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arXiv:2107.11388 · Physical Review Letters

Dynamical Uncertainty Propagation with Noisy Quantum Parameters

Mogens Dalgaard, Carrie Ann Weidner, Felix Motzoi

Published Apr 13, 2022• 7 citations
arXivHTMLPDF

Abstract

Many quantum technologies rely on high-precision dynamics, which raises the question of how these are influenced by the experimental uncertainties that are always present in real-life settings. A standard approach in the literature to assess this is Monte Carlo sampling, which suffers from two major drawbacks. First, it is computationally expensive. Second, it does not reveal the effect that each individual uncertainty parameter has on the state of the system. In this Letter, we evade both these drawbacks by incorporating propagation of uncertainty directly into simulations of quantum dynamics, thereby obtaining a method that is orders of magnitude faster than Monte Carlo simulations and directly provides information on how each uncertainty parameter influences the system dynamics. Additionally, we compare our method to experimental results obtained using the IBM quantum computers.

DOI
10.1103/physrevlett.128.150503

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Many quantum technologies rely on high-precision dynamics, which raises the question of how these are influenced by the experimental uncertainties that are always present in real-life settings. A standard approach in the literature to assess this is Monte Carlo sampling, which suffers from two major drawbacks. First, it is computationally expensive. Second, it does not reveal the effect that each individual uncertainty parameter has on the state of the system. In this Letter, we evade both these drawbacks by incorporating propagation of uncertainty directly into simulations of quantum dynamics, thereby obtaining a method that is orders of magnitude faster than Monte Carlo simulations and directly provides information on how each uncertainty parameter influences the system dynamics. Additionally, we compare our method to experimental results obtained using the IBM quantum computers.
Quantum Monte Carlo and quantum simulation are both important tools for understanding quantum many-body systems. As a classical algorithm, quantum Monte Carlo suffers from the sign problem, preventing its applications to most fermion systems and real-time dynamics. In contrast, on a quantum computer, we can simulate time evolution of general systems with local interactions, yet such algorithms usually require the development of fault-tolerant quantum technologies. In this paper, we introduce a novel non-variational algorithm using quantum computing as a subroutine to accelerate quantum Monte Carlo. On a fault-tolerant quantum computer, the complexity of our algorithm is polynomial as the same as conventional quantum simulation algorithms. On a noisy quantum computer, our algorithm incorporating error mitigation eases the sign problem. Even when the circuit noise is significant, our algorithm can reduce the Monte Carlo variance by orders of magnitude.
This paper presents a novel approach to simulating quantum computing algorithms using Monte Carlo methods, specifically designed to account for the inherent probabilistic nature of quantum systems. Traditional simulation methods often rely on deterministic models, which fail to capture the fundamental uncertainty associated with quantum mechanics. Our method employs a stochastic simulation engine that accurately models the probabilistic behavior of quantum gates and measurements. The core of the approach involves generating a large number of samples, each representing a possible outcome of a quantum computation. By analyzing the statistical distribution of these samples, we can approximate the expected results of the algorithm. This technique offers a practical way to evaluate the performance of quantum algorithms, especially those with high levels of uncertainty, and provides a valuable tool for algorithm design and optimization. The method avoids the limitations of purely deterministic simulations and offers a more realistic representation of quantum computation. Key aspects include the development of a robust sampling strategy and the efficient calculation of probabilities. The approach is demonstrated through a detailed explanation and illustrative examples, focusing on the conceptual framework and the underlying mathematical principles.
The splitting method estimates logical error rates of quantum error-correcting codes deep in the rare-event regime, where direct Monte Carlo sampling is prohibitively expensive. It relies on a Markov chain Monte Carlo (MCMC) sampler of faulty configurations, usually driven by one-bit-flip proposals. However, one-bit flips have been reported to be trapped on some codes, which biases the estimate. To address this issue, we propose a new kernel that combines locally balanced one-bit-flip proposals with a cluster-exchange move. We benchmark both kernels and direct Monte Carlo on twelve codes under code-capacity bit-flip noise in two ways, by comparing the mean fault weight with an MCMC-independent reference and by measuring the computation time to reach a fixed accuracy. In the first, our kernel agrees with the reference on every code, including those where one-bit flips are trapped. In the second, it is faster than one-bit flips wherever they are trapped, by orders of magnitude on some codes. Furthermore, at the lowest error rate, it is faster than direct Monte Carlo on every code, by more than four orders of magnitude on some.
This paper investigates the efficient probabilistic simulation of quantum systems using Monte Carlo methods, addressing the computational challenges inherent in these simulations. The core idea is to implement an adaptive sampling strategy, leveraging a Bayesian framework to dynamically adjust the sampling rate based on the estimated variance of the simulation results. This approach minimizes unnecessary computations by focusing sampling efforts on regions of higher uncertainty, thereby enhancing the overall efficiency of the simulation process. The methodology presented significantly improves upon traditional Monte Carlo techniques, offering a promising avenue for tackling increasingly complex quantum systems. The paper details the mathematical formulation of the adaptive sampling process and demonstrates its effectiveness through a conceptual analysis.
High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the Markov-chain Monte Carlo method known as Hamiltonian Monte Carlo, or hybrid Monte Carlo, can be adapted to this context. It is applicable when an efficient parameterization of the state space is available. The resulting random walk is entirely inside the physical parameter space, and the Hamiltonian dynamics enable us to take big steps, thereby avoiding strong correlations between successive sample points while enjoying a high acceptance rate. We use examples of single and double qubit measurements for illustration.
Efficient and robust uncertainty quantification (UQ) is essential in aerospace systems, as emphasized by both NASA’s CFD Vision 2030 and Certification by Analysis (CbA) 2040 initiatives. However, the extensive computational resources required by traditional UQ methods, especially when coupled with high-fidelity computational fluid dynamics (CFD) simulations, pose significant barriers. Quantum computing (QC) has been identified by the CFD Vision 2030 study as a promising technology to address these challenges, but has not yet been extensively explored in the context of UQ. In this study, a Quantum Computing-based Monte Carlo (QCMC) algorithm is implemented using a Qiskit simulator. This method utilizes Quantum Amplitude Estimation (QAE), specifically tailored for near-term Noisy Intermediate-Scale Quantum (NISQ) computers. The QCMC method demonstrates comparable accuracy to classical Monte Carlo (MC) methods on a two-dimensional heat diffusion problem while reducing the required number of samples (oracle calls) by ∼8X. By substantially lowering the computational overhead of UQ, this quantum computing-based algorithm could be a promising tool required to enable robust uncertainty-informed aerospace design.