## Quantum walks on a superconducting circuit

Quantum walks generate large-scale quantum superposed states. This allows for classically unavailable applications, such as simulating many-body quantum systems, and also yields quantum algorithms exponentially faster than classical computation. Yan *et al.* demonstrate quantum walks of one and two strongly correlated microwave photons in a one-dimensional array of 12 superconducting qubits with short-range interactions. The scalability of the superconducting platform could lead to large-scale implementations and the quantum simulation of complex systems.

*Science*, this issue p. 753

## Abstract

Quantum walks are the quantum analogs of classical random walks, which allow for the simulation of large-scale quantum many-body systems and the realization of universal quantum computation without time-dependent control. We experimentally demonstrate quantum walks of one and two strongly correlated microwave photons in a one-dimensional array of 12 superconducting qubits with short-range interactions. First, in one-photon quantum walks, we observed the propagation of the density and correlation of the quasiparticle excitation of the superconducting qubit and quantum entanglement between qubit pairs. Second, when implementing two-photon quantum walks by exciting two superconducting qubits, we observed the fermionization of strongly interacting photons from the measured time-dependent long-range anticorrelations, representing the antibunching of photons with attractive interactions. The demonstration of quantum walks on a quantum processor, using superconducting qubits as artificial atoms and tomographic readout, paves the way to quantum simulation of many-body phenomena and universal quantum computation.

Quantum walks (QWs) are the quantum mechanical analogs of classical random walks, describing the propagation of quantum walkers on a lattice (*1*, *2*). Different from classical random walks, QWs generate large-scale quantum superposed states and entanglement, allowing for classically unavailable applications, such as simulating many-body quantum biological, chemical, and physical systems (*3*), and for developing quantum algorithms exponentially faster than classical computation (*4*). QWs have been investigated in many physical systems, including photons (*5*, *6*), nuclear magnetic resonance (*7*), trapped ions (*8*, *9*), superconducting qubits (*10*), and neutral atoms (*11*, *12*). These studies are useful for further studies on quantum dynamical phenomena such as entangled state engineering (*13*), dynamical phase transition (*14*), and thermalization versus localization (*15*, *16*).

In systems with finite-range interactions, QWs of a single particle show the propagation of correlations within a linear light cone limited by the Lieb-Robinson bounds (*17*), implemented in a chain of seven ion spins for entanglement (*18*). Multiparticle QWs in interacting systems—for example, the Bose-Hubbard model (*19*)—are capable of performing universal quantum computation (*4*, *20*). Here we investigated the continuous-time QWs (*2*) of one and two strongly correlated microwave photons on a 12-qubit superconducting processor (Fig. 1) using superconducting qubits as artificial atoms with high-fidelity manipulation and tomographic readout. We observed fundamental quantum effects, including light cone–like propagation of quantum information of superposed states, in particular, entanglement between qubit pairs, and exotic behaviors of time-evolved correlations, representing photon antibunching with attractive interactions.

To preserve and observe the quantum features of QWs, a low-decoherence system capable of simultaneous readouts and high-precision full controls is required. In our experiment, QWs of photons are performed on a one-dimensional (1D) array of 12 coupled superconducting transmon qubits of the Xmon variety (*21*–*23*). This system can be described by the Bose-Hubbard model with the Hamiltonian (we set *h* divided by 2π) (*15*, *19*, *22*)

For one-photon QWs, we used 11 superconducting qubits by turning *24*) on the target superconducting qubit (Fig. 1E). Then, all qubits are biased to the working point (4.88 GHz) using the Z pulses, for the quench dynamics with a time t. After turning off the Z pulses to tune the qubits back to their idle points, the single-photon density distribution *25*). The density distributions of single-photon QWs are obtained by averaging 1000 repeated single-shot measurements for the initially centrally localized, leftmost-localized, and rightmost-localized states (Fig. 2, A to C, compared with the numerical simulations, Fig. 2, D to F, respectively). To study the high coherence generated in the QWs, we investigated the fidelity

Furthermore, to observe the time evolution of nonlocal correlations, we implemented full tomography measurements on all reduced two-qubit states. We first consider the two-site correlation function *d* is the distance between two qubits, which also reveals the light cone–like spread of correlations (*18*, *26*). Here, *27*)*18*).

Because longer-range interactions in our system can be neglected, compared with the nearest-neighbor ones, the propagation speed of quantum information is limited by the maximal group speed *17*). It exhibits a linear light cone with exponentially decaying tails analogous to the causal light cones arising in relativistic theories, which has been previously observed in different physical systems (*18*, *26*, *28*). For only nearest-neighbor interactions, the photon density propagation, after a local excitation at the *j*-th qubit, has a tighter bound *29*), where

Finally, we demonstrate the QWs of two indistinguishable photons (*30*) with an array of 12 superconducting qubits in the presence of strong attractive interactions (*12*) (Fig. 1F). Strongly correlated multiparticle QWs are believed to have higher quantum complexity than the single-particle case and are capable of realizing universal quantum computation (*4*). In addition, two-photon QWs can demonstrate the Hanbury Brown-Twiss (HBT) interference (*6*), and the dynamical behaviors of the QWs are sensitive to the particle statistics. In the Bose-Hubbard model, for weakly interacting photons *12*)

In Fig. 4, A and B, we show the time evolution of the density distribution after two identical photons placed at two adjacent central qubits (*30*). For *19*). However, the time evolutions of the density distributions are also similar to those of noninteracting bosons (see supplementary materials). In Fig. 4, C to H, we clearly observed anticorrelations compared with the free-boson cases, where composite probabilities concentrate around the diagonal of the normalized correlator *30*).

We experimentally demonstrated QWs of one and two strongly interacting photons in a 1D array of superconducting qubits with short-range interactions. We observed the light cone–like propagation of quantum information, especially entanglement, and the photon antibunching with the two-photon HBT interference. Our results would be scalable to a few tens of qubits beyond classical simulation and lay the foundation for further studies on many-body dynamical phenomena and universal quantum computation.

## Supplementary Materials

science.sciencemag.org/content/364/6442/753/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S20

Table S1

References

Movie S1

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**The authors thank the Laboratory of Microfabrication, University of Science and Technology of China, Institute of Physics CAS, and National Center for Nanoscience and Technology for supporting the sample fabrication. The authors also thank QuantumCTek Co., Ltd., for supporting the fabrication and the maintenance of room-temperature electronics.

**Funding:**This research was supported by the National Key Research and Development Program of China (grant nos. 2017YFA0304300, 2016YFA0302104, 2016YFA0301200, and 2017YFA0303703), the Chinese Academy of Science and its Strategic Priority Research Program (grant no. XDB28000000), Alibaba Cloud, and the Science and Technology Committee of Shanghai Municipality. This research was also supported by NSFC (grant nos. 11574380, 11774022, 11774406, 11874212, U1530401, 11890704, and U1801661), AFOSR (grant no. FA9550-14-1-0040), AOARD (grant no. FA2386-18-1-4045), ARO (grant no. W911NF-18-1-0358), JSPS, JST (Q-LEAP and CREST grant no. JPMJCR1676), and Anhui Initiative in Quantum Information Technologies.

**Author contributions:**X.Z., H.F., and J.-W.P. conceived the research. Y.-R.Z., M.G., Z.Y., K.X., J.Q.Y., F.N., H.F., and X.Z. designed the experiment. Y.Z. designed the sample. H.D., Z.Y., and H.R. prepared the sample. Z.Y., M.G., Y.W., S.L., and C.W. carried out the measurements. Y.W. developed the programming platform for measurements. Y.-R.Z. performed numerical simulations. Z.Y., Y.-R.Z., and M.G. analyzed the results. F.L., J.L., Y.X., C.G., L.S., and C.-Z.P. developed room-temperature electronics equipment. All authors contributed to discussions of the results and the development of the manuscript. X.Z. and J.-W.P. supervised the whole project.

**Competing interests:**None declared.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper or the supplementary materials.