We consider metric-aware quantum algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object.For example, the recently introduced quantum natural gradient approach uses the Fisher matrix as a metric tensor to correct the gradient vector for the codependence of the circuit parameters.We rigorously characterize and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries.Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.
We consider metric-aware quantum algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object.For example, the recently introduced quantum natural gradient approach uses the Fisher matrix as a metric tensor to correct the gradient vector for the codependence of the circuit parameters.We rigorously characterize and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries.Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.
The Variational Quantum Eigensolver (VQE) is one of the most promising algorithms for current quantum devices. It employs a classical optimizer to iteratively update the parameters of a variational quantum circuit in order to search for the ground state of a given Hamiltonian. The efficacy of VQEs largely depends on the optimizer employed. Recent studies suggest that Quantum Natural Gradient Descent (QNG) can achieve faster convergence than vanilla gradient descent (VG), but at the cost of additional quantum resources to estimate Fubini-Study metric tensor in each optimization step. The Fubini-Study metric tensor used in QNG is related to the entire quantum state space and does not incorporate information about the target Hamiltonian. To take advantage of the structure of the Hamiltonian and address the limitation of additional computational cost in QNG, we propose Hamiltonian-aware Quantum Natural Gradient Descent (H-QNG). In H-QNG, we propose to use the Riemannian pullback metric induced from the lower-dimensional subspace spanned by the Hamiltonian terms onto the parameter space. We show that H-QNG inherits the desirable features of both approaches: the low quantum computational cost of VG and the reparameterization-invariance of QNG. We also validate its performance through numerical experiments on molecular Hamiltonians, showing that H-QNG achieves faster convergence to chemical accuracy while requiring fewer quantum computational resources.
Abstract The Quantum Fisher Information Matrix (QFIM) quantifies how sensitive parameterized quantum states are to changes in their parameters. Recently, it has been used to improve variational quantum algorithm optimization through geometry-aware techniques. However, estimating the QFIM—particularly its off-block-diagonal elements—requires substantial resources. To address this, we introduce a novel protocol that efficiently computes these elements for commuting-block variational circuits. Our approach reduces the number of quantum state preparations from O(m2) O ( m 2 ) to O(L2) O ( L 2 ) , where m is the number of parameters and L the number of circuit layers. This also lowers classical measurement and post-processing requirements, improving computational efficiency.
Quantum natural gradient has emerged as a superior minimisation technique in quantum variational algorithms. Classically simulating the algorithm running on near-future quantum hardware is paramount in its study, as it is for all variational algorithms. In this case, state-vector simulation of the P-parameter/gate ansatz circuit does not dominate the runtime; instead, calculation of the Fisher information matrix becomes the bottleneck, involving O(P^3) gate evaluations, though this can be reduced to O(P^2) gates by using O(P) temporary state-vectors. This is similar to the gradient calculation subroutine dominating the simulation of quantum gradient descent, which has attracted HPC strategies and bespoke simulation algorithms with asymptotic speedups. We here present a novel simulation strategy to precisely calculate the quantum natural gradient in O(P^2) gates and O(1) state-vectors. While more complicated, our strategy is in the same spirit as that presented for gradients in Reference 6, and involves iteratively evaluating recurrent forms of the Fisher information matrix. Our strategy uses only "apply gate", "clone state" and "inner product" operations which are present in practically all quantum computing simulators. It is furthermore compatible with parallelisation schemes, like hardware acceleration and distribution. Since our scheme leverages a form of the Fisher information matrix for strictly unitary ansatz circuits, it cannot be simply extended to density matrix simulation of quantum natural gradient with non-unitary circuits.
Abstract Variational Quantum Algorithms (VQAs) are widely regarded as the most promising route to practical quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) hardware, but their trainability is threatened by the barren plateau phenomenon, in which cost-function gradients vanish exponentially with system size and circuit depth. Standard first-order optimizers, which treat the parameter space of a parameterized quantum circuit (PQC) as flat Euclidean space, are especially vulnerable to this effect and to the additional gradient suppression introduced by hardware noise. This extended treatment presents (i) a from-first-principles derivation of Quantum Natural Gradient (QNG) descent, tracing its lineage from Amari's classical information geometry [1] through the Fubini-Study metric to the Quantum Fisher Information Matrix (QFIM); (ii) a complete taxonomy of barren-plateau mechanisms (expressibility-induced, entanglement-induced, cost-function-locality-induced, and noise-induced); (iii) a comparative survey of the broader NISQ optimizer landscape (SPSA, Adam, COBYLA, Rotosolve, Quantum Natural SPSA, and reduced-cost QFIM estimators); and (iv) a controlled simulation study comparing QNG against vanilla gradient descent on a 4-qubit weighted MaxCut problem under three physically motivated NISQ noise models. Across 50 independent trials, QNG raised the convergence success rate from 30% to 95%, reduced the mean iteration count to convergence by a factor of roughly 6, and achieved a 16% net wall-clock speedup despite the added per-iteration cost of QFIM estimation. We give the full theoretical basis for these gains, an exhaustive discussion of their limitations, an A-to-Z glossary of the field's terminology, and a staged roadmap toward physical hardware validation. Figure placements throughout the document are annotated with detailed image-generation prompts so that the accompanying visual assets can be produced independently of this text.
Sampling noisy intermediate-scale quantum devices is a fundamental step that converts coherent quantum-circuit outputs to measurement data for running variational quantum algorithms that utilize gradient and Hessian methods in cost-function optimization tasks. This step, however, introduces estimation errors in the resulting gradient or Hessian computations. To minimize these errors, we discuss tunable numerical estimators, which are the finite difference (including their generalized versions) and scaled parameter-shift estimators [introduced in Phys. Rev. A 103, 012405 (2021)], and propose operational circuit-averaged methods to optimize them. We show that these optimized numerical estimators offer estimation errors that drop exponentially with the number of circuit qubits for a given sampling-copy number, revealing a direct compatibility with the barren-plateau phenomenon. In particular, there exists a critical sampling-copy number below which an optimized difference estimator gives a smaller average estimation error in contrast to the standard (analytical) parameter-shift estimator, which exactly computes gradient and Hessian components. Moreover, this critical number grows exponentially with the circuit-qubit number. Finally, by forsaking analyticity, we demonstrate that the scaled parameter-shift estimators beat the standard unscaled ones in estimation accuracy under any situation, with comparable performances to those of the difference estimators within significant copy-number ranges, and are the best ones if larger copy numbers are affordable.
Quantum computing has become a potential way to solve complex problems that conventional computers can’t handle, especially those involving strongly correlated quantum systems. Classical simulation of these systems becomes exponentially more challenging as the number of interacting particles rises, resulting in significant computational and memory limitations. As a result, classical computational approaches are unable to handle many significant physical and chemical systems, including molecular structures, biological molecules, and quantum materials. Quantum computers operate using the same quantum mechanical principles as these systems. Consequently, they provide a natural framework for exploring large Hilbert spaces and capturing complex quantum correlations. This thesis begins by discussing the fundamental principles of quantum mechanics that are applicable to computation, such as qubits, superposition, and entanglement. The construction of quantum circuits is discussed after a discussion of basic quantum gates and their matrix representations. These ideas are the foundation of quantum algorithms and offer the essential framework for comprehending the modeling and manipulation of complicated quantum states using quantum hardware. Building on this foundation, the thesis focuses on Variational Quantum Algorithms (VQAs), which offer a feasible hybrid quantum-classical approach to optimizing near-term quantum devices. In VQAs, a parameterized quantum circuit is used to approximate the ground-state energy of the quantum system. A classical optimizer then iteratively updates the variational parameters to minimize a cost function, usually the expectation value of the system’s Hamiltonian. A thorough analysis of the function of gradient-based optimization is conducted, encompassing the parameter-shift rule that is employed to calculate precise energy derivatives. The discussion is expanded to include major applications such as the Quantum Approximate Optimization Algorithm and developing approaches to quantum machine learning, emphasizing the importance of efficient and accurate gradient evaluation. A major challenge in the real-world application of VQAs is the high resource cost of gradient estimation using conventional direct measurement techniques. These techniques require repeated projective measurements and several executions of parameter-shifted quantum circuits, which lead to significant measurement overhead and deeper circuits. Quantum Non-Demolition Measurement (QNDM) is examined in this thesis as a potential framework for gradient evaluation to get around this limitation. Based on recent theoretical advances, QNDM introduces an ancillary detector qubit that coherently stores gradient information within its phase. Derivatives can then be estimated indirectly through detector measurements. This allows gradient information to be extracted without destroying the quantum state and with a greatly reduced number of measurements. The efficiency of QNDM is evaluated by extending the study to systems with bigger Hamiltonians and greater molecular complexity, as well as by comparing it in detail with direct measurement techniques utilizing benchmark molecular systems like LiH, Li2, and H2. Although both methods successfully converge to the ground-state energy, as shown by numerical results, QNDM consistently uses fewer quantum resources in terms of logical gate operations, measurement shots, and circuit executions. Additionally, as system size grows, QNDM’s resource advantage becomes more noticeable, suggesting that it has favorable scalability for bigger molecular systems. This thesis concludes that, while both direct measurement and quantum non-demolition measurement approaches allow for precise variational optimization, QNDM gives a significant decrease in quantum resources. These results establish QNDM as an effective and scalable method for developing quantum simulations of realistic physical and chemical systems and for executing variational quantum algorithms on near-term quantum hardware.