Probing the Kitaev honeycomb model on a neutral-atom quantum computer

TL;DR

Digital quantum simulations of Kitaev’s honeycomb model for two-dimensional fermionic systems using a reconfigurable atom-array processor are realized and pave the way for digital quantum simulations of complex fermionic systems for materials science, chemistry and high-energy physics.

Abstract

Quantum simulations of many-body systems are among the most promising applications of quantum computers. In particular, models based on strongly-correlated fermions are central to our understanding of quantum chemistry and materials problems, and can lead to exotic, topological phases of matter. However, due to the non-local nature of fermions, such models are challenging to simulate with qubit devices. Here we realize a digital quantum simulation architecture for two-dimensional fermionic systems based on reconfigurable atom arrays. We utilize a fermion-to-qubit mapping based on Kitaev's model on a honeycomb lattice, in which fermionic statistics are encoded using long-range entangled states. We prepare these states efficiently using measurement and feedforward, realize subsequent fermionic evolution through Floquet engineering with tunable entangling gates interspersed with atom rearrangement, and improve results with built-in error detection. Leveraging this fermion description of the Kitaev spin model, we efficiently prepare topological states across its complex phase diagram and verify the non-Abelian spin liquid phase by evaluating an odd Chern number. We further explore this two-dimensional fermion system by realizing tunable dynamics and directly probing fermion exchange statistics. Finally, we simulate strong interactions and study dynamics of the Fermi-Hubbard model on a square lattice. These results pave the way for digital quantum simulations of complex fermionic systems for materials science, chemistry, and high-energy physics.

Figures (14)

• Figure 1: Digital quantum simulations with reconfigurable atom arrays.a, The honeycomb lattice used in this work with 104 total atomic qubits and periodic boundary conditions along the shorter direction, forming a cylinder (see also Extended Data Fig. \ref{['fig:ED_ExperimentSequence']}a). The qubits are encoded in $^{87}$Rb atoms and entangling gates are realized through excitation to interacting Rydberg states. To encode fermion statistics, we prepare a long-range entangled state characterized by hexagonal plaquette operators $W_p{=}{\rm X}_1{\rm Z}_2{\rm Y}_3{\rm X}_4{\rm Z}_5{\rm Y}_6$ that commute with the unitary evolution and are therefore conserved. The encoded Majorana fermions live on vertices, at the ends of operator strings, and conventional (complex) fermions are formed by combining two Majoranas along a chosen link orientation. b, The long-range entangled state is prepared using mid-circuit measurement of ancilla qubits, and plaquettes are deterministically flipped to be +1 using conditional single-qubit gates (red circle). c, The Floquet evolution cycle consists of atom reconfiguration interspersed with tunable entangling gates and global basis changes. d, Accumulated entangling phase upon repeated application of $\exp(i\pi\theta {\rm [Z{\otimes}Z]/4})$ gates with a variable angle $\theta$ realized through fast, parameterized laser pulses (Extended Data Fig. \ref{['fig:ED_EntanglingGates']}). The values show extracted values of $\theta$, and error bars represent one standard deviation.
• Figure 1: Honeycomb lattice layout and experiment sequence.a, Mapping from honeycomb geometry into experimental array geometry used, indicating link orientations, an example column, and an example plaquette orientation. We use this mapping in order to shrink the number of rows needed, as well as put the atoms into a rectangular grid which is convenient for atom motion and local Raman operations. b, Loss radius definition for error detection based on atom loss. The central (white) atom is the reference point, and for loss radius of 0, we postselect on this atom being present at the end of the circuit. For a larger loss radius, we postselect on atoms within a certain local region being present. For string observables, we perform this procedure for all atoms within the string. c, High-level overview of the experimental sequence used in these experiments, including feedforward topological state preparation, Floquet evolution, and measurement steps. The Fermi-Hubbard quantum simulations in Fig. \ref{['fig:fig5']} have an additional part of the Floquet circuits for engineering the onsite density-density interactions.
• Figure 2: Measurement-based preparation of topological order.a, A long-range entangled state of the toric-code type is prepared using a depth-3 circuit, independent of the system size, accompanied by midcircuit readout of the ancilla qubits (I). The measurement results are random up to parity constraints along the periodic direction. A feedforward step realized through FPGA-triggered single-qubit Z rotations pairs the -1 outcomes (white hexagons). Finally, a parallel controlled-Y operation creates the weight-6 plaquettes (hexagons) and initializes the ZZ-link operators (ovals) to all be +1 (II). b, Expectation values of the weight-6 plaquettes and weight-2 ZZ-links across the array of 72 data and 32 ancilla qubits with cylindrical boundary conditions. The inset shows values averaged across the system. c, Parity expectation values of increasingly large loops, including loops that enclose 1, 2, 3, and 4 hexagons within a column and the loop around the cylinder (plotted for maximum decoding postselection, see Extended Data Fig. \ref{['fig:ED_FeedforwardMethods']}c). The largest operator is equivalent to the product of two loops enclosing the cylinder. Error bars represent 68% confidence intervals. d, The product of plaquette operators in each column is equivalent to loops enclosing the cylinder, which are +1 in the absence of errors. Postselection based on the parity of the measured ancilla values within each column improves the plaquette expectation values and the quality of the resulting fermion encoding.
• Figure 2: Tunable CPHASE gates and automated calibration procedures.a, Example entangling gate phase profiles for different entangling phases, with detuning included as a linear phase term. As the entangling phase becomes larger, the gate becomes longer. An entangling phase of $\theta{=}1$ is equivalent to the CZ gate profile. b, Theoretical parameters for different entangling phase gates, where the gate has a constant amplitude profile and a phase profile given by the cosine function $A\cos(\omega t{+}\varphi)$evered_high-fidelity_2023. We optimize these gates experimentally with parameter scans in the vicinity of theoretical values evered_high-fidelity_2023. c, Circuit used to benchmark and calibrate the entangling phase gates, adapted from the approach in mi_timecrystal_2022. One atom is prepared in $\ket{0}$ and the other in $\ket{+}_y {=}(\ket{0}{+}i\ket{1})/\sqrt{2}$ with a local Raman gate. Then a series of CPHASE gates are applied to the atom pair, which effectively rotates the phase of the initial $\ket{+}_y$ state. A precalculated final ${\rm Z}(\phi)$ gate ensures that both atoms return to $\ket{0}$ in the absence of errors. Errors, for example coming from a miscalibrated entangling phase or qubit loss/leakage during the gate, will reduce the probability of finding both atoms in $\ket{0}$ at the end of the circuit (the return probability). d, Data used to extract the entangling phase in Fig. \ref{['fig:fig1']}d, shown for two different CPHASE gates. For different numbers of gates applied, we scan the ${\rm Z}(\phi)$ gate before the final local $\pi$/2 pulse in order to extract the amount of phase accumulated. The trend shows how much phase has been accumulated over the 10 gates, and the reduction in the peak return probability is a result of gate errors. Note that for a smaller number of gates, we use the same Raman pulse sequence as for the 10 gate sequence and only reduce the number of CPHASE gates. e, Comparison of return probability after 20 CPHASE gates with this benchmarking method for different entangling phases, with the point $\theta{=}1$ being the CZ gate. We attribute the non-monotonic behavior to varying levels of calibration between the gates. These values can be compared to a return probability after 0 gates of 0.952(3) owing to errors from components separate from the CPHASE gates. We note that this calibration sequence is not a proper measure of fidelity, and we rather use this measurement to compare the different CPHASE gates to each other. The CZ gate was benchmarked right after taking this data with a fidelity of 99.4% in the global RB sequence utilized in evered_high-fidelity_2023 (the slightly lower fidelity than our typical 99.5% operation can be attributed to the increased scattering from the closer intermediate-state detuning used here). f, Example calculated Rydberg beam profile and the effect of adding two separate peak-correction holograms. The local peak corrections are added to the base holograms with variable weights on the Rydberg beam SLMs, in addition to correcting for more global Zernike aberrations. g, Example automatic calibration routine showing the homogenization of our Rydberg beam across an array. (top) The peak-to-peak variations in row intensity across the array during calibration, as measured by the differential light shift on the hyperfine qubit. (bottom) Example of how the defocus and one of the peak corrections change during this automated calibration procedure. h, Examples of the measured light shift across the rows of our atom array for the two Rydberg beams, comparing the uniformity before and after the automated calibration procedure. We also ensure that the columns are uniform, although they are naturally more homogeneous due to the beam geometry.
• Figure 3: Kitaev model on a honeycomb lattice.a, The honeycomb model, consisting of anisotropic spin interactions along the three directions of the honeycomb lattice ($J_{\rm X}{+}J_{\rm Y}{+}J_{\rm Z}{=}1$), exhibits topological order with three Abelian (A) phases and a single non-Abelian phase B. The toric code state in Fig. \ref{['fig:fig2']} is the fixed-point state A$_{\rm Z}^{\rm I}$ of the Abelian A$_{\rm Z}$ phase ($J_{\rm Z}{=}1$). b, Starting from this initial state, we prepare different states on the phase diagram. Top: numerically optimized sequence of two-qubit gates used to prepare the non-Abelian phase B. Each circuit layer includes CPHASE (CP) gates and global single-qubit rotations. Bottom: plaquette parity during the evolution, with postselection based on atom loss and decoding. The plaquette parity is lower than in Fig. \ref{['fig:fig2']} due to a longer sequence of gates and atom movements (Methods). c, Pauli strings of different lengths measured on the three studied states and averaged over the bulk of the system (Extended Data Fig. \ref{['fig:ED_MajoranaChern']}a-b). Error bars represent 68% confidence intervals. d, In the fermion representation, the link operators are proportional to nearest-neighbor Majorana hoppings, $K^{{\rm X/Y/Z}}_{ij}{\propto} ic_i c_j$. Longer Pauli strings constructed from their products result in longer-range hopping operators. e, The free-fermion parent Hamiltonian ${H}_{\textbf{k}}$ can be reconstructed from measured two-point Majorana correlations. f, The Chern number $\rm{C}$ is evaluated using the learned parent Hamiltonians of the string distributions in c, resulting in $\rm{C}$=0 in phase A and $\rm{C}$=1 in phase B; the robustness of this procedure is explored in Methods (Extended Data Fig. \ref{['fig:ED_MajoranaChern']}f-h). The non-Abelian phase B is characterized by the underlying topological order and an odd Chern number kitaev_anyons_2006.