# Publications by Tomi Johnson

## Chiral quantum walks

PHYSICAL REVIEW A 93 (2016) ARTN 042302

D Lu, JD Biamonte, J Li, H Li, TH Johnson, V Bergholm, M Faccin, Z Zimboras, R Laflamme, J Baugh, S Lloyd

## Nondestructive selective probing of phononic excitations in a cold Bose gas using impurities

PHYSICAL REVIEW A 91 (2015) ARTN 013611

D Hangleiter, MT Mitchison, TH Johnson, M Bruderer, MB Plenio, D Jaksch

## Capturing exponential variance using polynomial resources: applying tensor networks to nonequilibrium stochastic processes.

Physical review letters 114 (2015) 090602-

TH Johnson, TJ Elliott, SR Clark, D Jaksch

Estimating the expected value of an observable appearing in a nonequilibrium stochastic process usually involves sampling. If the observable's variance is high, many samples are required. In contrast, we show that performing the same task without sampling, using tensor network compression, efficiently captures high variances in systems of various geometries and dimensions. We provide examples for which matching the accuracy of our efficient method would require a sample size scaling exponentially with system size. In particular, the high-variance observable e^{-βW}, motivated by Jarzynski's equality, with W the work done quenching from equilibrium at inverse temperature β, is exactly and efficiently captured by tensor networks.

## Community Detection in Quantum Complex Networks

ArXiv (0)

M Faccin, P Migdał, TH Johnson, V Bergholm, JD Biamonte

Determining community structure is a central topic in the study of complex networks, be it technological, social, biological or chemical, in static or interacting systems. In this paper, we extend the concept of community detection from classical to quantum systems---a crucial missing component of a theory of complex networks based on quantum mechanics. We demonstrate that certain quantum mechanical effects cannot be captured using current classical complex network tools and provide new methods that overcome these problems. Our approaches are based on defining closeness measures between nodes, and then maximizing modularity with hierarchical clustering. Our closeness functions are based on quantum transport probability and state fidelity, two important quantities in quantum information theory. To illustrate the effectiveness of our approach in detecting community structure in quantum systems, we provide several examples, including a naturally occurring light-harvesting complex, LHCII. The prediction of our simplest algorithm, semiclassical in nature, mostly agrees with a proposed partitioning for the LHCII found in quantum chemistry literature, whereas our fully quantum treatment of the problem uncovers a new, consistent, and appropriately quantum community structure.

## Degree Distribution in Quantum Walks on Complex Networks

Physical Review X 3 (2013) 041007

M Faccin, T Johnson, J Biamonte, S Kais, P Migdał

We relate the average long time probability of finding a quantum walker at a node to that for a corresponding classical walk. The latter is proportional to the node degree. We replace the state dependence of quantum evolution by a partial order, bounding the quantumness of the walker in terms of the energy of the initial state and the spectral gap of the complex network. For a uniform initial state, we identify complex network classes and regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate. The importance of quantum effects, or the quantumness of a complex network, is given by the R\'enyi entropy of order 1/2 of the normalized weighted degrees, which can be upper bounded by the Shannon entropy.

## Solving search problems by strongly simulating quantum circuits

Scientific Reports Nature Publishing Group 3 (2013) 1235

TH Johnson, JD Biamonte, SR Clark, D Jaksch

Simulating quantum circuits using classical computers lets us analyse the inner workings of quantum algorithms. The most complete type of simulation, strong simulation, is believed to be generally inefficient. Nevertheless, several efficient strong simulation techniques are known for restricted families of quantum circuits and we develop an additional technique in this article. Further, we show that strong simulation algorithms perform another fundamental task: solving search problems. Efficient strong simulation techniques allow solutions to a class of search problems to be counted and found efficiently. This enhances the utility of strong simulation methods, known or yet to be discovered, and extends the class of search problems known to be efficiently simulable. Relating strong simulation to search problems also bounds the computational power of efficiently strongly simulable circuits; if they could solve all problems in $\mathrm{P}$ this would imply the collapse of the complexity hierarchy $\mathrm{P} \subseteq \mathrm{NP} \subseteq # \mathrm{P}$.

## Ab initio derivation of Hubbard models for cold atoms in optical lattices

Physical Review A: Atomic, Molecular and Optical Physics 87 (2013) 043613

R Walters, G Cotugno, TH Johnson, SR Clark, D Jaksch

We derive ab initio local Hubbard models for several optical lattice potentials of current interest, including the honeycomb and Kagom\'{e} lattices, verifying their accuracy on each occasion by comparing the interpolated band structures against the originals. To achieve this, we calculate the maximally-localized generalized Wannier basis by implementing the steepest-descent algorithm of Marzari and Vanderbilt [N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)] directly in one and two dimensions. To avoid local minima we develop an initialization procedure that is both robust and requires no prior knowledge of the optimal Wannier basis. The MATLAB code that implements our full procedure is freely available online at http://ccpforge.cse.rl.ac.uk/gf/project/mlgws/.

## Time Invariant Discord and Non-Markovianity

Physical Review A 87 (2013) 010103(R)

P Haikka, TH Johnson, S Maniscalco

We study non-Markovianity and information flow for qubits experiencing local dephasing with an Ohmic class spectrum. We demonstrate the existence of a temperature-dependent critical value of the Ohmicity parameter s for the onset of non-Markovianity and give a physical interpretation of this phenomenon by linking it to the form of the reservoir spectrum. We demonstrate that this link holds also for more general spectra. We unveil a class of initial states for which discord is forever frozen at a positive value. We connect time invariant discord to non-Markovianity and propose a physical system in which it could be observed.

## Breathing oscillations of a trapped impurity in a Bose gas

Europhysics Letters 98 (2012)

TH Johnson, M Bruderer, Y Cai, SR Clark, W Bao, D Jaksch

Motivated by a recent experiment (Catani J. et al., Phys. Rev. A, 85 (2012) 023623) we study breathing oscillations in the width of a harmonically trapped impurity interacting with a separately trapped Bose gas. We provide an intuitive physical picture of such dynamics at zero temperature, using a time-dependent variational approach. The amplitudes of breathing oscillations are suppressed by self-trapping, due to interactions with the Bose gas. Further, exciting phonons in the Bose gas leads to damped oscillations and non-Markovian dynamics of the width of the impurity, the degree of which can be engineered through controllable parameters. Our results, supported by simulations, reproduce the main features of the dynamics observed by Catani et al. despite the temperature of that experiment. Moreover, we predict novel effects at lower temperatures due to self-trapping and the inhomogeneity of the trapped Bose gas.

## Teaching physics in higher education: aims, experiences and strategies

Developing learning and teaching portfolio, University of Oxford (2012)

TH Johnson

I am lucky enough to have experienced higher education in several different formats, as both a learner and teacher. In this portfolio I present examples of how I have sought to build on these experiences, in particular those as a teacher. My ultimate aim is to improve my teaching, but this necessarily involves identifying teaching goals and methods, as well as understanding students’ perspectives and the learning process. The structure of this portfolio is as follows: I begin in section 2 by describing my teaching experiences, directly motivating the later sections. Section 3 contains a discussion of the aims of higher education teaching, using the University of Oxford undergraduate physics course as an example. I then analyse the responses of undergraduate students to a questionnaire that asked them to rate the importance of different sources of learning. This evaluation of learning from the students’ perspective is presented in section 4. In section 5, I look at the benefits of teaching through a mini project, discussing my observation of a project run by my mentor. Section 6 then discusses the design of problem sets through the lens of threshold concepts.

## Impurity transport through a strongly interacting bosonic quantum gas

Physical Review A 84 (2011) 023617

TH Johnson, SR Clark, M Bruderer, D Jaksch

Using near-exact numerical simulations, we study the propagation of an impurity through a one-dimensional Bose lattice gas for varying bosonic interaction strengths and filling factors at zero temperature. The impurity is coupled to the Bose gas and confined to a separate tilted lattice. The precise nature of the transport of the impurity is specific to the excitation spectrum of the Bose gas, which allows one to measure properties of the Bose gas nondestructively, in principle, by observing the impurity; here we focus on the spatial and momentum distributions of the impurity as well as its reduced density matrix. For instance, we show it is possible to determine whether the Bose gas is commensurately filled as well as the bandwidth and gap in its excitation spectrum. Moreover, we show that the impurity acts as a witness to the crossover of its environment from the weakly to the strongly interacting regime, i.e., from a superfluid to a Mott insulator or Tonks-Girardeau lattice gas, and the effects on the impurity in both of these strongly interacting regimes are clearly distinguishable. Finally, we find that the spatial coherence of the impurity is related to its propagation through the Bose gas.

## Dynamical simulations of classical stochastic systems using matrix product states

Physical Review E 82 (2010) 036702

TH Johnson, SR Clark, D Jaksch

We adapt the time-evolving block decimation (TEBD) algorithm, originally devised to simulate the dynamics of one-dimensional quantum systems, to simulate the time evolution of nonequilibrium stochastic systems. We describe this method in detail; a system’s probability distribution is represented by a matrix product state (MPS) of finite dimension and then its time evolution is efficiently simulated by repeatedly updating and approximately refactorizing this representation. We examine the use of MPS as an approximation method, looking at parallels between the interpretations of applying it to quantum state vectors and probability distributions. In the context of stochastic systems we consider two types of factorization for use in the TEBD algorithm: non-negative matrix factorization (NMF), which ensures that the approximate probability distribution is manifestly non-negative, and the singular value decomposition (SVD). Comparing these factorizations, we find the accuracy of the SVD to be substantially greater than current NMF algorithms. We then apply TEBD to simulate the totally asymmetric simple exclusion process (TASEP) for systems of up to hundreds of lattice sites in size. Using exact analytic results for the TASEP steady state, we find that TEBD reproduces this state such that the error in calculating expectation values can be made negligible even when severely compressing the description of the system by restricting the dimension of the MPS to be very small. Out of the steady state we show for specific observables that expectation values converge as the dimension of the MPS is increased to a moderate size.

## Phonon resonances in atomic currents through Bose-Fermi mixtures in optical lattices

Physical Review A 82 (2010) 043617

M Bruderer, TH Johnson, SR Clark, D Jaksch, A Posazhennikova, W Belzig

We present an analysis of Bose-Fermi mixtures in optical lattices for the case where the lattice potential of the fermions is tilted and the bosons (in the superfluid phase) are described by Bogoliubov phonons. It is shown that the Bogoliubov phonons enable hopping transitions between fermionic Wannier-Stark states; these transitions are accompanied by energy dissipation into the superfluid and result in a net atomic current along the lattice. We derive a general expression for the drift velocity of the fermions and find that the dependence of the atomic current on the lattice tilt exhibits negative differential conductance and phonon resonances. Numerical simulations of the full dynamics of the system based on the time-evolving block decimation algorithm reveal that the phonon resonances should be observable under the conditions of a realistic measuring procedure.

## Understanding trafﬁc jams using quantum algorithms

MPhys project, University of Oxford (2009)

TH Johnson

An algorithm developed for studying the time evolution of weakly entangled onedimensional (1D) quantum systems is extended to simulate stochastic classical systems. The accuracy and applicability of the algorithm are examined using a number of classical 1D lattice models, which are interpreted in terms of vehicular trafﬁc. Notably, the algorithm is successfully used to describe time evolution away from steady-state. Such situations are difﬁcult to simulate using current methods. Thus the algorithm opens up new possibilities for studying the dynamical behaviour of stochastic classical systems.

## What is a quantum simulator?

ArXiv (0)

TH Johnson, SR Clark, D Jaksch

Quantum simulators are devices that actively use quantum effects to answer questions about model systems and, through them, real systems. Here we expand on this definition by answering several fundamental questions about the nature and use of quantum simulators. Our answers address two important areas. First, the difference between an operation termed simulation and another termed computation. This distinction is related to the purpose of an operation, as well as our confidence in and expectation of its accuracy. Second, the threshold between quantum and classical simulations. Throughout, we provide a perspective on the achievements and directions of the field of quantum simulation.