James R. Wootton
ContactDepartment of PhysicsUniversity of Basel Klingelbergstrasse 82 CH4056 Basel, Switzerland

Research Interests
 Topological order
 Topological quantum computation
 Disordered Systems/Anderson Localization
 Entanglement Theory
Publications
Show all abstracts.1.  A repetition code of 15 qubits 
James R. Wootton and Daniel Loss. arXiv:1709.00990 (2017)
The repetition code is an important primitive for the techniques of quantum error correction. Here we implement repetition codes of at most 15 qubits on the 16 qubit \emph{ibmqx3} device. Each experiment is run for a single round of syndrome measurements, achieved using the standard quantum technique of using ancilla qubits and controlled operations. The size of the final syndrome is small enough to allow for lookup table decoding using experimentally obtained data. The results show strong evidence of error correction, and also give insight into the nature of noise in the device.
 
2.  A proposal for a minimal surface code experiment 
James R. Wootton, Andreas Peter, Janos R. Winkler, and Daniel Loss. Phys. Rev. A 96, 032338 (2017)
Current quantum technology is approaching the system sizes and fidelities required for quantum error correction. It is therefore important to determine exactly what is needed for proofofprinciple experiments, which will be the first major step towards faulttolerant quantum computation. Here we propose a surface code based experiment that is the smallest, both in terms of code size and circuit depth, that would allow errors to be detected and corrected for both the X and Z basis of a qubit. This requires 17 physical qubits initially prepared in a product state, on which 16 twoqubit entangling gates are applied before a final measurement of all qubits. A platform agnostic error model is applied to give some idea of the noise levels required for success. It is found that a true demonstration of quantum error correction will require fidelities for the preparation and measurement of qubits and the entangling gates to be above 99%.
 
3.  Poking holes and cutting corners to achieve Clifford gates with the surface code 
Ben Brown, Katharina Laubscher, Markus Kesselring, and James R. Wootton. Phys. Rev. X 7, 021029 (2017)
The surface code is currently the leading proposal to achieve faulttolerant quantum computation. Among its strengths are the plethora of known ways in which faulttolerant Clifford operations can be performed, namely, by deforming the topology of the surface, by the fusion and splitting of codes and even by braiding engineered Majorana modes using twist defects. Here we present a unified framework to describe these methods, which can be used to better compare different schemes, and to facilitate the design of hybrid schemes. Our unification includes the identification of twist defects with the corners of the planar code. This identification enables us to perform singlequbit Clifford gates by exchanging the corners of the planar code via code deformation. We analyse ways in which different schemes can be combined, and propose a new logical encoding. We also show how all of the Clifford gates can be implemented with the planar code without loss of distance using code deformations, thus offering an attractive alternative to ancillamediated schemes to complete the Clifford group with lattice surgery.
 
4.  Demonstrating nonAbelian braiding of surface code defects in a five qubit experiment 
James R. Wootton Quantum Science and Technology, Volume 2, Number 1 (2017)
Currently, the mainstream approach to quantum computing is through surface codes. One way to store and manipulate quantum information with these to create defects in the codes which can be moved and used as if they were particles. Specifically, they simulate the behaviour of exotic particles known as Majoranas, which are a kind of nonAbelian anyon. By exchanging these particles, important gates for quantum computation can be implemented. Here we investigate the simplest possible exchange operation for two surface code Majoranas. This is found to act nontrivially on only five qubits. The system is then truncated to these five qubits, so that the exchange process can be run on the IBM 5Q processor. The results demonstrate the expected effect of the exchange. This paper has been written in a style that should hopefully be accessible to both professional and amateur scientists.
 
5.  Conformal energy currents on the edge of a topological superconductor 
Chris Self, Jiannis K. Pachos, James. R Wootton, and Sofyan Iblisdir. Phys. Rev. B 95, 115141 (2017)
The boundary of a 2D topological superconductor can be modeled by a conformal field theory. Here we demonstrate the behaviors of this high level description emerging from a microscopic model at finite temperatures. To achieve that, we analyze the low energy sector of Kitaev's honeycomb lattice model and probe its energy current. We observe that the scaling of the energy current with temperature reveals the central charge of the conformal field theory, which is in agreement with the Chern number of the bulk. Importantly, these currents can discriminate between distinct topological phases at finite temperatures. We assess the resilience of this measurement of the central charge under coupling disorder, bulk dimerisation and defects at the boundary, thus establishing it as a favorable means of experimentally probing topological superconductors.
 
6.  Continuous error correction for Ising anyons 
Adrian Hutter and James Wootton. Phys. Rev. A 93, 042327 (2016)
Quantum gates in topological quantum computation are performed by braiding nonAbelian anyons. These braiding processes can presumably be performed with very low error rates. However, to make a topological quantum computation architecture truly scalable, even rare errors need to be corrected. Error correction for nonAbelian anyons is complicated by the fact that it needs to be performed on a continuous basis and further errors may occur while we are correcting existing ones. Here, we provide the first study of this problem and prove its feasibility, establishing nonAbelian anyons as a viable platform for scalable quantum computation. We thereby focus on Ising anyons as the most prominent example of nonAbelian anyons and show that for these a finite error rate can indeed be corrected continuously. There is a threshold error rate p_c>0 such that for all error rates p
 
7.  Quantum Memories at Finite Temperature 
Benjamin J. Brown, Daniel Loss, Jiannis Pachos, Chris Self, and James R. Wootton. Rev. Mod. Phys. 88, 045005 (2016)
To use quantum systems for technological applications we first need to preserve their coherence for macroscopic timescales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of nogo theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. Here we review the stateoftheart developments in this field in an informative and pedagogical way. We give the main principles of selfcorrecting quantum memories and we analyze several milestone examples from the literature of two, three and higherdimensional quantum memories.
 
8.  Active error correction for Abelian and nonAbelian anyons 
James R. Wootton and Adrian Hutter. Phys. Rev. A 93, 022318 (2016)
We consider a class of decoding algorithms that are applicable to error correction for both Abelian and nonAbelian anyons. This class includes multiple algorithms that have recently attracted attention, including the BravyiHaah RG decoder. They are applied to both the problem of single shot error correction (with perfect syndrome measurements) and that of active error correction (with noisy syndrome measurements). For Abelian models we provide a threshold proof in both cases, showing that there is a finite noise threshold under which errors can be arbitrarily suppressed when any decoder in this class is used. For nonAbelian models such a proof is found for the single shot case. The means by which decoding may be performed for active error correction of nonAbelian anyons is studied in detail. Differences with the Abelian case are discussed.
 
9.  Parafermions in a Kagome lattice of qubits for topological quantum computation 
Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. X 5, 041040 (2015)
Engineering complex nonAbelian anyon models with simple physical systems is crucial for topological quantum computation. Unfortunately, the simplest systems are typically restricted to Majorana zero modes (Ising anyons). Here we go beyond this barrier, showing that the ℤ4 parafermion model of nonAbelian anyons can be realized on a qubit lattice. Our system additionally contains the Abelian D(ℤ4) anyons as lowenergetic excitations. We show that braiding of these parafermions with each other and with the D(ℤ4) anyons allows the entire d=4 Clifford group to be generated. The error correction problem for our model is also studied in detail, guaranteeing faulttolerance of the topological operations. Crucially, since the nonAbelian anyons are engineered through defect lines rather than as excitations, nonAbelian error correction is not required. Instead the error correction problem is performed on the underlying Abelian model, allowing high noise thresholds to be realized.
 
10.  A family of stabilizer codes for D(Z_2) anyons and Majorana modes 
James R. Wootton J. Phys. A: Math. Theor. 48 215302 (2015)
We study and generalize the class of qubit topological stabilizer codes that arise in the Abelian phase of the honeycomb lattice model. The resulting family of codes, which we call `matching codes' realize the same anyon model as the surface codes, and so may be similarly used in proposals for quantum computation. We show that these codes are particularly well suited to engineering twist defects that behave as Majorana modes. A proof of principle system that demonstrates the braiding properties of the Majoranas is discussed that requires only three qubits.
 
11.  Improved HDRG decoders for qudit and nonAbelian quantum error correction 
Adrian Hutter, Daniel Loss, and James R. Wootton. New J. Phys. 17, 035017 (2015)  
12.  Error Thresholds for Abelian Quantum Double Models: Increasing the bitflip Stability of Topological Quantum Memory 
Ruben S. Andrist, James R. Wootton, and Helmut G. Katzgraber. Phys. Rev. A 91, 042331 (2015)  
13.  Decoding nonAbelian topological quantum memories 
James R. Wootton, Jan Burri, Sofyan Iblisdir, and Daniel Loss. Phys. Rev. X 4, 011051 (2014)  
14.  A simple decoder for topological codes 
James R. Wootton Entropy 2015, 17(4), 19461957  
15.  Enhanced thermal stability of the toric code through coupling to a bosonic bath 
Fabio L. Pedrocchi, Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. A 88, 062313 (2013)
We propose and study a model of a quantum memory that features
selfcorrecting properties and a lifetime growing arbitrarily with system size
at nonzero temperature. This is achieved by locally coupling a 2D L x L toric
code to a 3D bath of bosons hopping on a cubic lattice. When the stabilizer
operators of the toric code are coupled to the displacement operator of the
bosons, we solve the model exactly via a polaron transformation and show that
the energy penalty to create anyons grows linearly with L. When the stabilizer
operators of the toric code are coupled to the bosonic density operator, we use
perturbation theory to show that the energy penalty for anyons scales with
ln(L). For a given error model, these energy penalties lead to a lifetime of
the stored quantum information growing respectively exponentially and
polynomially with L. Furthermore, we show how to choose an appropriate coupling
scheme in order to hinder the hopping of anyons (and not only their creation)
with energy barriers that are of the same order as the anyon creation gaps. We
argue that a toric code coupled to a 3D Heisenberg ferromagnet realizes our
model in its lowenergy sector. Finally, we discuss the delicate issue of the
stability of topological order in the presence of perturbations. While we do
not derive a rigorous proof of topological order, we present heuristic
arguments suggesting that topological order remains intact when perturbative
operators acting on the toric code spins are coupled to the bosonic
environment.
 
16.  Novel Topological Phases and SelfCorrecting Memories in Interacting Anyon Systems 
James R. Wootton Phys. Rev. A 88, 062312 (2013)
Recent studies have shown that topological models with interacting anyonic
quasiparticles can be used as selfcorrecting quantum memories. Here we study
the behaviour of these models at thermal equilibrium. It is found that the
interactions allow topological order to exist at finite temperature, not only
in an extension of the ground state phase but also in a novel form of
topologically ordered phase. Both phases are found to support selfcorrection
in all models considered, and the transition between them corresponds to a
change in the scaling of memory lifetime with system size.
 
17.  An efficient Markov chain Monte Carlo algorithm for the surface code 
Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. A 89, 022326 (2014)
Minimumweight perfect matching (MWPM) has been been the primary classical
algorithm for error correction in the surface code, since it is of low runtime
complexity and achieves relatively low logical error rates [Phys. Rev. Lett.
108, 180501 (2012)]. A Markov chain Monte Carlo (MCMC) algorithm [Phys. Rev.
Lett. 109, 160503 (2012)] is able to achieve lower logical error rates and
higher thresholds than MWPM, but requires a classical runtime complexity which
is superpolynomial in L, the linear size of the code. In this work we present
an MCMC algorithm that achieves significantly lower logical error rates than
MWPM at the cost of a polynomially increased classical runtime complexity. For
error rates p close to the threshold, our algorithm needs a runtime complexity
which is increased by O(L^2) relative to MWPM in order to achieve a lower
logical error rate. If p is below an Ldependent critical value, no increase in
the runtime complexity is necessary any longer. For p>0, the logical error
rate achieved by our algorithm is exponentially smaller (in L) than that of
MWPM, without requiring an increased runtime complexity. Our algorithm allows
for tradeoffs between runtime and achieved logical error rates as well as for
parallelization, and can be also used to correct in the case of imperfect
stabilizer measurements.
 
18.  Quantum memories and error correction 
James R. Wootton Journal of Modern Optics, 59, 20 (2012)
Quantum states are inherently fragile, making their storage a major concern
for many practical applications and experimental tests of quantum mechanics.
The field of quantum memories is concerned with how this storage may be
achieved, covering everything from the physical systems best suited to the task
to the abstract methods that may be used to increase performance. This review
concerns itself with the latter, giving an overview of error correction and
selfcorrection, and how they may be used to achieve faulttolerant quantum
computation. The planar code is presented as a concrete example, both as a
quantum memory and as a framework for quantum computation.
 
19.  Effective quantum memory Hamiltonian from local twobody interactions 
Adrian Hutter, Fabio L. Pedrocchi, James R. Wootton, and Daniel Loss. Phys. Rev. A 90, 012321 (2014)
We study a 2D toric code embedded in a 3D Heisenberg ferromagnet (FM) in a
brokensymmetry state at finite temperature. Stabilizer operators of the toric
code are locally coupled to individual spins of the FM. The effects of the
lowenergy modes of the FM lead to an energy penalty for anyons that grows
linearly with $L$, the linear size of the toric code, and thus to a lifetime of
the quantum memory growing exponentially with $L$. We study the backaction of
the toric code onto the FM both analytically and with a MonteCarlo simulation
and show a tilting of the spins close to the code after a time $t_{r}$
independent of $L$. When $t>t_{r}$ two scenarios are conceivable. Either
magnetic pulses are applied to the FM at constant time intervals $t_{r}$ in
order to refresh the FM and thus maintain a $O(L)$ energy penalty for the
anyons, or the spins of the FM reach the new equilibrium position and the
chemical potential scales with $\ln(L)$ only. In both scenarios, this provides
a stable quantum memory with strictly local boundedstrength interactions in
three dimensions.
 
20.  Lifetime of topological quantum memories in thermal environment 
Abbas AlShimary, James R. Wootton, and Jiannis K. Pachos. New J. Phys. 15 025027 (2013)
Here we investigate the effect lattice geometry has on the lifetime of
twodimensional topological quantum memories. Initially, we introduce various
lattice patterns and show how the errortolerance against bitflips and
phaseflips depends on the structure of the underlying lattice. Subsequently,
we investigate the dependence of the lifetime of the quantum memory on the
structure of the underlying lattice when it is subject to a finite temperature.
Importantly, we provide a simple effective formula for the lifetime of the
memory in terms of the average degree of the lattice. Finally, we propose
optimal geometries for the Josephson junction implementation of topological
quantum memories.
 
21.  Selfcorrecting quantum memory with a boundary 
Adrian Hutter, James R. Wootton, Beat Röthlisberger, and Daniel Loss. Phys. Rev. A 86, 052340 (2012)
We study the twodimensional toric code Hamiltonian with effective longrange
interactions between its anyonic excitations induced by coupling the toric code
to external fields. It has been shown that such interactions allow to increase
the lifetime of the stored quantum information arbitrarily by making $L$, the
linear size of the memory, larger [Phys. Rev. A 82 022305 (2010)]. We show that
for these systems the choice of boundary conditions (open boundaries as opposed
to periodic boundary conditions) is not a mere technicality; the influence of
anyons produced at the boundaries becomes in fact dominant for large enough
$L$. This influence can be both beneficial or detrimental. In particular, we
study an effective Hamiltonian proposed in [Phys. Rev. B 83 115415 (2011)] that
describes repulsion between anyons and anyon holes. For this system, we find a
lifetime of the stored quantum information that grows exponentially in $L^2$
for both periodic and open boundary conditions, though the exponent in the
latter case is found to be less favourable. However, $L$ is upperbounded
through the breakdown of the perturbative treatment of the underlying
Hamiltonian.
 
22.  High threshold error correction for the surface code 
James R. Wootton and Daniel Loss. Phys. Rev. Lett. 109, 160503 (2012)
An algorithm is presented for error correction in the surface code quantum
memory. This is shown to correct depolarizing noise up to a threshold error
rate of 18.5%, exceeding previous results and coming close to the upper bound
of 18.9%. The time complexity of the algorithm is found to be polynomial with
error suppression, allowing efficient error correction for codes of realistic
sizes.
 
23.  A witness for topological order and stable quantum memories in abelian anyonic systems 
James R. Wootton J. Phys. A: Math. Theor. 45 395301 (2012)
We propose a novel parameter, the anyonic topological entropy, designed to
detect the error correcting phase of a topological memory. Unlike similar
quantities such as the topological entropy, the anyonic topological entropy is
defined using the states of the anyon occupations. As such, though the
parameter deals with phases and phase transitions that are quantum in nature,
it can be calculated solely from classical probability distributions. In many
cases, these calculations will be tractable using efficient classical
algorithms. The parameter therefore provides a new avenue for efficient studies
of anyonic systems.
 
24.  Incoherent dynamics in the toric code subject to disorder 
Beat Röthlisberger, James R. Wootton, Robert M. Heath, Jiannis K. Pachos, and Daniel Loss. Phys. Rev. A 85, 022313 (2012)
We numerically study the effects of two forms of quenched disorder on the
anyons of the toric code. Firstly, a new class of codes based on random
lattices of stabilizer operators is presented, and shown to be superior to the
standard square lattice toric code for certain forms of biased noise. It is
further argued that these codes are close to optimal, in that they tightly
reach the upper bound of error thresholds beyond which no correctable CSS codes
can exist. Additionally, we study the classical motion of anyons in toric codes
with randomly distributed onsite potentials. In the presence of repulsive
longrange interaction between the anyons, a surprising increase with disorder
strength of the lifetime of encoded states is reported and explained by an
entirely incoherent mechanism. Finally, the coherent transport of the anyons in
the presence of both forms of disorder is investigated, and a significant
suppression of the anyon motion is found.
 
25.  Longdistance spinspin coupling via floating gates 
Luka Trifunovic, Oliver Dial, Mircea Trif, James R. Wootton, Rediet Abebe, Amir Yacoby, and Daniel Loss. Phys. Rev. X 2, 011006 (2012)
The electron spin is a natural two level system that allows a qubit to be
encoded. When localized in a gate defined quantum dot, the electron spin
provides a promising platform for a future functional quantum computer. The
essential ingredient of any quantum computer is entanglementbetween electron
spin qubitscommonly achieved via the exchange interaction. Nevertheless,
there is an immense challenge as to how to scale the system up to include many
qubits. Here we propose a novel architecture of a large scale quantum computer
based on a realization of longdistance quantum gates between electron spins
localized in quantum dots. The crucial ingredients of such a longdistance
coupling are floating metallic gates that mediate electrostatic coupling over
large distances. We show, both analytically and numerically, that distant
electron spins in an array of quantum dots can be coupled selectively, with
coupling strengths that are larger than the electron spin decay and with
switching times on the order of nanoseconds.
 
26.  Towards unambiguous calculation of the topological entropy for mixed states 
James R. Wootton (University of Leeds) J. Phys. A: Math. Theor. 45 215309 (2012)
The topological entanglement entropy is an important quantity, used to determine whether or not states are topologically ordered. However, it can give misleading results when longrange nontopological correlations are present. To solve this problem, we propose a modified topological entropy which allows topologically ordered states to be identified with greater confidence. This also provides a deeper understanding of the topological entropy, and strengthens the links between condensed matter and information theory.
 
27.  Bringing Order through Disorder: Localization of Errors in Topological Quantum Memories 
James R. Wootton (University of Leeds) and Jiannis K. Pachos (University of Leeds). Presented as a contributed talk at QIP 2011 Phys. Rev. Lett. 107, 030503 (2011)
Anderson localization emerges in quantum systems when randomized parameters cause the exponential suppression of motion. Here we consider this phenomenon in topological models and establish its usefulness for protecting topologically encoded quantum information. For concreteness we employ the toric code. It is known that in the absence of a magnetic field this can tolerate a finite initial density of anyonic errors, but in the presence of a field anyonic quantum walks are induced and the tolerable density becomes zero. However, if the disorder inherent in the code is taken into account, we demonstrate that the induced localization allows the topological quantum memory to regain a finite critical anyon density and the memory to remain stable for arbitrarily long times. We anticipate that disorder inherent in any physical realization of topological systems will help to strengthen the fault tolerance of quantum memories.
 
28.  Engineering complex topological memories from simple Abelian models 
James R. Wootton (University of Leeds), Ville Lahtinen (University of Leeds), Jiannis K. Pachos (University of Leeds), and Benoit Doucot (LPTHE, Paris). Ann. Phys. 326, 2307 (2011)
In three spatial dimensions, particles are limited to either bosonic or fermionic statistics. Twodimensional systems, on the other hand, can support anyonic quasiparticles exhibiting richer statistical behaviours. An exciting proposal for quantum computation is to employ anyonic statistics to manipulate information. Since such statistical evolutions depend only on topological characteristics, the resulting computation is intrinsically resilient to errors. Socalled nonAbelian anyons are most promising for quantum computation, but their physical realization may prove to be complex. Abelian anyons, however, are easier to understand theoretically and realize experimentally. Here we show that complex topological memories inspired by nonAbelian anyons can be engineered in Abelian models. We explicitly demonstrate the control procedures for the encoding and manipulation of quantum information in specific lattice models that can be implemented in the laboratory. This bridges the gap between requirements for anyonic quantum computation and the potential of stateoftheart technology.
 
29.  Universal quantum computation with a nonAbelian topological memory 
James R. Wootton, Ville Lahtinen, and Jiannis K. Pachos. LNCS 5906, 5665 (2009)
An explicit lattice realization of a nonAbelian topological memory is
presented. The correspondence between logical and physical states is seen
directly by use of the stabilizer formalism. The resilience of the encoded
states against errors is studied and compared to that of other memories. A set
of nontopological operations are proposed to manipulate the encoded states,
resulting in universal quantum computation. This work provides insight into the
nonlocal encoding nonAbelian anyons provide at the microscopical level, with
an operational characterization of the memories they provide.
 
30.  Universal Quantum Computation with Abelian Anyon Models 
James R. Wootton and Jiannis K. Pachos. ENTCS 270, 209218 (2011)
We consider topological quantum memories for a general class of abelian anyon
models defined on spin lattices. These are nonuniversal for quantum
computation when restricting to topological operations alone, such as braiding
and fusion. The effects of additional nontopological operations, such as spin
measurements, are studied. These are shown to allow universal quantum
computation, while still utilizing topological protection. Our work gives an
insight into the relation between abelian models and their nonabelian
counterparts.
 
31.  Nonabelian statistics from an abelian model 
James R. Wootton, Ville Lahtinen, Zhenghan Wang, and Jiannis K. Pachos. Phys. Rev. B 78, 161102(R) (2008)
It is well known that the abelian $Z_2$ anyonic model (toric code) can be
realized on a highly entangled twodimensional spin lattice, where the anyons
are quasiparticles located at the endpoints of stringlike concatenations of
Pauli operators. Here we show that the same entangled states of the same
lattice are capable of supporting the nonabelian Ising model, where the
concatenated operators are elements of the Clifford group. The Ising anyons are
shown to be essentially superpositions of the abelian toric code anyons,
reproducing the required fusion, braiding and statistical properties. We
propose a string framing and ancillary qubits to implement the nontrivial
chirality of this model.
