Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

High-fidelity singlet-triplet S−T− qubits in inhomogeneous magnetic fields

Clement H. Wong, M. A. Eriksson, S. N. Coppersmith, and Mark Friesen
Phys. Rev. B 92, 045403 – Published 6 July 2015
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • S−T− QUBIT
  • DIRECT CURRENT PULSED-GATE OPERATIONS
  • GATE FIDELITIES
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We propose an optimized set of quantum gates for a singlet-triplet qubit in a double quantum dot with two electrons utilizing the S−T− subspace. Qubit rotations are driven by the applied magnetic field and a field gradient provided by a micromagnet. We optimize the fidelity of this qubit as a function of the magnetic fields, taking advantage of “sweet spots” where the rotation frequencies are independent of the energy level detuning, providing protection against charge noise. We simulate gate operations and qubit rotations in the presence of quasistatic noise from charge and nuclear spins as well as leakage to nonqubit states. Our results show that, for silicon quantum dots, gate fidelities greater than 99% should be realizable, for rotations about two nearly orthogonal axes.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    1 More
    • Received 8 October 2014
    • Revised 25 May 2015

    DOI:https://doi.org/10.1103/PhysRevB.92.045403

    ©2015 American Physical Society

    Authors & Affiliations

    Clement H. Wong, M. A. Eriksson, S. N. Coppersmith, and Mark Friesen

    • Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 92, Iss. 4 — 15 July 2015

    Reuse & Permissions
    Access Options
    CHORUS

    Article Available via CHORUS

    Download Accepted Manuscript
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      (a) Illustration of a nonuniform magnetic field Bm provided by a micromagnet (dark purple rectangle) fabricated above a double quantum dot, and a uniform external field Bext (blue arrow). Random, quasistatic Overhauser fields are also present, due to nuclear spins. (b) Singlet-triplet energy diagram, showing the dominant couplings between levels (arrows). A Bloch sphere representation of the S−T− qubit indicates the rotation axes associated with the different coupling terms. (c) (Top) Singlet-triplet energy diagram as a function of detuning ε. X rotations are performed at a detuning sweet spot (black circle at εX) where the qubit energy levels are parallel and the splitting is set by ΔBx. Z′ rotations occur in the far-detuned regime (εZ′≪0), with a rotation axis Z′ tilted slightly away from Z on the Bloch sphere. (Bottom) Illustration of typical pulse sequences for implementing X and Z rotations. Measurement of the singlet probability is done at the detuning value εm>0 in the (0,2) charge state. The Z protocol shows a Ramsey pulse sequence where the Z rotation is implemented using a three-step sequence [25] to correct for the tilt of the Z′ axis, as illustrated on the Bloch sphere.

      Reuse & Permissions
    • Figure 2
      Figure 2

      Optimization of X rotations. (a) Semilogarithmic plot of the state infidelity of an Xπ rotation from |1〉 to |0〉,1−Fs(Xπ), as a function of the applied longitudinal field Bz, for several values of the field gradient ΔBx, as indicated in the legend. (Inset) A similar plot showing the contributions to the state infidelity due to leakage, Pleak, and the combined effect of detuning and Overhauser field fluctuations, 1−P−′, for the case gμBΔBx=0.25 μeV. (b) A color density plot of 1−Fs(Xπ) for an Xπ rotation, as a function of Bz and ΔBx. The red star indicates the optimal working point gμB(ΔBx,Bz)=(0.25,0.75) μeV. (c) Larmor oscillations (X rotations), and the corresponding Gaussian decay envelope, [1±e−(t/T2(X))2]/2, obtained at the optimal working point, with T2*(X)=2ℏ/σh.

      Reuse & Permissions
    • Figure 3
      Figure 3

      Optimization of Z′ rotations. (a) Semilogarithmic plot of the state infidelity of a Zπ′ rotation, 1−Fs(Zπ′), as a function of the detuning, for nearly optimal values of gμBΔBx and gμBBz. The red markers indicate the correspondence with curves in Fig. 2, with the star indicating the optimal working point. (b) Z′ rotations performed at the starred point in (a), for εZ′=−1.5 meV.

      Reuse & Permissions
    • Figure 4
      Figure 4

      Singlet-triplet energy diagrams, including the qubit states |0〉 and |1〉 and leakage states |T0〉 and |T+〉, as a function of the detuning, for two different tunnel coupling models. Here, state |S′〉 lies outside the range of the plot. Solid lines: case (i), the constant tunnel coupling model, with tc=20μeV and magnetic field values gμB(ΔBx,Bz)=(0.25,0.75)μeV, which were optimized as described in the main text. On the right-hand side of the anticrossing, the qubit states correspond to |0〉≃|S〉 and |1〉≃|T−〉. Dashed lines: case (ii), the detuning-dependent tunnel coupling model, tc(ε)=t0exp(ε/ε0), with t0=20 μeV and ε0=1 meV. Here too, the magnetic field parameters gμB(ΔBx,Bz)=(0.3,0.7)μeV were optimized to achieve a high Xπ gate fidelity. (Inset) The singlet mixing terms (|cosη|,|sinη|), from Eq. (1). Note that |sinη|≃J(ε)/tc, the effective exchange energy, to a very good approximation.

      Reuse & Permissions
    • Figure 5
      Figure 5

      Probability P+ of leaking into state |T+〉 as a function of time t during X rotations, for the optimal field values gμB(ΔBx,Bz)=(0.25,0.75)μeV, plotted with the blue, solid line (the left hand axis). The envelope of the oscillations closely follows the probability PS of occupying state |S〉, as indicated by the maroon, dashed line (the right-hand axis).

      Reuse & Permissions
    • Figure 6
      Figure 6

      The real and imaginary parts of the χ matrix elements for the fully optimized Xπ and Zπ′ gates, obtained from Eq. (D2). Triangles indicate nonzero target values of the χ matrix elements.

      Reuse & Permissions
    • Figure 7
      Figure 7

      The numerically computed probability P− of occupying the state |T−〉 as a function of time t during X rotations in the absence of Overhauser field fluctuations, assuming the initial state |S〉. The dephasing envelope (dashed lines) is the Gaussian decay given by Eq. (E11, E12, E13), setting σh=0, with T2⊥*(X)=4.7μs.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Z′ rotations in the asymptotic regime ε→−∞, obtained from analytic solutions for the qubit dynamics arising from Eq. (F1). Here, we plot the probability P(−X) of being in the final state |−X〉 as a function of time t, for an initial state |X〉 and the optimal magnetic fields (ΔBx,Bz)=(0.25,0.75)μeV. The analytic solutions are numerically averaged over the Overhauser field fluctuations.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review B

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×