QuBound: Quantum Performance Bounds

• QuBound is a workflow that efficiently predicts computational performance bounds for quantum circuits by combining statistical decomposition with an LSTM-based encoder.
• It leverages historical noise and performance traces to separate deterministic trends from stochastic shot noise, yielding statistically sound confidence intervals.
• The method achieves over six orders of magnitude speedup and produces much tighter bounds compared to traditional noisy simulation and analytical prediction techniques.

QuBound is a workflow for efficient, accurate prediction of computational performance bounds for quantum circuits operating under dynamically fluctuating device noise. The methodology addresses a crucial need in quantum-centric supercomputing environments, where run-to-run device repeatability and correct functional performance are challenged by noise processes that fluctuate over short and long timescales. By leveraging historical performance and noise traces, QuBound derives statistically grounded, data-driven performance bounds, incorporating a neural sequence encoder (LSTM) to model device- and circuit-specific behaviors. Compared to existing learning-based and analytical prediction techniques, QuBound attains higher predictive compliance, significantly tighter bound ranges, and over six orders of magnitude speedup relative to full noisy simulation, making it suitable for scheduling, compilation, and management within next-generation quantum computing infrastructures (Li et al., 22 Jul 2025).

1. Workflow Architecture: Performance Decomposition and LSTM-based Prediction

QuBound consists of two principal stages: QuDECOM for performance decomposition and QuPRED for predictive modeling via a neural sequence encoder.

• QuDECOM (Performance Decomposition): Historical measurement traces (trP) and noise records (trN) are processed through time-series decomposition techniques (e.g., moving average or seasonal-trend decomposition) to isolate the trend component associated with device-related noise (including T₁/T₂ drift and time-correlated gate error rates) from the residual component representing stochastic measurement noise due to finite sampling (shots). Outliers in performance traces are removed using the interquartile range method, and the residual is then characterized statistically. Assuming approximate normality, lower and upper confidence limits (with confidence level CL) for the residual are computed as

$$L = \bar{X} - Z(CL)\cdot \sqrt{\sigma^2/n}, \quad U = \bar{X} + Z(CL)\cdot \sqrt{\sigma^2/n}$$

where $\bar{X}$ is the mean, $\sigma^2$ the variance, $n$ the effective sample size, and $Z(CL)$ the standard normal quantile corresponding to CL.

• QuPRED (ML-based Performance Predictor): A novel encoder embeds each stage of the quantum circuit and its noise context as a vector. For single-qubit gates, the vector consists of a 5-tuple: $$(\text{gate type}, \text{gate parameter}, T_1, T_2, \text{error rate})$$; for two-qubit gates, both qubits’ parameters are included. The sequence of such vectors forms $L_{(C,N)} = \{V_0, V_1, \dotsc, V_{m-1}\}$, representing the circuit’s operations and the qubit-wise noise context.

This sequence is fed into a Long Short-Term Memory (LSTM) network, which models temporal correlations and cross-stage error accumulation. The LSTM thus learns long-range dependencies between operations and the time-varying noise, predicting the (trend) central performance, which is then combined with the residual’s confidence interval from QuDECOM to yield final lower and upper bounds.

2. LSTM Encoder and Circuit-Noise Representation

Quantum circuits are inherently sequential, with each operation’s effect dependent on both circuit history and noise fluctuations. The QuPRED encoder produces a length-$m$ sequence, where $m$ is the number of primitive gate operations, and each vector $V_\ell$ encodes all relevant gate and noise parameters for that layer.

For a $n$-qubit system, each $V_\ell$ might be structured as:

• Gate type (encoded categorically)
• Gate parameter (e.g., rotation angle for $R_y(\theta)$; 0 for Clifford gates)
• $T_1$ and $T_2$ relaxation times for the participating qubits
• Gate error rates for both single- and two-qubit gates

The LSTM, known for modeling nontrivial temporal dependencies, maps $(V_0, V_1, ..., V_{m-1}) \mapsto$ predicted performance central value, robustly reflecting the cumulative effects of noise in the presence of arbitrary ordering and heterogeneity of gate types.

3. Statistical Bounds, Correction, and Validation

QuBound computes bounds as the sum of the central trend (from the LSTM) and statistically derived confidence intervals over the shot noise residual. Notably, the workflow distinguishes trend (device drift/noise) from the residual (intrinsic randomness of quantum sampling), yielding tighter and more realistic bounds compared to direct prediction or simulation. Table 1 summarizes the bound construction details.

Stage	Component	Method/Fml./Stat. Basis
QuDECOM	Trend	Moving average / additive decomposition
QuDECOM	Residual (Shot Noise)	Residual = trP – Trend, normal model, outlier removal
QuDECOM	Bound calculation	$L, U$ by normal mean CI over residuals, eqns above
QuPRED	Central prediction	Seq. circuit-noise LSTM encoder
QuBound	Final bounds	Central $\pm$ CI (from above)

Empirical evaluation on a variety of circuit types (GHZ, random benchmark [RB], variational quantum eigensolver [VQE]) using IBMQ devices (ibmq_mumbai, ibm_nairobi) and simulated traces demonstrates:

• All central predictions from QuBound lie within the bounds produced by decomposition
• Bound-compliance rates (BCR; proportion of measurement results within bounds) approach or equal 1.0, far surpassing simpler ML predictors (QUEST, linear regression)
• Bound ranges are over 10× narrower than best existing analytical approaches and produced with over $10^6$ speedup compared to noisy simulation

4. Mathematical Formulations and Performance Metrics

Crucial mathematical relationships from the QuBound framework include:

• Performance seen as a function of both circuit and noise:

$P_t = f(C, N_t)$

where $C$ is the circuit, $N_t$ the noise at time $t$.

• Empirical performance trace:

$\text{trP} = \{P_t | t \in [t_0, t_f]\}$

• Fidelity between measured and ideal output distributions:

$$F = \left(\sum_i \sqrt{p_{\text{ideal}, i} \cdot p_{\text{observed}, i}}\right)^2$$

• Bound range for sample $s$:

$$\text{Range}_{s, CL} = P_{\text{up}, s, CL} – P_{\text{low}, s, CL}$$

• Confidence interval over the residual:

$L = \bar{X} - Z(CL)\cdot \sqrt{D(X)/n}$

$U = \bar{X} + Z(CL)\cdot \sqrt{D(X)/n}$

where $D(X)$ is the sample variance of the residuals.

• Bound-compliance rate (BCR) measures fraction of observed samples $P_t$ such that $L \leq P_t \leq U$.

5. Empirical Results, Benchmarking, and Range Tightness

Experimental comparison was performed against state-of-the-art learning-based predictors (QUEST), linear regression, noisy simulation, and analytical bound estimators. For a wide range of circuits and backends, QuBound yields:

• Microsecond-scale bound prediction, with over $10^6$ speedup versus simulation (e.g., 15-qubit GHZ circuit prediction time of 50 $\mu$s versus 47 minutes for simulation)
• Bound ranges that are consistently much smaller (often 10$\times$ or more) than analytical estimates, yet with higher empirical coverage
• Central prediction values that not only lie within the bounds but consistently improve upon the best known learning-based predictions in close agreement with noisy simulation results

6. Practical Implications and Future Research

QuBound provides an operationally meaningful metric for quantum system management, including:

• Job Scheduling: Schedulers can pre-allocate jobs to devices with the predicted highest minimum fidelity, prioritizing performance over mere resource utilization.
• Compiler Transpilation: Transpiler strategies (e.g., qubit mapping or routing) can be tuned for maximal predicted performance under current drift and gate error parameters, rather than relying on worst-case bounds or long-term averages.
• Benchmarking and Reproducibility: By embedding both static circuit and temporally variable noise parameters, QuBound enables performance forecast and reproducibility studies across both simulated and real hardware.
• Proxy Circuit Matching: For unseen circuits, possible future improvements include proxy-based approaches to select historical traces of similar circuits based on explicit circuit structure, depth, and coupling, in order to transfer credible bounds.

Further potential exists for integrating richer hardware metrics into the encoder and for online, continual learning to adapt to hardware recalibration and fast drift profiles.

7. Limitations and Extension Prospects

While QuBound attains high accuracy and rapid inference for most tested circuits, limitations exist:

• Generalization to entirely novel circuit structures not observed in the training traces is ultimately limited by the diversity of the trace database and the representational power of the encoder.
• Application to circuits of highly irregular topology or gates with noise features not included in the existing 5-tuple representation may require encoder modification, possibly moving toward more elaborate structural (e.g., graph neural network) models.
• The methodology currently assumes the validity of normality in the residual for the confidence interval step; significant non-Gaussian behavior in shot noise or drift processes could require robust or nonparametric alternatives.

Enhancing circuit-noise encoding, incorporating more granular noise models, and online adaptation remain open research directions likely to further extend QuBound’s applicability in quantum-centric supercomputing contexts.

---

In summary, QuBound is a workflow comprising statistical performance decomposition and an LSTM-based sequence encoder to predict rapid, tight, and statistically credible performance bounds for noisy quantum circuits across diverse hardware and programming contexts. Its combination of statistical rigor, scalability, and fidelity awareness addresses core demands of emerging quantum system management and application workflows (Li et al., 22 Jul 2025).