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 $O(1/ε^2)$ . 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 $2^q$ low discrepancy points, one can prepare a higher-resolution net of size $2^Q$ , 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.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 $O(1/ε^2)$. 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 $2^q$ low discrepancy points, one can prepare a higher-resolution net of size $2^Q$, 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.

quant-pharXiv:2609.03625Sep 4, 20260 citations

Quasi Monte Carlo method for linear combination unitaries via classical postprocessing

Yuya Kawamata, Kosuke Mitarai, Keisuke Fujii

We propose the quasi Monte Carlo method for linear combination of unitaries via classical postprocessing (LCU-CPP) on quantum applications. The LCU-CPP framework has been proposed as an approach to reduce hardware resources, expressing a general target operator F ( A ) as F ( A ) = ∫ V f ( t ) G ( A , t ) d t , where each G ( A , t ) is proportional to a unitary operator. On a quantum device, Re [ Tr ( G ( A , t ) ρ ) ] can be estimated using the Hadamard test and then combined through classical integration, allowing for the realization of nonunitary functions with reduced circuit depth. While previous studies have employed the Monte Carlo method or the trapezoid rule to evaluate the integral in LCU-CPP, we show that the quasi Monte Carlo method can achieve even lower errors. In two numerical experiments, ground-state property estimation and Green’s function estimation, the quasi Monte Carlo method achieves the lowest errors with a number of Hadamard test shots per unitary that is practical for real hardware implementations. These results indicate that quasi Monte Carlo is an effective integration strategy within the LCU-CPP framework.We propose the quasi Monte Carlo method for linear combination of unitaries via classical postprocessing (LCU-CPP) on quantum applications. The LCU-CPP framework has been proposed as an approach to reduce hardware resources, expressing a general target operator F ( A ) as F ( A ) = ∫ V f ( t ) G ( A , t ) d t , where each G ( A , t ) is proportional to a unitary operator. On a quantum device, Re [ Tr ( G ( A , t ) ρ ) ] can be estimated using the Hadamard test and then combined through classical integration, allowing for the realization of nonunitary functions with reduced circuit depth. While previous studies have employed the Monte Carlo method or the trapezoid rule to evaluate the integral in LCU-CPP, we show that the quasi Monte Carlo method can achieve even lower errors. In two numerical experiments, ground-state property estimation and Green’s function estimation, the quasi Monte Carlo method achieves the lowest errors with a number of Hadamard test shots per unitary that is practical for real hardware implementations. These results indicate that quasi Monte Carlo is an effective integration strategy within the LCU-CPP framework.

arXiv:2509.14451Physical Review ResearchApr 18, 20260 citations

ε

typically requires a number of function evaluations scaling as

O(1/ε^2)

. 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

2^q

low discrepancy points, one can prepare a higher-resolution net of size

2^Q

, 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.