We present mathematical and conceptual foundations for the task of robust amplitude estimation using engineered likelihood functions (ELFs), a framework introduced by Wang et al. [PRX Quantum 2, 010346 (2021)] that uses Bayesian inference to enhance the rate of information gain in quantum sampling. These ELFs, which are obtained by choosing tunable parameters in a parametrized quantum circuit to minimize the expected posterior variance of an estimated parameter, play an important role in estimating the expectation values of quantum observables. We give a thorough characterization and analysis of likelihood functions arising from certain classes of quantum circuits and combine this with the tools of Bayesian inference to give a procedure for picking optimal ELF tunable parameters. Finally, we present numerical results to demonstrate the performance of ELFs.
We present mathematical and conceptual foundations for the task of robust amplitude estimation using engineered likelihood functions (ELFs), a framework introduced by Wang et al. [PRX Quantum 2, 010346 (2021)] that uses Bayesian inference to enhance the rate of information gain in quantum sampling. These ELFs, which are obtained by choosing tunable parameters in a parametrized quantum circuit to minimize the expected posterior variance of an estimated parameter, play an important role in estimating the expectation values of quantum observables. We give a thorough characterization and analysis of likelihood functions arising from certain classes of quantum circuits and combine this with the tools of Bayesian inference to give a procedure for picking optimal ELF tunable parameters. Finally, we present numerical results to demonstrate the performance of ELFs.
We develop a framework for characterizing and analyzing engineered likelihood functions (ELFs), which play an important role in the task of estimating the expectation values of quantum observables. These ELFs are obtained by choosing tunable parameters in a parametrized quantum circuit that minimize the expected posterior variance of an estimated parameter. We derive analytical expressions for the likelihood functions arising from certain classes of quantum circuits and use these expressions to pick optimal ELF tunable parameters. Finally, we show applications of ELFs in the Bayesian inference framework.
The number of measurements demanded by hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) is prohibitively high for many problems of practical value. Quantum algorithms that reduce this cost (e.g. quantum amplitude and phase estimation) require error rates that are too low for near-term implementation. Here we propose methods that take advantage of the available quantum coherence to maximally enhance the power of sampling on noisy quantum devices, reducing measurement number and runtime compared to VQE. Our scheme derives inspiration from quantum metrology, phase estimation, and the more recent alpha-VQE proposal, arriving at a general formulation that is robust to error and does not require ancilla qubits. The central object of this method is what we call the engineered likelihood function (ELF), used for carrying out Bayesian inference. We show how the ELF formalism enhances the rate of information gain in sampling as the physical hardware transitions from the regime of noisy intermediate-scale quantum computers into that of quantum error corrected ones. This technique speeds up a central component of many quantum algorithms, with applications including chemistry, materials, finance, and beyond. Similar to VQE, we expect small-scale implementations to be realizable on today's quantum devices.
The number of measurements demanded by hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) is prohibitively high for many problems of practical value.For such problems, realizing quantum advantage will require methods that dramatically reduce this cost.Previous quantum algorithms that reduce the measurement cost (e.g., quantum amplitude and phase estimation) require error rates that are too low for near-term implementation.Here we propose methods that take advantage of the available quantum coherence to maximally enhance the power of sampling on noisy quantum devices, reducing the measurement number and runtime compared to the standard sampling method of the VQE.Our scheme derives inspiration from quantum metrology, phase estimation, and the more recent "alpha-VQE" proposal, arriving at a general formulation that is robust to error and does not require ancilla qubits.The central object of this method is what we call the "engineered likelihood function" (ELF), used for carrying out Bayesian inference.We show how the ELF formalism enhances the rate of information gain in sampling as the physical hardware transitions from the regime of noisy intermediate-scale quantum computers to that of quantum error-corrected ones.This technique speeds up a central component of many quantum algorithms, with applications including chemistry, materials, finance, and beyond.Similar to the VQE, we expect small-scale implementations to be realizable on today's quantum devices.
We establish a unified statistical framework that underscores the crucial role statistical inference plays in Quantum Amplitude Estimation (QAE), a task essential to fields ranging from chemistry to finance and machine learning. We use this framework to harness Bayesian statistics for improved measurement efficiency with rigorous interval estimates at all iterations of Iterative Quantum Amplitude Estimation. We demonstrate the resulting method, Bayesian Iterative Quantum Amplitude Estimation (BIQAE), accurately and efficiently estimates both quantum amplitudes and molecular ground-state energies to high accuracy, and show in analytic and numerical sample complexity analyses that BIQAE outperforms all other QAE approaches considered. Both rigorous mathematical proofs and numerical simulations conclusively indicate Bayesian statistics is the source of this advantage, a finding that invites further inquiry into the power of statistics to expedite the search for quantum utility.
Bayesian inference is a widely used technique for real-time characterization of quantum systems. It excels in experimental characterization in the low data regime, and when the measurements have degrees of freedom. A decisive factor for its performance is the numerical representation of the Bayesian probability distributions. In this work, we explore advanced statistical methods for this purpose, and numerically analyze their performance against the state-of-the-art in quantum parameter learning. In particular, we consider sequential importance resampling, tempered likelihood estimation, Markov Chain Monte Carlo, random walk Metropolis (RWM), Hamiltonian Monte Carlo (HMC) and variants (stochastic gradients with and without friction, energy conserving subsampling), block pseudo-marginal Metropolis-Hastings with subsampling, hybrid HMC-RWM approaches, and Gaussian rejection filtering. We demonstrate advantages of these approaches over existing ones, namely robustness under multi-modality and high dimensionality. We apply these algorithms to the calibration of superconducting qubits from IBMQ, surpassing the standard quantum limit and achieving better results than Qiskit's default tools. In Hahn echo and Ramsey experiments, we reduce the uncertainty by factors of 10 and 3 respectively, without increasing the number of measurements; conversely, we match the performance of Qiskit's methods while using up to to 99.5% less experimental data. We additionally investigate the roles of adaptivity, dataset ordering and heuristics in quantum characterization. Our findings have applications in challenging quantum characterization tasks, namely learning the dynamics of open quantum systems.
We present BAE, a problem-tailored and noise-aware Bayesian algorithm for quantum amplitude estimation. In a fault tolerant scenario, BAE is capable of saturating the Heisenberg limit; if device noise is present, BAE can dynamically characterize it and self-adapt. We further propose aBAE, an annealed variant of BAE drawing on methods from statistical inference, to enhance robustness. Our proposals are parallelizable in both quantum and classical components, offer tools for fast noise model assessment, and can leverage preexisting information. Additionally, they accommodate experimental limitations and preferred cost trade-offs. We propose a robust benchmark for amplitude estimation algorithms and use it to test BAE against other approaches, demonstrating its competitive performance in both noisy and noiseless scenarios. In both cases, it achieves lower error than any other algorithm as a function of the cost. In the presence of decoherence, it is capable of learning when other algorithms fail.