Coogee’13

Sydney Quantum Information Theory Workshop

Ed Barnes

(Maryland, USA)

Electron-nuclear spin hyperfine dynamics in a quantum dot

The leading source of decoherence for electron spin qubits in III-V semiconductor quantum dots is the hyperfine interaction between the electron spin and the bath of nuclear spins residing in the dot. Gaining control over the nuclear spin polarization can not only mitigate this decoherence, but opens up the possibility of re-purposing the nuclear bath into a quantum memory resource. I will present a nonperturbative, closed-form solution of electron spin decoherence in the presence of a polarized nuclear bath that can be used to quantify the reduction in decoherence and to detect detailed properties of the bath polarization state. I will further present theoretical work that sheds light on how nuclear spin polarization is produced through the application of non-unitary controls on the electron spin.

PRL 107, 047601 (2011)

PRB 84, 155315 (2011)

Andrew Bolt

(Queensland, Australia)

Scalable Mesurement Based Fault-Tolerant Quantum Communication

I will present a fault-tolerant measurement based scheme for creating encoded Bell pair resources on foliated CSS codes. This is supplemented by an efficient decoding scheme for CSS Turbo codes based on Belief propagation. The advantage of using turbo codes is that we can achieve strong error correction with a relatively small overhead of physical resources.

Fernando Brandao

(ETH Zurich, Switzerland)

Approximation Guarantees for the Quantum Local Hamiltonian Problem and Limitations for Quantum PCPs

The local Hamiltonian problem consists in estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. It can be considered as a quantum generalization of constraint satisfaction problems

(CSPs) and has a key role in quantum complexity theory, being the first and most natural QMA-complete problem known. An interesting regime for the local Hamiltonian problem is that of extensive error, where one is interested in estimating the mean ground-state energy to constant accuracy. The problem is NP-hard by the PCP theorem, but whether it is QMA-hard is an important open question in quantum complexity theory. A positive solution would represent a quantum analogue of the PCP theorem. In this talk I will present new approximation guarantees for solving the local Hamiltonian problem within constant extensive error.

First Ill show how to obtain in NP a proof of small mean ground-state energy for 2-local Hamiltonians (only containing 2-body interactions) that depends on three parameters:

1. The average expansion of a partition of the sites of the Hamiltonian: Here we recover the well-known fact that hard instances must be based on highly expanding constraint graphs.

2. The degree of the constraint graph: Here I'll show that the problem is in NP for Hamiltonians on high degree graphs, with a product state giving an approximation to the groundstate energy of error inverse polynomial in the degree of the graph. In contrast, the PCP theorem holds true with error independent of the degree of the constraint graph of the CSP. This result also gives a rigorous justification to the folklore in condensed matter physics that mean field becomes exact once the dimension of the lattice grows.

3. The average entanglement in the ground state of the Hamiltonian:

Here we find that the problem is also in NP for Hamiltonians that satisfy a subvolume law for the entanglement entropy. This gives a new connection between the amount of entanglement in the ground- state of a local Hamiltonian and its computational complexity.

The approximation guarantees obtained put strong limitations on which type of Hamiltonians one could expect to base a possible quantum analogue of the PCP theorem.

Second I'll briefly present new polynomial-time algorithms for approximating the mean groundstate energy of two classes of Hamiltonians:

(i) 2-local Hamiltonians on any planar graph, solving an open problem of Bansal, Bravyi, and Terhal; and

(ii) dense k-local Hamiltonians for any con-stant k, solving an open problem of Gharibian and Kempe. Both algorithms generalize similar known results for CSPs.

The main idea behind the results is an information-theoretic argument inspired by recent work on the convergence of the Lasserre hierarchy for CSPs.

Based on joint work with Aram Harrow.

Reading list:

[1] Dorit Aharonov and Tomer Naveh. Quantum NP - A Survey. quant-ph/0210077

[2] Dorit Aharonov, Itai Arad, Zeph Landau, Umesh Vazirani. The

Detectability Lemma and Quantum Gap Amplification. arXiv:0811.3412

[3] Nikhil Bansal, Sergey Bravyi, Barbara M. Terhal. Classical

approximation schemes for the ground-state energy of quantum and

classical Ising spin Hamiltonians on planar graphs. arXiv:0705.1115

[4] Sevag Gharibian, Julia Kempe. Hardness of approximation for

quantum problems. arXiv:1209.1055

[5] Itai Arad. A note about a partial no-go theorem for quantum PCP.

arXiv:1012.3319.

[6] M. B. Hastings. Trivial Low Energy States for Commuting

Hamiltonians, and the Quantum PCP Conjecture. arXiv:1201.3387.

[7] Prasad Raghavendra, Ning Tan. Approximating CSPs with Global

Cardinality Constraints Using SDP Hierarchies. arXiv:1110.1064.

[8] Boaz Barak, Prasad Raghavendra, David Steurer. Rounding

Semidefinite Programming Hierarchies via Global Correlation.

arXiv:1104.4680.

Courtney Brell

(University of Sydney, Australia)

Many-body models based on quantum double algebras

We explore the use of quantum double algebras to define many-body spin models of interest for quantum information. This is motivated by the topologically ordered Kitaev quantum double models, which are an important testbed for ideas in topological quantum information processing and storage. We present two explicit examples of these constructions. The first is a generalization of the cluster state which can be used to implement an analogue of the topological cluster state computation scheme, simulating the braiding of topological defects in a Kitaev quantum double model. The second is a generalization of the color code capable of supporting non-abelian anyonic excitations. Although our explicit constructions are restricted to the quantum doubles of groups, we also consider extensions to the more general case.

Jacob Bridgeman

(University of Sydney, Australia)

MERA for Spin Chains with Continuously Varying Criticality

Using the multiscale entanglement renormalization ansatz, we examine a class of spin models believed to be described by c = 1 CFTs in the continuum limit. These theories are characterised by an exactly marginal primary field which is fixed under the rescaling operation. Thus perturbing a Hamiltonian by a marginal operator gives another scale invariant model leading to critical lines in the phase diagram, along which the dimension of the primary fields vary continuously. We demonstrate the ability to reproduce this behaviour in our spin model simulations.

In particular, we investigate two spin models, the Ashkin-Teller quantum antiferromagnet and the perturbed cluster state spin chain. The latter lies on the boundary of a D2 symmetry protected topological phase, containing the cluster state, within which the ground state remains a resource for MBQC.

In the thermodynamic limit, these models are believed to correspond to the orbifold boson and S1 boson CFTs respectively. Using the MERA to simulate these chains, we extract conformal data consistent with this.

Toby Cubitt

(UCM, Spain)

Undecidability of the spectral gap problem

The spectral gap of a many-body Hamiltonian -- the difference between the ground state energy (lowest eigenvalue) and lowest excited state (next-lowest eigenvalue, ignoring degeneracies) in the thermodynamic limit (limit of arbitrarily large system size) -- plays an important role in determining the physical properties of a many-body system. In particular, low-energy excitations of gapped systems behave as massive particles, whereas those of gapless systems are massless.

A number of famous open problems in mathematical physics concern whether or not particular many-body models are gapped. For example, the "Haldane conjecture" states that Heisenberg spin chains are gapped for integer spin, and gapless for half-integer spin. The seminal Lieb-Schultz-Mattis result proves the half-integer case; but, whilst there exists strong numerical evidence, there the integer case remains unproven. In 2-dimensions, there is a longstanding conjecture that the 2D AKLT model is gapped. In the related setting of quantum field theories, determining if Yang-Mills theory is gapped is one of the Millennium Prize Problems.

I will show that the general spectral gap problem is unsolvable. There exist translationally-invariant, two-body Hamiltonians on a 2D square lattice of finite-dimensional spins, for which the spectral gap problem is undecidable. Thus there exist gapless Hamiltonians for which the absence of a gap cannot be proven in any consitent mathematical framework. The proof draws together techniques from classical tiling problems, recent quantum Hamiltonian complexity results, and an even more recent PEPS Hamiltonian construction.

Based on ongoing work with David Perez-Garcia and Michael Wolf.

Background reading:

1. F.D.M. Haldane, Phys. Rev. Lett. 50, 1153 (1983)

2. E.H. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, 407 (1961)

3. I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki,

     Phys. Rev. Lett. 59, 799 (1987)

4. R. Berger, Memoirs Amer. Math. Soc. 66 (1966)

5. R. Robinson, Inventiones Math. 12, 177-209 (1971)

6. D. Gottesman, S. Irani, FOCS '09, 95 (2009)

7. C. Fernández-González, N. Schuch, M. M. Wolf, J. I. Cirac,

     D. Pérez-García, arXiv:1210.6613[quant-ph]

Sania Jevtic

(Imperial College London, UK)

The Quantum Steering Ellipsoid

A quantum steering ellipsoid may be used to faithfully represent a density matrix describing two qubits A and B. The ellipsoid is the geometric set of states that Bob can steer Alice's qubit to when he implements all possible measurements on his qubit. We show how the correlations between qubits A and B manifest themselves in this paradigm, giving simple conditions for when the state is entangled, or has discord. We will also present novel features of the two qubit state that are revealed by the steering ellipsoid formalism, and show that a state corresponding to an ellipsoid with non-zero volume contains a new type of correlation.

Isaac Kim

(Caltech, USA)

Application of conditional independence to gapped quantum many-body systems

It is widely known in the quantum information community that the states that satisfy strong subadditivity of entropy with equality have the form of quantum Markov chain. Based on a recent strengthening of strong subadditivity of entropy, I will describe how such structure can be exploited in the studies of gapped quantum many-body system. In particular, I will describe a diagrammatic trick to i) give a quantitative statement about the locality of entanglement spectrum ii) perturbatively bound changes of topological entanglement entropy under generic perturbation. If time permits, I will talk about some related open problems.

Background materials:

D. Petz, Monnotonicity of quantum relative entropy revisited. arXiv:quant-ph/0209053 (2002)

P. Hayden, R. Jozsa, D. Petz, A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality. arXiv:quant-ph/0304007

I. H. Kim, Operator extension of strong subadditivity of entropy. arXiv:1210.5190 (2012)

I. H. Kim, Perturbative analysis of topological entanglement entropy from conditional independence. arXiv:1210.2360 (2012)

I. H. Kim, Determining structure of real-space entanglement spectrum from approximate conditional independence. arXiv:1210.1831 (2012)

I especially recommend to read the first two papers by Petz and Hayden et al.

Olivier Landon-Cardinal

(Sherbrooke, Canada)

Local topological order inhibits thermal stability in 2D

We study the robustness of quantum information stored in the degenerate ground space of a local, frustration-free Hamiltonian with commuting terms on a 2D spin lattice. On one hand, a macroscopic energy barrier separating the distinct ground states under local transformations would protect the information from thermal fluctuations. On the other hand, local topological order would shield the ground space from static perturbations. Here we demonstrate that local topological order implies a constant energy barrier, thus inhibiting thermal stability.

Joint work with David Poulin. arXiv:1209.5750

Background reading :

[1] S. Bravyi and B. Terhal, New Journal of Physics, 11,

043029 (2009). arXiv:0810.1983

[2] S. Bravyi, D. Poulin, and B. Terhal, Phys. Rev. Lett.,

104, 050503 (2010). arXiv:0909.5200

[3] S. Bravyi, M. Hastings, and S. Michalakis, Journal of

Mathematical Physics, 51 (2010). arXiv:1001.0344

[4] J. Haah and J. Preskill, Phys. Rev. A, 86, 032308 (2012). arXiv:1011.3529

Daniel Loss

(Basel, Switzerland)

Local 3D spin Hamiltonian as a thermally stable surface code

We study a 2D toric code embedded in a 3D Heisenberg ferromagnet in a broken-symmetry state at finite temperature. Stabilizer operators of the toric code are locally coupled to individual spins of the ferromagnet. The effects of the Goldstone modes of the ferromagnet in the ordered phase lead to an energy penalty for anyons that grows linearly with $L$, the linear size of the toric code. This $O(L)$ energy barrier for logical errors leads to a lifetime of the quantum memory that grows exponentially with $L$, assuming that the toric code is weakly coupled to a thermal bath with temperature below the phase transition of the ferromagnet. We address the backaction effect of the toric code on the ferromagnet. This provides a stable quantum memory with strictly local bounded-strength interactions in less than four dimensions.

[1] Fabio L. Pedrocchi, Adrian Hutter, James R. Wootton, and Daniel Loss.

arXiv:1209.5289.

Ashley Montanaro

(Cambridge University, UK)

Three quantum learning algorithms

In full generality, computational learning is the task of identifying an unknown object, picked from a known set of objects. In this talk I will discuss three optimal, and varied, quantum learning algorithms. The first is an algorithm for a purely quantum task: learning stabilizer states. In this problem, we are given access to a number of copies of an unknown stabilizer state of n qubits, and would like to identify the state. I will present an efficient algorithm for this task which uses O(n) copies of the state, which is optimal and represents an exponential improvement over standard tomography. The second algorithm is a generalisation of one of the earliest algorithms in quantum computing, the Bernstein-Vazirani algorithm for learning linear functions over the finite field F_2. I will present an exact quantum algorithm which learns n-variate multilinear polynomials over arbitrary finite fields and achieves an O(n) speed-up over any possible classical algorithm. Finally, I will discuss recent work on a problem dubbed "search with wildcards". Here we wish to learn an unknown bit-string, given the ability to determine whether arbitrary subsets of the bits are equal to a provided query string. I will present a quantum algorithm for this task which is based on novel ingredients and achieves a quadratic speed-up over any possible classical algorithm.

This talk is based on the papers arXiv:1210.1148 (joint work with Andris Ambainis), arXiv:1105.3310, and ongoing joint work with Scott Aaronson, David Chen, Daniel Gottesman and Vincent Liew.

Background reading:

arXiv:1210.1148 and arXiv:1105.3310, and perhaps arXiv:0911.4724 by Martin Roetteler.

David Poulin

(Sherbrooke, Canada)

2D quantum memories

I will demonstrate some interesting properties of two dimensional quantum memories, including

-limitations on their information storage capacities, 

-the fact that only the size of the boundary of an error determines whether it is corrected, not its volume

-the fact that the code admit 1D logical operators

This last fact leads to a tradeoff between protection against thermal and quantum fluctuations, as will be explained in a talk by O. Landon-Cardinal at this workshop. Then, I will explain some of the ways in which these results can be extended. For instance, understanding deformations of 1D logical operators could explain the thermalization time of these memories. Also, extending some of these results to infinite dimensions could lead to new ways on understanding quantum LDPC codes.

References:

Tradeoffs for reliable quantum information storage in {2D} systems

S. Bravyi and D. Poulin and B. M. Terhal

Phys. Rev. Lett. 104  050503  (2010)

arXiv:0909.5200

Logical operator tradeoff for local quantum codes

J. Haah and J. Preskill

      (2010)

arXiv:1011.3529

A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes

S. Bravyi and B. M. Terhal

New J. Phys. 11  043029  (2009)

arXiv:0810.1983

Local topological order inhibits thermal stability in {2D}

O. Landon-Cardinal and D. Poulin

      (2012)

arXiv:1209.5750

Jiannis Pachos

(University of Leeds, UK)

Abelian Chern-Simons-Maxwell theory from a tight binding model of spinless fermions

Abelian Chern-Simons-Maxwell theory can emerge from the bosonisation of the 2+1-dimensional Thirring model that describes interacting Dirac fermions. I will show how the Thirring model manifests itself in the low energy limit of a two-dimensional tight binding model of spinless fermions. To establish that we employ a modification of Haldane's model, where the ``doubling" of fermions is rectified by adiabatic elimination. Subsequently, fermionic interactions are introduced that lead to the analytically tractable Thirring model. Importantly, topological order of the ground state can be demonstrated just by local density measurements of the lattice fermions. For specific values of the couplings one can probe the confining 2+1 QED or the deconfining Chern-Simons theory. The implementation of the model as well as the measurement protocol are accessible with current technology of cold atoms in optical lattices.

Terry Rudolph

(Imperial College London, UK)

An assorted bunch of small open questions Terry is stuck on

The talk will be about the title.

Norbert Schuch

(Aachen, Germany)

Bulk-boundary correspondence and entanglement Hamiltonians in the PEPS formalism

In many physical scenarios, bulk properties of a system are reflected at its boundary.  This is in particular true for topologically ordered systems which exhibit rich boundary physics.  Recently, Li and Haldane have shown that the bulk entanglement properties of certain quantum Hall wavefunctions can be explained from the spectrum of a one-dimensional "entanglement Hamiltonian" which is naturally associated to the boundary. In this talk, I will discuss how the framework of Projected Entangled Pair States (PEPS) -- an entanglement based way to locally describe quantum many-body systems -- provides a natural way to construct one-dimensional entanglement Hamiltonians, and how the locality of the Hamiltonian allows to identify phase transitions.  I will in particular also focus on topological models and explain how in such systems the entanglement Hamiltonian splits into a universal non-local (topological) and a non-universal local (trivial) part, which allows to identify topological phases from the boundary.

Important references:

* http://arxiv.org/abs/1103.3427 : Discusses how to use PEPS to construct 1D entanglement Hamiltonians

* http://arxiv.org/abs/1210.5601 : Discusses the structure of entanglement Hamiltonians for topological models, and in particular the decomposition into universal non-local and non-universal local part

Additional reading:

* http://arxiv.org/abs/0805.0332 : bulk-boundary duality + entanglement Hamiltonian for the fractional quantum Hall effect

* http://arxiv.org/abs/1202.0947 : applies the method to derive an entanglement Hamiltonian using the PEPS formalism to Resonating Valence Bond States

Sukhbinder Singh

(Macquarie University, Australia)

Tensor networks, symmetries and geometry

Tensor networks offer an efficient description of certain quantum many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. Examples of TNs include matrix product states [1], projected entangled-pair states [2], and the multi-scale entanglement renormalization ansatz (MERA) [3]. In the presence of a global symmetry on the lattice (given by a compact and reducible group G), such as conservation of total particle number (G=U(1)) or total spin (G=SU(2)), a more compact description can be given using tensor networks made of invariant tensors (namely, tensors that are invariant under the action of the group on their indices). I describe how a symmetric tensor network decomposes [4] as a linear superposition of spin networks [5], which allows for significant computational gains in the context of numerical simulations. I also discuss a potential connection between this decomposition and the recent interpretation [6] of the MERA as a holographic anti-de Sitter geometry.

References:

[1] S. Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995); M. Fannes, B. Nachtergaele, and R. Werner,Commun. Math. Phys. 144, 443 (1992).

[2] F. Verstraete and J. I. Cirac, arXiv:cond-mat/0407066v1; G. Sierra and M.A. Martin-Delgado,arXiv:cond-mat/9811170v3.

[3] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007). G. Vidal, Phys. Rev. Lett. 101, 110501 (2008).

[4] S. Singh, R.N.C. Pfeifer and G. Vidal, Phys. Rev. A 82, 050301 (2010).

[5] A spin network is a well-known object in mathematical physics and, especially, in loop quantum gravity, where it is used to describe states of quantum geometry, see for instance “C. Rovelli and L. Smolin, Phys. Rev. D 53, 5743 (1995)”.

[6] B. Swingle, Phys. Rev. D 86, 065007 (2012); G. Evenbly, G.Vidal, J Stat Phys 145, 891-918 (2011).

Guifre Vidal

(Perimeter Institute, Canada)

Characterization of topological order by studying ground states on an infinite cylinder

Only a few years ago, establishing whether a microscopic Hamiltonian on a two-dimensional lattice would give rise to emergent topological order was regarded as a very hard task. Thanks to progress in our understanding of many-body entanglement and improved numerical approaches, it is now possible to obtain a very refined characterization of the emergent anyon model, including a list of anyon charges, fusion rules, quantum dimensions, mutual statistics, and self statistics (and, where applicable, a characterization of the edge CFT). I will summarize the results of Ref. [L. Cincio and G. Vidal, "Characterizing topological order by studying the ground states of an infinite cylinder", http://arxiv.org/abs/1208.2623] based on obtaining a complete set of ground states on a finite torus (by studying an infinite cylinder with 2D DMRG), and will discuss extensions to obtain an explicit tensor network representation also for all the fractionalized quasi-particle excitations.

Background material:

Characterizing topological order by studying the ground states of an infinite cylinder

http://arxiv.org/abs/1208.2623

Quasi-particle Statistics and Braiding from Ground State Entanglement

http://arxiv.org/abs/1111.2342

Fern Watson

(Imperial College London, UK)

Overheads in Kitaev’s 2d toric code

In this talk we will review one proposal for topological quantum error correction: Kitaev’s 2d toric code. The toric code distance scales with lattice size, making a physically larger code more robust. However, a smaller code is desirable because the experimental challenges in creating and manipulating such a state also scale with the number of qubits in the code. The overhead is a balance between these two requirements; in other words the minimum code size that will protect the state with a given accuracy, for a known error rate. 

We consider different approaches to revealing the overhead, including both analytic approximations and numerically investigations. We find that for a large range of parameter space the overhead for the toric code is polylogarithmic in the desired fidelity.