SPAM Error: Quantum Prep & Measurement

• State Preparation and Measurement (SPAM) error is a key quantum issue where deviations in preparing states and measuring outcomes undermine the accuracy of quantum tomography and device benchmarking.
• Gauge freedom in SPAM tomography introduces undetermined parameters, with gauge-invariant metrics such as Δ(S) used to effectively detect correlated errors.
• Experimental protocols that employ multiple state preparations and diverse measurement settings leverage these diagnostics to robustly calibrate and benchmark quantum devices.

State Preparation and Measurement (SPAM) Error refers to the errors that originate from imperfections in both the preparation of quantum states and the quantum measurements used to infer system properties. Unlike gate errors, which are associated with the evolution of quantum systems via quantum operations, SPAM errors impact the initial and terminal stages of quantum algorithms and have become a dominant concern as quantum hardware improves. The treatment, detection, and mitigation of SPAM errors are central for precise quantum characterization, robust benchmarking, and the validation of quantum processors, especially as they undermine the accuracy of tomography, fidelity estimation, and error correction schemes.

1. Conceptual Basis of SPAM Error and Tomography

SPAM error arises in any quantum experiment when prepared states deviate from their intended form or when measurements fail to correctly distinguish quantum outcomes. In standard quantum state tomography, one assumes that the measurement process (i.e., the POVM elements) is perfectly known, which enables the estimation of the unknown quantum state $\rho$ via the measurement probabilities $p_i = \operatorname{Tr}(\rho \Pi_i)$. Conversely, in detector tomography, the states are assumed known, and the task is to infer POVM elements from measurement outcomes.

State Preparation and Measurement (SPAM) tomography generalizes these paradigms: neither the prepared quantum states nor the measurement apparatus are presumed known. For a $d$-dimensional system measured with $d$-outcome detectors, data is modeled as $$S_{\mu i} = \operatorname{Tr}(\rho_{(\mu)} \Sigma_i)$$, with $\rho_{(\mu)}$ unknown states and $\Sigma_i$ unknown measurement observables. The global equation $S = PW$ encapsulates all measured frequencies, with $P$ and $W$ parameterizing the unknowns in state preparations and measurement devices, respectively. The decomposition of $S$ into $P$ and $W$ exhibits intrinsic ambiguities—the so-called gauge freedom—since only their product is experimentally accessible (Jackson et al., 2015).

2. Gauge Parameters and Underdetermined Quantities

For a $d$-dimensional quantum system measured with $d$-outcome detectors, there exist $d^2 (d^2 - 1)$ gauge parameters that cannot be determined by any SPAM experiment, regardless of the number of distinct states or measurement settings. For instance, in the qubit case ($d=2$), there are 12 undetermined gauge parameters. Explicitly, the data $S$ is invariant under the simultaneous transformation $P \to PG^{-1}$, $W \to GW$, where $G$ is a $4 \times 4$ block matrix incorporating an invertible $3 \times 3$ matrix $H$ and vector $a$.

These gauge degrees of freedom represent both standard basis ambiguities and the so-called "blame gauges," which describe the freedom to reattribute imperfections between state preparation and measurement. As a result, no SPAM tomography reconstruction can assign all observed imperfections uniquely to either preparation or measurement. Instead, a family of physically equivalent solutions related by gauge transformations exists, all consistent with the measured data (Jackson et al., 2015).

3. Gauge-Invariant Quantities and Detection of Correlated Errors

To circumvent the ambiguity arising from gauge freedom, the analysis leverages gauge-invariant quantities derived from the measured data. In the two-dimensional (qubit) scenario, the measured data matrix $S$ is partitioned into four submatrices (for example, $Q_{11}, Q_{12}, Q_{21}, Q_{22}$, each $4 \times 4$). If SPAM errors between state and measurement are uncorrelated, a key relation holds: $$\Delta(S) \equiv Q_{11}^{-1} Q_{12} Q_{22}^{-1} Q_{21} = \mathbb{1}$$ A violation of $\Delta(S) = \mathbb{1}$ is a gauge-invariant indicator of correlated SPAM error. In cases where block inverses are singular, a nonsingular partial determinant $\nabla(S)$ can be constructed to assess the same property. Both $\Delta(S)$ and $\nabla(S)$ are constructed solely from direct measurements and are immune to gauge transformations, thus providing an experimentally accessible test for detecting SPAM correlations (Jackson et al., 2015).

4. Protocols and Experimental Implementation

The detection of SPAM correlations via gauge-invariant quantities forms the basis of both theoretical and experimental protocols. Experiments typically involve preparing multiple (at least five) different states and performing measurements with multiple (five or more) detector settings to form a sufficiently large data matrix. By partitioning $S$ and evaluating $\Delta(S)$ or $\nabla(S)$, experimenters can identify statistically significant deviations from the identity matrix (or zero, respectively), flagging the presence of correlations between SPAM errors. Propagation of binomial measurement uncertainty is used to assign error bars to these diagnostics.

These techniques are model-independent under the SPAM paradigm and do not require choosing fiducial states or detector settings to "fix the gauge." Instead, the measurements themselves serve as a self-consistent, direct probe of correlated error structure.

5. Implications for Quantum Device Characterization

Detecting and understanding correlated SPAM errors is essential for the reliable characterization and operation of quantum information devices. Correlations between state preparation and measurement errors undermine the validity of traditional tomography protocols and can lead to over- or underestimation of quantum process fidelities. In particular, unaccounted SPAM correlations can impact fault-tolerance thresholds, yielding misleading conclusions about the true error rates of quantum gates or processes. The gauge-invariant procedures enable robust diagnosis, and their integration into gate-set tomography or device calibration routines strengthens error assignment and fidelity benchmarking for both state preparation and measurement modules (Jackson et al., 2015).

6. Extensions and Future Directions

Several directions are identified for extending the theoretical and practical scope of SPAM error characterization:

• Higher-dimensional systems and multi-qubit devices: Generalization of gauge-invariant diagnostics to $d>2$ systems and composite multi-qubit architectures.
• Automated model selection: Development of automated statistical tests that utilize partial determinants to discriminate between models with uncorrelated and correlated SPAM errors.
• Generalized detection strategies: Incorporation of multi-outcome detector structures and continuously parametrized state or measurement settings.
• Integrated device characterization: Embedding SPAM diagnostics into operational protocols for scalable, fault-tolerant quantum computing, and refining error propagation and noise modeling for realistic experimental environments.

Such extensions are crucial for advancing robust quantum verification and error mitigation in both near-term and large-scale quantum technology development.

---

In summary, SPAM error analysis reveals fundamental gauge ambiguities in the simultaneous recovery of quantum state and measurement properties. The identification and measurement of gauge-invariant quantities provide a principled route to diagnosing correlated SPAM errors, with direct implications for quantum tomography, device certification, and the calibration of quantum hardware (Jackson et al., 2015).