Barren plateaus in quantum neural network training landscapes

Abstract

Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied.

Fig. 1

Cartoon of concentration of quantum observables. The sphere depicts the phenomenon of concentration of measure in quantum space: the fraction of states that fall outside a fixed angular distance from zero along any coordinate decreases exponentially in the number of qubits. This implies a flat plateau where observables concentrate on their average over Hilbert space and the gradient is exponentially small. The fact that only an exponentially small fraction of states fall outside of this band means that searches resembling random walks will have an exponentially small probability of exiting this “barren plateau”

Fig. 2

Structure of quantum circuits. a The generic subunit of circuits we study in this work, with a parameterized component U_l(θ_l) and non-parameterized unit W_l for each layer l. b Example schematic of the 1D random circuits used in our numerical experiments. The circuit begins with $R_Y\left(\frac{\pi }{4}\right)$ gates applied to all qubits followed by a specified number of layers of randomly chosen Pauli rotations applied to each qubit and then a 1D ladder of controlled Z gates. The initial $R_Y\left(\frac{\pi }{4}\right)$ gates are not repeated in each layer. The indices i and j in θ_i,j index the layer and qubit, respectively. For each layer and qub it P_i,j∈{X, Y, Z} and θ_i,j∈[0,2π) are sampled independently

Fig. 3

Exponential decay of variance. The sample variance of the gradient of the energy for the first circuit component of a two-local Pauli term $\left({\partial }_{{\theta }_{1,1}}E\right)$ plotted as a function of the number of qubits on a semi-log plot. As predicted, an exponential decay is observed as a function of the number of qubits, n, for both the expected value and its spread. The slope of the fit line is indicative of the rate of exponential decay as determined by the operator

Fig. 4

Convergence to 2-design limit. Here we show the sample variance of the gradient of the energy for the first circuit component of a two-local Pauli term $\left({\partial }_{{\theta }_{1,1}}E\right)$ plotted as a function of the number of layers, L, in a 1D quantum circuit. The different lines correspond to all even numbers of qubits between 2 and 24, with 2 qubits being the top line, and the rest being ordered by qubit number. The dotted black lines depict the 2-design asymptotes for this Hamiltonian as determined by our analytic results. This shows the convergence of the second moment as a function of the number of layers to a fixed value determined by the number of qubits