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Proceedings of the ACM on Programming Languages

Quantum Monte Carlo Estimation via Probabilistic Programming

Seungmin Jeon, Jaeho Choi, Jonguk Jeon, Kanguk Lee, Kyeongmin Cho, Sukyoung Ryu, Jeehoon Kang

Published Oct 1, 2026• 0 citations
Publisher

Abstract

Monte Carlo methods are fundamental to finance, system verification, and scientific simulation, but converge slowly: achieving an additive error of є requires O (1/є 2 ) samples. Quantum Amplitude Estimation (QAE) offers a quadratic speedup by encoding the target probabilistic model into a quantum circuit. However, constructing such a circuit demands low-level quantum expertise, and existing tools for this task all sacrifice at least one of generality, usability, or efficiency. To address these, we design QPPL (Quantum Probabilistic Programming Language), a simple imperative language, and a compiler that translates probabilistic programs into quantum circuits. The key insight is that the circuit construction amounts to specifying a probability distribution, precisely the task that probabilistic programming addresses. QPPL achieves generality by supporting joint distributions, conditional updates, dynamic probabilities, and real-valued expectations in a single language; usability by offering a sequential, imperative syntax with named variables and direct arithmetic that hides all quantum details; and efficiency by modularly compiling each construct into reversible circuit primitives, achieving scalable circuit synthesis. We prove that the compilation is semantics-preserving. On benchmarks spanning finance and probabilistic model checking, QPPL is the only tool that covers all benchmarks, while producing circuits with up to 8.8× fewer gates and 26× shallower depth than existing tools.

DOI
10.1145/3839463

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Monte Carlo methods are fundamental to finance, system verification, and scientific simulation, but converge slowly: achieving an additive error of є requires O (1/є 2 ) samples. Quantum Amplitude Estimation (QAE) offers a quadratic speedup by encoding the target probabilistic model into a quantum circuit. However, constructing such a circuit demands low-level quantum expertise, and existing tools for this task all sacrifice at least one of generality, usability, or efficiency. To address these, we design QPPL (Quantum Probabilistic Programming Language), a simple imperative language, and a compiler that translates probabilistic programs into quantum circuits. The key insight is that the circuit construction amounts to specifying a probability distribution, precisely the task that probabilistic programming addresses. QPPL achieves generality by supporting joint distributions, conditional updates, dynamic probabilities, and real-valued expectations in a single language; usability by offering a sequential, imperative syntax with named variables and direct arithmetic that hides all quantum details; and efficiency by modularly compiling each construct into reversible circuit primitives, achieving scalable circuit synthesis. We prove that the compilation is semantics-preserving. On benchmarks spanning finance and probabilistic model checking, QPPL is the only tool that covers all benchmarks, while producing circuits with up to 8.8× fewer gates and 26× shallower depth than existing tools.
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.
High-performance computing heavily relies on Monte Carlo (MC) simulation, but achieving high precision imposes massive computational loads due to its low convergence rate. To overcome this classical bottleneck, Quantum Amplitude Estimation (QAE) has emerged as a powerful algorithmic alternative, theoretically offering a quadratic speedup by offloading these workloads to quantum accelerators. However, mitigating Quantum Fourier Transform (QFT) discretization errors typically require Maximum Likelihood Estimation (MLE). MLE introduces prohibitive iterative latency and struggles with shot-noise in low-measurement regimes. To address this, we propose a comprehensive, hardware-efficient framework that integrates uniform state preparation with a novel Peak Estimation Refinement (PER) readout technique. PER exploits the symmetric conjugate eigenvalues of the quantum operator to provide deterministic,O(1){O}(\mathrm{1})O(1)probability extraction. This eliminates classical iterative optimization, achieving an approximately600×600 \times600×reduction in post-processing latency while inherently amplifying the signal-to-noise margin to resist stochastic fluctuations in low-shot environments. Validated via an RTL-to-quantum compilation flow and a 10 -qubitπ\piπ-estimation synthesis benchmark, we demonstrate an estimation error of 0.007 %, proving the viability of migrating complex probabilistic workloads to real-time quantum accelerators
Simulating the evolution of quantum circuits is a computationally intensive task, fundamentally limited by the need to explore the exponentially large Hilbert space associated with quantum systems. This work introduces a novel approach utilizing probabilistic programming to efficiently model and simulate quantum circuit dynamics. The core idea involves representing quantum circuit operations as probabilistic functions, enabling the use of Monte Carlo methods for estimating circuit evolution with significantly reduced computational overhead. This approach provides a flexible and scalable framework for exploring complex quantum systems, particularly those where traditional deterministic simulation methods become intractable. The method's effectiveness is demonstrated through theoretical analysis and a conceptual outline of implementation, highlighting potential benefits and future research directions. Key mathematical concepts employed include probability theory, Markov chains, and Monte Carlo integration. The primary goal is to offer a practical alternative to standard quantum simulation techniques, particularly for scenarios requiring approximate solutions or exploration of large parameter spaces.
Quantum amplitude estimation (QAE) can achieve a quadratic speedup for applications classically solved by Monte Carlo simulation. A key requirement to realize this advantage is efficient state preparation. If state preparation is too expensive, it can diminish the quantum advantage. Preparing arbitrary quantum states has exponential complexity with respect to the number of qubits, and thus, is not applicable. Currently known efficient techniques require problems based on log-concave probability distributions, involve learning an unknown distribution from empirical data, or fully rely on quantum arithmetic. In this paper, we introduce an approach to simplify state preparation, together with a circuit optimization technique, both of which can help reduce the circuit complexity for QAE state preparation significantly. We demonstrate the introduced techniques for a numerical integration example on real quantum hardware, as well as for option pricing under the Heston model, i.e., based on a stochastic volatility process, using simulation.
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.
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