Quantum machine learning has emerged as a promising method to improve near-term quantum computation devices. However, algorithmic classes such as variational quantum algorithms have been shown to suffer from barren plateaus due to vanishing gradients in their parameter spaces. We present an approach to quantum algorithm optimization that is based on trainable Fourier coefficients of Hamiltonian system parameters. Our ansatz applies to the extension of discrete quantum variational algorithms to analogue quantum optimal control schemes and is non-local in time. We demonstrate the viability of our ansatz on several objective functions using quantum natural gradient descent. In comparison to the temporally local discretization ans\atze in quantum optimal control and parametrized circuits, our ansatz exhibits faster and more consistent convergence with a distinct lack of barren plateaus. We propose our ansatz as a viable parametrization candidate for near-term quantum machine learning.
Quantum machine learning has emerged as a promising method to improve near-term quantum computation devices. However, algorithmic classes such as variational quantum algorithms have been shown to suffer from barren plateaus due to vanishing gradients in their parameter spaces. We present an approach to quantum algorithm optimization that is based on trainable Fourier coefficients of Hamiltonian system parameters. Our ansatz applies to the extension of discrete quantum variational algorithms to analogue quantum optimal control schemes and is non-local in time. We demonstrate the viability of our ansatz on several objective functions using quantum natural gradient descent. In comparison to the temporally local discretization ans\atze in quantum optimal control and parametrized circuits, our ansatz exhibits faster and more consistent convergence with a distinct lack of barren plateaus. We propose our ansatz as a viable parametrization candidate for near-term quantum machine learning.
Quantum machine learning has emerged as a promising utilization of near-term quantum computation devices. However, algorithmic classes such as variational quantum algorithms have been shown to suffer from barren plateaus due to vanishing gradients in their parameters spaces. We present an approach to quantum algorithm optimization that is based on trainable Fourier coefficients of Hamiltonian system parameters. Our is exclusive to the extension of discrete quantum variational algorithms to analog quantum optimal control schemes and is nonlocal in time. We demonstrate the viability of our on the objectives of compiling the quantum Fourier transform and preparing ground states of random problem Hamiltonians. In comparison to the temporally local discretization in quantum optimal control and parametrized circuits, our exhibits faster and more consistent convergence. We uniformly sample objective gradients across the parameter space and find that in our the variance decays at a nonexponential rate with the number of qubits, while it decays at an exponential rate in the temporally local benchmark . This indicates the mitigation of barren plateaus in our . We propose our as a viable candidate for near-term quantum machine learning. Published by the American Physical Society 2024
Variational quantum algorithms (VQAs) promise efficient use of near-term quantum computers. However, training VQAs often requires an extensive amount of time and suffers from the barren plateau problem where the magnitude of the gradients vanishes with increasing number of qubits. Here, we show how to optimally train VQAs for learning quantum states. Parameterized quantum circuits can form Gaussian kernels, which we use to derive adaptive learning rates for gradient ascent. We introduce the generalized quantum natural gradient that features stability and optimized movement in parameter space. Both methods together outperform other optimization routines in training VQAs. Our methods also excel at numerically optimizing driving protocols for quantum control problems. The gradients of the VQA do not vanish when the fidelity between the initial state and the state to be learned is bounded from below. We identify a VQA for quantum simulation with such a constraint that thus can be trained free of barren plateaus. Finally, we propose the application of Gaussian kernels for quantum machine learning.
Variational quantum algorithms, which combine highly expressive parameterized quantum circuits (PQCs) and optimization techniques in machine learning, are one of the most promising applications of a near-term quantum computer. Despite their huge potential, the utility of variational quantum algorithms beyond tens of qubits is still questioned. One of the central problems is the trainability of PQCs. The cost function landscape of a randomly initialized PQC is often too flat, asking for an exponential amount of quantum resources to find a solution. This problem, dubbed barren plateaus , has gained lots of attention recently, but a general solution is still not available. In this paper, we solve this problem for the Hamiltonian variational ansatz (HVA), which is widely studied for solving quantum many-body problems. After showing that a circuit described by a time-evolution operator generated by a local Hamiltonian does not have exponentially small gradients, we derive parameter conditions for which the HVA is well approximated by such an operator. Based on this result, we propose an initialization scheme for the variational quantum algorithms and a parameter-constrained ansatz free from barren plateaus.
Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren plateaus for problem-inspired ansatzes. Such ansatzes include the Quantum Alternating Operator Ansatz (QAOA), the Hamiltonian Variational Ansatz (HVA), and others. With our framework, we prove that avoiding barren plateaus for these ansatzes is not always guaranteed. Specifically, we show that the gradient scaling of the VQA depends on the degree of controllability of the system, and hence can be diagnosed through the dynamical Lie algebra g obtained from the generators of the ansatz. We analyze the existence of barren plateaus in QAOA and HVA ansatzes, and we highlight the role of the input state, as different initial states can lead to the presence or absence of barren plateaus. Taken together, our results provide a framework for trainability-aware ansatz design strategies that do not come at the cost of extra quantum resources. Moreover, we prove no-go results for obtaining ground states with variational ansatzes for controllable system such as spin glasses. Our work establishes a link between the existence of barren plateaus and the scaling of the dimension of g .
Barren plateaus appear to be a major obstacle to using variational quantum algorithms to simulate large-scale quantum systems or replace traditional machine learning algorithms. They can be caused by multiple factors such as expressivity, entanglement, locality of observables, or even hardware noise. We propose classical splitting of ansätze or parametrized quantum circuits to avoid barren plateaus. Classical splitting is realized by splitting an N qubit ansatz to multiple ansätze that consists of O(logN) qubits. We show that such an ansatz can be used to avoid barren plateaus. We support our results with numerical experiments and perform binary classification on classical and quantum datasets. Then, we propose an extension of the ansatz that is compatible with variational quantum simulations. Finally, we discuss a speed-up for gradient-based optimization and hardware implementation, robustness against noise and parallelization, making classical splitting an ideal tool for noisy intermediate scale quantum (NISQ) applications.
Barren Plateaus are a formidable challenge for hybrid quantum-classical algorithms that lead to flat plateaus in the loss function landscape making it difficult to take advantage of the expressive power of parameterized quantum circuits with gradient-based methods. Like in classical neural network models, parameterized quantum circuits suffer the same vanishing gradient issue due to large parameter spaces with non-convex landscapes. In this review, we present an overview of the different genesis for barren plateaus, mathematical formalisms of common themes around barren plateaus, and dives into gradients. The central objective is to provide a conceptual perspective between classical and quantum interpretations of vanishing gradients as well as dive into techniques involving cost functions, entanglement, and initialization strategies to mitigate barren plateaus. Addressing barren plateaus paves the way towards feasibility of many classically intractable applications for quantum simulation, optimization, chemistry, and quantum machine learning.