Emulating Anyonic Fractional Statistical Behavior in a Superconducting Quantum Circuit

Editors' Suggestion

Emulating Anyonic Fractional Statistical Behavior in a Superconducting Quantum Circuit

Y. P. Zhong, D. Xu, P. Wang, C. Song, Q. J. Guo, W. X. Liu, K. Xu, B. X. Xia, C.-Y. Lu, Siyuan Han, Jian-Wei Pan, and H. Wang

Phys. Rev. Lett. 117, 110501 – Published 7 September 2016

Abstract

Anyons are exotic quasiparticles obeying fractional statistics, whose behavior can be emulated in artificially designed spin systems. Here we present an experimental emulation of creating anyonic excitations in a superconducting circuit that consists of four qubits, achieved by dynamically generating the ground and excited states of the toric code model, i.e., four-qubit Greenberger-Horne-Zeilinger states. The anyonic braiding is implemented via single-qubit rotations: a phase shift of $π$ related to braiding, the hallmark of Abelian $1 / 2$ anyons, has been observed through a Ramsey-type interference measurement.

Received 23 March 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.110501

Physics Subject Headings (PhySH)

Anyons Quantum computation Quantum information architectures & platforms Quantum simulation Superconducting qubits

Quantum Information, Science & Technology

Authors & Affiliations

Y. P. Zhong^1,2, D. Xu¹, P. Wang¹, C. Song¹, Q. J. Guo¹, W. X. Liu¹, K. Xu¹, B. X. Xia², C.-Y. Lu^2,3,*, Siyuan Han⁴, Jian-Wei Pan^2,3,†, and H. Wang^1,2,‡

¹Department of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China
²CAS Center for Excellence and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
³CAS-Alibaba Quantum Computing Laboratory, Shanghai 201315, China
⁴Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA

^*cylu@ustc.edu.cn
^†pan@ustc.edu.cn
^‡hhwang@zju.edu.cn

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 117, Iss. 11 — 9 September 2016

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Illustration of the toric code model. Qubits, symbolized by balls with arrows, are located on the edges of a two-dimensional square lattice. The lattice is divided into two types of regions, the vertices (light blue) and the faces (light red), where $e$ and $m$ particles reside, respectively. A pair of $e$ ( $m$ ) particles can be created on neighboring vertices (faces) by applying a $Z$ ( $X$ ) rotation on a qubit. (b) Four qubits, labeled from $Q_{1}$ to $Q_{4}$ , of a vertex represent the minimal unit of the toric code. The panel illustrates the braiding action by moving an $m$ particle around an $e$ particle. (c) Schematic of the superconducting circuit featuring four qubits coupled to a central resonator. Arrangement of the four qubits correspond to those in (b). Also shown with each qubit are the control coil, which tunes the qubit frequency, and the integrated superconducting quantum interference device, which probes the qubit state.
Reuse & Permissions
Figure 2
(a) Illustration of the pulse sequence in three dimensions for the one-step generation of the four-qubit GHZ state $| ψ_{g} ⟩$ , where the three axes (frequency, time, and qubit index) are as labeled. For each color-coded qubit sequence line, the first 5-ns full width at half maximum (FWHM) Gaussian-shaped sinusoidal pulse realizes the $X / 2$ gate [15] (i.e., the $π / 2$ rotation around the $x$ axis in the $x − y − z$ reference frame defined at the interaction frequency as labeled by the dashed line), the square pulse tunes the qubit to the interaction frequency, and the second 4-ns FWHM sinusoidal pulse completes the phase-adjustment rotation $θ_{z^{'}}$ around the $z^{'}$ axis in each qubit’s own reference frame as defined in (d). (b) The same pulse sequence projected onto the 2D plane as defined by the frequency and time axes [see the shadow in (a)]. Quantum state tomography (QST) is performed at the end to map out $| ψ_{g} ⟩$ . (c) Real components of the density matrix $| ψ_{g} ⟩ ⟨ ψ_{g} |$ , where the prime sign on each $0^{'}$ or $1^{'}$ in the labels referring to the new frame is omitted for the clarity of the display. All imaginary components (data not shown) are measured to be no higher than 0.043. The polarization axis defining $| 0^{'} ⟩$ and $| 1^{'} ⟩$ is the $z^{'}$ axis as illustrated in (d). The state fidelity $0.574 \pm 0.019$ exceeds the threshold of 0.5, confirming its genuine four-partite entanglement [16, 17]. (d) Illustration of converting the reference frame $x − y − z$ defined at the interaction frequency to the new $x^{'} − y^{'} − z^{'}$ frame for qubit $j$ that picks up a dynamical phase $ϕ_{j}$ in the pulse sequence.
Reuse & Permissions
Figure 3
(a) Illustration of the pulse sequence in 2D for generation of the four-qubit GHZ state with the $e$ -anyonic excitation $| ψ_{e} ⟩$ . The sequence is similar to that shown in Fig. 2, except that an extra 10-ns FWHM sinusoidal pulse is appended, completing the $Z^{'}$ rotation on $Q_{1}$ , to excite $| ψ_{e} ⟩$ out of $| ψ_{g} ⟩$ . The QST pulses are not drawn. (b) The real components of $| ψ_{e} ⟩ ⟨ ψ_{e} |$ generated by the pulse sequence in (a), where the prime sign on each $0^{'}$ or $1^{'}$ in the labels referring to the new frame is omitted for the clarity of the display. All imaginary components (data not shown) are measured to be no higher than 0.044. The state fidelity of $| ψ_{e} ⟩$ is $0.516 \pm 0.010$ , indicating a genuine four-partite entanglement.
Reuse & Permissions
Figure 4
Correlation measurement demonstrating the anyonic braiding statistics. (a) The pulse sequence for measuring, for example, $P (γ)$ of braiding an $m$ particle around a half-filled vertex in the superposition state of $(| ψ_{g} ⟩ + | ψ_{e} ⟩) / \sqrt{2}$ : The first $Z^{'} / 2$ rotation following $θ_{z^{'}}$ creates the superposition, four simultaneous $X^{'}$ rotations finish the braiding operation $C_{loop}$ , the second $- Z^{'} / 2$ rotation returns the state to either $| ψ_{g} ⟩$ or $| ψ_{e} ⟩$ for the correlation measurement, and the final 10-ns FWHM sinusoidal pulse with varying amplitudes, $γ_{z^{'}}$ , rotates the state by an angle $γ$ ( $0 \sim π$ ) around the $z^{'}$ axis, following which the four-qubit joint readout is performed, yielding the 16 occupation probabilities, ${P_{0000}, P_{0001}, \dots, P_{1111}}$ . $⟨ P (γ) ⟩$ is calculated as $\sum_{i_{1}, i_{2}, i_{3}, i_{4}} (- 1)^{i_{1} + i_{2} + i_{3} + i_{4}} P_{i_{1}, i_{2}, i_{3}, i_{4}}$ , where $i_{j} = 0$ or 1 refers to the state of qubit $j$ . (b) Data of $P (γ)$ for the ground state $| ψ_{g} ⟩$ (blue dots) and the $e$ -anyonic excited state $| ψ_{e} ⟩$ (red dots). The solid lines are fits according to the equation $P (γ) \propto \cos (4 γ + φ)$ , yielding $φ s$ of $(0.033 \pm 0.008) π$ for $| ψ_{g} ⟩$ and $(1.049 \pm 0.008) π$ for $| ψ_{e} ⟩$ . (c) Data of $P (γ)$ for looping an $m$ particle around an empty vertex described by $| ψ_{g} ⟩$ (blue dots), an $e$ -anyonic vertex described by $| ψ_{e} ⟩$ (the red dots), and a half-filled vertex in the state of $(| ψ_{g} ⟩ + | ψ_{e} ⟩) / \sqrt{2}$ (black dots), with fits (the solid lines with corresponding colors) yielding $φ s$ of $(0.135 \pm 0.007) π$ , $(0.106 \pm 0.008) π$ , and $(0.983 \pm 0.007) π$ , respectively.
Reuse & Permissions

Physical Review Letters