We propose a sequential minimal optimization method for quantum-classical hybrid algorithms, which converges faster, robust against statistical error, and hyperparameter-free.Specifically, the optimization problem of the parameterized quantum circuits is divided into solvable subproblems by considering only a subset of the parameters.In fact, if we choose a single parameter, the cost function becomes a simple sine curve with period 2π , and hence we can exactly minimize with respect to the chosen parameter.Furthermore, even in general cases, the cost function is given by a simple sum of trigonometric functions with certain periods and hence can be minimized by using a classical computer.By repeatedly performing this procedure, we can optimize the parameterized quantum circuits so that the cost function becomes as small as possible.We perform numerical simulations and compare the proposed method with existing gradient-free and gradient-based optimization algorithms.We find that the proposed method substantially outperforms the existing optimization algorithms and converges to a solution almost independent of the initial choice of the parameters.This accelerates almost all quantum-classical hybrid algorithms readily and would be a key tool for harnessing near-term quantum devices.
We propose a sequential minimal optimization method for quantum-classical hybrid algorithms, which converges faster, robust against statistical error, and hyperparameter-free.Specifically, the optimization problem of the parameterized quantum circuits is divided into solvable subproblems by considering only a subset of the parameters.In fact, if we choose a single parameter, the cost function becomes a simple sine curve with period 2π , and hence we can exactly minimize with respect to the chosen parameter.Furthermore, even in general cases, the cost function is given by a simple sum of trigonometric functions with certain periods and hence can be minimized by using a classical computer.By repeatedly performing this procedure, we can optimize the parameterized quantum circuits so that the cost function becomes as small as possible.We perform numerical simulations and compare the proposed method with existing gradient-free and gradient-based optimization algorithms.We find that the proposed method substantially outperforms the existing optimization algorithms and converges to a solution almost independent of the initial choice of the parameters.This accelerates almost all quantum-classical hybrid algorithms readily and would be a key tool for harnessing near-term quantum devices.
Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.
Abstract In variational quantum algorithms, it is important to balance conflicting requirements of expressibility and trainability of a parameterized quantum circuit (PQC). However, appropriate PQC designs are not necessarily trivial. Here, we propose an algorithm for optimizing the PQC structure, where single-qubit gates are sequentially replaced by the optimal ones via diagonalization of a matrix whose elements are evaluated on slightly modified circuits. This replacement leads to a better approximation of target states with limited circuit depth. Furthermore, we clarify the existence of a barren plateau in the sequential optimization in terms of the spectrum concentration of the matrix, which defines the cost landscape with respect to changes in the target gate. Then, we rigorously show the concentration is no faster than polynomials in the number of qubits when an n-qubit PQC depth is O(logn) using local observables. Finally, numerical experiments are provided to show the convergence of our method which is faster than gradient-based optimizers. Our results provide evidences for sequential optimizers as better alternatives to optimize PQCs on near-term quantum devices.
The variational quantum eigensolver (VQE) is a hybrid algorithm to find the minimum eigenvalue/vector of the given Hamiltonian by optimizing a parameterized quantum circuit (PQC) using a classical computer. Sequential optimization methods, which are often used in quantum circuit tensor networks, are popular for optimizing the parameterized gates of PQCs. In this paper, we focus on the case where the components to be optimized are single-qubit gates, in which the analytic optimization of a single-qubit gate is sequentially performed. The analytical solution is given by diagonalization of a matrix whose elements are computed from the expectation values of observables specified by a set of predetermined parameters, which we refer to as the parameter configurations. In this paper, we first show that the optimization accuracy significantly depends on the choice of parameter configurations owing to the statistical errors in the expectation values. We then identify a metric that quantifies the optimization accuracy of a parameter configuration for all possible statistical errors, named configuration overhead/cost or C-cost. We theoretically provide the lower bound of C-cost and show that, for the minimum size of parameter configurations, the lower bound is achieved if and only if the parameter configuration satisfies the so-called equiangular line condition. Finally, we provide numerical experiments demonstrating that the optimal parameter configuration exhibits the best result in several VQE problems. We hope that this general statistical methodology will enhance the efficacy of sequential optimization of PQCs for solving practical problems with near-term quantum devices.
Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of s t o c h a s t i c gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with k measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of k . In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases k -shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain ``doubly stochastic'' gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.
Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with k measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of k. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values - originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances - and we show that in these cases k-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain "doubly stochastic" gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.
Parametric quantum circuits are usually trained by classical, local optimizers, which use the quantum computer only to evaluate the cost function, or its gradient, one point at a time. Quantum Hamiltonian Descent (QHD) takes a different route: in this quantum algorithm for continuous optimization, the parameters become the position of a quantum particle, with the cost function as its potential, and two time schedules control kinetic and potential energy so that the wavefunction explores the whole landscape and tunnels towards the global minimum. QHD, however, assumes a classically computable cost function, and its digital implementation is poorly understood: standard error bounds grow exponentially with the number of qubits per parameter, and the schedules are borrowed from classical optimization. Here we propose to train parametric circuits with QHD entirely on the quantum computer, through an online step in which the qubits encoding the parameters coherently control the circuit on a second register. We prove that, for cost functions with a smooth periodic extension, including those of parametric circuits, the simulation error admits a bound independent of the grid resolution, and that for circuits the bound grows at most linearly with the number of parameters. We also show that the ideal dynamics depends on the two schedules only through a single curve, which controls the adiabaticity of the search, and that the remaining freedom can be used to minimize the gate count. Simulations on benchmark functions and parametric circuits show that, at equal cost, schedules designed in this way markedly increase the probability of finding the global minimum.