**CoogeeĠ12**

**Sydney
Quantum Information Theory Workshop**

**John Baez **

**(CQT, Singapore)**

*Probabilities and Amplitudes.*

Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this turns out to involve creation and annihilation operators, coherent states and other well-known ideas - but with a few big differences. The stochastic analogue of quantum field theory is also used in population biology, and here the connection is well-known. But what does it mean to treat wolves as fermions or bosons?

For reading material, people can visit

http://math.ucr.edu/home/baez/networks/networks.html

**Sean Barrett **

**(Imperial College London,
UK)**

*Protected superconducting
devices*

**Hector Bombin
**

**(Perimeter Institute, Canada)**

*Strong Resilience of
Topological Codes to Depolarization*

The inevitable presence of decoherence effects in systems suitable for quantum computation necessitates an effective error correction scheme to protect information from noise. In this work we compute the stability of the toric code to depolarization by mapping the quantum problem onto a classical disordered eight-vertex Ising model. By studying the stability of the related ferromagnetic phase both via large-scale Monte Carlo simulations and via duality methods, we are able to demonstrate an increased error threshold of pc = 0.189(3) when noise correlations are taken into account. Remarkably, this agrees within error bars with the result for a different class of codes – topological color codes – where the mapping yields interesting new types of 8-vertex models with additional interactions. Our results indicate that error correction codes exploiting topological properties are both efficient and feasible for real world applications. This work is a collaboration with Ruben S. Andrist, Masayuki Ohzeki, Helmut G. Katzgraber, and M. A. Martin-Delgado.

**Andrew Doherty **

**(University of Sydney,
Australia)**

*The algebraic Bethe ansatz.*

**Guillaume Duclos-Cianci **

**(University of Sherbrooke, Canada)**

*2D Cellular Automaton
Fault-Tolerant Quantum Memory?*

The main goal of this talk is to introduce/discuss a new idea for a decoder for topological codes. In order to get there, I will first briefly review what is known about topological codes and self-correction in quantum systems [1-8] . Second, as there is no hope for self-correction in 2D with stabilizer codes, I will review what is our current state of knowledge regarding active decoding of 2D topological codes [9-14]. Third, and most importantly, I will motivate the need for an integrated/on-chip decoder. In order to achieve this feat, I will propose a new idea: a decoder that simulates self-correction through artificial confinement [15] of defects. The ultimate goal is to implement this simulator as a cellular automaton. We hope this results in a physically realistic, constant-time complexity, fault-tolerant decoder. I will present encouraging preliminary data in the QEC setting.

This list of reference is by no means exhaustive:

[1] Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code, Sergey Bravyi, Jeongwan Haah,

arXiv:1112.3252v1 [quant-ph]

[2] Logical operator tradeoff for local quantum codes, Jeongwan Haah, John Preskill,

arXiv:1011.3529v2 [quant-ph]

[3] Tradeoffs for reliable quantum information storage in 2D systems, Sergey Bravyi, David Poulin, Barbara Terhal,

Phys. Rev. Lett. 104 050503 (2010), (arXiv:0909.5200v1 [quant-ph])

[4] A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes, Sergey Bravyi, Barbara Terhal,

New J. Phys. 11 (2009) 043029 (arXiv:0810.1983v2 [quant-ph])

[5] On thermal stability of topological qubit in Kitaev's 4D model, R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki,

Open Syst. Inf. Dyn. 17 (2010) 1, (arXiv:0811.0033v1 [quant-ph])

[6] Topological fault-tolerance in cluster state quantum computation, Robert Raussendorf, Jim Harrington, Kovid Goyal,

New Journal of Physics 9, 199 (2007), arXiv:quant-ph/0703143v1

[7] Topological quantum memory, Eric Dennis, Alexei Kitaev, Andrew Landahl, John Preskill,

J. Math. Phys. 43, 4452-4505 (2002), (arXiv:quant-ph/0110143v1)

[8] Fault-tolerant quantum computation by anyons, A. Yu. Kitaev

Annals Phys. 303 (2003) 2-30, (arXiv:quant-ph/9707021v1)

[9] Efficient Decoding of Topological Color Codes, Pradeep Sarvepalli, Robert Raussendorf,

arXiv:1111.0831v1 [quant-ph]

[10] Towards practical classical processing for the surface code, Austin G. Fowler, Adam C. Whiteside, Lloyd C. L. Hollenberg,

arXiv:1110.5133v1 [quant-ph]

[11] Fault-tolerant quantum computing with color codes, Andrew J. Landahl, Jonas T. Anderson, Patrick R. Rice,

arXiv:1108.5738v1 [quant-ph]

[12] Universal topological phase of 2D stabilizer codes, H. Bombin, G. Duclos-Cianci, D. Poulin,

arXiv:1103.4606v1 [quant-ph]

[13] A renormalization group decoding algorithm for topological quantum codes, G. Duclos-Cianci, D. Poulin

Information Theory Workshop (ITW), 2010 IEEE ,1-5 (arXiv:1006.1362v1 [quant-ph])

[14] Fast Decoders for Topological Quantum Codes, G. Duclos-Cianci, D. Poulin,

Phys. Rev. Lett. 104 050504 (2010) (arXiv:0911.0581v2 [quant-ph])

[15] Toric-boson model: Toward a topological quantum memory at finite temperature,

Phys. Rev. B 79, 245122 (2009) (arXiv:0812.4622v3 [quant-ph])

**Steve Flammia
**

**(University of Washington,
USA)**

*Direct Fidelity Estimation from
Few Pauli Coefficients*

**Jeongwan**** Haah **

**(Caltech, USA)**

*Quantum memory on
topological spin glass*

Based
on the joint work with Sergey Bravyi, IBM Watson.

We show that any topologically ordered local stabilizer model of spins in three dimensional lattices that lacks string logical operators can be used as a reliable quantum memory against thermal noise. It is shown that any local process creating a topologically charged particle separated from other particles by a distance $R$ must cross an energy barrier of height $c \log R$. This property makes the model glassy. We devise an efficient decoding algorithm that should be used at the final read-out, and prove a lower bound on the memory time until which the fidelity between the outcome of the decoder and the initial state is close to 1. The memory time increases as $L^{\beta}$ where $L$ is the system size and $\beta$ the inverse temperature, as long as $L < L^\star \sim e^\beta$. Hence, the optimal memory time scales as $e^{\beta^2}$. Our bound applies when the system interacts with thermal bath via a Markovian master equation. We give an example of 3D local stabilizer codes that satisfies all of our assumptions. We numerically verify for this example that our bound is tight up to constants.

arXiv:1101.1962 arXiv:1105.4159 arXiv:1112.3252

**Robert Koenig **

**(IBM, USA)**

*Disorder-assisted error
correction in Majorana chains*

It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions - the simplest toy model of a quantum memory. Disorder takes the form of a random site-dependent chemical potential. The corresponding one-particle problem is a one-dimensional Anderson model with disorder in the hopping amplitudes. We focus on the zero-temperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an error-correction step. Assuming dynamical localization of the one-particle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the one-particle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.

This is joint work with Sergey Bravyi, available at

http://arxiv.org/abs/1108.3845 .

Background reading:

Wootton & Pachos, PRL 107, 030503 (2011)

Stark et al., PRL 107, 030504 (2011)

**Tobias Osborne **

**(Leibniz Universitat
Hannover, Germany)**

*A class of quantum field
states inside the `physical corner of hilbert space'*

I'll discuss a higher-dimensional formulation of the recently introduced variational class of quantum field states known as continuous matrix-product states (cMPS) through a path integral representation. Symmetry constraints, namely, rotation and translation invariance, imply that the auxiliary system involved in their definition must be Lorentz-invariant. I'll then argue that these states provide a parametrisation of the physically accessible corner of quantum field hilbert space. Also, an argument for how expectation values may be calculated efficiently and entropy/area laws will be presented. These properties suggest that such states will allow powerful analytical and numerical approaches to describe ground-state, finite-temperature, finite-fermion density, and real-time physics of strongly interacting quantum fields in two and higher spatial dimensions.

**David Poulin
**

**(Sherbrooke,
Canada)**

*Quantum Hammersley-Clifford
theorem*

**Jiannis**** Pachos **

**(University of Leeds, UK)**

*Seeing topological order*

A cold atom perspective.

**Norbert Schuch
**

**(Caltech, USA)**

*An order parameter for
symmetry-protected topological order*

One-dimensional systems can exhibit non-trivial symmetry-protected topological phases which are characterized by the inequivalent ways in which the symmetry acts on the entanglement in the state. We ask the question whether it is possible to distinguish different phases by a physical measurement, i.e., whether it is possible to detect the action of the symmetry on the entanglement, and we show how to construct an order parameter which allows to identify symmetry-protected topological phases. The order parameter consists of string-like operators, but differs from conventional string order parameters in that it depends only on the symmetry but not on the state. We verify our framework through numerical simulations for the SO(3) invariant spin-1 bilinear-biquadratic model which exhibits a dimerized and a Haldane phase, and find that the estimator works very well not only for the dimerized and the Haldane phase, but is also returns a distinct signature for gapless phases.

Background literature:

[1] N. Schuch, D. Perez-Garcia, and I. Cirac,

Classifying quantum phases using Matrix Product States and PEPS.

Phys. Rev. B 84, 165139 (2011); arXiv:1010.3732.

[2] X. Chen, Z. Gu, and X. Wen,

Classification of Gapped Symmetric Phases in 1D Spin Systems.

Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745.

[3] J. Haegeman, D. Perez-Garcia, I. Cirac, and N. Schuch,

in preparation.

**Frank Verstraete
**

**(University of Vienna,
Austria)**

*Quantum Chi-Squaried and Goodness of Fit Testing*

K. Temme, F. Verstraete, arXiv:1112.6343.

**Guifre**** Vidal **

**(Perimeter Institute, Canada)**

*Criticality, impurities and
real-space renormalization*

In the context of using the multi-scale entanglement renormalization ansatz (MERA) to describe critical quantum many-body systems, I will explain how to exploit translation invariance in an impurity system (which is not translation invariant!). More generally, a "principle of minimal influence" in the renormalization group flow allows to define a simple ansatz, depending on a small number of tensors, from which one can extract the critical properties of impurity, boundary, interface and Y-junction critical systems on an infinite lattice.