Quantum error correction with silicon spin qubits

Quantum computing leverages superposition and entanglement to accelerate computation, but these properties are highly susceptible to errors from relaxation and dephasing, leading to accuracy loss as systems scale. Quantum error correction distributes information across multiqubit entangled states to detect and correct errors. Foundational demonstrations of QEC have been achieved in other platforms (nuclear magnetic resonance, trapped ions, NV centers, superconducting circuits), establishing its importance as a benchmark. Silicon spin qubits have rapidly advanced, showing long coherence times, high-fidelity universal gates, high-temperature operation, and generation of three-qubit entanglement. Despite this progress, implementing QEC in silicon—requiring at least three coupled qubits and either a three-qubit gate or fast, high-fidelity measurement for feedback—remained unachieved. This work addresses that gap by demonstrating a three-qubit phase-flip correcting code in a silicon triple-quantum-dot device using a coherent, single-step resonantly driven iToffoli gate for measurement-free correction, targeting the dominant phase-type errors in silicon spin qubits.

Prior QEC has been demonstrated across multiple architectures: NMR, trapped ions (including repetitive QEC), NV centers in diamond, and superconducting circuits, highlighting the feasibility and value of QEC as a benchmark for qubit platforms. In silicon spin qubits, recent milestones include fault-tolerant-level control fidelities, two-qubit processors with >99% fidelities, operation at elevated temperatures, and creation/tomography of three-qubit entangled states. Theoretical and gate-decomposition results for multi-qubit gates (e.g., Toffoli via CNOT networks) exist, as well as proposals for resonantly driven Toffoli-class gates in silicon via exchange-coupling-mediated conditional spectroscopy. These works motivate a silicon-based QEC realization but also underscore challenges such as limited coherence relative to complex multi-gate sequences, and the need for fast, high-fidelity measurement for feedback-based correction in scalable fault-tolerant schemes.

Device and qubits: A gate-defined triple quantum dot in an isotopically natural Si/SiGe heterostructure was fabricated with three layers of overlapping aluminum gates, and a micromagnet providing a local magnetic field gradient. One electron is confined under each plunger gate (P1, P2, P3) with interdot tunnel couplings controlled by barrier gates (B2, B3). Charge sensing is performed via a nearby quantum dot using RF reflectometry. An in-plane external magnetic field B_ext = 0.607 T sets Zeeman splittings (~20 GHz), well above the thermal energy (~0.8 GHz at 40 mK), enabling initialization/readout via energy-selective tunneling, shuttling, and controlled rotations. The three spin-1/2 qubits (Q1, Q2, Q3) are the Zeeman-split states of the confined electrons.

Single- and two-qubit control: Single-qubit rotations use resonant microwave pulses that electrically displace the dot to induce electric-dipole spin resonance (EDSR). Two-qubit controlled-phase (CZ) gates are implemented by adiabatically pulsing nearest-neighbor exchange couplings J12 and J23 via barrier gates B2 and B3. Operation near charge-symmetry is maintained using virtual gates to suppress capacitive crosstalk. Characteristic coherence times are T1 ≈ 22 ms, T2* ≈ 1.8 μs, and Hahn-echo T2 ≈ 43 μs.

Encoding/decoding and GHZ verification: A three-qubit system with data qubit Q2 and ancilla qubits Q1 and Q3 is used. Encoding maps an equatorial input state on Q2 into a three-qubit GHZ-class state using two CNOTs decomposed into native CZ gates with interleaved dynamical-decoupling Y pulses to mitigate quasi-static phase noise. Three-qubit quantum state tomography validates GHZ-class states across input phases φ, with state fidelities exceeding the GHZ-class witness threshold.

Three-qubit iToffoli gate: Instead of synthesizing a Toffoli from many CNOTs (which would suffer coherence-limited fidelity), a single-step resonantly driven iToffoli gate is implemented. Simultaneous nearest-neighbor exchange couplings lift degeneracy among the four Q2 transitions, shifting Q2’s resonance by ω0 + S1 J12 + S3 J23 (Si = ±1/2), enabling a controlled-controlled-rotation when |J12| ≈ |J23|. Spectroscopy of Q2 under four ancilla configurations (Q1Q3 ∈ {↓↓, ↓↑, ↑↓, ↑↑}) at |J12| = |J23| = 4.5 MHz confirms the expected frequency separation. A resonant π pulse at the |Q1Q3 = ↑↑⟩-conditioned transition implements the iToffoli, chosen for highest visibility. Truth-table characterization prepares each computational-basis state, applies the iToffoli, and measures three-spin populations; readout errors are corrected using calibrated fidelities. Off-resonant rotations are synchronized by choosing appropriate Rabi frequency. Population transfer swaps |↓↓↓⟩ and |↓↑↑⟩ with others largely unaffected, yielding a population-transfer fidelity Tr(U_expt U_ideal†)/8 = 0.96. Pulse duration/timing are calibrated to eliminate unwanted Q2 phase accumulation; dephasing/phase on ancillas does not affect the correction outcome.

Phase-flip QEC protocol: The unitary U encodes the data-qubit state |ψ⟩ = α|+⟩ + β|−⟩ into α|+++⟩ + β|−−−⟩ (|±⟩ are X-eigenstates). After phase-flip noise Z(p) (implemented as a known single-qubit phase rotation by θ with p = sin^2(θ/2)) on targeted qubits, decoding maps error syndromes onto ancillas. Correction flips Q2 only when Q1Q3 = |↑↑⟩ by applying π pulses on ancillas followed by the iToffoli, restoring Q2 to the original state while ancillas encode error information. One- and three-qubit error scenarios are tested. Process tomography of Q2 estimates process fidelity versus error strength. Ancilla joint-state probabilities are measured to verify error detection.

Dephasing mitigation test: To probe correction of quasi-static phase noise (dominant in natural Si due to 29Si nuclear spin fluctuations), the encoded state is held for varying waiting time between U and U†. State fidelities versus tw are compared for physical, corrected, and uncorrected encoded qubits. Data acquisition uses averaging over thousands of repetitions with interleaved frequency calibrations. Decays are fit to F(t) = (1 + a exp(-(t/T2)^n))/2 to quantify dephasing behavior.

• Three-qubit GHZ-state generation: Across input azimuthal angles φ, all measured states exceed the GHZ-class witness threshold of 0.75, with an average GHZ state fidelity of 0.866.
• Single-step resonantly driven iToffoli gate: Conditional spectroscopy under |J12| = |J23| = 4.5 MHz resolves four Q2 transitions by ancilla configuration. The iToffoli swaps |↓↓↓⟩ and |↓↑↑⟩ populations with minimal effect on others, achieving a population-transfer fidelity Tr(U_expt U_ideal†)/8 = 0.96. Full process tomography confirms Toffoli-class behavior.
• One-qubit phase-flip correction: When intentional phase errors are applied to only Q2, the uncorrected data-qubit process fidelity oscillates with θ (p = sin^2(θ/2)), while the corrected fidelity becomes insensitive to θ, demonstrating successful correction. A slight fidelity drop at θ = 0 after correction is attributed to iToffoli infidelity projected onto the Q2 subspace.
• Error detection via ancillas: Joint measurements of Q1 and Q3 for no-error and full-flip cases show ancilla outcomes correctly reflect the error on the encoded state.
• Multi-qubit error model (equal error rate p on all qubits): The uncorrected process fidelity decreases linearly with p, whereas the corrected fidelity exhibits the QEC hallmark quadratic dependence. Fitting yields negligible first-order sensitivity (first-order term 0.0 ± 0.1) and an improvement threshold of p < 0.429 ± 0.003, consistent with the ideal expectation F(p) = 1 − 3p^2 + 2p^3.
• Dephasing mitigation: For quasi-static dephasing, the corrected encoded qubit shows a suppressed initial decay slope compared to the uncorrected encoded qubit, indicating mitigation of intrinsic phase noise. Fits to F(t) = (1 + a exp(-(t/T2)^n))/2 give representative parameters: physical qubit a = 0.492 ± 0.005, T2 = 1.44 ± 0.02 μs, n = 1.90 ± 0.08; corrected encoded qubit a = 0.432 ± 0.008, T2 = 1.36 ± 0.04 μs, n = 2.1 ± 0.2; uncorrected encoded qubit a = 0.464 ± 0.007, T2 = 1.12 ± 0.03 μs, n = 1.68 ± 0.08.
• Device coherence: Measured T1 ≈ 22 ms, T2* ≈ 1.8 μs, Hahn-echo T2 ≈ 43 μs support high-fidelity single- and two-qubit operations and enable the QEC demonstration.

The work demonstrates, for the first time in silicon spin qubits, a full three-qubit phase-flip error-correcting code using a coherent, measurement-free correction via a resonantly driven iToffoli gate. This directly addresses the challenge of implementing QEC on a silicon platform where fast, high-fidelity projective readout and real-time feedback remain difficult. The key QEC hallmark—suppression of first-order sensitivity to uniformly distributed phase errors, yielding quadratic fidelity dependence on error probability—is experimentally observed. Error detection is verified through ancilla outcomes, and the protocol mitigates quasi-static dephasing, the dominant intrinsic phase noise in natural Si.

The observed performance is primarily limited by the iToffoli gate fidelity and finite coherence, leading to corrected fidelities below those of an ideal uncorrected qubit and setting a practical improvement threshold (p < ~0.43). Nonetheless, the results validate the feasibility of QEC primitives in silicon and suggest that with improved coherence, faster/high-fidelity readout, and refined multi-qubit control, silicon spin qubits can progress toward larger-scale, fault-tolerant architectures.

This study achieves three central milestones in silicon spin qubits: generation of high-fidelity three-qubit GHZ-class states, realization of an efficient single-step resonantly driven iToffoli gate, and demonstration of a three-qubit phase-flip error-correcting code that suppresses first-order phase errors and mitigates quasi-static dephasing. These advances establish the viability of QEC primitives in a CMOS-compatible platform.

Future directions include transitioning to fast, high-fidelity measurement and feedback (e.g., singlet-triplet readout with microsecond-scale timescales) to enable flexible, fault-tolerant correction, improving coherence and multi-qubit gate fidelities (especially the three-qubit gate), and leveraging scalable device designs and cryo-electronics to implement more sophisticated error-correcting codes in larger silicon-based quantum processors.

• Measurement-based feedback not implemented: The correction uses a coherent iToffoli gate rather than projective ancilla readout and fast feedback, which are required for fully fault-tolerant operation.
• Gate infidelities limit performance: The iToffoli fidelity and accumulated errors during encoding/decoding reduce the corrected process fidelities, leading to improvement only below a threshold error rate (p < ~0.43).
• Coherence constraints: Intrinsic dephasing (quasi-static nuclear spin noise) and finite T2* constrain multi-qubit gate sequences; corrected fidelities remain below those of an ideal uncorrected qubit.
• Scope limited to phase-flip errors: The work focuses on phase-flip correction; bit-flip correction would require additional single-qubit rotations but is not demonstrated here.
• Slow spin measurement and initialization via energy-selective tunneling hinder rapid error detection/correction cycles needed before phase coherence is lost.