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arXiv:2308.10791

A Block-Ring connected Topology of Parameterized Quantum Circuits

Wenjie Liu, Qingshan Wu

Published Aug 21, 2023• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2308.10791

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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.
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.
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.
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.
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 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.
Variational quantum algorithms, which utilize Parametrized Quantum Circuits (PQCs), are promising tools to achieve quantum advantage for optimization problems on near-term quantum devices. Their PQCs have been conventionally constructed from parametrized rotational angles of single-qubit gates around predetermined set of axes, and two-qubit entangling gates, such as CNOT gates. We propose a method to construct a PQC by continuous parametrization of both the angles and the axes of its single-qubit rotation gates. 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 toπ\piπ. 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 Hamiltonians for condensed matter physics, quantum chemistry and combinatorial optimization. Because free-axis selection allows designing PQCs without specifying their single-qubit rotational axes, it may significantly improve the handiness of PQCs.