Phase estimation is known to be a robust method for single-qubit gate calibration in quantum computers, while Bayesian estimation is widely used in devising optimal methods for learning in quantum systems. We present a Bayesian method that adaptively chooses a control phase and the time of coherent evolution. The uniqueness of the approach is to use analytics to implement an efficient numerical algorithm that offers exact calculation of the Bayesian probability and associated utility functions, even in the presence of noise. We determine the utility of control parameter values using functions of the prior probability of the phase that quantify expected knowledge gain in terms of either expected narrowing of the posterior or expected information gain. We find that by maximizing the of expected gain we obtain phase estimates having standard deviation a factor of 1.43 larger than the Heisenberg limit using a classical sequential strategy for noise-free experiments. In the presence of dephasing, we find near-optimal performance with respect to the Cramér-Rao bound. We also demonstrate some robustness of the estimates to noise that is not accounted for in the model of the estimator, making the methods suitable for calibrating operations in quantum computers. The rate of gain can be adjusted to account for times in the experiment other than the coherent evolution of the unknown phase, such as times required for state preparation or readout. The combined features of the method make it well suited to single-qubit gate calibration in trapped-ion or neutral atom experiments and promises a wide range of other applications by offering near-optimal phase estimation that straightforwardly accounts for experimental imperfections.
Phase estimation is known to be a robust method for single-qubit gate calibration in quantum computers, while Bayesian estimation is widely used in devising optimal methods for learning in quantum systems. We present a Bayesian method that adaptively chooses a control phase and the time of coherent evolution. The uniqueness of the approach is to use analytics to implement an efficient numerical algorithm that offers exact calculation of the Bayesian probability and associated utility functions, even in the presence of noise. We determine the utility of control parameter values using functions of the prior probability of the phase that quantify expected knowledge gain in terms of either expected narrowing of the posterior or expected information gain. We find that by maximizing the of expected gain we obtain phase estimates having standard deviation a factor of 1.43 larger than the Heisenberg limit using a classical sequential strategy for noise-free experiments. In the presence of dephasing, we find near-optimal performance with respect to the Cramér-Rao bound. We also demonstrate some robustness of the estimates to noise that is not accounted for in the model of the estimator, making the methods suitable for calibrating operations in quantum computers. The rate of gain can be adjusted to account for times in the experiment other than the coherent evolution of the unknown phase, such as times required for state preparation or readout. The combined features of the method make it well suited to single-qubit gate calibration in trapped-ion or neutral atom experiments and promises a wide range of other applications by offering near-optimal phase estimation that straightforwardly accounts for experimental imperfections.
Accurate calibration of control parameters in quan-tum gates is crucial for high-fidelity operations, yet it represents a significant time and resource challenge, necessitating periods of downtime for quantum computers. Robust Phase Estimation (RPE) [8] has emerged as a practical and effective calibration technique aimed at tackling this challenge. It combines a provably efficient number of control pulses with a classical post-processing algorithm to estimate the phase accumulated by a quantum gate. We introduce Bayesian Robust Phase Estimation (BRPE), an in-novative approach that integrates Bayesian parameter estimation into the classical post-processing phase to reduce the sampling overhead. Our numerical analysis shows that BRPE markedly reduces phase estimation errors, requiring approximately 50% fewer samples than standard RPE. Specifically, in an ideal, noise-free setting, it achieves up to a 96% reduction in average absolute estimation error for a fixed sample cost of 88 shots when compared to RPE. Under a depolarizing noise model, it attains up to a 47% reduction for a fixed cost of 1 76 shots. Additionally, we adapt BRPE for Ramsey spectroscopy applications and successfully implement it experimentally in a trapped ion system.
Characterizing the time over which quantum coherence survives is critical for any implementation of quantum bits, memories, and sensors. The usual method for determining a quantum system’s decoherence rate involves a suite of experiments probing the entire expected range of this parameter, and extracting the resulting estimation in postprocessing. Here we present an adaptive multiparameter Bayesian approach, based on a simple analytical update rule, to estimate the key decoherence timescales ( T1 , T2∗ , and T2 ) and the corresponding decay exponent of a quantum system in real time, using information gained in preceding experiments. This approach reduces the time required to reach a given uncertainty by a factor up to an order of magnitude, depending on the specific experiment, compared to the standard protocol of curve fitting. A further speedup of a factor approximately 2 can be realized by performing our optimization with respect to sensitivity as opposed to variance. Published by the American Physical Society 2024
Bayesian estimation approaches, which are capable of combining the information of experimental data from different likelihood functions to achieve high precisions, have been widely used in phase estimation via introducing a controllable auxiliary phase. Here, we present a non-adaptive Bayesian phase estimation (BPE) algorithms with an ingenious update rule of the auxiliary phase designed via active learning. Unlike adaptive BPE algorithms, the auxiliary phase in our algorithm is determined by a pre-established update rule with simple statistical analysis of a small batch of data, instead of complex calculations in every update trails. As the number of measurements for a same amount of Bayesian updates is significantly reduced via active learning, our algorithm can work as efficient as adaptive ones and shares the advantages (such as wide dynamic range and perfect noise robustness) of non-adaptive ones. Our algorithm is of promising applications in various practical quantum sensors such as atomic clocks and quantum magnetometers.
Abstract Quantum effects like entanglement and coherent amplification can be used to drastically enhance the accuracy of quantum parameter estimation beyond classical limits. However, challenges such as decoherence and time-dependent errors hinder Heisenberg-limited amplification. We introduce Quantum Signal-Processing Phase Estimation algorithms that are robust against these challenges and achieve optimal performance as dictated by the Cramér-Rao bound. These algorithms use quantum signal transformation to decouple interdependent phase parameters into largely orthogonal ones, ensuring that time-dependent errors in one do not compromise the accuracy of learning the other. Combining provably optimal classical estimation with near-optimal quantum circuit design, our approach achieves a standard deviation accuracy of 10−4 radians for estimating unwanted swap angles in superconducting two-qubit experiments, using low-depth ( < 10) circuits. This represents up to two orders of magnitude improvement over existing methods. Theoretically and numerically, we demonstrate the optimality of our algorithm against time-dependent phase errors, observing that the variance of the time-sensitive parameter φ scales faster than the asymptotic Heisenberg scaling in the small-depth regime. Our results are rigorously validated against the quantum Fisher information, confirming our protocol’s ability to achieve unmatched precision for two-qubit gate learning.
The calibration of high-quality two-qubit entangling gates is an essential component in engineering large-scale, fault-tolerant quantum computers. However, many standard calibration techniques are based on randomized circuits that are only quadratically sensitive to calibration errors. As a result, these approaches are inefficient, requiring many experimental shots to achieve acceptable performance. In this work, we demonstrate that robust phase estimation can enable high-precision, Heisenberg-limited estimates of coherent errors in multi-qubit gates. Equipped with an efficient estimator, the calibration problem may be reduced to a simple optimization loop that minimizes the estimated coherent error. We experimentally demonstrate our calibration protocols by improving the operation of a two-qubit controlled-Z gate on a superconducting processor, and we validate the improved performance with gate set tomography. Our methods are applicable to gates in other quantum hardware platforms such as ion traps and neutral atoms, and on other multi-qubit gates, such as CNOT or iSWAP.
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary Ntot times, the variance of the estimates can scale as O(1/Ntot2), compared to the best ``classical'' strategy with O(1/Ntot). The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without inter-probe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O(1/Ntot2). Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analyses the results of previous steps.