Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum‐Classical Algorithms

By Sukin Sim, Peter D. Johnson, Alán Aspuru-Guzik. Advanced Quantum Technologies, 2019

1,132 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/95a8c1f501de77e5019a4094ab405044d2b96ccc

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Citations in the corpus

Diagnosing Barren Plateaus with Tools from Quantum Optimal Control (2021, methodology): “In this context, one can quantify the expressibility of an ansatz by comparing the distribution of unitaries obtained from U(θ) to the maximally expressive uniform (Haar) distribution UH [65].”
Transfer learning in hybrid classical-quantum neural networks (2019, background): “One of the possible quantum generalizations of feedforward neural networks can be given in terms of variational quantum circuits [17, 24, 30, 30, 33, 34, 39, 41, 43].”
Effect of data encoding on the expressive power of variational quantum-machine-learning models (2020, background): “The majority of quantum machine learning papers concerned with questions of expressivity and universality [19–22], however, interpret these concepts from a quantum information perspective, which asks whether a circuit can express any quantum computation, not any function in the inputs.”
Barren plateaus in variational quantum computing (2024, neutral): “For example, the optimizer might get stuck in a suboptimal local minimum 12–18 , a solution might not exist in the parameter hyperspace 19,20 , or the landscape might exhibit BPs — the focus of this Review.”
Variational quantum algorithms (2020, methodology): “Other expressibility measures can be considered as well [61], and the expressibility of different ansatze is investigated further in [62].”
Estimating the gradient and higher-order derivatives on quantum hardware (2020, background): “[5] Sukin Sim, Peter D Johnson, and Alan Aspuru-Guzik.”
Connecting ansatz expressibility to gradient magnitudes and barren plateaus (2021, background): “These measures generalize the notion of expressibility introduced in [47] where the expressiblity was defined in terms of ερU for ρ = |0〉〈0|.”
Quantum variational algorithms are swamped with traps (2022, neutral): “This is particularly exciting given the variety of no-go results via statistical learning theory, which demonstrate that in the worst case these models are not trainable via stochastic gradient-based methods [9, 52, 24, 51].”