IEEE International Conference on Quantum Computing and Engineering (QCE)
The maximum-likelihood quantum amplitude estimation algorithm provides the best tradeoff between accuracy and circuit depth among quantum solutions for integral estimation
Marco Maronese, Massimiliano Incudini, Luca Asproni, Enrico Prati
The approximation of an integral with numerical integration methods is a relevant problem in many fields, from finance to chemistry. Monte Carlo methods are a family of methods widely used for numerical integration but which are not always applicable to industrial tasks. Instead, there are quantum computing methods such as Quantum Amplitude Estimation able to offer a speed-up compared to the Monte Carlo methods for numerical integration which promise a significant impact, especially in finance. The Quantum Amplitude Estimation algorithm is a fundamental quantum algorithm with the potential to achieve a quadratic speed-up. Unfortunately, being competitive over classical Monte Carlo methods would require hardware performances currently not available. Therefore, alternative methods have been developed in order to exploit fewer resources while maintaining an advantageous theoretical scaling. We present a comparison between the standard QAE algorithm and two NISQ-friendly versions of the QAE on a numerical integration task. The Monte Carlo technique of Metropolis-Hastings instead is used as a classical benchmark. The scaling of the different algorithms are evaluated in terms of estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the quantum solutions. The conclusion highlights that an alternative approach is preferable with respect to employng the phase estimation routine. Indeed, the maximum likelihood estimation guarantees the best trade-off between the length of the quantum circuits and the precision in the integral estimation.
The approximation of an integral with numerical integration methods is a relevant problem in many fields, from finance to chemistry. Monte Carlo methods are a family of methods widely used for numerical integration but which are not always applicable to industrial tasks. Instead, there are quantum computing methods such as Quantum Amplitude Estimation able to offer a speed-up compared to the Monte Carlo methods for numerical integration which promise a significant impact, especially in finance. The Quantum Amplitude Estimation algorithm is a fundamental quantum algorithm with the potential to achieve a quadratic speed-up. Unfortunately, being competitive over classical Monte Carlo methods would require hardware performances currently not available. Therefore, alternative methods have been developed in order to exploit fewer resources while maintaining an advantageous theoretical scaling. We present a comparison between the standard QAE algorithm and two NISQ-friendly versions of the QAE on a numerical integration task. The Monte Carlo technique of Metropolis-Hastings instead is used as a classical benchmark. The scaling of the different algorithms are evaluated in terms of estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the quantum solutions. The conclusion highlights that an alternative approach is preferable with respect to employng the phase estimation routine. Indeed, the maximum likelihood estimation guarantees the best trade-off between the length of the quantum circuits and the precision in the integral estimation.
The Quantum Amplitude Estimation (QAE) algorithm is a major quantum algorithm designed to achieve a quadratic speed-up. Until fault-tolerant quantum computing is achieved, being competitive over classical Monte Carlo (MC) remains elusive. Alternative methods have been developed so as to require fewer resources while maintaining an advantageous theoretical scaling. We compared the standard QAE algorithm with two Noisy Intermediate-Scale Quantum (NISQ)-friendly versions of QAE on a numerical integration task, with the Monte Carlo technique of Metropolis–Hastings as a classical benchmark. The algorithms were evaluated in terms of the estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the solutions, respectively. The effectiveness of the two QAE alternatives was tested on an 11-qubit trapped-ion quantum computer in order to verify which solution can first provide a speed-up in the integral estimation problems. We concluded that an alternative approach is preferable with respect to employing the phase estimation routine. Indeed, the Maximum Likelihood estimation guaranteed the best trade-off between the length of the quantum circuits and the precision in the integral estimation, as well as greater resistance to noise.
Quantum Amplitude Estimation (QAE) -- a technique by which the amplitude of a given quantum state can be estimated with quadratically fewer queries than by standard sampling -- is a key sub-routine in several important quantum algorithms, including Grover search and Quantum Monte-Carlo methods. An obstacle to implementing QAE in near-term noisy intermediate-scale quantum (NISQ) devices has been the need to perform Quantum Phase Estimation (QPE) -- a costly procedure -- as a sub-routine. This impediment was lifted with various QPE-free methods of QAE, wherein Grover queries of varying depths / powers (often according to a "schedule") are followed immediately by measurements and classical post-processing techniques like maximum likelihood estimation (MLE). Existing analyses as to the optimality of various query schedules in these QPE-free QAE schemes have hitherto assumed noise-free systems. In this work, we analyse QPE-free QAE under common noise models that may afflict NISQ devices and report on the optimality of various query schedules in the noisy regime. We demonstrate that, given an accurate noise characterization of one's system, one must choose a schedule that balances the trade-off between the greater ideal performance achieved by higher-depth circuits, and the correspondingly greater accumulation of noise-induced error.
Option pricing is a fundamental problem in quantitative finance, where the accuracy and efficiency of valuation methods directly affect investment and risk management decisions. While the Black-Scholes model provides a closed-form solution for European options, numerical approaches such as Monte Carlo simulation are widely used when analytical methods are intractable. However, Monte Carlo suffers from slow convergence with complexity O(1/ε2).Quantum Amplitude Estimation (QAE), an algorithm in quantum computing, promises a quadratic speedup with complexity O(1/ε). This paper implements QAE, specifically Iterative Amplitude Estimation (IAE), for European call option pricing and compares its performance against Monte Carlo simulation and the Black-Scholes model. Experiments are conducted on both Qiskit simulators and IBM Quantum hardware (ibm_brisbane).Results show that QAE achieves estimates close to Monte Carlo and analytical benchmarks, with mean payoff deviations of less than 0.01. While QAE produces wider confidence intervals due to noise and discretization, the findings demonstrate its potential scalability advantage for future quantum finance applications.
Numerical integration with Monte Carlo methods is a central computational task in many scientific and industrial applications, including financial derivative pricing and risk management. Classical Monte Carlo algorithms are computationally demanding: achieving an accuracy ε typically requires a number of function evaluations scaling as O(1/ε2). Quantum-accelerated Monte Carlo methods based on quantum amplitude estimation can in principle quadratically improve this dependence. However, \textit{quasi}-Monte Carlo methods have not been explored in the quantum context. In this work, we introduce a quantum quasi-Monte Carlo algorithm that combines low-discrepancy nets with quantum amplitude estimation. The proposed method prepares the quasi-random point set coherently in superposition. The method does not yield an asymptotic improvement over classical quasi-Monte Carlo, since the total error separates into a discretization error, determined by the finite net, and a quantum estimation error. Instead, we explore a pre-asymptotic advantage window: for a target accuracy that would classically require 2q low discrepancy points, one can prepare a higher-resolution net of size 2Q, with Q>q, in superposition and reach the same accuracy using significantly fewer function queries. This window can be controlled by tuning the circuit resolution and amplitude-estimation parameters, making the approach relevant for practical regimes where the number of queries is finite rather than asymptotically large.
This paper introduces quantum computing methods for Monte Carlo simulations in power systems which are expected to be exponentially faster than their classical computing counterparts. Monte Carlo simulations is a fundamental method, widely used in power systems to estimate key parameters of unknown probability distributions, such as the mean value, the standard deviation, or the value at risk. It is, however, very computationally intensive. Approaches based on Quantum Amplitude Estimation can offer a quadratic speedup, requiring orders of magnitude less samples to achieve the same accuracy. This paper explains three Quantum Amplitude Estimation methods to replace the Classical Monte Carlo method, namely the Iterative Quantum Amplitude Estimation (IQAE), Maximum Likelihood Amplitude Estimation (MLAE), and Faster Amplitude Estimation (FAE), and compares their performance for three different types of probability distributions for power systems.
Amplitude Estimation (AE) is a critical subroutine in many quantum algorithms, allowing for a quadratic speedup in various applications like those involving estimating statistics of various functions as in financial Monte Carlo simulations. Much work has gone into devising methods to efficiently estimate the amplitude of a quantum state without expensive operations like the Quantum Fourier Transform (QFT), which is especially prohibitive given the constraints of current NISQ devices. Newer methods have reduced the number of operations required on a quantum computer and are the most promising near-term implementations of the AE subroutine. While it remains to be seen the exact circuit requirements for a quantum advantage in applications relying on AE, it is necessary to continue to benchmark the algorithm's performance on current quantum computers and the circuit costs associated with such subroutines. Given these considerations, we expand on results from previous experiments in using Maximum Likelihood Estimation (MLE) to approximate the amplitude of a quantum state and provide empirical upper bounds on the current feasible circuit depths for AE on a superconducting quantum computer. Our results show that MLE using optimally-compiled circuits can currently outperform naive sampling for up to 3 Grover Iterations with a circuit depth of 131, which is higher than reported in other experimental results. This functional benchmark is one of many that will be continually monitored against current quantum hardware to measure the necessary progress towards quantum advantage.