Sydney Quantum Information Theory Workshop
(Max Planck Institute for Quantum Optics, Germany)
Topological order on the lattice: Dualities and perspectives.
2D topological phases on the lattice subject to very general symmetry requirements are described by Levin and Wen's string-net models [LW]. These are based on physical renormalisation group ideas, and in the mathematics of category theory. I will report on an ongoing project [BA, BMCA, BCKA] to recast these models in a language closer to Kitaev's original topological lattice models [K], including the toric code, which are essentially gauge theories. The gauge-like reinterpretation of the string-nets allows us to give a precise notion of electric-magnetic duality underlying topological orders, as well as to understand more deeply the properties of the anyonic excitations. Further applications and perspectives will be touched upon.
The project includes so far produced the following:
[BA] O. Buerschaper and M. Aguado, Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models, Phys. Rev. B 80, 155136 (2009), arXiv:cond-mat-0907.2670.
[BMCA] O. Buerschaper, J.M. Mombelli, M. Christandl and M. Aguado, A hierarchy of topological tensor network states, arXiv:1007.5283 [cond-mat.str-el].
[BCKA] O. Buerschaper, M. Christandl, L. Kong and M. Aguado, Electric-magnetic duality and topological order in the lattice, arXiv:1006.5823 [cond-mat.str-el].
Background literature about the models involved:
[K] A.Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals Phys. 303, 2 (2003), arXiv:quant-ph/9707021.
[LW] M. A. Levin, X.-G. Wen, String-net condensation. A physical mechanism for topological phases, Phys. Rev. B71, 045110 (2005), arXiv:cond-mat/0404617.
(University of Washington, USA)
Turning Time Into Space for Codes and Profit
I will describe a new code construction which turns any stabilizer code into a subsystem code with local check operators (joint work with Jonathan Shi.) This construction is derived from measurement-based quantum computing and is related to a recent model of adiabatic quantum computing called adiabatic quantum transistors (joint work with Steve Flammia and Gregory Crosswhite.) I will then show how to use this construction to saturate the Bravyi-Terhal distance bounds for spatially local codes in three dimensions, having a distance that scales as an area. Hamiltonians constructed from these codes are therefore natural candidates for self-correcting quantum memories.
"Adiabatic Cluster State Quantum Computing"
Dave Bacon, Steven T. Flammia
"Stabilizer Formalism for Operator Quantum Error Correction"
(Imperial College London, UK)
Percolation, topological order, and the loss error threshold for FTQC
We will describe recent results from an ongoing project (with Tom Stace and others) which examines the robustness of Kitaev's surface codes, and related FTQC schemes (due to Raussendorf and coworkers) to loss errors. The key insight is that, in a topologically ordered system, the quantum information is encoded in delocalized degrees of freedom that can be "deformed" to avoid missing physical qubits. This allows one to relate error correction and fault tolerance thresholds to percolation thresholds. Furthermore, stabilizer operators can be deformed in a similar way, which means that surface codes retain their robustness to arbitrary types of error, even when significant numbers of qubits are lost.
We present numerical evidence, utilizing these insights, to show that (1) the surface code can tolerate up to 50 percent loss errors, and (2) Raussendorfs FTQC scheme can tolerate up 25 percent loss errors. The numerics indicate both schemes retain good performance when loss and computational errors are simultaneously present. Finally we will describe extensions to other error models, in particular the case where logic gates can fail but in a heralded manner.
Phys. Rev. Lett. 102, 200501 (2009)
Thresholds for Topological Codes in the Presence of Loss
Thomas M. Stace, Sean D. Barrett, and Andrew C. Doherty
Phys. Rev. A 81, 022317 (2010)
Error correction and degeneracy in surface codes suffering loss
Thomas M. Stace and Sean D. Barrett
Phys. Rev. Lett. 105, 200502 (2010)
Fault Tolerant Quantum Computation with Very High Threshold for Loss Errors
Sean D. Barrett and Thomas M. Stace
Phys. Rev. Lett. 105, 250502 (2010)
Fault Tolerant Quantum Computation with Nondeterministic Gates
Ying Li, Sean D. Barrett, Thomas M. Stace, and Simon C. Benjamin
Two good background papers on Kitaev's surface codes and Raussendorf's
FTQC scheme are:
Topological quantum memory
Authors: Eric Dennis, Alexei Kitaev, Andrew Landahl, John Preskill
Topological fault-tolerance in cluster state quantum computation
Authors: Robert Raussendorf, Jim Harrington, Kovid Goyal
Bulk Fault Tolerant Quantum Computing with Boundary Addressability
Abstract: Making quantum computers simpler to control is a challenge for scalability. By combining measurement free quantum error correction with global control I will show that fault tolerant computation in a d-dimensional array of qubits needs only (d-1)-dimensional addressing resolution. Measurements and individual control of qubits are required only at the boundaries of the computer, or can be done in a homogenous manner in effect reducing the number of required control modes by a factor equal to the number of logical qubits. This model alleviates the heavy physical conditions on current qubit candidates imposed by addressability and could be useful for implementions in 3D with nearest neighbour interactions or in 2D with next-nearest neighbour interactions.
"Fault Tolerance with Noisy and Slow Measurements and Preparation,"
Gerardo A. Paz-Silva, Gavin K. Brennen, and Jason Twamley, Phys. Rev.
Lett. 105, 100501 (2010), arXiv:1002.1536
"Bulk fault-tolerant quantum information processing with boundary
addressability," Gerardo A. Paz-Silva, Gavin K. Brennen, and Jason
Twamley, New J. Phys (in press), arXiv:1008.1634
(University College London, UK)
Bell Inequalities for Beginners
Linear functions, loopholes, and how to post-select data without causing one.
This work began as an investigation into whether the cosmetic similarities between Bell inequality experiments and measurement-based quantum computing might reflect a deeper connection. What we find is that the ``computational viewpoint'' is remarkably well-suited to the study of Bell inequalities, and provides a very simple way of characterising the correlations permitted by local hidden variable (LHV) theories. While mathematically equivalent to previous work (see Werner and Wolf's paper cited below for the original derivation of this set of correlations) this provides a cleaner and simpler operational definition, which, in particular, make ''loopholes'' easy to characterise.
Loopholes in Bell inequality experiments arise when imperfections in the experimental setup mean that it does not concord precisely with the assumptions under which Bell inequalities are derived. Often those flaws mean that local hidden variable theories would be able to access correlations which in a strict Bell setup would be forbidden to them, and hence a quantum mechanical reproduction of that correlation cannot refute local hidden variable theories - a "loophole" in the argument.
The simplicity of the characterisation of the LHV correlations in our model allow us to pin down explicit mechanisms by which loopholes arise. For example, a well-known loophole is the detector loophole, where, due to inefficient detectors, data must be post-selected - only when both detectors fire can the data be used. The origin of the detector loophole can be cleanly understood within our framework, and constructing LHV models which fake an imperfect quantum detector while maximally violating Bell inequalities (to the bounds previously identified by Garg and Mermin) is straightforward. We see that there are many ways in which post-selection of data can cause loopholes.
In addition to providing a simpler (and quantum information friendly) way to understand previous results (hence "for beginners"), we can use our model for new investigations. For example, we can characterise post-selection strategies where no loopholes arise - and consider the effect of this post-selection upon quantum correlations. We find that there is a broad family of non-loophole inducing post-selection strategies which one can adopt. Surprisingly, we see that while not expanding the region of correlations accessible by LHV theories, such post-selection can expand the region of correlations accessible by quantum theories. In other words, performing this post-selection allows quantum measurements to achieve correlations which were previously impossible, without creating a loophole. This effect becomes apparent in the multi-partite setting (the smallest example we have is for 6 qubits), and does not enhance the bi-partite CHSH inequality, therefore, it is currently unclear whether this post-selection will aid current Bell inequality experiments. However, the larger multi-partite region now includes new types of quantum correlation previously overlooked in the Bell inequality setting, most notably (post-selected simulations of) the adaptive measurements which arise in measurement-based quantum computation. We expect that these results will give new insights into measurement-based quantum computation and related areas and will be valuable in the search for information theoretic characterisations of the set of quantum correlations which go beyond the bi-partite setting.
These results were developed in collaboration with Matty Hoban.
Previous work in this direction
J. Anders and D.E. Browne, arXiv:0805.1002
M. Hoban et al, arXiv:1009.5213
Main reference for these results
M. Hoban and D. E. Browne, in preparation (hopefully on the arxiv very soon)
Multi-partite Bell inequalities, Werner and Wolf, Phys. Rev. A 64, 032112 (2001)
Detector Loophole, Garg and Mermin, Phys. Rev. D 35, 3831–3835 (1987)
(University of New Hampshire, USA)
Anyon condensation and commutative algebras
The fact that condensations of anyons in a topological order are controlled by certain commutative algebras in the corresponding modular category allows us to describe all possible condensation patterns and the resulting new topological orders. In particular it can be shown that for a given topological order there are only a finite number of condensation patterns and that new topological orders emerging in maximal condensations are all equivalent.
A good background reading is a review like article
F.A. Bais, J.K. Slingerland, "Condensate induced transitions between topologically ordered phases."
The stability of the spectral gap for frustration-free systems under local perturbations
We generalize the recent result on stability of topological quantum order for Hamiltonians that are sums of commuting projectors, to sums of unfrustrated projections (i.e. each one has energy zero in the ground state space) with a constant energy gap to the first excited states. We extend the notion of topological quantum order to "local topological quantum order" (L-TQO), a condition necessary for stability of such Hamiltonians. Moreover, it can be shown that L-TQO implies the area law (up to a logarithmic correction) for the entanglement entropy of the corresponding ground states, even in the absence of a spectral gap. Some early results suggest that parent Hamiltonians of MPS and PEPS satisfy the conditions for stability. Several open questions on the applicability of this result will be presented.
Topological Quantum Order: Stability under local perturbations. S. Bravyi, M. Hastings, S. Michalakis (2010) http://arxiv.org/abs/1001.0344
A short proof of stability; http://arxiv.org/abs/1001.4363
(Perimeter Institute, Canada)
Quantum computational capability of a two-dimensional valence bond solid phase
I will talk about quantum computation by harnessing many-body correlations of a condensed-matter system. Quantum phases of naturally-occurring systems exhibit rich nature as manifestation of their many-body correlations, in contrast to our persistent technological challenge to build at will such correlations artificially from scratch. Here we show theoretically that quantum correlations exhibited in the two-dimensional valence bond solid phase of a quantum antiferromagnet, modeled by Affleck, Kennedy, Lieb, and Tasaki as a precursor of spin liquids and topological orders, are sufficiently complex yet structured enough to simulate universal quantum computation when every single spin can be measured individually. This unveils that an intrinsic complexity of naturally-occuring 2D quantum systems -- which has been a long-standing challenge for traditional computers -- could be tamed as a computationally valuable resource, regardless of our constraint not to create newly entanglement during computation. Our constructive protocol leverages a novel way to herald the correlations suitable for deterministic quantum computation through a random sampling, and may be extensible to other ground states of various 2D valence bond phases beyond the AKLT state.
The background materials are arXiv:1009.3491 and references therein.
Local equivalence of topological order: Kitaev's code and color codes
We demonstrate that distinct topological codes can be mapped onto each other by local transformations. The existence of such a local mapping can be interpreted as saying that these codes belong to the same topological phase.When used as quantum error correcting codes, the local mapping also enable us to use any decoding algorithm suitable for one of these codes to decode other codes in the same topological phase. We illustrate this idea with the topological color code and the topological subsystem color code that are found to be locally equivalent to two copies of Kitaev's toric code. We are therefore able to decode these two codes that had no previously known efficient decoding algorithm, and find error thresholds comparable to previously estimated optimal values. These local mappings could have additional use for fault-tolerant quantum computation. In particular, one could in principle take advantage of the features (transversal gates, topological gates, etc.) of all the codes that are locally equivalent by switching between them during the computation in a fault tolerant fashion.
(TU Darmstadt, Germany)
Holonomic quantum computing using symmetry-protected topological order
I will talk about two related approaches to quantum computation using the Haldane phase of spin-chains. The first is a measurement-based approach in which ground states of such spin chains are used to encode single qubits and logical gates are performed by measurement. Departing from an idealized computational model based on the matrix-product state representation of the Affleck-Kennedy-Lieb-Tasaki (AKLT) chain , we show that it is possible to use the same computational model elsewhere in the Haldane phase by simulating a renormalization group transformation which, conveniently, flows toward the AKLT chain . In the second, we successively adapt this measurement-based scheme to one involving only adiabatic transformations, in the manner of adiabatic state swapping (ASS) of Bacon and Flammia . This reveals that the real computational power of the Haldane phase can be understood just in terms of various rotational symmetries, not any particular matrix-product state representation. Moreover, the scheme can be implemented in a wide variety of settings, and requires only nearest-neighbor, two-body interactions, making it a promising computational architecture in the near term.
 G.K. Brennen and A. Miyake, PRL 101, 010502 (2008).
 S.D. Bartlett, G.K. Brennen, A.Miyake, and J.M. Renes, PRL 105,
 D. Bacon and S.T. Flammia, Phys. Rev. Lett. 103, 120504 (2009).
Understanding quantum phases using Matrix Product States and PEPS
We apply the framework of Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) and their associated parent Hamiltonians to the classification of quantum phases in one and higher dimensions. In the case of one-dimensional systems, we show that phases are characterized solely by the ground state degeneracy. Subsequently, we consider the effect of symmetries and show how the representation theory of the symmetry group allows to classify the phases of systems both with and without symmetry breaking. Finally, we discuss how to extend our framework to two-dimensional systems and give a classification of two-dimensional quantum phases in the neighborhood of a number of important cases, such as systems with unique ground states, local symmetry breaking, and topological order. As a central tool, we introduce a method which allows to transform any system using a Hamiltonian flow to an RG fixed point without the need for actual renormalization.
Branching MERA: beyond the boundary law for ground state entanglement entropy
[in collaboration with Glen Evenbly]
Many ground states of two-dimensional lattice models obey an “area law” (actually, a boundary law) for entanglement entropy. However, some two-dimensional models, including Fermi liquids and Bose metals, are significantly more entangled, with their ground states displaying logarithmic multiplicative corrections to the boundary law. On the other hand, tensor network states (PEPS, 2D MERA, etc) can only reproduce a strict boundary law, and the question of how to classically simulate these more robustly entangled ground states have remained open for several years. In this talk I will answer this question. Recall that the MERA can be regarded as a quantum circuit with a very peculiar property: the reduced density matrix of an outgoing qubit can be efficiently computed with a classical computer. I will explain how to generalize this quantum circuit so as to produce a branching MERA, capable of violating the boundary law for entanglement entropy while remaining efficiently simulatable by a classical computer.