Impact of leakage on the dynamics of a ST$_0$ qubit implemented in a Double Quantum Dot device

Abstract

Spin qubits in quantum dots are a promising technology for quantum computing due to their fast response time and long coherence times. An electromagnetic pulse is applied to the system for a specific duration to perform a desired rotation. To avoid decoherence, the amplitude and gate time must be highly accurate. In this work, we aim to study the impact of leakage during the gate time evolution of a spin qubit encoded in a double quantum dot device. We prove that, in the weak interaction regime, leakage introduces a shift in the phase of the time evolution operator, causing over- or under-rotations. Indeed, controlling the leakage terms is useful for adjusting the time needed to perform a quantum computation and increasing the coherence time of the readout process. This is crucial for running fault-tolerant algorithms and is beneficial for Quantum Error Mitigation techniques.

Figures (5)

• Figure 1: Graphical representation of the possible states of (a) an ensemble of two TLSs (or qubits) and (b) an ensemble of two three level systems (or qutrits) with the transitions between the states (blue and red lines). The energy of each state is given by the respective diagonal term of the Hamiltonian, $\text{H}_0$, and the transitions are related to the off-diagonal terms of the Hamiltonian, $\text{H}_I$. Blue (solid) lines represent rotations of the qubit, i.e. in the SU(2) subspace, and red (dashed) lines represent leakage transitions to higher energy levels and between them.
• Figure 2: Free evolution of (a) $\ket{S}$ and (b) $\ket{T_0}$ states in presence of leakage.The values of the transverse magnetic fields have been taken relative to the normalized value of $J = 1$, such that the red (dotted) line refers to transverse fields at 2% and the blue (dash-dotted) line to 10%. We can observe that the population remains constant when there is not magnetic transverse fields since the states are eigenstates of the Hamiltonian (Eq. \ref{['Eq: Toy Model']}), however in presence of them the population varies over time.
• Figure 3: Schematic representation of the computational states' energy shift due to transitions to higher energy levels using perturbation theory.
• Figure 4: Comparison of the time evolution of the states (a) $\ket{+}$ when a rotation around $\hat{z}$ is performed for $\text{J}=1$ and (b) $\ket{S}$ when we apply a magnetic field such as $\delta b_z = 1$. The gray (solid) line is the evolution without leakage terms described by Eq.10, the blue (dashed) and the green (dashed-dotted) lines are the evolution for the values of the transverse magnetic field $b_{x,y}=\delta b_{x,y} = 0.02$ and $b_{x,y}=\delta b_{x,y} = 0.1$, respectively.
• Figure 5: Schematic representation of the energy levels of a DQD system with one electron in each QD. The Lie algebra is given by SU(2) $\otimes$ SU(2) $=$ U(1) $\oplus$ SU(3), i.e. the singlet and the triplets states. The Hamiltonian of the system is given in Eq.\ref{['Eq: Hamil6x6']} with $\bm{B}= (0,0,B_z)$ and $\delta \bm{B} = (0,0,0)$