One qubit as a universal approximant

Abstract

A single-qubit circuit can approximate any bounded complex function stored in the degrees of freedom defining its quantum gates. The single-qubit approximant presented in this work is operated through a series of gates that take as their parameterization the independent variable of the target function and an additional set of adjustable parameters. The independent variable is re-uploaded in every gate while the parameters are optimized for each target function. The output state of this quantum circuit becomes more accurate as the number of re-uploadings of the independent variable increases, i. e., as more layers of gates parameterized with the independent variable are applied. In this work, we provide two different proofs of this claim related to both Fourier series and the Universal Approximation Theorem for Neural Networks, and we benchmark both methods against their classical counterparts. We further implement a single-qubit approximant in a real superconducting qubit device, demonstrating how the ability to describe a set of functions improves with the depth of the quantum circuit. This work shows the robustness of the re-uploading technique on Quantum Machine Learning.

Fragmented imaginary-time evolution for intermediate-scale quantum signal processors

“…In Ref. [61], it was observed that gate sequence R 2 (x, ω, Φ 2 ) := 4(q+1) , can encode certain finite Fourier series into its matrix components. Here, not only do we formally prove that for any target series but also we provide an explicit, efficient recipe for finding the adequate choice of pulses Φ 2 .…”

“…( 18) is then no longer valid and one must assess the query complexity on a case-bycase basis. This can for instance be tackled numerically by variationally optimising the gate sequence R 2 (x, ω, Φ 2 ) to block-encode the periodic extension of f [61]. Either way, clearly, if a normalized Fourier expansion is a-priori available, one can skip steps 1 to 3 in Alg.…”

“…e iωxZ e −iκY with ξ = {ζ, η, ϕ, κ} can be conveniently represented as [61] 2 (x, ω, Φ 2 ) as the partial product up to the m-th term of R 2 (x, ω, Φ 2 ). Take ω 0 = 0, ω k = 1/2 for all k = 0 even, and ω k = −1/2 for all odd k. Thus, a complex constant, i.e.…”

Style-based quantum generative adversarial networks for Monte Carlo events

“…This approach is promising, though its utility depends on the existence of large-scale quantum computers with low gate errors and enough qubits to perform quantum error correction. More recent proposals focus on defining a quantum neural network (QNN), or parameterized quantum circuit [13][14][15][16], which then can be trained to implement a function class [17][18][19]; these proposals can be implemented on current NISQ-era devices. For example, several QNNs have been proposed for pattern classification [20][21][22][23] or data compression [24][25][26][27].…”

Direct implementation of a perceptron in superconducting circuit quantum hardware

“…In the classical domain, expressivity is guaranteed by the universal approximation theorem, which requires a nonlinear activation function for the perceptrons in the network. In the quantum setting, it has been shown that similar notions of universal approximation exist for functions mapping gate parameters to the state prepared by the quantum circuit [16]. However, currently there is no universally accepted way to extend the concept of classical neural networks to quantum systems [17,18].…”