Variational quantum algorithms dominate gate-based applications of modern quantum processors. The so-called layerwise trainability conjecture appears in various works throughout the variational quantum computing literature. The conjecture asserts that a quantum circuit can be trained piecewise, e.g., that a few layers can be trained in sequence to minimize an objective function. Here, we prove this conjecture false. Counterexamples are found by considering objective functions that are exponentially close (in the number of qubits) to the identity matrix. In the finite setting, we found abrupt transitions in the ability of quantum circuits to be trained to minimize these objective functions. Specifically, we found that below a critical (target-gate-dependent) threshold, circuit training terminates close to the identity and remains near to the identity for subsequently added blocks trained piecewise. A critical layer depth will abruptly train arbitrarily close to the target, thereby minimizing the objective function. These findings shed light on the divide-and-conquer trainability of variational quantum circuits and apply to a wide collection of contemporary literature.
Variational quantum algorithms dominate gate-based applications of modern quantum processors. The so-called layerwise trainability conjecture appears in various works throughout the variational quantum computing literature. The conjecture asserts that a quantum circuit can be trained piecewise, e.g., that a few layers can be trained in sequence to minimize an objective function. Here, we prove this conjecture false. Counterexamples are found by considering objective functions that are exponentially close (in the number of qubits) to the identity matrix. In the finite setting, we found abrupt transitions in the ability of quantum circuits to be trained to minimize these objective functions. Specifically, we found that below a critical (target-gate-dependent) threshold, circuit training terminates close to the identity and remains near to the identity for subsequently added blocks trained piecewise. A critical layer depth will abruptly train arbitrarily close to the target, thereby minimizing the objective function. These findings shed light on the divide-and-conquer trainability of variational quantum circuits and apply to a wide collection of contemporary literature.
The quantum approximate optimization algorithm (QAOA) is the most studied gate-based variational quantum algorithm today. We train QAOA one layer at a time to maximize overlap with an n qubit target state. Doing so we discovered that such training always saturates---called training saturation---at some depth p∗, meaning that past a certain depth, overlap cannot be improved by adding subsequent layers. We formulate necessary conditions for saturation. Numerically, we find layerwise QAOA reaches its maximum overlap at depth p∗=n for the problem of state preparation. The addition of coherent dephasing errors to training removes saturation, recovering robustness to layerwise training. This study sheds new light on the performance limitations and prospects of QAOA.
Exploring quantum applications of near-term quantum devices is a rapidly growing field of quantum information science with both theoretical and practical interests. A leading paradigm to establish such near-term quantum applications is variational quantum algorithms (VQAs). These algorithms use a classical optimizer to train a parameterized quantum circuit to accomplish certain tasks, where the circuits are usually randomly initialized. In this work, we prove that for a broad class of such random circuits, the variation range of the cost function via adjusting any local quantum gate within the circuit vanishes exponentially in the number of qubits with a high probability. This result can unify the restrictions on gradient-based and gradient-free optimizations in a natural manner and reveal extra harsh constraints on the training landscapes of VQAs. Hence a fundamental limitation on the trainability of VQAs is unraveled, indicating the essential mechanism of the optimization hardness in the Hilbert space with exponential dimension. We further showcase the validity of our results with numerical simulations of representative VQAs. We believe that these results would deepen our understanding of the scalability of VQAs and shed light on the search for near-term quantum applications with advantages.
Variational quantum algorithms (VQAs) generally rely on classical optimization to train parameterized quantum circuits. This training seeks to minimize an objective function, and its efficiency is central to the practical success of these algorithms. However, globally minimizing such training objectives over the circuit parameters is known to be NP-hard, limiting the prospect of general guarantees for efficient training. In this Letter, we prove that even the weaker task of finding a local minimum of such VQA training objectives is strongly NP-hard, including when the objective admits efficient classical evaluation. Moreover, we show that this hardness persists even for the task of finding a parameter vector within ℓp-distance strictly less than π/2 of some local minimizer, for every p≥1. Our central technical result is that approximating a local minimizer of a Hermitian trigonometric polynomial is strongly NP-hard. By explicitly constructing quantum circuits whose training objectives reproduce these hard instances, we obtain a polynomial-time reduction to VQA training. Our results establish a fundamental computational barrier to variational quantum training: even reaching the vicinity of a local minimum remains hard in the worst case.
The Quantum Approximate Optimization Algorithm (QAOA) is an extensively studied variational quantum algorithm utilized for solving optimization problems on near-term quantum devices. A significant focus is placed on determining the effectiveness of training the n-qubit QAOA circuit, i.e., whether the optimization error can converge to a constant level as the number of optimization iterations scales polynomially with the number of qubits. In realistic scenarios, the landscape of the corresponding QAOA objective function is generally non-convex and contains numerous local optima. In this work, motivated by the favorable performance of Bayesian optimization in handling non-convex functions, we theoretically investigate the trainability of the QAOA circuit through the lens of the Bayesian approach. This lens considers the corresponding QAOA objective function as a sample drawn from a specific Gaussian process. Specifically, we focus on two scenarios: the noiseless QAOA circuit and the noisy QAOA circuit subjected to local Pauli channels. Our first result demonstrates that the noiseless QAOA circuit with a depth of O~(logn) can be trained efficiently, based on the widely accepted assumption that either the left or right slice of each block in the circuit forms a local 1-design. Furthermore, we show that if each quantum gate is affected by a q-strength local Pauli channel with the noise strength range of 1/poly(n) to 0.1, the noisy QAOA circuit with a depth of O(logn/log(1/q)) can also be trained efficiently. Our results offer valuable insights into the theoretical performance of quantum optimization algorithms in the noisy intermediate-scale quantum era.
Recent advancements in machine learning have revolutionised research across various fields. Despite their success, conventional learning techniques are hindered by their significant computational resources and energy requirements. Prompted by recent experimental breakthroughs in quantum computing, variational quantum machine learning (QML) – machine learning integrated with variational quantum circuits (VQCs) – has emerged as a promising alternative. Nonetheless, the theoretical framework underpinning the advantages of variational QML is still rudimentary. Specifically, the training of VQCs faces several challenges, such as the barren plateau problem, where the gradient diminishes exponentially with an increasing qubits. A related issue arises in variational QML training, where the convergence rate is exponentially small. In this thesis, we present theoretically guaranteed solutions to these challenges. First, we construct innovative circuit architectures to address the vanishing gradient problem in deep VQCs. We propose quantum controlled-layer and quantum ResNet structures, demonstrating that the expected gradient norm's lower bound is unaffected by the increase in qubits and circuit depth. Next, we introduce an initialization strategy to mitigate the vanishing gradient issue in general deep quantum circuits. We prove that Gaussian-initialized parameters ensure the gradient norm's decay rate remains inversely polynomial despite the increase in qubit numbers and circuit depth. Finally, we propose a novel and effective theory for analysing the training of quantum neural networks with moderate depths. We prove that, under certain randomness conditions in the circuits and datasets, training converges linearly with a rate inversely proportional to the dataset size. Our approach surpasses previous results, achieving exponentially larger convergence rates with modest depth, or conversely, requiring exponentially less depth for equivalent rates.
Variational Quantum Algorithms (VQAs) are a leading paradigm for near-term quantum computing, yet their training suffers from sensitivity to circuit depth, initialization, and landscape pathologies such as barren plateaus. We study \emph{progressive depth training} (PDT) -- a layerwise curriculum that trains a shallow circuit before appending new layers -- and identify a fundamental obstacle: fixed entangling gates (CNOTs) in hardware-efficient ansätze cause \emph{initialization shock}, an energy spike when new layers are added. We propose \emph{identity-paired progressive depth training} (IP-PDT), which appends forward/inverse block pairs -- each consisting of a standard rotation+CNOT block followed by its reverse -- that compose to the identity at initialization. Because the adjacent CNOT rings cancel, the effective circuit retains only \textit{a single entangling layer} surrounded by \textit{overparameterized local rotations}. We prove a simple \textit{Reachable Set Saturation Theorem}: under this construction the variational manifold expands exactly once (when post-entangler rotations are first introduced) and then \emph{saturates}; all subsequent depth increases provide pure overparameterization of single-qubit unitaries. Despite this saturation, progressive addition of rotation parameters can continue to improve optimization outcomes -- a phenomenon we term \emph{trainability beyond expressibility}. We formalize IP-PDT as a continuation method on nested manifolds, prove monotone energy guarantees under an acceptance rule, and connect energy error to ground-state fidelity through spectral-gap inequalities. A detailed resource analysis shows that IP-PDT achieves lower total gate cost than both baselines by eliminating most CNOT gates.