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.
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.
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.
Expressibility is a crucial factor of a Parameterized Quantum Circuit (PQC). In the context of Variational Quantum Algorithms (VQA) based Quantum Machine Learning (QML), a QML model composed of highly expressible PQC and sufficient number of qubits is theoretically capable of approximating any arbitrary continuous function. While much research has explored the relationship between expressibility and learning performance, as well as the number of layers in PQCs, the connection between expressibility and PQC structure has received comparatively less attention. In this paper, we analyze the connection between expressibility and the types of quantum gates within PQCs using a Gradient Boosting Tree model and SHapley Additive exPlanations (SHAP) values. Our analysis is performed on 1,615 instances of PQC derived from 19 PQC topologies, each with 2-18 qubits and 1-5 layers. The findings of our analysis provide guidance for designing highly expressible PQCs, suggesting the integration of more RX or RY gates while maintaining a careful balance with the number of CNOT gates. Furthermore, our evaluation offers an additional evidence of expressibility saturation, as observed by previous studies.
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.
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~(PQCs) constitute a central building block of variational quantum algorithms~(VQAs) and quantum machine learning~(QML) methods. Existing ansatz designs often adopt hardware-agnostic or simplified 1D chain/ring entanglement patterns. However, as quantum hardware continues to develop, native 2D connectivity patterns, such as planar superconducting-qubit architectures, are becoming increasingly important. Inspired by this hardware structure, we construct a native 2D pairwise ansatz and compare its expressibility and trainability with representative 1D ansatze at identical layer depths, despite their different circuit depths. For the fixed 16-qubit system, the 2D ansatz has the smallest KL divergence at L=1 and 2, and its second-order frame potential approaches the theoretical lower bound more rapidly at shallow layer counts than the frame potentials of the three 1D ansatze. We also evaluate the gradient variance of the Pauli-Z-string expectation value ⟨Z0⊗⋯⊗Z15⟩ with respect to the first Ry angle. For this Pauli-Z string and fixed parameter, the gradient variance is smaller for the 2D circuit at L=1--4. The differences narrow at L=5, and the four ansatze yield statistically compatible variances at L=6.