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 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.
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.
Variational quantum algorithms are one of the promising methods to obtain quantum advantage in the noisy intermediate scale quantum computers era.They rely on a classical optimization procedure, a cost function and a parameterized quantum circuit to build the solution of a particular problem.Most of the work regarding the quantum circuits part is based on heuristic propositions for the circuit structure and reside only within the borders of VQA applications.In this context, the main objective of our work was the characterization of entanglement generation and distribution of generated states for different PQCs structures.Applying the mean entanglement considering the Scott entanglement measures and the expressibility quantifier, we studied the behavior of 5 possible connectivities between qubits that appear in the contemporary quantum computers: No connections, linear, ring, star and all-to-all, for different number of qubits and circuit concatenations (layers).For two circuit architectures with different local parameterizations, we discussed how entanglement and expressibility are connected, showing that the entanglement generation for only 1 layer is influential for the expressibility evolution as a function of the number of layers.Circuits generating mean and standard deviation for entanglement closer to the uniformly distributed states at 1 layer will have a steeper evolution of expressibility.This result is affected by the local parameterization and number of qubits.We then compared the circuits generated entanglement with the entanglement of paradigmatic states EP R n , GHZ n and W n to understand the entanglement characteristics of the different connections.The results showed how the different connectivities will influence parameterized quantum circuits for applications in VQAs and also that these can present the behavior of pseudorandom quantum circuits.
Parameterized quantum circuits (PQCs) serve as a flexible framework for implementing various quantum algorithms, particularly in quantum machine learning and optimization tasks. However, their training is challenging due to both theoretical constraints and the inherent limitations in current quantum hardware. A judicious choice of PQC architecture is crucial for accurately representing the solution space, and for ensuring convergence in training. Several works introduced measures of expressibility and entangling capability for PQCs, and compared these metrics across different circuit structures, providing insights into how they impact performance and efficiency. However, these investigations often assumed idealized all-to-all qubit connectivity, which is not applicable to most hardware architectures. In practice, additional transpilation steps are performed prior to execution on hardware. The potential impact of better alignment of the PQC structure with the underlying native topology has in fact motivated introduction of hardware-efficient ansatze. In our work, we extend the methodology of measuring PQC efficiency to include more realistic quantum hardware models. We examine several popular hardware topologies, including hexagonal, square, circular, linear, and all-to-all. We evaluate the expressibility and entangling capability of circuits with different connectivity types and various arrangements of rotation gates. We assess the training performance of these PQC types in the context of a variational compiling procedure and identify the preferred gate patterns. This approach provides a more realistic evaluation of PQC performance, thus offering practical insights into their applications on existing quantum devices.
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.
Pseudorandom circuits generate quantum states and unitary operators which are approximately distributed according to the unitarily invariant Haar measure. We explore how several design parameters affect the efficiency of pseudorandom circuits, with the goal of identifying relevant tradeoffs and optimizing convergence. The parameters we explore include the choice of single- and two-qubit gates, the topology of the underlying physical qubit architecture, the probabilistic application of two-qubit gates, as well as circuit size, initialization, and the effect of control constraints. Building on the equivalence between pseudorandom circuits and approximate t-designs, a Markov matrix approach is employed to analyze asymptotic convergence properties of pseudorandom second-order moments to a 2-design. Quantitative results on the convergence rate as a function of the circuit size are presented for qubit topologies with a sufficient degree of symmetry. Our results may be useful towards optimizing the efficiency of random state and operator generation.
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.