Mitigating errors in state preparation and measurement with noncomputational states

Technical Summary: Mitigating Errors in State Preparation and Measurement with Noncomputational States

Problem Statement
Current error mitigation techniques for quantum computing often rely on noise-aware methods that require faithful characterization of hardware noise channels. However, a fundamental limitation exists when attempting to separate state-preparation and measurement (SPAM) errors from gate errors or to distinguish state-preparation errors from measurement errors. In standard frameworks restricted to the computational subspace, specific degrees of freedom in the noise model are "unlearnable" (mathematically identified as the cut space of a cycle graph). Consequently, joint mitigation of SPAM errors can lead to inaccuracies, such as the over-compensation observed in the Twirled Readout Error Extinction (TREX) technique. When the set of qubits affected by state-preparation errors differs from the support of the observable being measured, TREX can produce unphysical expectation values (e.g., values exceeding 1 or dropping below -1). Furthermore, existing self-consistent noise characterization protocols, while sufficient for unitary circuits, do not reveal the "ground truth" for all parameters (specifically state-preparation errors), limiting their applicability to dynamic circuits with mid-circuit measurements (MCMs) and classical feed-forward.

Methodology
The authors propose leveraging non-computational states (specifically the $|2\rangle$ state in superconducting transmon qubits) as an additional resource to break the symmetry that renders state-preparation errors unlearnable.

1. Characterization of State-Preparation Errors: The protocol utilizes a RabiEF experiment (Rabi oscillations in the $|1\rangle \leftrightarrow |2\rangle$ subspace) to directly measure the state-preparation fidelity ($f_{sp}$) of a passive reset. By waiting for thermalization (passive reset), the qubit enters a thermal state with a known population in the excited state. The RabiEF signal allows for the precise estimation of the probability $p_{sp}$ of finding the qubit in $|1\rangle$, effectively isolating the state-preparation error from measurement noise.
2. Split Mitigation of SPAM Errors: The authors demonstrate that by independently knowing $f_{sp}$, one can correct the TREX mitigation factor. Instead of normalizing the raw expectation value $\langle O \rangle_{raw}$ by a factor $\langle Z^\star \rangle$ that conflates state and measurement errors, the raw value is normalized by a product of the known state-preparation fidelities and the measurement correction factor. This "split mitigation" prevents over-correction.
3. Noise Model for Mid-Circuit Measurements (MCM): The paper introduces a comprehensive noise model for MCMs that includes three error channels: state errors (bit flips on the qubit), assignment errors (bit flips on the classical bit), and correlated errors (simultaneous flips). This model is learned using Measurement Cycle Benchmarking (MCB).
4. Full Protocol: The proposed workflow involves:
   • Learning state-preparation fidelity ($f_{sp}$) via RabiEF (using either slow passive resets or fast active qutrit resets).
   • Learning measurement noise fidelities ($f_a, f_s, f_c$) via MCB.
   • Combining these to fully specify the noise model, allowing for the separation of SPAM errors in dynamic circuits.
   • Applying these parameters to error mitigation schemes like Probabilistic Error Cancellation (PEC).

Key Results

• Hardware Demonstration: Experiments on an IBM Quantum Eagle device showed that standard TREX mitigation produced unphysical expectation values for global observables (e.g., $X^{\otimes n}$) on 4, 6, and 8-qubit GHZ circuits due to unaccounted state-preparation errors. The proposed "Split SPAM" method, utilizing the independently measured $f_{sp}$, corrected these values to fall within physically valid bounds and aligned with theoretical noise models.
• Simulation of Dynamic Circuits: Numerical simulations demonstrated that the proposed noise learning protocol successfully characterizes noise in dynamic circuits. Specifically, a three-qubit teleportation circuit was simulated with noise, and PEC was applied using the learned, split SPAM parameters. The mitigated state fidelities significantly exceeded unmitigated values (median >98% vs. <96%), confirming that the method effectively separates and mitigates state-preparation and measurement errors independently.
• Robustness to Reset Types: Simulations indicated that while fast qubit resets (binary classification) can lead to unphysical estimates of $p_{sp}$ due to residual $|2\rangle$ population, fast qutrit resets (discriminating $|0\rangle, |1\rangle, |2\rangle$) allow for accurate state-preparation characterization without the time penalty of slow passive resets.

Significance and Claims
The paper claims that by utilizing non-computational states, it is possible to overcome "no-go theorems" regarding noise learning that apply when the framework is restricted to the computational subspace. The primary significance lies in:

1. Ground Truth Establishment: The ability to fully constrain noise models by learning the ground truth of state-preparation errors, which was previously unlearnable.
2. Dynamic Circuit Applicability: The method enables accurate error mitigation for dynamic circuits involving mid-circuit measurements and classical feed-forward, a regime where previous self-consistent protocols were insufficient due to unconstrained gauge degrees of freedom.
3. Physical Validity: The approach prevents the generation of unphysical expectation values in error mitigation, ensuring that results remain within the bounds of physical reality.
4. Hardware Agnosticism: While demonstrated on superconducting qubits, the authors assert the method is applicable to any hardware platform where non-computational states are addressable (e.g., trapped ions, solid-state spin qutrits).

The work concludes that this approach opens the door to improved error mitigation for both final and mid-circuit measurements, facilitating more reliable utility-scale quantum experiments.