SPAQ: Quantum Calibration Analysis Framework

• SPAQ is a framework that models quantum calibration as a stochastic process, analyzing reliability and availability of calibration stacks.
• The framework uses nonparametric, binomial-based statistics to evaluate temporal properties and quantify time-to-failure with confidence intervals.
• Simulation studies demonstrate that adjusting calibration schedules based on SPAQ's analysis can significantly improve system availability and detect hidden dependencies.

SPA for Quantum Calibration (SPAQ) is a statistical-model-checking–based framework for analyzing and tuning quantum device calibration procedures, with a particular focus on directed acyclic graph–based calibration schemes such as Google’s Optimus. It extends SPA for Processor Analysis from classical processors to quantum calibration by treating a calibration stack as a stochastic system that generates execution traces, then evaluating temporal and probabilistic properties of those traces with explicit confidence levels and proportions. In this formulation, calibration is not only a control problem but also a reliability and availability problem: time-to-failure, hidden node dependencies, parameter thresholds, and calibration-induced downtime become analyzable objects rather than heuristic concerns (Mazurek et al., 16 Jul 2025).

1. Conceptual scope and historical placement

SPAQ is motivated by a specific systems-level difficulty in quantum hardware: calibration procedures act on analog control parameters that drift, interdependencies between calibrations accumulate as devices scale, and calibration itself consumes uptime. The framework is designed for settings in which parameters are affected by behavior that is described as non-linear, non-Gaussian, and non-Markovian, making simple Gaussian or low-order parametric assumptions unreliable for operational analysis (Mazurek et al., 16 Jul 2025).

The framework inherits its basic philosophy from SPA, but changes the object of analysis. In SPA, the system of interest is a classical processor and its workloads; in SPAQ, the system is a quantum calibration procedure running on hardware or on a simulator. The primary objects are node-level calibration events such as check_state, check_data, and calibrate, together with their timestamps, parameter values, and pass/fail outcomes. This shift makes the calibration schedule itself the subject of formal analysis rather than merely the container for lower-level estimation routines (Mazurek et al., 16 Jul 2025).

A central implication is that SPAQ is not itself a gate-characterization protocol, a tomography method, or an optimizer for a single pulse. It is an evaluation and tuning framework for calibration stacks. Related work situates this role more clearly. QubiC organizes automated calibration as reusable protocol nodes defined by an experiment template, analysis model, loss function, and optimizer (Xu et al., 2021). Self-consistent gate-set calibration constructs linearly informative sequences and iteratively tunes coherent error generators without fully trusted references (Cerfontaine et al., 2019). SPAM-aware process tomography adds explicit SPAM calibration data and gauge regularization to process reconstruction (Blume-Kohout et al., 2024). Blind calibration infers calibration parameters and quantum states jointly from simple tomographic data (Jeanette et al., 9 Jan 2025). Fast-feedback IOC/DOC protocols use shot-by-shot or event-driven updates for in-situ drift mitigation, including during quantum error correction (Magann et al., 8 Dec 2025). This suggests SPAQ occupies a supervisory layer above such primitives, providing statistically rigorous analysis of when, how often, and in what dependency structure they should be invoked.

2. Statistical model checking formulation

SPAQ models a calibration run as a stochastic process that produces traces

$\tau = (e_0, e_1, \ldots, e_n),$

where each event encodes node identity, operation type, outcome, timestamp, and relevant parameters. A property $\varphi$ is a predicate over traces or trace fragments. Each sampled execution yields a Bernoulli variable

$X_i(\varphi) \in \{0,1\},$

and the key unknown is the satisfaction probability

$F = \Pr[X_i(\varphi)=1].$

SPAQ refers to $F$ as the proportion of runs satisfying the property (Mazurek et al., 16 Jul 2025).

The framework supports both hypothesis testing and confidence-interval estimation. Typical questions include whether a node’s time-to-failure exceeds a threshold in at least some fraction of cases, or whether one node fails within a bounded horizon after another node’s calibration or failure with probability above a specified level. Confidence level $C$ and target proportion $F$ are user-set quantities; the 5th and 95th percentiles are treated as especially useful operational summaries for lower and upper lifetime bounds (Mazurek et al., 16 Jul 2025).

The framework uses temporal-logic-style properties without fixing a single logic syntax. Examples described for SPAQ include statements of the form:

• $\mathrm{MTTF} > C$
• $A > \mathrm{performance} > B$
• “event1 occurs $$\rightarrow \Pr[\text{event2 occurs within } C \text{ cycles}] \le \text{threshold}$$”

For binary properties, SPAQ uses nonparametric, binomial-based statistics. The Clopper–Pearson construction provides exact confidence bounds on $\varphi$0:

$\varphi$1

where $\varphi$2 is the number of trials, $\varphi$3 the number of satisfying trials, and $\varphi$4 the binomial CDF (Mazurek et al., 16 Jul 2025). This avoids distributional assumptions about the underlying calibration observables. The framework is explicitly motivated by empirical settings in which time-to-failure histograms can be heavily skewed.

3. DAG-based calibration model

SPAQ is developed around DAG-based calibration schemes, especially Optimus. In this model, nodes are calibration experiments and edges represent dependencies. Each node exposes three operations.

Operation	Function	Typical role
check_state	Cheap timeout-based check	Decide whether a deeper check is needed
check_data	Faster in-spec / out-of-spec experiment	Detect failure of current calibration
calibrate	More expensive optimization or reset	Restore node to optimal parameter values

When a node’s check_data fails, Optimus recursively checks and recalibrates dependencies in depth-first order before recalibrating the failing node itself. SPAQ ingests logs from exactly this kind of workflow and treats node events as the raw material for statistical analysis (Mazurek et al., 16 Jul 2025).

This representation supports several kinds of system-level quantities. For an individual node, SPAQ can estimate time-to-failure, check/calibration cost, and the effect of parameter jumps. For the graph as a whole, it can examine co-failure patterns, hidden edges, and availability penalties induced by particular timeout schedules or dependency structures. Availability is defined as

$\varphi$5

This moves calibration engineering from static hand-tuning toward explicit, testable scheduling policies (Mazurek et al., 16 Jul 2025).

The simulator used to demonstrate SPAQ instantiates this graph model for a trapped-ion single-qubit $\varphi$6 gate and associated dependencies. The implemented nodes include an $\varphi$7 gate node, a drive-frequency node, a pulse-time node, a state-initialization node, and generic nodes A and B. Parameters drift according to a logistic-like function, and calibration resets them to optimal values with added noise. Simulations were run for 100 independent runs of 100,000 cycles each (Mazurek et al., 16 Jul 2025).

4. Operational properties, inference targets, and case studies

The most important SPAQ observables are time-to-failure, availability, cross-node failure correlations, and parameter-threshold effects. Time-to-failure is defined as the interval between a successful calibration event and the next failure event of the same node. If $\varphi$8 denotes this random variable, the 5th percentile $\varphi$9 satisfies

$X_i(\varphi) \in \{0,1\},$0

SPAQ uses order-statistics and binomial reasoning to produce confidence intervals for such quantiles (Mazurek et al., 16 Jul 2025).

Cross-node properties are encoded as bounded-horizon cause–effect statements. A typical hidden-dependency query asks whether node $X_i(\varphi) \in \{0,1\},$1 fails within $X_i(\varphi) \in \{0,1\},$2 cycles of node $X_i(\varphi) \in \{0,1\},$3 with probability greater than $X_i(\varphi) \in \{0,1\},$4 at $X_i(\varphi) \in \{0,1\},$5 confidence. A threshold query asks whether a parameter shift in node A larger than $X_i(\varphi) \in \{0,1\},$6 leads to failure of node B at its next check with probability at least $X_i(\varphi) \in \{0,1\},$7 at $X_i(\varphi) \in \{0,1\},$8 confidence. These are operationalized by extracting trace fragments, turning them into Bernoulli trials, and then applying exact binomial inference (Mazurek et al., 16 Jul 2025).

Three demonstration studies summarize the intended use pattern.

Study	SPAQ finding	Reported effect
Delayed checks	5th-percentile TTF bounds support postponing early checks	Availability improved from 76.2% to 84.1%
Inter-node compensation	Large changes in A predict B failure	Availability improved from 92.2% to 93.4%
Hidden connections	Correlated failures reveal missing dependency	Availability improved from 80.6% to 81.4%

In the delayed-check study, a high-frequency run first collected fine-grained TTF data. SPAQ then estimated a 95% confidence interval for each node’s 5th-percentile TTF and used the lower bound to delay the first check_data call after calibration, since failure was unlikely in that initial interval. Calibration time itself remained similar, but check_data overhead decreased, producing a substantial availability gain (Mazurek et al., 16 Jul 2025).

In the inter-node compensation study, SPAQ showed that large changes in node A’s parameter, rather than generic A failures, were statistically associated with subsequent failure of node B. The resulting redesign merged A and B into a joint node AB. In the hidden-connection study, SPAQ evaluated pairwise bounded-horizon correlations and identified a significant relation between top_2 and bottom_2, despite their nominal independence in the DAG. Adding an edge from top_2 to bottom_2 increased the total number of checks but still improved availability, because the hidden shared failure mode was addressed sooner (Mazurek et al., 16 Jul 2025).

5. Relation to broader quantum calibration methodologies

SPAQ is complementary to, rather than competitive with, lower-level calibration and characterization techniques. QubiC demonstrates an automated superconducting-qubit workflow built from multi-dimensional loss-based optimization of single-qubit gates and a full XY-plane measurement method for two-qubit CNOT calibration, using an FPGA-based control stack and a DAG-like protocol organization (Xu et al., 2021). In such a setting, SPAQ can analyze timeout choices, node ordering, and availability consequences of that workflow.

Self-consistent calibration of quantum gate sets provides another complement. It constructs pulse sequences that probe linearly independent coherent error generators, forms a sensitivity matrix, and iteratively tunes hardware parameters by minimizing a residual vector

$X_i(\varphi) \in \{0,1\},$9

often with Levenberg–Marquardt updates (Cerfontaine et al., 2019). That procedure supplies a structured, self-consistent local calibrator; SPAQ can evaluate when and how often that calibrator should be triggered, and which sequence sets or dependency assumptions matter most at the system level.

SPAM-aware tomography methods play a similar supporting role. “Easy better quantum process tomography” augments standard QPT with a SPAM-only experiment,

$F = \Pr[X_i(\varphi)=1].$0

and a process experiment,

$F = \Pr[X_i(\varphi)=1].$1

then extracts a SPAM error superoperator and uses gauge-regularized correction of the process estimate (Blume-Kohout et al., 2024). Blind calibration, by contrast, treats the state and calibration parameters as jointly unknown and solves a bilinear inverse problem from simple tomographic data,

$F = \Pr[X_i(\varphi)=1].$2

using alternating minimization under purity and physicality constraints (Jeanette et al., 9 Jan 2025). These methods strengthen node-level diagnosis; SPAQ strengthens system-level policy selection.

Fast-feedback protocols make the connection to online control even more direct. IOC implements shot-by-shot stochastic-approximation updates using indefinite-outcome circuits, while DOC updates parameters after accumulating failures from definite-outcome circuits, including syndrome extraction in the $F = \Pr[X_i(\varphi)=1].$3 code (Magann et al., 8 Dec 2025). Autonomous drift control in semiconductor quantum-dot devices likewise uses repeated charge-stability diagrams, transition-line tracking, and feedback to stabilize operating points and extract spatially resolved noise diagnostics (Rao et al., 31 Dec 2025). A plausible implication is that SPAQ can serve as the statistical evaluation layer for such streaming controllers, determining whether their gain schedules, trigger thresholds, and graph-level interactions actually improve availability.

6. Limitations, open problems, and future directions

SPAQ’s strengths are tied to the quality and quantity of calibration-log data. The framework is nonparametric and distribution-free at the inference layer, but it still requires enough failures or threshold-crossing events to support statistically meaningful conclusions. This becomes difficult for ultra-stable nodes, where failures may be too rare to estimate quantiles or cross-node probabilities efficiently (Mazurek et al., 16 Jul 2025).

The current demonstration is simulation-based. The simulator uses a trapped-ion example with explicit node models and drift behavior, but real-device deployment requires comprehensive logging of node operations, parameters, and timing. The framework is described as lightweight on the analysis side relative to exhaustive state-space model checking, yet the dominant cost remains collecting sufficient device or simulator data. Rare-event techniques such as importance sampling are not part of the present formulation (Mazurek et al., 16 Jul 2025).

Several extensions are explicitly identified. Dynamic calibration graphs would adapt the DAG structure to workload, keeping heavily used operations tightly calibrated while pruning rarely used paths. Dynamic thresholds would continuously adjust acceptable parameter ranges and timeout schedules. Richer SMC properties and hyperproperties could express relations across multiple executions or across multiple devices. Real-device deployment is presented as straightforward in principle, with existing integration examples based on standardized event logging (Mazurek et al., 16 Jul 2025).

In the broader calibration literature, these directions align with increasingly autonomous architectures. QubiC emphasizes reusable protocol templates and closed-loop optimization (Xu et al., 2021). TERNS emphasizes real-time stabilization and noise spectroscopy in large quantum-dot arrays (Rao et al., 31 Dec 2025). Fast-feedback IOC/DOC protocols emphasize shot-level or syndrome-level parameter control (Magann et al., 8 Dec 2025). SPAQ contributes the layer that asks whether such mechanisms measurably improve system availability, reveal hidden dependencies, and justify specific scheduling or graph-structural decisions with explicit confidence guarantees.