Combinatorics

Physics

Topology (electrical circuits)

Quantum phase estimation algorithm

Pure mathematics

Qubit

Dimension (graph theory)

Mathematics

Quantum error correction

Algorithm

Statistical physics

Quantum algorithm

Quantum circuit

Parameterized complexity

Computer science

Quantum computer

Quantum mechanics

Quantum gate

Hilbert space

Quantum

Nature Communications

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.

Barren plateaus in quantum neural network training landscapes