Exploring the world of Bell inequalities requires numerical methods that scale effectively with the number of parties, inputs, and outputs. Such methods can be used to study large-scale Bell inequalities beyond the reach of analytic methods, to design experiments for violating more complicated Bell inequalities, and to enable applications of Bell inequality violation. Such applications include randomness generation and multi-agent coordination with restricted communication. However, current general purpose optimizers such as the see-saw method struggle to handle Bell inequalities with only a dozen inputs and outputs. We introduce an optimizer for the quantum value of a Bell inequality based on batched gradient descent (BGD). More precisely, our BGD optimizer is a differentiable search over feasible quantum strategies in which states and projective measurements are generated from unconstrained variables. The Bell expression is evaluated by a direct tensor contraction without forming the prohibitively large Bell operator. This formulation makes parallel random restarts inexpensive and is thus naturally implemented by a GPU. We evaluate our optimizer against the see-saw method and find a significant speedup for a wide variety of Bell inequality families. Implemented on a GPU, our BGD optimizer can optimize Bell inequalities in our numerical experiments with more than a thousand inputs or outputs within minutes. For these large-scale Bell inequalities, we also efficiently compute the classical value by enumerating possible strategies using a GPU or expressing the optimization as a mixed-integer linear program. Our optimizers can serve as numerical evaluation tools for Bell experiments and multi-agent coordination problems modeled by Bell inequalities with many inputs and outputs, a common feature of real-world scenarios such as high frequency trading and distributed systems.
Exploring the world of Bell inequalities requires numerical methods that scale effectively with the number of parties, inputs, and outputs. Such methods can be used to study large-scale Bell inequalities beyond the reach of analytic methods, to design experiments for violating more complicated Bell inequalities, and to enable applications of Bell inequality violation. Such applications include randomness generation and multi-agent coordination with restricted communication. However, current general purpose optimizers such as the see-saw method struggle to handle Bell inequalities with only a dozen inputs and outputs. We introduce an optimizer for the quantum value of a Bell inequality based on batched gradient descent (BGD). More precisely, our BGD optimizer is a differentiable search over feasible quantum strategies in which states and projective measurements are generated from unconstrained variables. The Bell expression is evaluated by a direct tensor contraction without forming the prohibitively large Bell operator. This formulation makes parallel random restarts inexpensive and is thus naturally implemented by a GPU. We evaluate our optimizer against the see-saw method and find a significant speedup for a wide variety of Bell inequality families. Implemented on a GPU, our BGD optimizer can optimize Bell inequalities in our numerical experiments with more than a thousand inputs or outputs within minutes. For these large-scale Bell inequalities, we also efficiently compute the classical value by enumerating possible strategies using a GPU or expressing the optimization as a mixed-integer linear program. Our optimizers can serve as numerical evaluation tools for Bell experiments and multi-agent coordination problems modeled by Bell inequalities with many inputs and outputs, a common feature of real-world scenarios such as high frequency trading and distributed systems.
In recent decades, a great variety of research and applications concerning Bell nonlocality have been developed with the advent of quantum information science. Providing that Bell nonlocality can be revealed by the violation of a family of Bell inequalities, finding maximal Bell violation (MBV) for unknown quantum states becomes an important and inevitable task during Bell experiments. In this paper we introduce a self-guided method to find MBVs for unknown states using a stochastic gradient ascent algorithm (SGA), by parametrizing the corresponding Bell operators. For three investigated systems (two qubit, three qubit, and two qutrit), this method can ascertain the MBV of general two-setting inequalities within 100 iterations. Furthermore, we prove SGA is also feasible when facing more complex Bell scenarios, e.g., d-setting d-outcome Bell inequality. Moreover, compared to other possible methods, SGA exhibits significant superiority in efficiency, robustness, and versatility.
Gaussian process regression (GPR) is a nonparametric supervised learning method, widely valued for its ability to quantify uncertainty. Despite its advantages and broad applications, classical GPR implementations face significant scalability challenges, as they involve matrix operations with a cubic complexity in relation to the dataset size. This computational challenge is further compounded by the demand of optimizing the Gaussian Process (GP) model over its hyperparameters, rendering the total computational cost prohibitive for data intensive problems. To address this issue, we propose a quantum gradient descent algorithm to optimize the GP model. Taking advantage of recent advances in quantum algorithms for linear algebra, our algorithm achieves exponential speedup in computing the gradients of the log marginal likelihood. The entire gradient descent process is integrated into the quantum circuit. Through runtime analysis and error bounds, we demonstrate that our approach significantly improves the scalability of GPR model optimization, making it computationally feasible for large-scale applications.
Abstract Quantum state tomography (QST) is a widely employed technique for characterizing the state of a quantum system. However, it is plagued by two fundamental challenges: computational and experimental complexity grows exponentially with the number of qubits, rendering experimental implementation and data post-processing arduous even for moderately sized systems. Here, we introduce gradient-descent (GD) algorithms for the post-processing step of QST in discrete- and continuous-variable systems. To ensure physically valid state reconstruction at each iteration step of the algorithm, we use various density-matrix parameterizations: Cholesky decomposition, Stiefel manifold, and projective normalization. These parameterizations have the added benefit of enabling a rank-controlled ansatz, which simplifies reconstruction when there is prior information about the system. We benchmark the performance of our GD-QST techniques against state-of-the-art methods, including constrained convex optimization, conditional generative adversarial networks, and iterative maximum likelihood estimation. Our comparison focuses on time complexity, iteration counts, data requirements, state rank, and robustness against noise. We find that rank-controlled ansatzes in our stochastic mini-batch GD-QST algorithms effectively handle noisy and incomplete data sets, yielding significantly higher reconstruction fidelity than other methods. Simulations achieving full-rank seven-qubit QST in under three minutes on a standard laptop, with 18 GB of RAM and no dedicated GPU, highlight that GD-QST is computationally more efficient and outperforms other techniques in most scenarios, offering a promising avenue for characterizing noisy intermediate-scale quantum devices. Our Python code for GD-QST algorithms is publicly available at github.com/mstorresh/GD-QST .
We investigate device-independent certification of global randomness based on Bell inequality violations in bipartite scenarios with three outcomes per party. Our goal is to determine whether multi-outcome measurements allow one to surpass the amount of randomness achievable with binary outputs in realistic scenarios. We begin by analyzing several known Bell expressions and evaluating their robustness against noise for randomness certification. We then introduce a systematic method for generating new Bell expressions within structured families and perform a large-scale numerical study. We find that a substantial number of inequalities certify significant amounts of min-entropy. In particular, we identify simple inequalities that achieve near-maximal global randomness while involving a reduced number of measurement settings, thus improving the balance between certified randomness and number of inputs. Moreover, the vast majority of nontrivial certificates exhibit robustness against realistic noise, maintaining positive certified randomness away from the ideal regime. These results demonstrate that strong device-independent randomness expansion in multi-outcome scenarios is not restricted to carefully engineered inequalities, but arises generically within suitably constructed families of Bell expressions.
The rapid growth of machine learning (ML) has been accompanied by a significant challenge: the scaling of optimization algorithms to handle the immense parameter spaces of modern models, such as deep neural networks. Traditional optimization methods, including gradient descent and its variants, often encounter difficulties in efficiently exploring these spaces, leading to slow training times and potential stagnation. This paper investigates the application of quantum-inspired optimization techniques to address this scaling problem. We explore algorithms inspired by quantum annealing and other quantum phenomena, adapting their core principles to classical computation. Specifically, we propose a novel optimization framework utilizing a stochastic search strategy based on quantum superposition and measurement, designed to navigate high-dimensional parameter spaces more effectively. The efficacy of this approach is evaluated through theoretical analysis and conceptual simulations, demonstrating its potential to accelerate the training of large-scale machine learning models. We demonstrate how these methods can improve convergence speed and potentially escape local minima more effectively than conventional techniques. The core contribution lies in providing a theoretical foundation and a practical framework for applying quantum-inspired concepts to optimization within the context of machine learning.
Training large-scale neural networks with traditional stochastic gradient descent (SGD) methods often faces significant challenges, including slow convergence rates and high computational costs. This paper proposes a novel approach, Quantum-Inspired Stochastic Gradient Descent (QISGD), designed to address these limitations. QISGD leverages principles inspired by quantum measurement processes, specifically employing a random projection operator to approximate the gradient. This technique aims to provide a more efficient and potentially faster way to update the network's parameters during training. The theoretical underpinnings of QISGD are presented, along with a detailed explanation of the algorithm's components. We demonstrate the potential benefits of QISGD through a conceptual analysis and outline a framework for future research and implementation. The core innovation lies in translating quantum measurement intuition into a practical optimization strategy for classical neural networks. The goal is to improve the training efficiency of large networks while retaining the benefits of SGD.