Supervised learning of time-independent Hamiltonians for gate design

Abstract

We present a general framework to tackle the problem of finding time-independent dynamics generating target unitary evolutions. We show that this problem is equivalently stated as a set of conditions over the spectrum of the time-independent gate generator, thus translating the task into an inverse eigenvalue problem. We illustrate our methodology by identifying suitable time-independent generators implementing Toffoli and Fredkin gates without the need for ancillae or effective evolutions. We show how the same conditions can be used to solve the problem numerically, via supervised learning techniques. In turn, this allows us to solve problems that are not amenable, in general, to direct analytical solution, providing at the same time a high degree of flexibility over the types of gate-design problems that can be approached. As a significant example, we find generators for the Toffoli gate using only diagonal pairwise interactions, which are easier to implement in some experimental architectures. To showcase the flexibility of the supervised learning approach, we give an example of a non-trivial four-qubit gate that is implementable using only diagonal, pairwise interactions.

Approximate supervised learning of quantum gates via ancillary qubits

“…These results provide further evidence in support of the power and flexibility of the supervised learning approach presented in, 15,17 which clearly applies to the cases where ancillary degrees of freedom are exploited during the evolution.…”

“…In the context of quantum information science, machine learning techniques have been showcased as a flexible tool to solve complex optimisation tasks in different areas. [7][8][9][10][11][12][13][14][15][16][17] In particular, supervised learning techniques were recently demonstrated to solve gate design problems. 15,17 Here, by gate design problem we mean the task of identifying a time-independent Hamiltonian generating a target evolution, under a series of restrictions imposed on the allowed Hamiltonian terms.…”

“…It was recently shown in Ref. 17 that, by exploiting supervised learning techniques and ideas, it is possible to flexibly tackle arbitrary gate design problems. A mathematical framework to further improve the efficiency of the numerics by providing improved ansatz for the optimisations can be formalised and used.…”

“…In this paper, we illustrate the methodology sketched in Ref. 17 and give further examples of time-independent Hamiltonians found via supervised learning optimisation, exploring cases in which ancillary qubits are used to catalyse the evolution, but are traced out at the end of the dynamics in such a way that the effective evolution over the system qubits is the desired one.…”

A high-bias, low-variance introduction to Machine Learning for physicists

“…Another interesting area of research that is likely to grow is asking if and how quantum computers can help improve state-of-the art ML algorithms (Arunachalam and de Wolf, 2017; Benedetti et al , 2016, 2017; Bromley and Rebentrost, 2018; Ciliberto et al , 2017; Daskin, 2018; Innocenti et al , 2018; Mitarai et al , 2018; Perdomo-Ortiz et al , 2017; Rebentrost et al , 2017; Schuld et al , 2017; Schuld and Killoran, 2018; Schuld et al , 2015). Concrete examples that seek to extend some of the basic ideas and methods we introduced in this review to the quantum computing realm include: algorithms for quantum-assisted gradient descent (Kerenidis and Prakash, 2017; Rebentrost et al , 2016), classification (Schuld and Petruccione, 2017), and Ridge regression (Yu et al , 2017).…”