Sequential optimal selection of a single-qubit gate and its relation to barren plateau in parameterized quantum circuits

Abstract

We propose an algorithm for variational quantum algorithms (VQAs) to optimize the structure of parameterized quantum circuits (PQCs) efficiently. The algorithm optimizes the PQC structure on-the-fly in VQA by sequentially replacing a single-qubit gate with the optimal one to minimize the objective function. To directly find the optimal gate, our method uses the factorization of matrices whose elements are evaluated on a set of the slightly-modified circuits. The matrix factorization enables us to not only unify the existing sequential methods for further extension but also provide rigorous proofs of their limitation and potential in comparison with conventional gradient-based optimizers. Firstly, when the circuits are sufficiently deep, the sequential methods encounter a barren plateau that the spectrum of the matrix concentrates on a single value exponentially fast with respect to the number of qubits. Secondly, if the objective functions are local observables, they can avoid barren plateaus as long as the depth of the n-qubit PQCs is $O(\log n)$. Although the family of these optimizers does not directly employ gradients of the objective function, our results establish their connection with conventional optimizations providing a consistent picture of the barren plateau. We also perform numerical experiments showing the advantages over conventional VQAs and confirm the successful optimization getting over the barren plateau in the ground state problem of the mixed field Ising model up to 12 qubits.

25 pages, 9 figures

References in corpus (33)

• Variational Quantum Algorithms
• The Variational Quantum Eigensolver: a review of methods and best practices
• Noise-Induced Barren Plateaus in Variational Quantum Algorithms
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Layerwise learning for quantum neural networks
• General parameter-shift rules for quantum gradients
• Exploring entanglement and optimization within the Hamiltonian Variational Ansatz
• Effect of barren plateaus on gradient-free optimization
• Quantum circuit architecture search for variational quantum algorithms
• Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers
• Quantum simulations of materials on near-term quantum computers
• Differentiable Quantum Architecture Search
• Hardware-efficient variational quantum algorithms for time evolution
• Large gradients via correlation in random parameterized quantum circuits
• Variational Quantum State Eigensolver
• Quantum error mitigation as a universal error-minimization technique: applications from NISQ to FTQC eras
• Adaptive Variational Quantum Dynamics Simulations
• Machine learning of noise-resilient quantum circuits
• Expressibility of the alternating layered ansatz for quantum computation
• The Variational Power of Quantum Circuit Tensor Networks
• A case study of variational quantum algorithms for a job shop scheduling problem
• Classically optimized Hamiltonian simulation
• Correlation-Informed Permutation of Qubits for Reducing Ansatz Depth in VQE
• Applications of Quantum Computing for Investigations of Electronic Transitions in Phenylsulfonyl-carbazole TADF Emitters
• Variational Quantum Solutions to the Advection-Diffusion Equation for Applications in Fluid Dynamics
• Mutual information-assisted Adaptive Variational Quantum Eigensolver
• Quantum Analytic Descent
• Computing Ground State Properties with Early Fault-Tolerant Quantum Computers
• Benchmarking adaptive variational quantum eigensolvers
• Simulating Time Evolution with Fully Optimized Single-Qubit Gates on Parameterized Quantum Circuits
• Post-Hartree-Fock method in Quantum Chemistry for Quantum Computer
• Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints
• Optimal Parameter Configurations for Sequential Optimization of Variational Quantum Eigensolver

Cited by in corpus (6)

• Barren Plateaus in Variational Quantum Computing
• Fast gradient-free optimization of excitations in variational quantum eigensolvers
• Greedy Gradient-free Adaptive Variational Quantum Algorithms on a Noisy Intermediate Scale Quantum Computer
• Expressivity of deterministic quantum computation with one qubit
• Multi-channel convolutional neural quantum embedding
• Efficient Estimation and Sequential Optimization of Cost Functions in Variational Quantum Algorithms