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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Noise resilience of Bayesian quantum phase estimation tested on a Si quantum photonic chip

Antonio A. Gentile, Stefano Paesani, Raffaele Santagati, Jianwei Wang, Nathan Wiebe, David P. Tew, Jeremy L. O’Brien, Mark G. Thompson

Published Jun 1, 2017• 2 citations
Publisher

Abstract

Summary form only given. Quantum Phase Estimation (PE) is a fundamental building block in the framework of Quantum Computing. Accurate estimation of the true eigenphase φ0of a known eigenstate |φ〉 is fundamental for the implementation of many promising quantum algorithms. The interest in PE is also due to the modest quantum hardware requirements of the Iterative Phase Estimation Algorithm (IPEA) implementation [1], successfully implemented on small-scale devices [2]. However, IPEA makes hard decisions at each step of the algorithm, relying on majority voting schemes for its robustness against noise. It has been observed how non-error-corrected machines may enter a regime where error-rates for IPEA diverge quickly, making this approach impractical [3]. New Bayesian approaches, such as the Rejection Filtering Phase Estimation (RFPE), have been recently proposed to overcome these limitations [4]. Here, we report the implementation of RFPE in a photonic device embedding state-of-art integrated elements. Spontaneous Four Wave Mixing sources generate photon pairs. State preparation and evolution are accomplished via integrated Mach-Zehnder interferometers and thermo-optical phase shifters operating in the hundreds of kHz regime. Fast reconfigurability is key for RFPE, as the algorithm adaptively chooses the parameters for each experiment. Arbitrary CU(2) controlled operations are made possible by an entanglement-based scheme, integrated on this chip for the first time.

DOI
10.1109/cleoe-eqec.2017.8087408

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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 $N_{\mathrm{tot}}$ times, the variance of the estimates can scale as $O(1/{N^2_{\mathrm{tot}}})$, compared to the best ``classical'' strategy with $O(1/{N_{\mathrm{tot}}})$. 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 $O(1/{N^2_{\mathrm{tot}}})$. 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.

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Bayesian quantum phase estimation beyond the eigenstate approximation: sample complexity, Cramér–Rao bounds, and adaptive measurement design for non-eigenstate and mixed-state inputs on NISQ devices

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Background Quantum phase estimation (QPE) is a foundational subroutine in quantum algorithms that ranges from Shor’s factoring scheme to variational quantum eigensolvers and quantum metrology protocols. Classical post-processing replacements for the inverse quantum Fourier transform (iQFT) —specifically Kitaev-style iterative estimation and Bayesian update schemes—substantially reduce circuit depth, making them practical for noisy intermediate-scale quantum (NISQ) hardware. Despite this promise, the statistical performance of these methods under the conditions that actually arise in practice—non-eigenstate inputs characterised by eigenstate overlap p ≡ O = | ψ | ϕ | 2 < 1 and mixed-state purity γ ≡ P = Tr ( ρ 2 ) < 1 —remains uncharacterised in the literature. Methods We develop a rigorous information-theoretic framework for Bayesian QPE (BQPE) beyond the eigenstate assumption. Starting from the binary measurement likelihood attenuated by the joint factor O P , we derive exact classical Fisher information expressions and apply the Cramér–Rao inequality to obtain tight variance lower bounds. We then construct an adaptive measurement protocol based on von Mises posterior updates, prove its asymptotic efficiency in the frequentist sense, and establish matching upper and lower bounds on sample complexity. All theoretical results are validated against 2,000 -trial Monte Carlo simulations and cross-checked against published nitrogen-vacancy (NV) centre and photonic experimental data. Results We prove that the fundamental variance floor for any unbiased frequentist estimator is ( 1 − p γ ) / ( N p γ n ̄ 2 ) 1 / ( N ( p γ ) 2 n ̄ 2 ) , where N

Frontiers in Quantum Science and Technology•Aug 11, 2026•0 citations
Summary form only given. Quantum Phase Estimation (PE) is a fundamental building block in the framework of Quantum Computing. Accurate estimation of the true eigenphase φ0of a known eigenstate |φ〉 is fundamental for the implementation of many promising quantum algorithms. The interest in PE is also due to the modest quantum hardware requirements of the Iterative Phase Estimation Algorithm (IPEA) implementation [1], successfully implemented on small-scale devices [2]. However, IPEA makes hard decisions at each step of the algorithm, relying on majority voting schemes for its robustness against noise. It has been observed how non-error-corrected machines may enter a regime where error-rates for IPEA diverge quickly, making this approach impractical [3]. New Bayesian approaches, such as the Rejection Filtering Phase Estimation (RFPE), have been recently proposed to overcome these limitations [4]. Here, we report the implementation of RFPE in a photonic device embedding state-of-art integrated elements. Spontaneous Four Wave Mixing sources generate photon pairs. State preparation and evolution are accomplished via integrated Mach-Zehnder interferometers and thermo-optical phase shifters operating in the hundreds of kHz regime. Fast reconfigurability is key for RFPE, as the algorithm adaptively chooses the parameters for each experiment. Arbitrary CU(2) controlled operations are made possible by an entanglement-based scheme, integrated on this chip for the first time.
Quantum phase estimation is a fundamental subroutine in many quantum algorithms, including Shor's factorization algorithm and quantum simulation. However, so far results have cast doubt on its practicability for near-term, nonfault tolerant, quantum devices. Here we report experimental results demonstrating that this intuition need not be true. We implement a recently proposed adaptive Bayesian approach to quantum phase estimation and use it to simulate molecular energies on a silicon quantum photonic device. The approach is verified to be well suited for prethreshold quantum processors by investigating its superior robustness to noise and decoherence compared to the iterative phase estimation algorithm. This shows a promising route to unlock the power of quantum phase estimation much sooner than previously believed.
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 NtotN_{\mathrm{tot}}Ntot​ times, the variance of the estimates can scale as O(1/Ntot2)O(1/{N^2_{\mathrm{tot}}})O(1/Ntot2​), compared to the best ``classical'' strategy with O(1/Ntot)O(1/{N_{\mathrm{tot}}})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)O(1/{N^2_{\mathrm{tot}}})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.
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 N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . 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 interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . 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 analizes the results of previous steps. Published by the American Physical Society 2024
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.
Abstract We introduce a new statistical and variational approach to the phase estimation algorithm (PEA). Unlike the traditional and iterative PEAs which return only an eigenphase estimate, the proposed method can determine any unknown eigenstate–eigenphase pair from a given unitary matrix utilizing a simplified version of the hardware intended for the iterative PEA (IPEA). This is achieved by treating the probabilistic output of an IPEA-like circuit as an eigenstate–eigenphase proximity metric, using this metric to estimate the proximity of the input state and input phase to the nearest eigenstate–eigenphase pair and approaching this pair via a variational process on the input state and phase. This method may search over the entire computational space, or can efficiently search for eigenphases (eigenstates) within some specified range (directions), allowing those with some prior knowledge of their system to search for particular solutions. We show the simulation results of the method with the Qiskit package on the IBM Q platform and on a local computer.
Background Quantum phase estimation (QPE) is a foundational subroutine in quantum algorithms that ranges from Shor’s factoring scheme to variational quantum eigensolvers and quantum metrology protocols. Classical post-processing replacements for the inverse quantum Fourier transform (iQFT) —specifically Kitaev-style iterative estimation and Bayesian update schemes—substantially reduce circuit depth, making them practical for noisy intermediate-scale quantum (NISQ) hardware. Despite this promise, the statistical performance of these methods under the conditions that actually arise in practice—non-eigenstate inputs characterised by eigenstate overlap p ≡ O = | ψ | ϕ | 2 < 1 and mixed-state purity γ ≡ P = Tr ( ρ 2 ) < 1 —remains uncharacterised in the literature. Methods We develop a rigorous information-theoretic framework for Bayesian QPE (BQPE) beyond the eigenstate assumption. Starting from the binary measurement likelihood attenuated by the joint factor O P , we derive exact classical Fisher information expressions and apply the Cramér–Rao inequality to obtain tight variance lower bounds. We then construct an adaptive measurement protocol based on von Mises posterior updates, prove its asymptotic efficiency in the frequentist sense, and establish matching upper and lower bounds on sample complexity. All theoretical results are validated against 2,000 -trial Monte Carlo simulations and cross-checked against published nitrogen-vacancy (NV) centre and photonic experimental data. Results We prove that the fundamental variance floor for any unbiased frequentist estimator is ( 1 − p γ ) / ( N p γ n ̄ 2 ) 1 / ( N ( p γ ) 2 n ̄ 2 ) , where N