With the advent of hybrid quantum classical algorithms using parameterized quantum circuits the question of how to optimize these algorithms and circuits emerges. In this paper we show that the number of single-qubit rotations in parameterized quantum circuits can be decreased without compromising the expressibility or entangling capability of the circuit. We also compare expressibility and entangling capability across different number of qubits in parameterized quantum circuits. We also consider a parameterized photonics quantum circuit, without any single-qubit rotations, which yields an expressibility and entangling capability comparable to the best regular parameterized quantum circuits.
With the advent of hybrid quantum classical algorithms using parameterized quantum circuits the question of how to optimize these algorithms and circuits emerges. In this paper we show that the number of single-qubit rotations in parameterized quantum circuits can be decreased without compromising the expressibility or entangling capability of the circuit. We also compare expressibility and entangling capability across different number of qubits in parameterized quantum circuits. We also consider a parameterized photonics quantum circuit, without any single-qubit rotations, which yields an expressibility and entangling capability comparable to the best regular parameterized quantum circuits.
Abstract Parameterized quantum circuits (PQCs) play an essential role in the performance of many variational quantum algorithms. One challenge in implementing such algorithms is choosing an effective circuit that well represents the solution space while maintaining a low circuit depth and parameter count. To characterize and identify expressible, yet compact, circuits, several descriptors are proposed, including expressibility and entangling capability, that are statistically estimated from classical simulations. These descriptors are computed for different circuit structures, varying the qubit connectivity and selection of gates. From these simulations, circuit fragments that perform well with respect to the descriptors are identified. In particular, a substantial improvement in performance of two‐qubit gates in a ring or all‐to‐all connected arrangement, compared to that of those on a line, is observed. Furthermore, improvement in both descriptors is achieved by sequences of controlled X‐rotation gates compared to sequences of controlled Z‐rotation gates. In addition, it is investigated how expressibility “saturates” with increased circuit depth, finding that the rate and saturated value appear to be distinguishing features of a PQC. While the correlation between each descriptor and algorithm performance remains to be investigated, methods and results from this study can be useful for algorithm development and design of experiments.
It is essential to select efficient topology of parameterized quantum circuits (PQCs) in variational quantum algorithms (VQAs). However, there are problems in current circuits, i.e. optimization difficulties caused by too many parameters or performance is hard to guarantee. How to reduce the number of parameters (number of single-qubit rotation gates and 2-qubit gates) in PQCs without reducing the performance has become a new challenge. To solve this problem, we propose a novel topology, called Block-Ring (BR) topology, to construct the PQCs. This topology allocate all qubits to several blocks, all-to-all mode is adopt inside each block and ring mode is applied to connect different blocks. Compared with the pure all-to-all topology circuits which own the best power, BR topology have similar performance and the number of parameters and 2-qubit gate reduced from 0(n^2) to 0(mn) , m is a hyperparameter set by ourselves. Besides, we compared BR topology with other topology circuits in terms of expressibility and entangling capability. Considering the effects of different 2-qubit gates on circuits, we also make a distinction between controlled X-rotation gates and controlled Z-rotation gates. Finally, the 1- and 2-layer configurations of PQCs are taken into consideration as well, which shows the BR's performance improvement in the condition of multilayer circuits.
Parameterized quantum circuits are central to the development of variational quantum algorithms in the NISQ era. A key feature of these circuits is their ability to generate an expressive set of quantum states, enabling the approximation of solutions to diverse problems. The expressibility of such circuits can be assessed by analyzing the ensemble of states produced when their parameters are randomly sampled, a property closely tied to quantum complexity. In this work, we compare different classes of parameterized quantum circuits with a prototypical family of universal random circuits to investigate how rapidly they approach the asymptotic complexity defined by the Haar measure. We find that parameterized circuits exhibit faster convergence in terms of the number of gates required, as quantified through expressibility and majorization-based complexity measures. Moreover, the topology of qubit connections proves crucial, significantly affecting entanglement generation and, consequently, complexity growth. The majorization criterion emerges as a valuable complementary tool, offering distinct insights into the behavior of random state generation in the considered circuit families.
Parameterized quantum circuits (PQCs), which are essential for variational quantum algorithms, have conventionally been optimized by parameterized rotational angles of single-qubit gates around predetermined set of axes. We propose a new method to optimize a PQC by continuous parameterization of both the angles and the axes of its single-qubit rotations. The method is based on the observation that when rotational angles are fixed, optimal axes of rotations can be computed by solving a system of linear equations whose coefficients can be determined from the PQC with small computational overhead. The method can be further simplified to select axes freely from continuous parameters with rotational angles fixed to π. We show the simplified free-axis selection method has better expressibility against other structural optimization methods when measured with Kullback-Leibler (KL) divergence. We also demonstrate PQCs with free-axis selection are more effective to search the ground states of Heisenberg models and molecular Hamiltoni- ans. Because free-axis selection allows designing PQCs without specifying their single-qubit rotational axes, it may significantly improve the handiness of PQCs.
Abstract While scalable error correction schemes and fault tolerant quantum computing seem not to be universally accessible in the near sight, the efforts of many researchers have been directed to the exploration of the contemporary available quantum hardware. Due to these limitations, the depth and dimension of the possible quantum circuits are restricted. This motivates the study of circuits with parameterized operations that can be classically optimized in hybrid methods as variational quantum algorithms, enabling the reduction of circuit depth and size. The characteristics of these Parameterized Quantum Circuits (PQCs) are still not fully understood outside the scope of their principal application, motivating the study of their intrinsic properties. In this work, we analyse the generation of random states in PQCs under restrictions on the qubits connectivities, justified by different quantum computer architectures. We apply the expressibility quantifier and the average entanglement as diagnostics for the characteristics of the generated states and classify the circuits depending on the topology of the quantum computer where they can be implemented. As a function of the number of layers and qubits, circuits following a Ring topology will have the highest entanglement and expressibility values, followed by Linear/All-to-all almost together and the Star topology. In addition to the characterization of the differences between the entanglement and expressibility of these circuits, we also place a connection between how steep is the increase on the uniformity of the distribution of the generated states and the generation of entanglement. Circuits generating average and standard deviation for entanglement closer to values obtained with the truly uniformly random ensemble of unitaries present a steeper evolution when compared to others.
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.