Hartmut Neven
Hartmut Neven is an Vice President of Engineering at Google. He is the founder and manager of the Quantum Artificial Intelligence lab. The objective of the lab is to fabricate quantum processors and develop novel quantum algorithms to dramatically accelerate computational tasks for machine intelligence. Previously, Hartmut was head of the Visual Search team. His team developed the visual search service which today is used by a large number of Google products including Image Search, Google Photos, YouTube, Street View and Google Goggles. His teams won a number of competitions designed to establish the best visual recognition software for faces (FERET 1996, FRVT 2002), objects (ImageNet 2014) and text (ICDAR 2013). Hartmut was also a co-founder of project Glass and led the team that built the first prototype. Prior to joining Google, Hartmut started two computer vision companies, the second one was acquired by Google in 2006. Hartmut obtained his Ph.D. in 1996 with a thesis on "Dynamics for vision-guided autonomous mobile robots". Then he became a research professor for computer science and theoretical neuroscience at the University of Southern California.
Authored Publications
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Stable quantum-correlated many-body states through engineered dissipation
Sara Shabani
Dripto Debroy
Jerome Lloyd
Alexios Michailidis
Andrew Dunsworth
Bill Huggins
Markus Hoffmann
Alexis Morvan
Josh Cogan
Ben Curtin
Guifre Vidal
Bob Buckley
Tom O'Brien
John Mark Kreikebaum
Rajeev Acharya
Joonho Lee
Ningfeng Zhu
Shirin Montazeri
Sergei Isakov
Jamie Yao
Clarke Smith
Rebecca Potter
Sean Harrington
Jeremy Hilton
Paula Heu
Alexei Kitaev
Alex Crook
Fedor Kostritsa
Kim Ming Lau
Dmitry Abanin
Trent Huang
Aaron Shorter
Steve Habegger
Gina Bortoli
Charles Rocque
Vladimir Shvarts
Alfredo Torres
Anthony Megrant
Charles Neill
Michael Hamilton
Dar Gilboa
Lily Laws
Nicholas Bushnell
Ramis Movassagh
Mike Shearn
Wojtek Mruczkiewicz
Desmond Chik
Leonid Pryadko
Xiao Mi
Brooks Foxen
Frank Arute
Alejo Grajales Dau
Yaxing Zhang
Lara Faoro
Alexander Lill
JiunHow Ng
Justin Iveland
Marco Szalay
Orion Martin
Juhwan Yoo
Michael Newman
William Giang
Alex Opremcak
Amanda Mieszala
William Courtney
Andrey Klots
Wayne Liu
Pavel Laptev
Charina Chou
Paul Conner
Rolando Somma
Vadim Smelyanskiy
Benjamin Chiaro
Grayson Young
Tim Burger
ILYA Drozdov
Agustin Di Paolo
Jimmy Chen
Marika Kieferova
Michael Broughton
Negar Saei
Juan Atalaya
Markus Ansmann
Pavol Juhas
Murray Ich Nguyen
Yuri Lensky
Roberto Collins
Élie Genois
Jindra Skruzny
Igor Aleiner
Yu Chen
Reza Fatemi
Leon Brill
Ashley Huff
Doug Strain
Monica Hansen
Noah Shutty
Ebrahim Forati
Dave Landhuis
Kenny Lee
Ping Yeh
Kunal Arya
Henry Schurkus
Cheng Xing
Cody Jones
Edward Farhi
Raja Gosula
Andre Petukhov
Alexander Korotkov
Ani Nersisyan
Christopher Schuster
George Sterling
Kostyantyn Kechedzhi
Trond Andersen
Alexandre Bourassa
Kannan Sankaragomathi
Vinicius Ferreira
Science, 383 (2024), pp. 1332-1337
Preview abstract
Engineered dissipative reservoirs have the potential to steer many-body quantum systems toward correlated steady states useful for quantum simulation of high-temperature superconductivity or quantum magnetism. Using up to 49 superconducting qubits, we prepared low-energy states of the transverse-field Ising model through coupling to dissipative auxiliary qubits. In one dimension, we observed long-range quantum correlations and a ground-state fidelity of 0.86 for 18 qubits at the critical point. In two dimensions, we found mutual information that extends beyond nearest neighbors. Lastly, by coupling the system to auxiliaries emulating reservoirs with different chemical potentials, we explored transport in the quantum Heisenberg model. Our results establish engineered dissipation as a scalable alternative to unitary evolution for preparing entangled many-body states on noisy quantum processors.
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Optimizing quantum gates towards the scale of logical qubits
Andrew Dunsworth
Anthony Megrant
Charles Neill
Desmond Chik
Alejo Grajales Dau
Yaxing Zhang
Vadim Smelyanskiy
Jimmy Chen
Will Livingston
Yu Chen
Vlad Sivak
Alexander Korotkov
Trond Andersen
Alexandre Bourassa
Nature Communications, 15 (2024), pp. 2442
Preview abstract
A foundational assumption of quantum error correction theory is that quantum gates can be scaled to large processors without exceeding the error-threshold for fault tolerance. Two major challenges that could become fundamental roadblocks are manufacturing high-performance quantum hardware and engineering a control system that can reach its performance limits. The control challenge of scaling quantum gates from small to large processors without degrading performance often maps to non-convex, high-constraint, and time-dynamic control optimization over an exponentially expanding configuration space. Here we report on a control optimization strategy that can scalably overcome the complexity of such problems. We demonstrate it by choreographing the frequency trajectories of 68 frequency-tunable superconducting qubits to execute single- and two-qubit gates while mitigating computational errors. When combined with a comprehensive model of physical errors across our processor, the strategy suppresses physical error rates by ~3.7× compared with the case of no optimization. Furthermore, it is projected to achieve a similar performance advantage on a distance-23 surface code logical qubit with 1057 physical qubits. Our control optimization strategy solves a generic scaling challenge in a way that can be adapted to a variety of quantum operations, algorithms, and computing architectures.
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Dynamics of magnetization at infinite temperature in a Heisenberg spin chain
Tomaž Prosen
Vedika Khemani
Rhine Samajdar
Jesse Hoke
Sarang Gopalakrishnan
Andrew Dunsworth
Bill Huggins
Markus Hoffmann
Alexis Morvan
Josh Cogan
Ben Curtin
Guifre Vidal
Bob Buckley
Tom O'Brien
John Mark Kreikebaum
Rajeev Acharya
Joonho Lee
Ningfeng Zhu
Shirin Montazeri
Sergei Isakov
Jamie Yao
Clarke Smith
Rebecca Potter
Sean Harrington
Jeremy Hilton
Paula Heu
Alexei Kitaev
Alex Crook
Fedor Kostritsa
Kim Ming Lau
Dmitry Abanin
Trent Huang
Aaron Shorter
Steve Habegger
Steven Martin
Gina Bortoli
Seun Omonije
Richard Ross Allen
Charles Rocque
Vladimir Shvarts
Alfredo Torres
Anthony Megrant
Charles Neill
Michael Hamilton
Dar Gilboa
Lily Laws
Nicholas Bushnell
Kyle Anderson
Ramis Movassagh
David Rhodes
Mike Shearn
Wojtek Mruczkiewicz
Desmond Chik
Leonid Pryadko
Xiao Mi
Brooks Foxen
Frank Arute
Alejo Grajales Dau
Yaxing Zhang
Lara Faoro
Alexander Lill
Gordon Hill
JiunHow Ng
Justin Iveland
Marco Szalay
Orion Martin
Juan Campero
Juhwan Yoo
Michael Newman
William Giang
Gonzalo Garcia
Alex Opremcak
Amanda Mieszala
William Courtney
Andrey Klots
Wayne Liu
Pavel Laptev
Paul Conner
Rolando Somma
Vadim Smelyanskiy
Benjamin Chiaro
Grayson Young
Tim Burger
ILYA Drozdov
Agustin Di Paolo
Jimmy Chen
Marika Kieferova
Hung-Shen Chang
Michael Broughton
Negar Saei
Juan Atalaya
Markus Ansmann
Pavol Juhas
Murray Ich Nguyen
Yuri Lensky
Roberto Collins
Élie Genois
Jindra Skruzny
Yu Chen
Reza Fatemi
Leon Brill
Seneca Meeks
Ashley Huff
Doug Strain
Monica Hansen
Noah Shutty
Ebrahim Forati
Doug Thor
Dave Landhuis
Kenny Lee
Ping Yeh
Kunal Arya
Henry Schurkus
Cheng Xing
Cody Jones
Edward Farhi
Vlad Sivak
Raja Gosula
Andre Petukhov
Clint Earle
Alexander Korotkov
Ani Nersisyan
Christopher Schuster
George Sterling
Trond Andersen
Alexandre Bourassa
Salvatore Mandra
Kannan Sankaragomathi
Vinicius Ferreira
Science, 384 (2024), pp. 48-53
Preview abstract
Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the one-dimensional Heisenberg model were conjectured as to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we studied the probability distribution of the magnetization transferred across the chain’s center, P(M). The first two moments of P(M) show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments ruled out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide insights into universal behavior in quantum systems.
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Quantum Computation of Stopping power for Inertial Fusion Target Design
Andrew Baczewski
Alec White
Dominic Berry
Alina Kononov
Joonho Lee
Proceedings of the National Academy of Sciences, 121 (2024), e2317772121
Preview abstract
Stopping power is the rate at which a material absorbs the kinetic energy of a charged particle passing through it - one of many properties needed over a wide range of thermodynamic conditions in modeling inertial fusion implosions. First-principles stopping calculations are classically challenging because they involve the dynamics of large electronic systems far from equilibrium, with accuracies that are particularly difficult to constrain and assess in the warm-dense conditions preceding ignition. Here, we describe a protocol for using a fault-tolerant quantum computer to calculate stopping power from a first-quantized representation of the electrons and projectile. Our approach builds upon the electronic structure block encodings of Su et al. [PRX Quantum 2, 040332 2021], adapting and optimizing those algorithms to estimate observables of interest from the non-Born-Oppenheimer dynamics of multiple particle species at finite temperature. We also work out the constant factors associated with a novel implementation of a high order Trotter approach to simulating a grid representation of these systems. Ultimately, we report logical qubit requirements and leading-order Toffoli costs for computing the stopping power of various projectile/target combinations relevant to interpreting and designing inertial fusion experiments. We estimate that scientifically interesting and classically intractable stopping power calculations can be quantum simulated with
roughly the same number of logical qubits and about one hundred times more Toffoli gates than is required for state-of-the-art quantum simulations of industrially relevant molecules such as FeMoCo or P450.
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Suppressing quantum errors by scaling a surface code logical qubit
Jeremy Hilton
Anthony Megrant
Michael Newman
Vadim Smelyanskiy
Jimmy Chen
Juan Atalaya
Yu Chen
Kenny Lee
Cody Jones
Nature (2023)
Preview abstract
Practical quantum computing will require error rates that are well below what is achievable with
physical qubits. Quantum error correction [1, 2] offers a path to algorithmically-relevant error rates
by encoding logical qubits within many physical qubits, where increasing the number of physical
qubits enhances protection against physical errors. However, introducing more qubits also increases
the number of error sources, so the density of errors must be sufficiently low in order for logical
performance to improve with increasing code size. Here, we report the measurement of logical qubit
performance scaling across multiple code sizes, and demonstrate that our system of superconducting
qubits has sufficient performance to overcome the additional errors from increasing qubit number.
We find our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3
logical qubits on average, both in terms of logical error probability over 25 cycles and logical error
per cycle (2.914%±0.016% compared to 3.028%±0.023%). To investigate damaging, low-probability
error sources, we run a distance-25 repetition code and observe a 1.7 × 10−6 logical error per round
floor set by a single high-energy event (1.6 × 10−7 when excluding this event). We are able to
accurately model our experiment, and from this model we can extract error budgets that highlight
the biggest challenges for future systems. These results mark the first experimental demonstration
where quantum error correction begins to improve performance with increasing qubit number, and
illuminate the path to reaching the logical error rates required for computation.
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Purification-Based Quantum Error Mitigation of Pair-Correlated Electron Simulations
Christian Gogolin
Vincent Elfving
Fotios Gkritsis
Oumarou Oumarou
Gian-Luca R. Anselmetti
Masoud Mohseni
Andrew Dunsworth
William J. Huggins
Markus Rudolf Hoffmann
Alexis Morvan
Josh Godfrey Cogan
Ben Curtin
Guifre Vidal
Bob Benjamin Buckley
Trevor Johnathan Mccourt
Thomas E O'Brien
John Mark Kreikebaum
Rajeev Acharya
Joonho Lee
Ningfeng Zhu
Shirin Montazeri
Sergei Isakov
Jamie Yao
Clarke Smith
Rebecca Potter
Sean Harrington
Jeremy Patterson Hilton
Alex Crook
Fedor Kostritsa
Kim Ming Lau
Dmitry Abanin
Trent Huang
Aaron Shorter
Steve Habegger
Richard Ross Allen
Vladimir Shvarts
Alfredo Torres
Stefano Polla
Anthony Megrant
Charles Neill
Michael C. Hamilton
Dar Gilboa
Lily MeeKit Laws
Nicholas Bushnell
Kyle Anderson
Ramis Movassagh
Mike Shearn
Wojtek Mruczkiewicz
Desmond Chun Fung Chik
Xiao Mi
Brooks Riley Foxen
Frank Carlton Arute
Alejandro Grajales Dau
Yaxing Zhang
Lara Faoro
Alexander T. Lill
Jiun How Ng
Justin Thomas Iveland
Marco Szalay
Orion Martin
Juhwan Yoo
Michael Newman
William Giang
Alex Opremcak
William Courtney
Andrey Klots
Wayne Liu
Pavel Laptev
Paul Conner
Rolando Diego Somma
Vadim Smelyanskiy
Benjamin Chiaro
Grayson Robert Young
Tim Burger
Ilya Drozdov
Jimmy Chen
Marika Kieferova
Michael Blythe Broughton
Juan Atalaya
Markus Ansmann
Pavol Juhas
Murray Nguyen
Daniel Eppens
Roberto Collins
Jindra Skruzny
Igor Aleiner
Yu Chen
Reza Fatemi
Leon Brill
Ashley Anne Huff
Doug Strain
Ebrahim Forati
Dave Landhuis
Kenny Lee
Ping Yeh
Kunal Arya
Cody Jones
Edward Farhi
Andre Gregory Petukhov
Alexander Korotkov
Ani Nersisyan
Christopher Schuster
Kostyantyn Kechedzhi
Trond Ikdahl Andersen
Alexandre Bourassa
Kannan Aryaperumal Sankaragomathi
Nature Physics (2023)
Preview abstract
An important measure of the development of quantum computing platforms has been the simulation of increasingly complex physical systems. Prior to fault-tolerant quantum computing, robust error mitigation strategies are necessary to continue this growth. Here, we study physical simulation within the seniority-zero electron pairing subspace, which affords both a computational stepping stone to a fully correlated model, and an opportunity to validate recently introduced ``purification-based'' error-mitigation strategies. We compare the performance of error mitigation based on doubling quantum resources in time (echo verification) or in space (virtual distillation), on up to 20 qubits of a superconducting qubit quantum processor. We observe a reduction of error by one to two orders of magnitude below less sophisticated techniques (e.g. post-selection); the gain from error mitigation is seen to increase with the system size. Employing these error mitigation strategies enables the implementation of the largest variational algorithm for a correlated chemistry system to-date. Extrapolating performance from these results allows us to estimate minimum requirements for a beyond-classical simulation of electronic structure. We find that, despite the impressive gains from purification-based error mitigation, significant hardware improvements will be required for classically intractable variational chemistry simulations.
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Quantum Simulation of Exact Electron Dynamics can be more Efficient than Classical Mean-Field Methods
Dominic W. Berry
David Reichman
Shu Fay Ung
Andrew Baczewski
William J. Huggins
Joonho Lee
Andrew Zhao
Nature Communications, 14 (2023), pp. 4058
Preview abstract
Quantum algorithms for simulating electronic ground states are slower than popular classical mean-field algorithms such as Hartree-Fock and density functional theory, but offer higher accuracy. Accordingly, quantum computers have been predominantly regarded as competitors to only the most accurate and costly classical methods for treating electron correlation. However, here we tighten bounds showing that certain first quantized quantum algorithms enable exact time evolution of electronic systems with exponentially less space and polynomially fewer operations in basis set size than conventional real-time time-dependent Hartree-Fock and density functional theory. Although the need to sample observables in the quantum algorithm reduces the speedup, we show that one can estimate all elements of the k-particle reduced density matrix with a number of samples scaling only polylogarithmically in basis set size. We also introduce a more efficient quantum algorithm for first quantized mean-field state preparation that is likely cheaper than the cost of time evolution. We conclude that quantum speedup is most pronounced for finite temperature simulations and suggest several practically important electron dynamics problems with potential quantum advantage.
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Measurement-induced entanglement and teleportation on a noisy quantum processor
Vedika Khemani
Matteo Ippoliti
Andrew Dunsworth
Bill Huggins
Markus Hoffmann
Alexis Morvan
Josh Cogan
Ben Curtin
Guifre Vidal
Bob Buckley
Tom O'Brien
John Mark Kreikebaum
Rajeev Acharya
Joonho Lee
Ningfeng Zhu
Shirin Montazeri
Sergei Isakov
Jamie Yao
Clarke Smith
Rebecca Potter
Jeremy Hilton
Paula Heu
Alexei Kitaev
Alex Crook
Fedor Kostritsa
Kim Ming Lau
Dmitry Abanin
Trent Huang
Aaron Shorter
Steve Habegger
Gina Bortoli
Seun Omonije
Charles Rocque
Vladimir Shvarts
Alfredo Torres
Anthony Megrant
Charles Neill
Michael Hamilton
Dar Gilboa
Lily Laws
Nicholas Bushnell
Ramis Movassagh
Mike Shearn
Wojtek Mruczkiewicz
Desmond Chik
Leonid Pryadko
Xiao Mi
Brooks Foxen
Frank Arute
Alejo Grajales Dau
Yaxing Zhang
Alexander Lill
JiunHow Ng
Justin Iveland
Marco Szalay
Orion Martin
Juhwan Yoo
Michael Newman
William Giang
Alex Opremcak
Amanda Mieszala
William Courtney
Andrey Klots
Wayne Liu
Pavel Laptev
Paul Conner
Rolando Somma
Vadim Smelyanskiy
Jesse Hoke
Benjamin Chiaro
Grayson Young
Tim Burger
ILYA Drozdov
Agustin Di Paolo
Jimmy Chen
Marika Kieferova
Michael Broughton
Negar Saei
Juan Atalaya
Markus Ansmann
Pavol Juhas
Murray Ich Nguyen
Yuri Lensky
Daniel Eppens
Roberto Collins
Jindra Skruzny
Yu Chen
Reza Fatemi
Leon Brill
Ashley Huff
Doug Strain
Monica Hansen
Noah Shutty
Ebrahim Forati
Dave Landhuis
Kenny Lee
Ping Yeh
Kunal Arya
Henry Schurkus
Cheng Xing
Cody Jones
Edward Farhi
Raja Gosula
Andre Petukhov
Alexander Korotkov
Ani Nersisyan
Christopher Schuster
George Sterling
Kostyantyn Kechedzhi
Trond Andersen
Alexandre Bourassa
Kannan Sankaragomathi
Vinicius Ferreira
Nature, 622 (2023), 481–486
Preview abstract
Measurement has a special role in quantum theory: by collapsing the wavefunction, it can enable phenomena such as teleportation and thereby alter the ‘arrow of time’ that constrains unitary evolution. When integrated in many-body dynamics, measurements can lead to emergent patterns of quantum information in space–time that go beyond the established paradigms for characterizing phases, either in or out of equilibrium. For present-day noisy intermediate-scale quantum (NISQ) processors, the experimental realization of such physics can be problematic because of hardware limitations and the stochastic nature of quantum measurement. Here we address these experimental challenges and study measurement-induced quantum information phases on up to 70 superconducting qubits. By leveraging the interchangeability of space and time, we use a duality mapping to avoid mid-circuit measurement and access different manifestations of the underlying phases, from entanglement scaling to measurement-induced teleportation. We obtain finite-sized signatures of a phase transition with a decoding protocol that correlates the experimental measurement with classical simulation data. The phases display remarkably different sensitivity to noise, and we use this disparity to turn an inherent hardware limitation into a useful diagnostic. Our work demonstrates an approach to realizing measurement-induced physics at scales that are at the limits of current NISQ processors.
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Quantum Computation of Molecular Structure using Data from Challenging-to-Classically-Simulate Nuclear Magnetic Resonance Experiments
David Fushman
Yuan Su
Thomas E O'Brien
Vadim Smelyanskiy
PRX Quantum, 3 (2022)
Preview abstract
We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging to classically simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined on a membrane as a benchmark of our method. We isolate small spin clusters, demonstrate the convergence of our learning algorithm on one such example, and then investigate the learnability of these clusters as we cross the ergodic to nonergodic phase transition by suppressing the dipolar interaction. We see a clear correspondence between a drop in the multifractal dimension measured across many-body eigenstates of these clusters, and a transition in the structure of the Hessian of the learning cost function (from degenerate to learnable). Our hope is that such quantum computations might enable the interpretation and development of new NMR techniques for analyzing molecular structure.
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Noise-resilient Majorana Edge Modes on a Chain of Superconducting Qubits
Zijun Chen
Brooks Foxen
Masoud Mohseni
Emily Mount
Joao Basso
Andrew Dunsworth
William J. Huggins
Yuan Su
Markus Rudolf Hoffmann
Alexis Morvan
Guifre Vidal
Bob Benjamin Buckley
Thomas E O'Brien
John Mark Kreikebaum
Rajeev Acharya
Joonho Lee
Shirin Montazeri
Sergei Isakov
Jamie Yao
Rebecca Potter
Jeremy Patterson Hilton
Alexei Kitaev
Alex Crook
Fedor Kostritsa
Kim Ming Lau
Dmitry Abanin
Trent Huang
Steve Habegger
Alexa Rubinov
Vladimir Shvarts
Anthony Megrant
Charles Neill
Dar Gilboa
Nicholas Bushnell
Mike Shearn
Wojtek Mruczkiewicz
Xiao Mi
Frank Carlton Arute
Alejandro Grajales Dau
Yaxing Zhang
Lara Faoro
Justin Thomas Iveland
Marco Szalay
Orion Martin
Juhwan Yoo
Michael Newman
William Giang
Alex Opremcak
William Courtney
Andrey Klots
Wayne Liu
Pavel Laptev
Paul Conner
Vadim Smelyanskiy
Benjamin Chiaro
Bernardo Meurer Costa
Michael Blythe Broughton
Juan Atalaya
Daniel Eppens
Roberto Collins
Igor Aleiner
Yu Chen
Reza Fatemi
Leon Brill
Ashley Anne Huff
Doug Strain
Ebrahim Forati
Dave Landhuis
Kenny Lee
Ping Yeh
Kunal Arya
Michel Henri Devoret
Cody Jones
Edward Farhi
Andre Gregory Petukhov
Alexander Korotkov
Christopher Schuster
Kostyantyn Kechedzhi
Trond Ikdahl Andersen
Alexandre Bourassa
Kannan Aryaperumal Sankaragomathi
Science (2022) (to appear)
Preview abstract
Inherent symmetry of a quantum system may protect its otherwise fragile states. Leveraging such protection requires testing its robustness against uncontrolled environmental interactions. Using 47 superconducting qubits, we implement the kicked Ising model which exhibits Majorana edge modes (MEMs) protected by a $\mathbb{Z}_2$-symmetry. Remarkably, we find that any multi-qubit Pauli operator overlapping with the MEMs exhibits a uniform decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This finding allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Spectroscopic measurements further indicate exponentially suppressed hybridization between the MEMs over larger system sizes, which manifests as a strong resilience against low-frequency noise. Our work elucidates the noise sensitivity of symmetry-protected edge modes in a solid-state environment.
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