Robust Phase Estimation (RPE)

Updated 1 January 2026

Robust Phase Estimation (RPE) is a set of protocols that achieve high-precision, resource-efficient phase determination with proven robustness against additive SPAM errors.
It utilizes geometric sequence design and arctangent post-processing to unwrap phase estimates, maintaining Heisenberg-limited scaling under bounded errors.
Recent extensions integrate Bayesian inference and adaptive smoothing to enhance performance in multidimensional and real-time phase tracking across quantum and classical applications.

Robust Phase Estimation (RPE) is a class of quantum phase estimation protocols and classical robust estimation methodologies developed for high-precision, resource-efficient, and error-tolerant phase determination in quantum information processing, optical metrology, interferometry, and signal analysis. RPE protocols are distinguished by their immunity to additive errors in state preparation and measurement (SPAM), provable Heisenberg-limited scaling, and general applicability across hardware platforms and estimation contexts. Originating in the quantum calibration literature, RPE now encompasses extensions to robust smoothing, multidimensional phase-tracking, and real-time adaptive phase estimation.

1. Theoretical Foundations and Robustness Guarantees

Core RPE protocols, as introduced by Kimmel, Low, and Yoder, combine non-adaptive, geometric-sequence-based experiment design with interval-nesting, arctangent post-processing, and explicit error propagation bounds (Kimmel et al., 2015). The signature feature is robustness to bounded additive errors in measurement and preparation—notably, the procedure provably maintains correctness for all SPAM/gate errors $|\delta|<1/\sqrt{8} \approx 0.354$ at every step.

This robustness arises from the analysis of observed probabilities as

$p_0(A, k) = \frac{1}{2}[1+\cos(kA)] + \delta_0(k), \quad p_+(A, k) = \frac{1}{2}[1+\sin(kA)] + \delta_+(k),$

and from geometric interval-shrinking, where the phase estimate at round $j$ lies within an interval of width $\pi/2^j$ . Correctness and Heisenberg-limited scaling are maintained as long as the additive error at every $k$ satisfies $|\delta_0(k)|,\, |\delta_+(k)| < 1/\sqrt{8}$ , even in the presence of substantial SPAM or coherent error (Kimmel et al., 2015, Meier et al., 2019).

2. Protocol Architecture and Sequence Design

The canonical RPE protocol comprises a sequence of rounds indexed by $k=1,\ldots, K$ , with each round corresponding to $N_k = 2^{k-1}$ applications of the gate under calibration. Each round utilizes two experiments:

Prepare an initial quantum state, apply the gate $N_k$ times, measure in a basis (e.g., $Z$ ).
Prepare an orthogonal state, repeat as above, but measure in a different basis (e.g., $X$ or $Y$ ).

The outcome frequencies yield empirical estimators for $\cos(N_k \theta)$ and $\sin(N_k \theta)$ , from which a “wrapped” estimate of $N_k \theta$ is obtained: $\phi_k = \arctan2(2p_+(N_k) - 1,\, 2p_0(N_k) - 1).$ The inferred phase is then unwrapped and refined using interval-nesting, ensuring that the estimate at each step remains consistent with the previous estimate within $\pm \pi/N_k$ (Rudinger et al., 2017, Meier et al., 2019).

Extensions generalize this approach to multi-qubit gates by using suitable pairs of input states and measurement projections (Russo et al., 2020, Rudinger et al., 10 Feb 2025).

3. Error Models, Self-Consistency, and Failure Detection

RPE’s performance hinges on the additive error in the projected probabilities. Experimentally relevant sources include:

Measurement error (e.g., photon-counting misclassification): bounded by the threshold for “bright/dark” distinction.
State-preparation error: imperfect initialization maps directly to a constant error term.
Gate error (e.g., dephasing, amplitude errors): manifests as a length-dependent error term.

Failure occurs if any $|\delta(k)|$ exceeds the $1/\sqrt{8}$ bound at some round. Advanced RPE protocols include internal, data-driven self-consistency checks to flag such hidden failure modes. These checks, as formalized by Russo et al., include:

Plausible interval intersection.
Consecutive and local consistency intervals.
Uniform-local and angular-historical intersections.
Probability-historical checks, which compare predicted vs. measured sinusoidal statistics (Russo et al., 2020).

These checks allow RPE-based calibration routines to self-terminate at the last generation before a threshold violation, preserving the Heisenberg-scaling error guarantee.

4. Algorithmic Scalings and Numerical Performance

RPE achieves Heisenberg-limited precision: for gate time budget $T = O(2^K)$ , the estimator’s standard deviation satisfies $\sigma(\hat{\theta}) = O(1/T)$ . Sample complexity to reach error $\epsilon$ is $O(1/\epsilon)$ in aggregate gate applications, with total shot complexity scaling as $O(M \log(1/\epsilon))$ for $M$ repeated shots per circuit (Kimmel et al., 2015, Rudinger et al., 2017, Hurant et al., 2024).

Typical laboratory implementations report:

Absolute uncertainties $<4\times 10^{-4}$ radians using as few as $176$ total samples per phase (Rudinger et al., 2017).
Fractional error $<1\%$ with $O(100)$ gate applications, much fewer than full tomography or Rabi scans (Meier et al., 2019).
Robustness to coherent SPAM errors (leakage up to $13\%$ and subspace mixing up to $5\%$ ) (Russo et al., 2020).
Substantial reductions in required samples (up to $96\%$ under ideal conditions) using Bayesian post-processing (BRPE) (Hurant et al., 2024).

5. Extensions: Bayesian, Smoothing-Based, and Multidimensional Approaches

Bayesian Robust Phase Estimation (BRPE)

Recent work integrates Bayesian inference into the post-processing, leading to Bayesian Robust Phase Estimation. BRPE uses the same experiments as standard RPE but sequentially updates the posterior using experiment-specific likelihoods, extracting maximum a posteriori phase estimates and confidence metrics. Empirical analyses demonstrate:

Up to $96\%$ reduction in average estimation error at fixed low sample count under ideal (noise-free) settings.
Retained improvement ( $\sim$ 47\%) under realistic depolarizing noise and SPAM models.
Empirical scaling laws for standard deviation as a function of number of shots, with performance converging to (or slightly surpassed by) standard RPE at large sample sizes (Hurant et al., 2024).

Robust Smoothing in Optical Phase Tracking

In continuous and quantum-optical phase estimation, robust fixed-interval smoothers explicitly hedge against parameter uncertainty (coupling rates, measurement gains) by solving coupled Riccati inequalities or IQC-constrained minimax problems (Roy et al., 2013, Roy et al., 2013, Roy et al., 2013, Roy et al., 2013). These methods guarantee bounded mean-squared error (MSE) under norm-bounded parameter perturbations and outperform nominal Kalman/RTS schemes in the presence of modeling uncertainties.

Multidimensional and Real-Time RPE Algorithms

Generalizations of RPE have been developed for phase assignment in multidimensional, pseudo-periodic signals (e.g., dynamical trajectories, motion-capture, biosignals). The ROPE algorithm, for instance, segments the signal into cycles using data-driven delimiters, then robustly matches current observation to stored cycles for phase assignment. Empirical benchmarks on chaotic, biomechanical, and physiological datasets demonstrate sub-radian phase error and robustness to noise/drift, outperforming PCA-based and Hilbert-transformed approaches (Spallone et al., 5 Sep 2025).

6. Applications Across Quantum and Classical Domains

RPE protocols are now central in:

Quantum gate calibration and control parameter tuning (single- and multi-qubit) (Kimmel et al., 2015, Rudinger et al., 2017, Rudinger et al., 10 Feb 2025, Hurant et al., 2024).
Certification of energy differences in quantum Hamiltonian simulation and chemistry (robust, ancilla-free energy gap estimation) (Russo et al., 2020).
Optical metrology (robust phase, amplitude, and squeezing estimation) (Roy et al., 2013, Roy et al., 2013).
Robust phase-offset estimation in network synchronization and delay attack contexts using EM-detection and shift-invariant vector-location estimators (Karthik et al., 2016).
Signal processing, interferometric phase recovery, and biologically plausible real-time rhythm analysis (Flores et al., 2019, Spallone et al., 5 Sep 2025).

RPE’s error tolerance and sample-efficiency support its deployment in automated calibration routines, SPAM-prone environments, and hardware-adaptive feedback contexts.

7. Practical Considerations, Limitations, and Future Directions

Effective deployment of RPE requires bounding worst-case additive errors, judicious selection of the maximum sequence length $L$ , and tuning the number of samples per sequence in accordance with the robustness margin. Failure diagnostics via self-consistency checks are essential to avoid overestimating achievable precision under unmodeled noise.

Current limitations of RPE include:

Sensitivity thresholds: performance degrades sharply if additive errors exceed $1/\sqrt{8}$ in any sequence (Meier et al., 2019).
Fixed-parameter focus: generalization to multi-parameter or time-varying gate calibration is straightforward in principle but unoptimized in practice (Kimmel et al., 2015, Meier et al., 2019).
The necessity of offline experiment design (except in fully Bayesian or adaptive variants) (Hurant et al., 2024).
Lack of fully general theoretical sample-complexity bounds for advanced (Bayesian/adaptive) extensions.

Open directions comprise rigorous convergence theory for multidimensional robust phase tracking, advanced Bayesian adaptive designs, extensions to nontrivial gate sets and high-dimensional systems, and tighter characterization of performance under correlated or adversarial noise models.

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