An initialization strategy for addressing barren plateaus in parametrized quantum circuits

Grant, Edward; Wossnig, Leonard; Ostaszewski, Mateusz; Benedetti, Marcello

doi:10.22331/q-2019-12-09-214

Cited by 379 publications

(295 citation statements)

References 61 publications

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“…point for the optimization of p + 1 layers and by setting α p+1 such that L p+1 (α p+1 ) is an identity. This strategy also avoids barren plateaus [27,28] and helps to mitigate the problem of local minima, as we discuss in Appendix C of Supplementary Material (SM). Next, we argue that C is closely connected to the accuracy of the eigensystem.…”

Section: Resultsmentioning

confidence: 99%

Variational quantum state diagonalization

et al. 2019

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Variational hybrid quantum-classical algorithms are promising candidates for near-term implementation on quantum computers. In these algorithms, a quantum computer evaluates the cost of a gate sequence (with speedup over classical cost evaluation), and a classical computer uses this information to adjust the parameters of the gate sequence. Here we present such an algorithm for quantum state diagonalization. State diagonalization has applications in condensed matter physics (e.g., entanglement spectroscopy) as well as in machine learning (e.g., principal component analysis). For a quantum state ρ and gate sequence U , our cost function quantifies how far U ρU † is from being diagonal. We introduce novel short-depth quantum circuits to quantify our cost. Minimizing this cost returns a gate sequence that approximately diagonalizes ρ. One can then read out approximations of the largest eigenvalues, and the associated eigenvectors, of ρ. As a proof-of-principle, we implement our algorithm on Rigetti's quantum computer to diagonalize one-qubit states and on a simulator to find the entanglement spectrum of the Heisenberg model ground state.

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Section: Resultsmentioning

confidence: 99%

Variational quantum state diagonalization

et al. 2019

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“…In particular, with an expressible circuit, choosing a random starting point for a VQE optimization is not a reasonable approach. Methods for circumventing this issue have been proposed, suggesting ways to improve the “optimizability” of a parameterized quantum circuit. These techniques indicate a tension between being expressible and being optimizable (i.e., having sufficient variation in the cost function).…”

Section: Discussionmentioning

confidence: 99%

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

2019

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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.

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“…This is in contrast to McClean et al 9 , who showed vanishing gradients for deep PQCs. They noted that randomly initializing θ for a V(θ) that forms a 2-design leads to a barren plateau, i.e., with the gradient vanishing exponentially in the number of qubits, n. Their work implied that researchers must develop either clever parameter initialization strategies 32,33 or clever PQCs ansatzes 4,34,35 . Similarly, our work implies that researchers must carefully weigh the balance between trainability and operational relevance when choosing C.…”

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confidence: 99%

Cost function dependent barren plateaus in shallow parametrized quantum circuits

et al. 2021

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Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is $${\mathcal{O}}(\mathrm{log}\,n)$$ O ( log n ) . Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.

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An initialization strategy for addressing barren plateaus in parametrized quantum circuits

Cited by 379 publications

References 61 publications

Variational quantum state diagonalization

Variational quantum state diagonalization

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

Cost function dependent barren plateaus in shallow parametrized quantum circuits

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