Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.
Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.
In this paper, we focus on the task of optimizing the parameters in Parametrized Quantum Circuits (PQCs). While popular algorithms, such as Simultaneous Perturbation Stochastic Approximation (SPSA), limit the number of circuit-execution to two per iteration, irrespective of the number of parameters in the circuit, they have their own challenges. These methods use central-differences to calculate biased estimates of directional derivatives. We show, both theoretically and numerically, that this may lead to instabilities in \emph{training} the PQCs. To remedy this, we propose Stochastic Shadow Descent (\texttt{SSD}), which uses random-projections (or \emph{shadows}) of the gradient to update the parameters iteratively. We eliminate the bias in directional derivatives by employing the Parameter-Shift Rule, along with techniques from Quantum Signal Processing, to construct a quantum circuit that parsimoniously computes \emph{unbiased estimates} of directional derivatives. Finally, we prove the convergence of the \texttt{SSD} algorithm, provide worst-case bounds on the number of iterations, and numerically demonstrate its efficacy.
We present a novel method for determining gradients of parameterised quantum circuits (PQCs) in hybrid quantum-classical machine learning models by applying the multivariate version of the simultaneous perturbation stochastic approximation (SPSA) algorithm. The gradients of PQC layers can be calculated with an overhead of two evaluations per circuit per forward-pass independent of the number of circuit parameters, compared to the linear scaling of the parameter shift rule. These are then used in the backpropagation algorithm by applying the chain rule. We compare our method to the parameter shift rule for different circuit widths and batch sizes, and for a range of learning rates. We find that, as the number of qubits increases, our method converges significantly faster than the parameter shift rule and to a comparable accuracy, even when considering the optimal learning rate for each method.
Optimisation of training still faces challenges due to an ill-conditioned loss surface and noisy gradient signals; at the same time, limited computational resources also restrict training options. Quantum Gradient Descent as a Hybrid Optimisation Strategy for Neural Network Training. The proposed noise-resilient quantum gradient descent framework maps selected gradient components to parameterised quantum circuits, estimates directional derivatives via parameter-shift measurements, and returns stable updates to the classical optimizer. It is not directly substituted for backpropagation; instead, it is a type of controllable gradient module that can be added to sensitive layers or low-dimensional projection heads. Add adaptive shot scheduling, damped momentum and layer-wise trust constraints to reduce estimator variance and prevent unstable quantum updates. Representative classification tasks are selected as the subjects of experiments for convergence speed, validation accuracy, gradient fidelity, measurement cost and noise robustness. Based on the above analysis, the new optimizer outperforms standard SGD in terms of final accuracy by 1.6% - 3.2% when using the same number of epochs; at the same time, it reduces the median gradient variance by 28.4% and maintains stable convergence even with an increase in simulated depolarizing noise from 0.001 to 0.015. Based on the above, it can be seen that Quantum Gradient Descent is not an all-purpose optimizer but rather a target-specific training accelerator for gradient-sensitive modules that can trade off sampling cost for more informative descent directions in measurement-aware update design.
Variational quantum circuits characterise the state of a quantum system through the use of parameters that are optimised using classical optimisation procedures that typically rely on gradient information. The circuit-execution complexity of estimating the gradient of expectation values grows linearly with the number of parameters in the circuit, thereby rendering such methods prohibitively expensive. In this paper, we address this problem by proposing a novel Quantum-Gradient Sampling algorithm that requires the execution of at most two circuits per iteration to update the optimisable parameters, and with a reduced number of shots. Furthermore, our proposed method achieves similar asymptotic convergence rates to classical gradient descent, and empirically outperforms gradient descent, randomised coordinate descent, and SPSA.
In the context of Noisy Intermediate-Scale Quantum (NISQ) computing, parameterized quantum circuits (PQCs) are a promising paradigm for quantum sensing, optimal control, optimization, and machine learning on near-term quantum hardware. Gradient-based methods are crucial for analyzing PQCs and offer significant advantages in the convergence rates of Variational Quantum Algorithms (VQAs) compared to gradient-free methods. However, existing gradient estimation techniques-such as Finite Difference, Parameter Shift Rule, Hadamard Test, and Direct Hadamard Test-can yield suboptimal circuits for certain PQCs. To address these limitations, we introduce the Flexible Hadamard Test, which can invert the roles of ansatz generators and observables in first-order gradient estimation, enabling the use of measurement optimization techniques to improve circuit efficiency. For higher-order derivatives, we propose thek-fold Hadamard Test, which evaluates higher-order partial derivatives using a single quantum circuit. Additionally, we present Quantum Automatic Differentiation (QAD)-a unified gradient estimation framework that adaptively selects the most efficient method for each parameter in a PQC, departing from the standard practice of using a single method uniformly. Through extensive numerical experiments, we demonstrate that our proposed methods yield up to a linear-factor reduction in circuit execution cost for real VQA applications. These advances accelerate quantum algorithm training and provide practical utility in the NISQ era.
Variational quantum algorithms, such as the variational quantum eigensolver (VQE), have a very high cost due to the large number of measurement outcomes that must be acquired from the quantum device during training. This is due in large part to the use of the parameter shift rule for estimating the gradient at each step, which requires two circuit evaluations per parameter. It was recently shown that if all the gate generators in the ansatz commute with each other, then the complete gradient vector can be found from a single evaluation of an augmented circuit (Bowles et al., 2023), and this has the potential to significantly reduce the quantum cost. Unfortunately, this is quite a severe restriction, as it prevents the ansatz from having a nontrivial dynamical Lie algebra (DLA). In order to obtain a more expressive ansatz than permitted by the commutation requirement, we construct it iteratively and train it in a layerwise fashion using the parallel gradient method. For VQE training of an 8-qubit transverse field Ising model (TFIM) on an ideal simulator, we show that this approach is able to reach a lower energy using fewer circuit evaluations than the parameter shift rule with a more conventional ansatz.