Demonstration of Discrete-Time Quantum Walks and Observation of Topological Edge States in a Superconducting Qutrit Chain

Abstract

Quantum walk serves as a versatile tool for universal quantum computing and algorithmic research. However, the implementation of discrete-time quantum walks (DTQWs) with superconducting circuits is still constrained by some limitations such as operation precision, circuit depth and connectivity. With improved hardware efficiency by using superconducting qutrits (three-level systems), we experimentally demonstrate a scalable DTQW in a superconducting circuit, observing the ballistic spreading of quantum walk in a qutrit chain. The usage of qutrits in our implementation allows hardware efficiently encoding of the walker position and the coin degree of freedom. By exploiting the flexibility and intrinsic symmetries of qutrit-based DTQWs, we successfully prepare two topological phases in the chain. For the first time, particle-hole-symmetry-protected edge states, bounded at the interface between these two topological phases, are observed in the superconducting platform. Measured parameter dependencies further validate the properties of edge states. The scalability and gate-control compatibility of the demonstrated DTQWs enable a versatile tool for superconducting quantum computing and quantum simulation.

Figures (4)

• Figure 1: DTQW in a 1D chain of a superconducting chip.a, Optical micrograph of the 19-transmon chain. Each qutrit is equipped with an independent XY control line and flux bias line, except for the leftmost qutrit, which only has a flux bias line. Each shift qubit (SQ) is equipped with a flux bias line. Both the qutrits and the SQs are coupled to a transmission line through their own $\lambda/4$ readout resonator for readout measurement. b, The coin operation $R_{\theta}$. The tossing is executed by an SU(2) gate within the subspace $\left\{\ket{e}, \ket{f}\right\}$ of the qutrit. The subscript $\theta$ denotes the rotation angle about an axis in the equatorial plane. c, The shift operation $S_{\gg e}$. This operation resembles the operation of a classical shift register, moving the $\ket{e}$ state to the right by one position. It is realized through two sequential swap operations, which are executed by frequency tuning of an SQ. The extended width of the $\ket{e}$ state of the SQ indicates the frequency tuning. d, The zig-zag spatial distribution of the frequency of the transmons. $Q_i$ labels $i$th qutrit (orange triangles), and blue circles are SQs. e, An effective quantum logical circuit for a 3-step DTQW. In the $n$-th step of the walk, coin tosses performed synchronously on the first $n$ qutrits constitute the coin operation described by Eq. (\ref{['coin']}). Similarly, synchronous SWAP pairs constitute the shift operation described by Eq. (\ref{['walker']}).
• Figure 2: DTQW results for edge state and non-edge state.a, The measured excitation distribution of DTQWs with $\theta_\pm=\pm\pi/4$ and the initial state $\ket{\phi_{co}^0}$. The inset is the obtained walk step dependence of the diffusion distance $D(t)=\sqrt{\sum_x x^2 p(x,t)}$, where $p(x,t)$ is the population (equivalent to probability) at position $x$. b, The calculated results for $\theta_\pm=\pm\pi/4$ and the initial state $\ket{\phi_{co}^0}$ without considering gate errors. c, The measured excitation distribution of DTQWs with $\theta_\pm=\pm\pi/4$ and the initial state $\ket{\phi_{ce}^0}$. d, The calculated results for $\theta_\pm=\pm\pi/4$ and the initial state $\ket{\phi_{ce}^0}$ without considering gate errors.
• Figure 3: Populations ($P_{\text{edge}}$) around $x=0$ with the initial state at $\ket{\phi_{ce}^0}$, fixed $\theta_{+}=\pi/4$ and varying $\theta_{-}$, for 5-step (grayish-blue) and 8-step (cyan) DTQWs respectively. Symbols are experimental data (circles for 5-step DTQWs; squares for 8-step DTQWs), and lines are theoretical calculation (dashed line for 5-step DTQWs; dotted line for 8-step DTQWs). Error bars denote the standard deviation of seven replicates.
• Figure 4: Populations ($P_{\text{edge}}$) around $x=0$ with the initial state at $\ket{\phi_{ce}^0}$, fixed relationship $\theta_{+}=-\theta_{-}$, and varying $\theta_{+}$ for 5-step (grayish-blue) and 8-step (cyan) DTQWs, respectively. Symbols are experimental data (open circles for 5-step DTQWs; open squares for 8-step DTQWs), and lines are theoretical calculation (dashed line for 5-step DTQWs; dotted line for 8-step DTQWs). Numerical simulations confirm that the behavior for $\theta_{+} < 0$ is symmetric with respect to $\theta_{+} > 0$, and thus it is not shown. Error bars denote the standard deviation of seven replicates