Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms (1905.10876v1)

Abstract: Parameterized quantum circuits play an essential role in the performance of many variational hybrid quantum-classical (HQC) algorithms. One challenge in implementing such algorithms is to choose an effective circuit that well represents the solution space while maintaining a low circuit depth and number of parameters. To characterize and identify expressible, yet compact, parameterized circuits, we propose several descriptors, including measures of expressibility and entangling capability, that can be statistically estimated from classical simulations of parameterized quantum circuits. We compute these descriptors for different circuit structures, varying the qubit connectivity and selection of gates. From our simulations, we identify circuit fragments that perform well with respect to the descriptors. In particular, we quantify the 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. Furthermore, we quantify the improvement in expressibility and entangling capability achieved by sequences of controlled X-rotation gates compared to sequences of controlled Z-rotation gates. In addition, we investigate how expressibility "saturates" with increased circuit depth, finding that the rate and saturated-value appear to be distinguishing features of a parameterized quantum circuit template. While the correlation between each descriptor and performance of an algorithm remains to be investigated, methods and results from this study can be useful for both algorithm development and design of experiments for general variational HQC algorithms.

• The paper introduces expressibility and entangling capability as key descriptors for evaluating PQC performance using state distribution and Meyer-Wallach metrics.
• Numerical simulations show that expressibility saturates beyond a certain circuit depth and that two-qubit gate configurations, especially CRX, significantly influence outcomes.
• The insights guide the design of resource-efficient PQCs for NISQ devices while addressing optimization challenges such as barren plateaus.

Overview of "Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms"

The paper "Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms" presents a comprehensive analysis of parameterized quantum circuits (PQCs) essential for variational hybrid quantum-classical (HQC) algorithms, particularly in the context of Noisy Intermediate-Scale Quantum (NISQ) devices. The authors introduce a framework to characterize the expressibility and entangling capability of these circuits, aiming to optimize their performance while maintaining a compact form.

Key Insights

1. Descriptors for PQCs:
   • The paper proposes two primary descriptors: expressibility and entangling capability. Expressibility is defined as a circuit's ability to generate a representative set of states in the Hilbert space. It is quantified by measuring the deviation in state distribution from that of Haar-random states. Entangling capability is assessed using the Meyer-Wallach entanglement measure, providing a global measure of multi-partite entanglement for a given circuit.
2. Numerical Analysis:
   • Classical simulations of various circuit templates are conducted to compute these descriptors across multiple configurations of single-qubit and two-qubit gates. The paper highlights the performance of two-qubit gate configurations such as nearest-neighbor, ring topology, and all-to-all connectivities.
3. Expressibility Saturation:
   • A key observation is the phenomenon of expressibility "saturation," where increasing the circuit depth beyond a certain point does not significantly increase expressibility. This feature suggests a trade-off between circuit depth and expressibility, providing a useful criterion for selecting circuit layers.
4. Comparison of Two-Qubit Gates:
   • The paper outlines differences in expressibility and entangling capability between circuits using controlled-Z rotations (CRZ) versus controlled-X rotations (CRX), with CRX gates generally demonstrating superior performance due to their non-commutative nature.
5. Practical Implications:
   • For experimental implementation on NISQ devices, circuits with favorable expressibility and entangling capability but lower resource demands (e.g., fewer parameters, shorter depth) are preferable. The insights from the paper could guide the design and selection of PQCs to match the capabilities and constraints of specific quantum devices.

Implications and Future Directions

The research underscores the necessity of optimizing PQCs in HQC algorithms for enhanced performance on NISQ devices. By accurately characterizing and comparing circuit templates, the proposed framework provides a foundation for future algorithm development and experimental quantum computing applications.

The findings signal new avenues for exploring the correlations between expressibility and algorithm performance. Questions remain regarding the optimal expressibility required for different applications and how subspace expressibility might come into play when leveraging symmetry in quantum problems.

The paper also suggests potential improvements in circuit optimization, emphasizing the need to balance expressibility and optimizability to counter the adverse effects of barren plateaus in the parameter landscape.

Ultimately, this work lays the groundwork for more systematic approaches to the design of parameterized quantum circuits, catering to the evolving requirements of quantum computing technologies.