A two-qubit controlled-not (cnot) gate, realized by a controlled-phase (C-phase) gate combined with single-qubit gates, has been experimentally implemented recently for quantum-dot spin qubits in isotopically enriched silicon, a promising solid-state system for practical quantum computation. In the experiments, the single-qubit gates have been demonstrated with fault-tolerant control fidelity, but the infidelity of the two-qubit C-phase gate is, primarily due to the electrical noise, still higher than the required error threshold for fault-tolerant quantum computation (FTQC). Here, by taking the realistic system parameters and the experimental constraints on the control pulses into account, we construct experimentally realizable high-fidelity cnot gates robust against electrical noise with the experimentally measured
1/f1.01 noise spectrum and also against the uncertainty in the interdot tunnel coupling amplitude. Our fine-tuned optimal cnot gate has about two orders of magnitude improvement in gate infidelity over the ideal C-phase gate constructed without considering any noise effect. Furthermore, within the same control framework, high-fidelity and robust single-qubit gates can also be constructed, paving the way for large-scale FTQC.
Single-qubit rotations and two-qubit CNOT operations are crucial ingredients for universal quantum computing. Although high-fidelity single-qubit operations have been achieved using the electron spin degree of freedom, realizing a robust CNOT gate has been challenging because of rapid nuclear spin dephasing and charge noise. We demonstrate an efficient resonantly driven CNOT gate for electron spins in silicon. Our platform achieves single-qubit rotations with fidelities greater than 99%, as verified by randomized benchmarking. Gate control of the exchange coupling allows a quantum CNOT gate to be implemented with resonant driving in ~200 nanoseconds. We used the CNOT gate to generate a Bell state with 78% fidelity (corrected for errors in state preparation and measurement). Our quantum dot device architecture enables multi-qubit algorithms in silicon.
Building on the previous results of the Weyl chamber steering method, we demonstrate how to generate high-fidelity controlled-NOT (CNOT) gates by direct application of certain physically relevant Hamiltonians with fixed coupling constants containing Rabi terms. Such Hamiltonians are often used to describe two superconducting qubits driven by local rf pulses. It is found that in order to achieve 100% fidelity in a system with capacitive coupling of strength
g, one Rabi term suffices. We give the exact values of the physical parameters needed to implement such CNOT gates. The gate time and all possible Rabi frequencies are found to be $t=\ensuremath{\pi}∕(2g)$ and ${\ensuremath{\Omega}}_{1}∕g=\sqrt{64{n}^{2}\ensuremath{-}1},n=1,2,3,\dots{}$. Generation of a perfect CNOT gate in a system with inductive coupling, characterized by additional constant
k, requires the presence of both Rabi terms. The gate time is again $t=\ensuremath{\pi}∕(2g)$, but now there is an infinite number of solutions, each of which is valid in a certain range of
k and is characterized by a pair of integers
(n,m), $\frac{{\ensuremath{\Omega}}_{1,2}}{g}=\sqrt{16{n}^{2}\ensuremath{-}{(\frac{k\ensuremath{-}1}{2})}^{2}}\ifmmode\pm\else\textpm\fi{}\sqrt{16{m}^{2}\ensuremath{-}{(\frac{k+1}{2})}^{2}}$. We distinguish two cases, depending on the sign of the coupling constant: (i) the antiferromagnetic case $(k\ensuremath{\geqslant}0)$ with $n\ensuremath{\geqslant}m=0,1,2,\dots{}$ and (ii) the ferromagnetic case $(k\ensuremath{\leqslant}0)$ with
n>m=0,1,2,…. We conclude with consideration of fidelity degradation by switching to resonance. Simulation of time evolution based on the fourth-order Magnus expansion reveals characteristics of the gate similar to those found in the exact case, with slightly shorter gate time and shifted values of the Rabi frequencies.
We model a controlled-NOT (CNOT) gate mediated by exchange interactions in a realistic Ge/GeSi hole spin-qubit device. We use for that purpose a computationally efficient framework based on a compact dressed basis of many-body singlet and triplet states, which enables fast time-dependent simulations that would be prohibitive within a conventional configuration-interaction approach. We discuss the dominant sources of infidelity in ideal (noise-free) devices. On the one hand, the nonadiabaticity of the tunnel barrier ramps, enhanced by the transverse component of the exchange interaction, imposes limits on the speed of the CNOT gate. On the other hand, the AC Stark shift of the conditional resonance and the off-resonant driving of the complementary transition can significantly degrade the gate fidelity and call for a careful optimization of the driving conditions. Finally, we investigate the impact of quasi-static charge noise on the fidelity of the CNOT gate. We find that the capture of charges near the qubits can strongly disrupt the conditional operation, while the nanometer-scale motion of charges results in a loss of fidelity that sensitively depends on the position and strength of the corresponding dipole. Our results quantify the requirements on barrier control, microwave driving, and charge-noise stability for high-fidelity two-qubit operations in Ge/GeSi hole-spin qubits.
Electron spins trapped in quantum dots have recently proven to be a promising technology for the implementation of qubits, already demonstrating high fidelity single- and two-qubits gates. The next step towards fault-tolerant quantum computing is to increase the number of so-called spin qubits on the processor. However, this poses several challenges, one of which being how to implement single-qubit gates in multi-qubit systems, with high fidelity. This work focused on two aspects. The first one was to identify the physical phenomena and limitations that hinder the realisation of high fidelity single-qubit gates in multi-qubit environments. The second objective was to investigate, implement and optimise methods in order to minimise the impact of these perturbing phenomena. The identified perturbing effects include unwanted driving between a pulse and the neighbouring qubits, as well as frequency shifts induced by drive-spin coupling and non-ideal pulse in experimental setups. Crosstalk was addressed using pulse shaping (i.e. amplitude or phase modulation of the driving pulse to tailor its spectral characteristics and reduce the energy transmitted at unwanted frequencies). The performance of shapes taken from both NMR and signal processing literature was investigated, for various pulse parameters, through an extensive grid search. In addition, a frequency correction algorithm was devised: off-resonant drive frequencies are selected so that the shifted qubits are resonantly driven. It currently accounts for two phenomena: the AC Stark and Bloch-Siegert shifts. The algorithm furthermore tracks the changes caused by the time-dependent characteristics of the shaped pulses. The proposed frequency correction algorithm was shown to almost entirely negate the effects of the considered shifts, leading to unitary fidelity improvements by up to 55%. In addition, pulse shaping was demonstrated to noticeably improve the fidelity of simultaneous single-qubit rotations compared to unshaped driving. Rotations with fidelities as high as 99.98% were obtained for pi/2 rotations on two-qubits systems. Moreover, shapes whose Fourier transform is narrow and sharp, associated with low Rabi frequencies, were demonstrated to generally provide the highest fidelities of the tested configurations. Lastly, the trends and guidelines highlighted by these results were shown to scale to systems with larger numbers of qubits. The correction techniques investigated in this work have proven promising for the implementation of high fidelity single-qubit gates in multi-qubit systems. In particular, the guidelines for selecting a well-performing pulse shape should also be useful for the design of optimised driving schemes, regardless of the number of qubits involved. Additionally, the proposed frequency shift correction algorithm is expected to be able to handle arbitrary shifts, and so to be easily adaptable to use in experiments.
Abstract Semiconductor spin qubits demonstrated single-qubit gates with fidelities up to 99.9 % benchmarked in the single-qubit subspace. However, tomographic characterizations reveal non-negligible crosstalk errors in a larger space. Additionally, it was long thought that the two-qubit gate performance is limited by charge noise, which couples to the qubits via the exchange interaction. Here, we show that coherent error sources such as a limited bandwidth of the control signals, diabaticity errors, microwave crosstalk, and non-linear transfer functions can equally limit the fidelity. We report a simple theoretical framework for pulse optimization that relates erroneous dynamics to spectral concentration problems and allows for the reuse of existing signal shaping methods on a larger set of gate operations. We apply this framework to common gate operations for spin qubits and show that simple pulse shaping techniques can significantly improve the performance of these gate operations in the presence of such coherent error sources. The methods presented in the paper were used to demonstrate two-qubit gate fidelities with F > 99.5 % in Xue et al (2022 Nature 601 343). We also find that single and two-qubit gates can be optimized using the same pulse shape. We use analytic derivations and numerical simulations to arrive at predicted gate fidelities greater than 99.9% with duration less than, 4 / ( Δ E z ) where Δ E z is the difference in qubit frequencies.