Fidelity Analysis of Adiabatically Driven Donor Spins as Two-Qubit and Ququart Systems

Technical Summary: Fidelity Analysis of Adiabatically Driven Donor Spins as Two-Qubit and Ququart Systems

Problem Statement
Quantum computing platforms typically utilize qubits (two-level systems), yet many physical implementations, such as donor spin systems in silicon, possess a native Hilbert space of higher dimensionality. While these systems can be treated as qudits (d-level systems), there is an ongoing debate regarding the feasibility and advantage of exploiting this extended dimensionality versus encoding multiple qubits within the same manifold. Specifically, for Silicon-Phosphorus (Si:P) donor systems, the electron spin and nuclear spin create a four-level computational manifold (a ququart). However, operating these systems is complicated by charge noise, which is particularly sensitive at the "ionization point" where the electron is shared between the donor and the interface. This sensitivity can induce leakage outside the computational basis and reduce gate fidelity, especially when using Electron Spin Resonance (ESR) pulses. The paper addresses whether a native ququart approach offers superior performance compared to an encoded two-qubit approach under realistic noise conditions.

Methodology
The authors perform a numerical fidelity analysis of a Si:P donor spin system, modeling it as a four-level ququart. The system is driven by two mechanisms:

1. Electron Spin Resonance (ESR): Used for pure electron spin transitions. These are highly sensitive to charge noise at the ionization point but are more robust near the interface.
2. Electron Dipole Spin Resonance (EDSR): Used for "flip-flop" transitions involving simultaneous electron and nuclear spin flips. These are most efficient at the ionization point.

To mitigate the sensitivity of ESR to charge noise, the authors implement adiabatic ramps to displace the electron. The electron is moved to the interface (where ESR is robust) for ESR operations and returned to the ionization point (where EDSR is efficient) for EDSR operations. Three ramp profiles are compared: linear, raised cosine, and a "K-adiabatic" ramp designed to slow down near avoided crossings to minimize non-adiabatic leakage.

The study compares two gate sets acting on the same four-level manifold:

• Native Ququart Clifford Group ($C_4$): Operates directly on the $d=4$ Hilbert space.
• Encoded Two-Qubit Clifford Group ($C_2^{\otimes 2}$): Treats the four levels as two logical qubits ($|0\rangle \to |00\rangle$, etc.).

To evaluate performance, the authors employ leakage-aware Randomized Benchmarking (RB). Standard RB assumes errors remain within the computational subspace, which is invalid for this system due to charge-noise-induced leakage. The leakage-aware protocol tracks population transfer out of the computational basis and calculates lower-bound error rates ($\varepsilon_{PT}^{LB}$) and fidelity bounds.

Key Results

1. Adiabatic Ramps and Noise Mitigation: The study confirms that operating ESR pulses at the ionization point results in very low fidelity due to charge noise. Displacing the electron to the interface during ESR operations significantly improves fidelity. The "K-adiabatic" ramp shape yields the highest fidelity, reducing oscillations in survival probability compared to linear or raised cosine ramps.
2. Resource Efficiency: Statistical analysis of the gate decompositions reveals that the native $C_4$ group requires fewer physical resources (generators, ESR/EDSR pulses, and displacement ramps) on average than the encoded $C_2^{\otimes 2}$ group. The encoded approach exhibits more outliers and higher resource counts.
3. Fidelity Comparison: Under realistic charge noise (modeled as a two-level fluctuator), the native ququart implementation ($C_4$) consistently achieves 40–50% lower lower-bound error rates compared to the encoded two-qubit implementation ($C_2^{\otimes 2}$). This advantage is attributed to the reduced circuit complexity and fewer operations required for $C_4$, which minimizes the cumulative exposure to noise and leakage during ramps and pulses.
4. Ramp Duration Optimization: While varying the ramp duration ($\tau$) showed that the error reduction for $C_4$ over $C_2^{\otimes 2}$ remains consistent, the absolute error is influenced by the trade-off between non-adiabatic leakage (favored by slower ramps) and noise exposure (favored by faster ramps).

Significance and Claims
The paper concludes that donor spin qudits benefit from genuine qudit operation rather than imposed encoded qubit operation. The results indicate that treating the Si:P system as a native ququart is not only more resource-efficient but also more robust against charge noise, leading to significantly higher gate fidelities. The authors assert that this structural advantage (fewer operations leading to less noise accumulation) is the primary driver of the performance discrepancy, rather than specific driving conditions.

The methodology presented, particularly the use of leakage-aware RB and adiabatic displacement strategies, is claimed to be extendable to other donor species and qudit platforms. The work supports the viability of higher-dimensional systems as computational units for future fault-tolerant quantum processing units (QPUs), suggesting that the "qudit approach" offers a distinct pathway for improving quantum information processing in donor-based architectures.