Cost function dependent barren plateaus in shallow parametrized quantum circuits

Abstract

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.

Contextual Subspace Variational Quantum Eigensolver

“…One concern for standard VQE as well as for CS-VQE is that the ansatz may suffer from the barren plateau problem [47][48][49], where the gradient of the cost function (in this case expected energy) vanishes exponentially with the system size. It is hoped that for standard VQE, using physically-motivated ansätze like UCC may avoid the barren plateau problem, so since we can use projections of the same ansätze for CS-VQE, this same hope transfers to our case.…”

Qubit-efficient encoding schemes for binary optimisation problems

“…global operator, the nonlinearity of C 1 ( θ) and C 2 ( θ) combined with the O(2 nq ) number of terms involved also do not fulfil the necessary conditions for the proof in Ref. [9]. A more thor-ough investigation of the barren plateaus for nonlinear cost function is left for future work.…”

“…In addition, for cost functions comprising of a linear combination of a Poly(n q ) number of global observables, Ref. [9] predicts the existence of barren plateaus even for shallow circuits. Despite the fact that each observable considered in this work is a projector, i.e.…”

Expressibility of the alternating layered ansatz for quantum computation

“…In this paper, we focus on the third approach given by Ref. [8] that provides a method for devising a specific structure of the HEA ansatz, called the Alternating Layered Ansatz (ALT), which in fact provably does not suffer from the vanishing gradient problem. By definition, the class of ALT is included in that of HEA; the difference is that, while a HEA consists of multiple layers of single-qubit rotation gates and entanglers that in principle combines all qubits in each layer, the entangling gates contained in an ALT is restricted to entangle only local qubits in each layer.…”

“…By definition, the class of ALT is included in that of HEA; the difference is that, while a HEA consists of multiple layers of single-qubit rotation gates and entanglers that in principle combines all qubits in each layer, the entangling gates contained in an ALT is restricted to entangle only local qubits in each layer. With this setting, the authors in [8] derived a strict lower bound of the variance of the gradient for an ALT with its parameters randomly chosen (more precisely, the ensemble of unitary matrices corresponding to each circuit block is 2-design) under the condition that the cost function is local (that is, the cost function is composed of local functions of only a small number of local qubits). By using this lower bound, it was also shown that the vanishing gradient problem can be resolved if the number of layers is of the order O(poly(log n)) where n is the total number of qubits, or roughly speaking if the circuit is shallow.…”