Quantum gates in coupled quantum dots controlled by coupling modulation

TL;DR

This work tackles universal quantum control in two coupled double quantum dots by encoding qubits in a quasi-degenerate central pair and using time-dependent harmonic modulation of tunnel and exchange couplings to realize single- and two-qubit gates.Analytical approaches based on rotating-wave approximations and leakage suppression provide clear conditions for gate implementations, which are then validated against numerically exact simulations of the full Hamiltonian.Single-qubit gates are achieved via resonant modulation of the tunnel coupling at the Zeeman-gap frequency, while two-qubit entangling gates arise from biharmonic modulation of the exchange interaction at the sums and differences of the one-qubit frequencies, with gate metrics characterized by Makhlin invariants.Numerical results show small leakage and infidelity under realistic detuning and field variations, with one-qubit gates operating on nanosecond scales and two-qubit gates on the order of hundreds of nanoseconds, indicating a feasible path toward scalable, all-electrical quantum computing in semiconductor QDs.

Abstract

We studied the dynamics of a pair of single-electron double quantum dots (DQD) under longitudinal and transverse static magnetic fields and time-dependent harmonic modulation of their interaction couplings. We propose to modulate the tunnel coupling between the QDs to produce one-qubit gates and the exchange coupling between DQDs to generate entangling gates, the set of operations required for quantum computing. We developed analytical approximations to set the conditions to control the qubits and applied them to numerical calculations to test the accuracy and robustness of the analytical model. The results shows that the unitary evolution of the two-electron state performs the designed operations even under conditions shifted from the ideal ones.

Figures (5)

• Figure 1: Two exchange-coupled single-electron flopping-mode qubits. (a) Scheme of the cross section of the device and its physical model in terms of electrostatically-defined potential wells and barriers. Double QDs (L1, R1) and (L2,R2) are subject to longitudinal and transverse magnetic fields produced by the micromagnets MM. (b) Scheme of energy levels of each single-electron DQD in absence of exchange coupling. From left to right the various mechanism involved in the setup: the coupling $t_c$ shifts bonding and antibonding states, the longitudinal magnetic field $B_z$ breaks spin degeneracy and fix the quantization axis and, finally, the inhomogeneous component $B_x$ hybridize $|\downarrow_b\rangle$ and $|\uparrow_a\rangle$. Inset: energy labels as a function of $t_c$ showing quasi-degeneration at $t_c=g\mu_B B_z/2$.
• Figure 2: Left: Infidelity $I$ for $H$ and $Z$ gates as a function of $\hbar\omega_x/\eta_c$ for level detuning $\epsilon=0$, 0.2 and 0.4 $\mu$eV. Right: Infidelity for $H$ and $Z$ gates as a function of $\hbar\Delta\omega_x/\eta_c$ for values of detuning $\epsilon$ increasing from zero (lower curve) to 0.8 $\mu$eV (higher curve) through steps of 0.2 $\mu$eV as indicated by the upward arrow. $\Delta \hbar\omega_x/\eta_c$ are shifts from the central dip $\hbar\omega_x/\eta_c$ on the left panel ($B_x=150$ mT, and $\eta_c=0.643\ \mu$eV).
• Figure 3: Numerically calculated leakage from the dynamics of the exact Hamiltonian corresponding to Hadamard $H$ and $Z$ gates as a function of the detuning $\epsilon$ with parameters set from the analytically defined gates. The longitudinal and transverse magnetic fields are $B_z=600$ mT and $B_x=150$ mT, respectively. Insets depict the leakage as a function of shifts from resonance (in units of $\eta_c$) for the two gates at $\epsilon=1\ \mu$eV as pointed by the arrows. The gray shaded areas correspond to the same ranges shown in Fig. \ref{['fig:H-Z-gates']}.
• Figure 4: (a) Occupation of state $|10\rangle$ calculated as a function of frequency of modulation $\omega_2$, after unitary evolution from $|\psi(0)\rangle=(|00\rangle+|01\rangle)/\sqrt{2}$, with biharmonic coupling $J(t)=J_0+J_1 (\sin\omega_1 t+ \sin\omega_2 t)$, Eq. (\ref{['eq:J-modulation']}) with $J_1=J_2$ for a CNOT-equivalent entangling gate, for transverse fields $(B_x^{(1)},B_x^{(2)})=$ (120, 60) mT, giving resonant frequencies $\Omega_1/2\pi=2.52$ GHz and $\Omega_2/2\pi=0.84$ GHz; amplitudes of control are $J_0=2\pi\times 10$ MHz and $J_1=J_2=2\pi\times 5$ MHz. (b)-(c) Occupation of state $|10\rangle$ y $|11\rangle$ as a function of the evolution time $t$ in nanoseconds. The frequency $\omega_1$ is resonant at $\Omega_1/2\pi=2.52$ GHz, while $\omega_2$ if off-resonant: $\omega_2/2\pi=0.84$, 0.841, 0.843 and 0.845 GHz. Initial state $|\phi_+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|01\rangle)$.
• Figure 5: Calculated infidelity $I$ and leakage ${\cal L}$ for the two-qubit entangling gate. (Above) $I$ as function of the $J_0^2/J_1$. The dashed straight line is the fitting $a x^n$, where $x=J_0^2/J_1$ and the parameters are: $a=3\times 10^{-6}$ MHz$^{-2}$, $n=2.0$. The Inset shows ${\cal L}$ as function of the $J_0$. (Below) ${\cal L}$ as function of the $\omega_2$ for the different values of the $J_0$ and $J_1$. The dashed line in the middle of the graph corresponds to the resonance point: $\omega_2=\Omega_2$.