Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus
TLDR
In this article , the authors derive a fundamental relationship between expressibility and trainability of quantum circuits, and show that highly expressible quantum circuits exhibit flatter cost landscapes and therefore will be harder to train.Abstract:
Parameterized quantum circuits serve as ans\"{a}tze for solving variational problems and provide a flexible paradigm for programming near-term quantum computers. Ideally, such ans\"{a}tze should be highly expressive so that a close approximation of the desired solution can be accessed. On the other hand, the ansatz must also have sufficiently large gradients to allow for training. Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability. This is done by extending the well established barren plateau phenomenon, which holds for ans\"{a}tze that form exact 2-designs, to arbitrary ans\"{a}tze. Specifically, we calculate the variance in the cost gradient in terms of the expressibility of the ansatz, as measured by its distance from being a 2-design. Our resulting bounds indicate that highly expressive ans\"{a}tze exhibit flatter cost landscapes and therefore will be harder to train. Furthermore, we provide numerics illustrating the effect of expressiblity on gradient scalings, and we discuss the implications for designing strategies to avoid barren plateaus. read more
Citations
More filters
Journal ArticleDOI
Challenges and opportunities in quantum machine learning
TL;DR: In this paper , the authors highlight differences between quantum and classical machine learning, with a focus on quantum neural networks and quantum deep learning, and discuss opportunities for quantum advantage with quantum machine learning.
Posted Content
Layer VQE: A Variational Approach for Combinatorial Optimization on Noisy Quantum Computers
TL;DR: This article presents a large-scale numerical study, simulating circuits with up to 40 qubits and 352 parameters, that demonstrates the potential of an iterative layer VQE (L-VQE) approach and shows that L-VZE is more robust to finite sampling errors and has a higher chance of finding the solution as compared with standard VZE approaches.
Posted Content
Equivalence of quantum barren plateaus to cost concentration and narrow gorges.
TL;DR: In this article, the authors investigate the connection between three different landscape features that have been observed for PQCs: (1) exponentially vanishing gradients (called barren plateaus), (2) exponential cost concentration about the mean, and (3) the exponential narrowness of minina.
Journal ArticleDOI
Generalization in quantum machine learning from few training data
TL;DR: In this article , the authors provided a comprehensive study of generalization performance in QML after training on a limited number N of training data points, and showed that the generalization error of a QML model with T trainable gates scales at worst as $$\sqrt{T/N}$$.
Posted Content
Diagnosing barren plateaus with tools from quantum optimal control
Martín Larocca,Piotr Czarnik,Kunal Sharma,Gopikrishnan Muraleedharan,Patrick J. Coles,Marco Cerezo +5 more
TL;DR: In this paper, the authors employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren plateaus for problem-inspired ansatzes, such as the Quantum Alternating Operator Ansatz (QAOA), the Hamiltonian Variational Anatz (HVA), and others.
References
More filters
Journal ArticleDOI
Double-slit photoelectron interference in strong-field ionization of the neon dimer.
Maksim Kunitski,Nicolas Eicke,Pia Huber,Jonas Köhler,S. Zeller,J. Voigtsberger,Nikolai Schlott,K. Henrichs,H. Sann,Florian Trinter,Lothar Ph. H. Schmidt,Anton Kalinin,Markus Schöffler,Till Jahnke,Manfred Lein,Reinhard Dörner +15 more
TL;DR: The authors show the double-slit interference effect in the strong-field ionization of neon dimers by employing COLTRIMS method to record the momentum distribution of the photoelectrons in the molecular frame.
Journal ArticleDOI
Supplementary information for "Quantum supremacy using a programmable superconducting processor"
Frank Arute,Kunal Arya,Ryan Babbush,Dave Bacon,Joseph C. Bardin,Rami Barends,Rupak Biswas,Sergio Boixo,Fernando G. S. L. Brandão,David A. Buell,B. Burkett,Yu Chen,Zijun Chen,Ben Chiaro,Roberto Collins,William Courtney,Andrew Dunsworth,Edward Farhi,Brooks Foxen,Austin G. Fowler,Craig Gidney,Marissa Giustina,R. Graff,Keith Guerin,Steve Habegger,Matthew P. Harrigan,Michael J. Hartmann,Alan Ho,Markus R. Hoffmann,Trent Huang,Travis S. Humble,Sergei V. Isakov,Evan Jeffrey,Zhang Jiang,Dvir Kafri,Kostyantyn Kechedzhi,Julian Kelly,Paul V. Klimov,Sergey Knysh,Alexander N. Korotkov,Fedor Kostritsa,David Landhuis,Mike Lindmark,Erik Lucero,Dmitry I. Lyakh,Salvatore Mandrà,Jarrod R. McClean,Matt McEwen,Anthony Megrant,Xiao Mi,Kristel Michielsen,Masoud Mohseni,Josh Mutus,Ofer Naaman,Matthew Neeley,Charles Neill,Murphy Yuezhen Niu,Eric Ostby,Andre Petukhov,John Platt,Chris Quintana,Eleanor Rieffel,Pedram Roushan,Nicholas C. Rubin,Daniel Sank,Kevin J. Satzinger,Vadim Smelyanskiy,Kevin Sung,Matthew D. Trevithick,Amit Vainsencher,Benjamin Villalonga,Theodore White,Z. Jamie Yao,Ping Yeh,Adam Zalcman,Hartmut Neven,John M. Martinis +76 more
TL;DR: In this paper, an updated version of supplementary information to accompany "Quantum supremacy using a programmable superconducting processor", an article published in the October 24, 2019 issue of Nature, is presented.
Journal ArticleDOI
Quantum Computing in the NISQ era and beyond
TL;DR: Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future as mentioned in this paper, which will be useful tools for exploring many-body quantum physics, and may have other useful applications.
Journal ArticleDOI
A variational eigenvalue solver on a photonic quantum processor
Alberto Peruzzo,Jarrod R. McClean,Peter Shadbolt,Man-Hong Yung,Xiao-Qi Zhou,Peter J. Love,Alán Aspuru-Guzik,Jeremy L. O'Brien +7 more
TL;DR: The proposed approach drastically reduces the coherence time requirements and combines this method with a new approach to state preparation based on ansätze and classical optimization, enhancing the potential of quantum resources available today and in the near future.
Journal ArticleDOI
Coupled-cluster theory in quantum chemistry
Rodney J. Bartlett,Monika Musiał +1 more
TL;DR: In this article, the essential aspects of coupled-cluster theory are explained and illustrated with informative numerical results, showing that the theory offers the most accurate results among the practical ab initio electronic-structure theories applicable to moderate-sized molecules.