Quantum error correction with silicon spin qubits

Large-scale quantum computers require quantum error correction (QEC) to protect quantum information during computation. Silicon-based spin qubits, compatible with mature nanofabrication technologies, are promising for scaling up quantum computers. While high-quality one- and two-qubit systems have been demonstrated, QEC, requiring three or more coupled qubits, remains a challenge. This research addresses this challenge by demonstrating a three-qubit phase-correcting code in silicon, a significant step towards building fault-tolerant quantum computers. The fragility of quantum information due to decoherence errors from energy relaxation and dephasing necessitates QEC. QEC protocols distribute quantum information across a larger multiqubit entangled state to detect and correct errors. This approach has been demonstrated in various platforms, including nuclear magnetic resonance, trapped ions, nitrogen vacancy centers, and superconducting circuits. Silicon-based spin qubits have shown rapid progress in long coherence times, high-fidelity universal quantum gates, high-temperature operation, and the generation of three-qubit entanglement, positioning them as a strong contender for quantum computing.

Significant advancements have been made in the field of silicon-based spin qubits, including the achievement of long coherence times, high-fidelity universal quantum gates, and high-temperature operation. Several research groups have demonstrated the generation of three-qubit entanglement, a crucial component for quantum error correction. Previous work has showcased QEC in various platforms such as nuclear magnetic resonance, trapped ions, nitrogen-vacancy centers, and superconducting circuits. However, demonstrating QEC in silicon-based spin qubits, while leveraging the advantages of silicon's compatibility with established nanofabrication technology, remained an open challenge until this work.

The researchers used a gate-defined triple quantum dot in a silicon/silicon-germanium (Si/SiGe) heterostructure. Three layers of overlapping aluminum gates controlled the triple-dot confinement, and a micro-magnet provided a local magnetic field gradient. The Zeeman-split spin-1/2 states of single electrons served as spin qubits. Single-qubit rotations were performed using resonant microwave pulses, inducing electric-dipole spin resonance. The two-qubit controlled phase (CZ) gate was implemented by adiabatically pulsing the exchange couplings. The virtual gate technique suppressed capacitive crosstalk. The system exhibited an average T1 relaxation time of 22 ms, an inhomogeneous dephasing time T2* of 1.8 μs, and a Hahn echo dephasing time T2 of 43 μs. The three-qubit phase-flip code was synthesized. Encoding involved using CNOT gates (decomposed into CZ gates and decoupling pulses), while decoding was the inverse. A single-step resonantly driven iToffoli gate, implemented using a resonant π pulse with simultaneous nearest-neighbor exchange couplings, performed the correction. The iToffoli gate's properties were characterized using three-spin projective measurement and quantum process tomography. The phase-flip correcting code was implemented, and its performance was evaluated with intentional one-qubit errors and dephasing. Quantum state tomography was used to characterize the generated three-qubit entangled states, specifically checking for genuine GHZ-class states. The fidelity of the iToffoli gate was assessed using quantum process tomography. Process fidelity was used to evaluate the performance of the error correction with intentional one-qubit errors introduced as phase rotations, and dephasing was evaluated by introducing waiting times between encoding and decoding.

The researchers successfully generated various three-qubit entangled states, including GHZ states, with fidelities above 0.75, demonstrating genuine three-qubit entanglement. They implemented an efficient single-step resonantly driven iToffoli gate with a population transfer fidelity of 0.96. The three-qubit phase-flip correcting code effectively mitigated one-qubit phase-flip errors. The process fidelity remained relatively insensitive to error probability (p) up to the first order, showing a quadratic dependence on p, a crucial property of QEC. The error correction improved fidelity for p < 0.429. The code also mitigated the effects of quasi-static phase noise (due to fluctuating nuclear spins), suppressing the initial slope of fidelity decay. Measurements on the ancilla qubits correctly reflected errors in the encoded state, demonstrating error detection capabilities. The experiment confirmed the mitigation of single-qubit phase-flip errors and the suppression of dephasing effects on the encoded qubit state.

The successful implementation of the three-qubit phase-correcting code in silicon demonstrates the feasibility of QEC in this promising platform for quantum computing. The results show that the code can effectively mitigate single-qubit phase-flip errors and reduce the impact of quasi-static phase noise. The use of a single-step resonantly driven iToffoli gate provides a significant improvement in efficiency compared to alternative approaches. However, the current implementation is limited to single-qubit errors. Future work will need to address the correction of multi-qubit errors, which are more common in real-world quantum computers. The observed quadratic dependence of fidelity on error probability confirms a key characteristic of QEC, suggesting improvements are achievable with advancements in gate fidelities and coherence times. The choice to focus on phase-flip correction is justified by the significantly longer T1 times compared to dephasing times in the system, highlighting the immediate relevance of this approach.

This work demonstrates the generation of three-qubit entangled states, a highly efficient single-step resonantly driven iToffoli gate, and the fundamental properties of a three-qubit quantum error-correcting code in silicon. While scaling up to larger systems requires a feedback-based approach limited by slow spin measurement and initialization, improvements like singlet-triplet readout, along with advancements in device design and gate fidelities, suggest the future potential for more sophisticated QEC in large-scale silicon-based quantum processors.

The current implementation focuses on correcting single-qubit phase-flip errors. Real-world quantum computers experience multiple simultaneous errors on all qubits, making this a simplification. The fidelity of the error correction is limited by the fidelity of the iToffoli gate, which, if improved, could lead to higher overall performance. The slow spin measurement and initialization based on energy-selective tunneling pose a challenge for correcting errors before decoherence completely destroys the quantum information.