Fourier expansion in variational quantum algorithms

Abstract

The Fourier expansion of the loss function in variational quantum algorithms (VQA) contains a wealth of information, yet is generally hard to access. We focus on the class of variational circuits, where constant gates are Clifford gates and parameterized gates are generated by Pauli operators, which covers most practical cases while allowing much control thanks to the properties of stabilizer circuits. We give a classical algorithm that, for an $N$-qubit circuit and a single Pauli observable, computes coefficients of all trigonometric monomials up to a degree $m$ in time bounded by $\mathcal{O}(N{2}^m)$. Using the general structure and implementation of the algorithm we reveal several novel aspects of Fourier expansions in Clifford+Pauli VQA such as (i) reformulating the problem of computing the Fourier series as an instance of multivariate boolean quadratic system (ii) showing that the approximation given by a truncated Fourier expansion can be quantified by the $L^2$ norm and evaluated dynamically (iii) tendency of Fourier series to be rather sparse and Fourier coefficients to cluster together (iv) possibility to compute the full Fourier series for circuits of non-trivial sizes, featuring tens to hundreds of qubits and parametric gates.

arXiv comment10+5 pages, code available at https://github.com/idnm/FourierVQA, comments welcome; v2: +refs, -typos

References in corpus (24)

• Quantum Computing in the NISQ era and beyond
• Quantum Machine Learning
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Efficient classical simulation of slightly entangled quantum computations
• Barren plateaus in quantum neural network training landscapes
• Noisy intermediate-scale quantum (NISQ) algorithms
• Improved Simulation of Stabilizer Circuits
• Quantum Chemistry in the Age of Quantum Computing
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Quantum algorithms for quantum chemistry and quantum materials science
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Simulating quantum computation by contracting tensor networks
• Training variational quantum algorithms is NP-hard
• Improved classical simulation of quantum circuits dominated by Clifford gates
• Simulation of quantum circuits by low-rank stabilizer decompositions
• Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps
• What limits the simulation of quantum computers?
• Theory of overparametrization in quantum neural networks
• Classical algorithms for quantum mean values
• Universal Effectiveness of High-Depth Circuits in Variational Eigenproblems
• Variational quantum eigensolver techniques for simulating carbon monoxide oxidation
• Multidimensional Fourier series with quantum circuits
• Partitioning Quantum Chemistry Simulations with Clifford Circuits

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• Barren Plateaus in Variational Quantum Computing
• Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance
• Does provable absence of barren plateaus imply classical simulability?
• Classically estimating observables of noiseless quantum circuits
• Real-time operator evolution in two and three dimensions via sparse Pauli dynamics
• Ensemble-learning error mitigation for variational quantum shallow-circuit classifiers
• Pauli path simulations of noisy quantum circuits beyond average case
• Barren plateaus are swamped with traps
• Efficient quantum-enhanced classical simulation for patches of quantum landscapes
• Fourier Analysis of Variational Quantum Circuits for Supervised Learning
• Regularizing quantum loss landscapes by noise injection
• Typical Machine Learning Datasets as Low-Depth Quantum Circuits
• The Dual Role of Low-Weight Pauli Propagation: A Flawed Simulator but a Powerful Initializer for Variational Quantum Algorithms
• Efficient Online Quantum Circuit Learning with No Upfront Training
• Interpolation-based coordinate descent method for parameterized quantum circuits
• Extended parameter shift rules with minimal derivative variance for parameterized quantum circuits
• Exploiting biased noise in variational quantum models