors. The presented approach is in principle scalable to a larger number of modes, and arbitrary spatial dimensions.
Introduction
Simulating quantum physics with a device which itself is quantum mechanical, a notion Richard Feynman originated^{1}, would be an unparallelled computational resource. However, the universal quantum simulation of fermionic systems is daunting due to their particle statistics^{2}, and Feynman left as an open question whether it could be done, because of the need for physically implementing nonlocal control. Quantum simulation of fermionic models is highly desirable, as computing the properties of interacting particles is classically difficult. Determining static properties with quantum Monte Carlo techniques is already complicated due to the sign problem^{3}, arising from anticommutation, and dynamic behaviour is even harder.
The key to quantum simulation is mapping a model Hamiltonian onto a physical system. When the physical system natively mimics the model, the mapping can be direct and simulations can be performed using analogue techniques. Already, fermionic models have been simulated at scale using large clouds of natively fermionic gases^{4,5}. A complementary approach is digital quantum simulation^{6}. It allows for constructing arbitrary interactions, and holds the promise that it can be implemented on an errorcorrected quantum computer, but at the cost of many gates. However, the digital approach is in its infancy—so far, the only experiment is the simulation of a spin Hamiltonian in ion traps^{7}—because it requires complex sequences of logic gates, especially for nonlocal control, which hinge on carefully constructed interactions between subsets of qubits in a larger system; a demanding task for any platform. A digital fermionic simulation can therefore be regarded as a hard test.
Here, we explore fermionic interactions with digital techniques^{6} in a superconducting circuit. Focusing on the Hubbard model^{8,9}, we perform time evolutions with constant interactions as well as a dynamic phase transition with up to four fermionic modes encoded in four qubits, using the Jordan–Wigner transformation^{10}. The implemented digital approach is universal and allows for the efficient simulation of fermions. The required number of gates scales only polynomially with the number of modes^{9}, even with physical nearestneighbour qubit coupling only. Moreover, the model system is not limited to the dimensionality of the physical system, allowing for the simulation of fermionic models in two and three spatial dimensions^{9,11}. We use in excess of 300 singlequbit and twoqubit gates, to implement fermionic models that require fully, yet separately tunable , and interactions. We reach global fidelities that are limited by gate errors in an intuitive error model. These results are made possible by recent advances in architecture and control of superconducting qubits^{12,13,14}. Our experiment is a critical step on the path to creating an analoguedigital quantum simulator—we foresee one using discrete fermionic modes combined with discrete^{15} or continuous^{16} bosonic modes, highlights the digital approach and is a demonstration of digital quantum simulation in the solid state.
Results
Implementing the Hubbard model with gates
At low temperatures, classes of fermionic systems can be accurately described by the Hubbard model. Here hopping (strength V) and repulsion (strength U) compete (see Fig. 1a), capturing the rich physics of manybody interactions such as insulating and conducting phases in metals^{17,18}. The generic Hubbard Hamiltonian is given by: , with b the fermionic annihilation operator and i,j running over all adjacent lattice sites. The first term describes the hopping between sites and the last term the onsite repulsion. It is insightful to look at a fermionic twomode example,
We can express the fermionic operators in terms of Pauli and ladder operators using the Jordan–Wigner transformation^{10}: and , where the σ_{z} term ensures anticommutation. In essence, we use nonlocal control and map a local fermionic Hamiltonian to a local spin Hamiltonian. The qubits act as spins, and carry the fermionic modes (Fig. 1a,b). A fermionic mode is either occupied or unoccupied, and spinless—the spin degree of freedom is implemented here by using four modes to simulate two sites with two spins. We note that for higher spatial dimensions this approach is still viable, the only difference is that the local fermionic Hamiltonian now maps to a nonlocal spin Hamiltonian, which can be efficiently implemented as recently shown^{9,11}. Using the above transformation, the Hamiltonian becomes
which can be implemented with separately tunable , and interactions. Here we use the convention to map an excited fermionic mode 1〉 (excited logical qubit) onto a qubit’s physical groundstate g〉, and a vacuum fermionic mode 0〉 (ground logical qubit) onto a qubit’s physical excited state e〉.
Our experiments use a superconducting ninequbit multipurpose processor, see Fig. 1b. Device details can be found in ref. 19. The qubits are the crossshaped structures^{20} patterned out of an aluminium film on a sapphire substrate. They are arranged in a linear chain with nearestneighbour coupling. Qubits have individual control, using microwave and frequencydetuning pulses (top), and readout is done through dispersive measurement (bottom)^{21}. By frequency tuning of the qubits, interactions between adjacent pairs can be separately turned on and off. This system allows for implementing nonlocal gates, as it has a high level of controllability, and is capable of performing highfidelity gates^{12,22}. Importantly, single and twoqubit gate fidelities are maintained when scaling the system to larger numbers of qubits, as shown by the consistency of errors with the fivequbit device^{12}.
The basic element used to generate all the interactions is a simple generalization of the controlledphase (CZ) entangling gate (Fig. 2a,b). We implement a statedependent frequency pull by holding one qubit steady in frequency and bringing a second qubit close to the avoided level crossing of ee〉 and gf〉 using an adiabatic trajectory^{23}. By tuning this trajectory, we can implement a tunable CZ_{φ} gate. During this operation, adjacent qubits are detuned away in frequency to minimize parasitic interactions. The practical range for φ is 0.5–4.0 rads; below this range, parasitic interactions with other qubits become relevant, and above this range population starts to leak into higherenergy levels (see Supplementary Note 5 and refs 12, 19). Using singlequbit gates and two entangling gates, we can implement the tunable interactions, as shown in Fig. 2c. In this gate construction, the πpulses naturally suppress dephasing^{24}.
Verifying operator anticommutativity
First, we have experimentally verified that the encoded fermionic operators anticommute, see Fig. 3, by implementing the following anticommutation relation . The latter can be separated into two nontrivial Hermitian terms: (Fig. 3a) and (Fig. 3b). Their associated unitary evolution, for the first one, has been implemented using gates with strength φ=π. The measured process matrices (χ) for these terms are determined using quantum process tomography, and constrained to be physical (Supplementary Note 2). We find that the processes are close to the ideal, with fidelities Tr(χ_{ideal}χ)=0.95, 0.96. As the Hermitian terms sum up to zero, their unitary evolutions combine to the identity (Fig. 3c). We find that the sequence of both processes yields in fact the identity, as expected for anticommutation, with a fidelity of 0.91.
Simulations with two fermionic modes
We now discuss the simulation of fermionic models. We use the Trotter approximation^{25} to digitize the evolution of Hamiltonian , with each part implemented using single and twoqubit gates (ℏ=1). We benchmark the simulation by comparing the experimental results with the exact digital outcome. Discretization unavoidably leads to deviations, and the digital errors are quantified in Supplementary Note 4.
We start by visualizing the kinetic interactions between two fermionic modes. The construction of the Trotter step is shown in Fig. 4a and directly follows from the Hamiltonian in equation (2). The step consists of the , and terms, constructed from terms and singlequbit rotations. We simulate the evolution during time Δt by setting φ_{xx}=φ_{yy}=VΔt and φ_{z}=φ_{zz}=UΔt/2, and using V=U=1. We evolve the system to a time of T=5.0, and increase the number of steps (Δt=T/n, with n=1,...,8). The data show hallmark oscillations, Fig. 4b, indicating that the modes interact and exchange excitations. We find that the endstate fidelity, taken at the same simulated time, decreases approximately linearly by 0.054 per step (Fig. 4c).
The above example shows that fermionic simulations, clearly capturing the dynamics arising from interactions, can be performed digitally using singlequbit gates and the tunable CZ_{φ} gate. Moreover, increasing the number of steps improves the time resolution, but at the price of increasing errors. A crucial result is that the perstep decrease in the endstate fidelity is consistent with the gate fidelities. Using the typical values of 7.4 × 10^{−3} entangling gate error and 8 × 10^{−4} singlequbit gate error as previously determined for this platform^{12}, we arrive at an expected Trotter step process error of 0.07, considering the step consists of six entangling gates and 28 singlequbit gates (including X, Y rotations as well as idles). In addition, we have determined the Trotter step gate error in a separate interleaved randomized benchmarking experiment (Supplementary Note 3), and found a process error of 0.074, which is consistent with the observed perstep state error. We find that the process fidelity is thus a useful estimate, even though the simulation fidelity depends on the state and implemented model.
Simulations with three and four fermionic modes
Simulations of fermionic models with three and four modes are shown in Fig. 5. The threemode Trotter step and its pulse sequence are shown in Fig. 5a,b. An implementation of the gate is highlighted: the top qubit (red) is passive and detuned away, the middle qubit (blue) is tuned to an optimal frequency for the interaction, and the bottom qubit (green) performs the adiabatic trajectory. πpulses on the passive qubit suppress dephasing and parasitic interactions. Figure 5c shows the simulation results for V=1, U=0 (hopping only) and V=1, U=1 (with onsite repulsion). Input state generation is shown in Supplementary Note 1. The simulation data (closed symbols) follows the exact digital outcome (open symbols), accumulating a perstep error of 0.15 (Fig. 5f) and gradually populating other states (black symbols). The fidelity is the relevant figure of merit; the perstep error being the same for different model parameters indicates that the simulation outcomes are distinct.
For the fourmode experiment, we simulate an asymmetric variation on the Hubbard model. Here the repulsive interaction is between the middle modes only (right well in Fig. 1a), while the hopping terms are kept equal. Asymmetric models are used in describing anisotropic fermionic systems^{26}. In addition, the simulation can be optimized: gate count is reduced by the removal of interaction between the top and bottom modes, and the Trotter expansion can be rewritten in terms of odd and even steps such that the starting and ending singlequbit gates cancel (Supplementary Note 6). The Trotter step is shown in Fig. 5d. The results are plotted in Fig. 5e. We find that the state fidelity decreases by 0.17 for the fourmode simulation, see Fig. 5f.
The three and fourmode experiments underline that fermionic models can be simulated digitally with large numbers of gates. The threemode simulation uses in excess of 300 gates. We perform three Trotter steps, and per step we use: 12 entangling gates, 53 microwave π and π/2 gates, 19 idle gates, 3 singlequbit phase gates and for the nonparticipating qubit during the entangling operation: 12 frequencydetuning gates where phases need to be accurately tracked. Using the above typical errors for gates, we arrive at an estimated process error of 0.16 for the threemode simulation, and an error of 0.15 for the fourmode simulation (per fourmode Trotter step: 10 entangling gates and 98 singlequbit gates). The process errors are close to the observed drop in state fidelity. The data are summarized in Table 1. Importantly, these results strongly suggest that the simulation errors scale with the number of gates, not qubits (modes), which is a crucial aspect of scalably implementing models on our platform. Therefore, the appreciable drop in total fidelity is currently the optimal for any quantum platform considering the large number of gates that we have implemented in this experiment. Moreover, the precision achieved in our experiment allows us to observe the expected fermionic behaviour at every Trotter step of the implemented protocol.
Timevarying interactions
We now address the simulation of fermionic systems with timedependent interactions. In Fig. 6a, we show an experiment where we ramp the hopping term V from 0 to 1 while keeping the onsite repulsion U at 1, essentially changing the system from an insulating to a metallic phase. This transition is simulated for two modes using two Trotter steps, see inset, and with one step for three modes. For the latter case, we take the average of V over the relevant time domain. The data are shown in Fig. 6b,c, and clearly mirror the dynamics of the hopping term. At time smaller than 1.0, the system is frozen and the mode occupations are virtually unchanged, reflecting the insulating state. Interactions become visible when hopping is turned on, effectively melting the system, and follow the generic features of the exact digital outcome (dashed). The simulation fidelities lie around 0.9–0.95 for two modes and 0.7–0.8 for three modes, see Fig. 6d. These fidelities are around or somewhat below those for time evolution with constant interactions, presumably due to control errors related to parasitic qubit interactions, which also lead to the populating of other states (black symbols). The dynamic simulation highlights the possibilities of exploring parameter spaces and transitions with few steps.
Discussion
We have demonstrated the digital quantum simulation of fermionic models. Simulation fidelities are close to the expected values, and with improvements in gates and architecture, the construction of larger testbeds for fermionic systems appears viable. Moreover, a future implementation of quantum error correction in combination with these techniques will enable the efficient and scalable digital quantum simulation of fermionic models. Bosonic modes can be elegantly introduced by adding linear resonators to the circuit, establishing a fermionboson analoguedigital system^{15,16} as a distinct paradigm for quantum simulation.
Methods
Experimental details
Experiments are performed in a wet dilution refrigerator with a base temperature of 20 mK. Qubit frequencies are chosen in a staggered pattern to minimize unwanted interaction. Typical qubit frequencies are 5.5 and 4.8 GHz. Exact frequencies are optimized based on the qubits’ e〉 and f〉 state spectra along the fully tunable trajectory of the CZ_{φ}gate, as well as on minimizing the interactions between nextnearest neighbouring qubits. Used qubits are Q1–Q4 in ref. 19. Data are corrected for measurement fidelity, typical measurement errors are 0.01 for qubits Q1 and Q3 and 0.04 for Q2 and Q4 (refs 19, 27).
State fidelity
The state fidelity is computed using , which is equal to 〈Ψ_{ideal}Ψ〉^{2} to first order. Here P_{k,ideal} and P_{k} are mode occupations and k runs over the computational basis. The consistency with measured process fidelities, and the scaling of the simulation fidelity with steps justify this approach.
Additional information
How to cite this article: Barends, R. et al. Digital quantum simulation of fermionic models with a superconducting circuit. Nat. Commun. 6:7654 doi: 10.1038/ncomms8654 (2015).
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Acknowledgements
We thank A.N. Korotkov for discussions. We acknowledge support from Spanish MINECO FIS201236673C0302; Ramón y Cajal Grant RYC201211391; UPV/EHU UFI 11/55 and EHUA14/04; Basque Government IT47210; a UPV/EHU PhD grant; PROMISCE and SCALEQIT EU projects. Devices were made at the UC Santa Barbara Nanofabrication Facility, a part of the NSFfunded National Nanotechnology Infrastructure Network, and at the NanoStructures Cleanroom Facility.
Author information
Author notes
 J. Kelly
Present address: Google Inc., Santa Barbara, California 93117, USA.
Affiliations
Google Inc., Santa Barbara, California 93117, USA.
 R. Barends
 , A. G. Fowler
 , E Jeffrey
 , D. Sank
 , J. Y. Mutus
 , Yu Chen
 , P. Roushan
 & John M. Martinis
Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, Bilbao E48080, Spain.
 L. Lamata
 , L. GarcíaÁlvarez
 & E. Solano
Department of Physics, University of California, Santa Barbara, California 93106, USA.
 J. Kelly
 , A Megrant
 , T. C. White
 , B. Campbell
 , Z. Chen
 , B. Chiaro
 , A. Dunsworth
 , I.C. Hoi
 , C. Neill
 , P. J. J. O’Malley
 , C. Quintana
 , A. Vainsencher
 , J. Wenner
 & John M. Martinis
Department of Materials, University of California, Santa Barbara, California 93106, USA.
 A Megrant
IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, Bilbao 48013, Spain.
 E. Solano
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Contributions
R.B., L.L. and L.G.Á. designed the experiment, with J.M.M. and E.S. providing supervision. L.G.Á., L.L. and E.S. provided the theoretical framework. R.B. and L.L. cowrote the manuscript with J.M.M. and E.S. The experiment and data were performed and analysed by R.B., J.K., L.L. and L.G.Á. R.B. and J.K. designed the device. J.K., R.B. and A.M. fabricated the sample. All authors contributed to the fabrication process, experimental setup and manuscript preparation.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to R. Barends or L. Lamata.
Supplementary information
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 1.
Supplementary Information
Supplementary Figures 16, Supplementary Notes 16 and Supplementary References
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