# Abstracts

**Monday**

09:00—09:55 Dan Browne (UCL) [ pdf | video ]

Decoding Homology. A lexicon for the uninitiated.

Homology is an important branch of modern mathematics, but is rarely known by physicists or computer scientists. In quantum error correction, homology played an important role in the development of topological surface codes such as the toric code, and homological ideas underly the way such codes protect quantum information. Indeed, almost every aspect of quantum error correction in surface codes can be most compactly expressed in homological terms. For this reason, homological terminology is common in literature on topological codes. However, textbooks on homology are usually targeted at mathematicians with a focus on advanced techniques of little relevance to topological error correction and there are few entry level treatments of the subject. Furthermore, homology terminology contains false friends (e.g. “chains”) and similar sounding words with crucially different meanings (homology, homotopy, homomorphism, homeomorphism) which can by confusing. Homology thus presents a barrier for many newcomers in quantum error correction research wishing to study key papers on topological codes.

Fortunately, the toric code itself presents a beautiful illustration of many of the key concepts in homology. In this tutorial lecture, I will use it to introduce these key ideas and terminology in homology theory. I will focus on some of the terminology pitfalls, and give an intuitive and example-based introduction to the subject, which I hope will help in particular those wishing to decode research papers written using standard quantum error correction techniques (e.g. the stabilizer formalism) but employing homology terminology and those embarking on a rigorous treatment of the subject elsewhere.

The lecture will be most accessible to those familiar some group theory techniques which arise in quantum error correction (e.g. the stabilizer formalism, the quotient group) and the basics of the toric code.

10:00—10:25 Xiaotong Ni (MPQ) [ pdf | video ]

A Non-Commuting Stabilizer Formalism

The Pauli stabilizer formalism (PSF) is the major tool for constructing and understanding quantum error correction codes, partially due to the fact that many properties of the Pauli stabilizer codes can be efficiently computed. In this work, we consider a new generalization of the PSF by adding S = diag(1, i) to the Pauli group. While keeping its spirit (defining codes by groups) and in many cases the computational efficiency, our formalism contains features significantly different from the PSF (e.g. the possibility of a non-abelian stabilizer group, describing more topological phases). The understanding of this new formalism may allow us to construct new quantum codes or have a better understanding on topological phases. The full paper is available at arXiv:1404.5327.

11:00—11:40 Peter van Loock (Mainz) [ pdf | video ]

Quantum error correction for long-distance quantum communication

The first and most common approach to quantum communication across large distances, circumventing the effect of an optical transmission loss exponentially growing with distance, is the quantum repeater. However, the standard quantum repeater based on local quantum memories and two-way classical communication is extremely slow, producing low rates and requiring long-lasting memories. An obvious remedy here is to replace quantum error detection (as employed in a standard quantum repeater) by quantum error correction. We shall give an overview over recent proposals for an encoded quantum repeater, with a particular emphasis on its ultrafast manifestation (with rates independent of the total distance) using quantum codes against photon losses and one-way classical communication. We also discuss the possibility of implementing such ultrafast long-distance quantum communication with linear optics.

11:45—12:25 Andreas Walraff (ETH Zürich) [ pdf | video ]

Deterministic Quantum Teleportation with Feed-Forward in a Solid State System

14:00—14:55 Bryan Eastin (Northrup Grumman) [ pdf | video ]

Fault-tolerant Quantum Computing

In this tutorial I will review the basics of fault-tolerant quantum computing. Topics to be covered include the stabilizer formalism, quantum error correction, fault-tolerance, universality, magic-state distillation, and thresholds.

15:00—15:40 Hector Bombin (Copenhagen) [ pdf | video ]

Color codes (are fun)

I will discuss recent surprising developments in connection with color codes: optimal transversal gates, gauge color codes, gauge fixing, single-shot error correction and dimensional jumps. Putting them together, fault-tolerant quantum computation can be achieved with constant time overhead on the number of logical gates, up to efficient global classical computation, using only local quantum operations in a 3D lattice.

16:15—16:55 Steve Flammia (Sydney) [ pdf | video ]

Sparse Quantum Codes from Quantum Circuits

We describe a general method for turning Clifford quantum circuits into sparse quantum subsystem codes. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local. Applying our construction to certain concatenated stabilizer codes yields families of subsystem codes with constant-weight generators and with minimum distance d = n^{1-ε}, where ε = O (1/√ log n). For spatially local codes in D dimensions we nearly saturate a bound due to Bravyi and Terhal and achieve d = n^{1-ε-1/D}. Previously the best code distance achievable with constant-weight generators in any dimension, due to Freedman, Meyer and Luo, was O (√ n log n) for a stabilizer code.This is joint work with Dave Bacon, Aram Harrow, and Jonathan Shi available at arXiv:1411.3334.

17:00—17:40 Tomas Jochym-O’Connor (IQC) [ pdf | video ]

Exploring the complementarity between quantum privacy and error correction

Private quantum channels are characterized by their ability to protect quantum information. The complementary channel to a private quantum subspace channel is an error-correctable subspace channel, providing a clear notion of complementarity between privacy and error correction. However, for a large set of channels, private subsystems can exist without the presence of private subspaces. Moreover, for a class of private subsystem channels, their complementary counterparts are no longer quantum error-correcting and the duality between error correction and privacy can only be recovered when extending the action of the channels to larger Hilbert spaces. This work uncovers the underlying mathematical structure in the duality between error correction and privacy and therein shows algebraic conditions for private quantum channels, analogous to the Knill-Laflamme conditions for quantum error correction.

17:45—18:10 Aleksander Kubica (Caltech) [ pdf | video ]

(In)equivalence of color code and toric code

A quantum error-correcting code with fault-tolerantly implementable non-Clifford logical gates is an indispensable ingredient for realization of universal quantum computation. At the moment, the topological color codes are the only known examples of topological stabilizer codes with transversally implementable non-Clifford logical gates. Thus, understanding similarities and differences between the topological color codes and ordinary topological quantum codes, such as the toric code, is an important problem. Here we prove that the topological color code on a d-dimensional closed manifold (without bound- aries) is equivalent to multiple decoupled copies of the d-dimensional toric code up to local unitary transformations and adding/removing ancillas. Our result not only generalizes the proven equivalence for d = 2, but also provides an explicit recipe how to decouple independent components of the topological color code, highlighting the importance of “colorability” in the construction of the code. This result implies that the topological color code and the toric code belong to the same quantum phase according to the definition widely accepted in condensed matter physics community. Moreover, for a d-dimensional topological color code with boundaries, which admits transversal implementation of non-Clifford logical gates, we find that the code is equivalent to multiple copies of the toric code in d dimensions which are welded together along the (d − 1)-dimensional boundaries. In particular, for d = 2, we show that the color code, defined on a lattice with three boundaries of distinct colors, is equivalent to a single copy of the toric code on a square lattice with boundaries, which is “folded” along a diagonal axis. Our findings may lead to a systematic method of composing known quantum codes to construct new codes with larger set of fault-tolerant logical gates. Our work also provides new insights into the classification of topological quantum field theories with boundaries in two or higher dimensions.

**Tuesday**

09:00—09:55 Sergey Bravyi (IBM) [ pdf | video ]

Maximum likelihood decoding in the surface code

In this talk I will describe two implementations of the optimal error correction algorithm known as the Maximum Likelihood (ML) decoder for the 2D surface code with a noiseless syndrome extraction. First, I will show how to implement the ML decoder exactly in time O (n^{2}), where n is the number of physical qubits. Our implementation exploits a reduction from the ML decoding problem to simulation of matchgate quantum circuits. This reduction however requires a special noise model with independent bit-flip and phase-flip errors. Secondly, I will show how to implement the ML decoder approximately for more general noise models using matrix product states (MPS). Our implementation has a running time O (n χ^{3}) where χ is a small integer parameter that controls the approximation precision. The key step of our algorithm, borrowed from the DMRG method, is a subroutine for contracting a tensor network on a two-dimensional grid. The subroutine uses MPS with a bond dimension chi to approximate the sequence of tensors arising in the contraction. We benchmark the MPS-based decoder against the standard minimum weight matching decoder observing a significant reduction of the logical error probability.

10:00—10:25 Andrew Landahl (Sandia) [ pdf | video ]

Quantum computing by color-code lattice surgery

In this talk, I will explain how to use lattice surgery to enact a universal set of fault-tolerant quantum operations with color codes. Along the way, I will also show how to improve existing surface-code lattice-surgery methods. Lattice-surgery methods use fewer qubits and the same time or less than associated defect-braiding methods. Per code distance, color-code lattice surgery uses approximately half the qubits and the same time or less than surface-code lattice surgery. Color-code lattice surgery can also implement the Hadamard and phase gates in a single transversal step-much faster than surface-code lattice surgery can. I will show that against uncorrelated circuit-level depolarizing noise, color-code lattice surgery uses fewer qubits to achieve the same degree of fault-tolerant error suppression as surface-code lattice-surgery when the noise rate is low enough and the error suppression demand is high enough.

11:00—11:40 Fernando Pastawski (Caltech) [ pdf | video ]

Fault-tolerant logical gates in quantum error-correcting codes

Recently, Bravyi and König have shown that there is a trade-off between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant-depth geometrically-local circuit and are thus fault-tolerant by construction. In particular, they shown that, for local stabilizer codes in D spatial dimensions, locality preserving gates are restricted to a set of unitary gates known as the D-th level of the Clifford hierarchy. In this talk I will present a version of their result applicable to subsystem codes and explain several extensions arising thereof.

11:45—12:25 Beni Yoshida (Caltech) [ pdf | video ]

On thermal stability of topological order

In this talk, we discuss whether topological order may exist at nonzero temperature in three spatial dimensions and try to establish the connection between thermally stable topological order and self-correcting quantum memory.

14:00—14:55 Ken Brown (Gatech) [ pdf | video ]

Fighting Hamiltonians with Hamiltonians

In this tutorial talk, I will discuss the role of quantum control for reducing errors in quantum gates. The errors analyzed will be due to both coherent interactions of the system and the environment and limited control of the system Hamiltonian. The emphasis will be on how techniques like dynamic decoupling, dynamically corrected gates, and composite pulse sequences can be used to both reduce the error and effectively whiten the noise. This results in gates that are compatible with standard assumptions used in fault-tolerant quantum error correction.

15:00—15:40 Lorenza Viola (Dartmouth) [ pdf | video ]

A general transfer-function approach to noise filtering in open-loop quantum control

Hamiltonian engineering via unitary open-loop quantum control provides a versatile and experimentally validated framework for precisely manipulating a broad class of non-Markovian dynamical evolutions of interest, with applications ranging from dynamical decoupling and dynamically corrected quantum gates to noise spectroscopy and quantum simulation. In this context, transfer-function techniques directly motivated by control engineering have proved invaluable for obtaining a transparent picture of the controlled dynamics in the frequency domain and for quantitatively analyzing control performance. In this talk, I will show how to construct a general filter-function approach, which overcomes the limitations of the existing formalism. The key insight is to identify a set of “fundamental filter functions”, whose knowledge suffices to construct arbitrary filter functions in principle and to determine the minimum “filtering order” that a given control protocol can guarantee. Implications for dynamical error correction and noise identification will be discussed.

16:15—16:55 Daniel Lidar (USC) [ pdf | video ]

Quantum annealing correction

Devices designed to implement quantum annealing with hundreds of qubits are commercially available and are the subject of intense scrutiny. So far there is no clear evidence of a quantum speedup despite careful benchmarking and results pointing to a constructive role that quantum effects play in the output of such devices. This lack of speedup should perhaps come as no surprise in light of the absence of error correction. Separately from the benchmarking efforts we have developed methods for quantum annealing correction (QAC), which combine ideas from stabilizer codes and error suppression via energy penalties. Quantum annealing correction in currently available hardware presents a host of challenges due to limited control and connectivity. This talk will present an overview of the theory and experimental results have obtained for QAC, which point to a substantial performance enhancement of quantum annealers on both 1D and 2D Ising problems. Time permitting, the talk will also address how QAC can be used to test and reject popular classical models of quantum annealing hardware.

References:

K.P. Pudenz, T. Albash, and D.A. Lidar, “Quantum Annealing Correction for Random Ising Problems”, arXiv:1408.4382

K.P. Pudenz, T. Albash, and D.A. Lidar, “Error Corrected Quantum Annealing with Hundreds of Qubits”, Nature Comm. 5, 3243 (2014).

T.F. Ronnow, Z. Wang, J. Job, S.V. Isakov, D. Wecker, J.M. Martinis, D.A. Lidar, and M. Troyer, “Defining and Detecting Quantum Speedup”, Science 345, 420 (2014).

17:00—17:40 Leonid Pryadko (Riverside) [ pdf | video ]

Irreducible normalizer operators and existence of a decoding threshold

I will discuss a technique for constructing lower bounds for the minimum-energy decoding threshold with quantum error correcting codes. It is based on enumerating “irreducible” undetectable operators that can not be decomposed into a disjoint product of undetectable operators. Bounding the number of such operators of a given weight produces a lower bound for decoding threshold with any model of uncorrelated errors. In the case of limited-weight quantum LDPC codes with logarithmic or larger distances, one obtains an explicit analytical expression combining probabilities of erasures, depolarizing errors, and phenomenological syndrome measurement errors. This threshold estimate is parametrically better than the existing analytical bound based on percolation.

17:45—18:10 Alexey Kovalev (Nebraska) [ pdf | video ]

Parafermion stabilizer codes

We define and study parafermion stabilizer codes which can be viewed as generalizations of Kitaev’s one dimensional model of unpaired Majorana fermions. Parafermion stabilizer codes can protect against low-weight errors acting on a small subset of parafermion modes in analogy to qudit stabilizer codes. Examples of several smallest parafermion stabilizer codes are given. A locality preserving embedding of qudit operators into parafermion operators is established which allows one to map known qudit stabilizer codes to parafermion codes. We also present a local 2D parafermion construction that combines topological protection of Kitaev’s toric code with additional protection relying on parity conservation. More information about this work can be found in the preprint arXiv:1409.4724.

**Wednesday**

09:00—09:55 David DiVincenzo (Aachen) [ ppt | video ]

What’s QEC for Solid State Physics?

“Surface code” is on the lips of many a solid-state device physicist these days. I will document this phenomenon with some examples, from the commonplace (CNOT to ancillas, then measure) to the more recondite (direct parity measurement, intrinsic leakage of DFS qubits). I will give some examples from current work in quantum-dot qubits. Mighty efforts are underway to improve laboratory fidelities, which are however neither quantitatively nor methodologically complete. Leakage reduction units are starting to come over the horizon, but QEC could probably help more with this. There are correspondingly mighty plans on the drawing board to collect and process all the data that the surface code implies. I will show what small parts of these plans have come to fruition; QEC should also do some work to determine what is really the best thing to do with this avalanche of data, when it comes. I will also touch on some examples where solid-state physics definitely gives back to QEC, with Fibonacci quantum codes being one example.

10:00—10:25 Naomi Nickerson (Imperial) [ pdf | video ]

Freely Scalable Quantum Technologies using Cells of 5 to 50 Qubits with Very Lossy and Noisy Photonic Links

In this talk, we present a detailed analysis of a network architecture for fault tolerant quantum computation, where small modules are connected through probabilistic entanglement generation using optical links. Our approach combines ideas of networked entanglement purification and a surface code architecture with a new approach to entanglement generation in the regime of high photon loss. This paper is the first to tackle the problem of photon loss, and through detailed simulations it is the first to demonstrate that reasonable speed is possible in the network approach. By speed we simply mean the rate at which basic operations are performed, analogous to the clock rate on a conventional machine. Speed is of fundamental practical importance one cannot wait decades for a calculation to complete, even if that calculation is now ‘possible’ in the formal sense. Using local gates performed with fidelities consistent with current technology and a network noise threshold of 13.3%, we find that interlinks attempting entanglement at a rate of 2MHz but suffering 98% photon loss can result in kilohertz computer clock speeds. This is of particular relevance for network architectures since the entanglementsharing links upon which they depend have so far operated only at Hertz rates. Here we have shown this can be improved by orders of magnitude, and without compromising on fault tolerance. These methods are applicable to many experimental architectures, such as trapped ions, superconducting circuits and NV centres in diamond. Thus we hope that this will be of interest to both researchers in the theory of implementations of quantum computing and experimentalists in this area. The preprint is available at arXiv:1406.0880.

11:00—11:40 Jean-Pierre Tillich (INRIA) [ pdf | video ]

Turning error-reducing quantum turbo codes into error-correcting codes

Obtaining a satisfying quantum generalisation of classical turbo-codes has proven to be a non-trivial task. This is due to the fact that it was shown that the fundamental object that is used to build turbo-codes, namely recursive and non-catastrophic convolutional encoders do not exist in the quantum setting. There are basically two approaches that have been tried to overcome this limitation. The first attempts used non-recursive and non-catastrophic encoders but the error-correction performance obtained in this way are by no means comparable with the outstanding error-correction performances of classical turbo-codes. This is due to the fact that the non-recursiveness of the quantum convolutional encoder entails a constant upper-bound on the minimum distance of the turbo-codes obtained in this way.

Another approach (Abbara & Tillich, ITW 2011) used catastrophic and recursive encoders together with a slight change in the encoder structure in order to allow successful iterative decoding. The codes obtained from this approach turned out to be excellent error-reducing codes, but it was also shown that the modification in the turbo-code structure which helped the first steps of iterative decoding also prevents iterative decoding to converge to zero noise levels and thus prevent these codes to be error-correcting codes.

We will show here that by considering a suitable multilayer version of the last construction of turbo-codes, the residual noise-level after iterative decoding can be reduced to an arbitrary small fraction and that we can obtain in this way very good error-correcting codes with performance comparable to classical turbo-codes or to the toric code of Kitaev for instance. However, unlike Kitaev’s construction which encodes only two quits, these new codes enjoy a fixed rate. Moreover, by analyzing iterative decoding of such codes on the erasure channel, we have shown that the residual noise-level after iterative decoding converges very quickly to zero with the number of layers and indeed the number of layers that are needed for error-correcting performances is only two or three for any reasonable code length. This is joint work with Mamdouh Abbara and Iryna Andriyanova.

11:45—12:25 Rüdiger Urbanke (EPF Lausanne) [ pdf | video ]

Finite Length — The Final Frontier

TBA

14:00—14:25 Thomas Schulte-Herbrüggen (TU Munich) [ pdf | video ]

Quantum Error Correction and Error Avoidance by Optimal Control

Quantum systems and control engineering has become increasingly useful for steering quantum devices with high precision. This also includes error-correcting gate sequences. Here we report on the implementation of quantum error correction of spin registers in a nitrogen vacancy centre (NV) in diamond enabled by optimal control. It allows for read-out fidelities of about 99% in spite of local spin-flip errors. The same tools also pave the way to entangle several NV centres with high fidelity over several cycles of pulsed gates. Yet, the experimental set-up comes with challenges such as non-alligned quantisation axes and cross-talk by spectral crowding thus asking for optimal-control techniques. The results are put into a broader context in view of combining error avoidance (by protected subspace engineering) with active error correction. To this end, current developments of dissipative state-space engineering can be put to good use.

14:30—14:55 Holger Frydrych (TU Darmstadt) [ pdf | video ]

Fully pulse-controlled gate operations on coupled flux qubit chains

In this work, we study a strongly coupled chain of superconducting flux qubits. Strong couplings promise fast and reliable implementations of multi-qubit gates, however, to successfully complete a gate operation, we must be able to isolate the affected qubits from the remaining chain. Unfortunately, designing strong couplings which can be switched on and off at a moment’s notice remains an experimental challenge. Instead, we propose a selective dynamical decoupling scheme to suppress any coupling in the chain when required. This scheme requires only local X and Y pulses, which are implemented with the help of a microwave emitter. With this approach, the couplings on the chain can be always on, and due to the freedom of selectively suppressing certain couplings, we can implement two-qubit gates with high fidelity and can even support parallel gate operations. We demonstrate how the available control can be used to implement single-qubit rotations and the two-qubit CNS gate, a combination of CNOT and SWAP. We then present numerical simulations which make use of multiple applications of the CNS gate to entangle all the qubits in the chain. Although we find that the scalability of our approach is strongly dependent on the quickness with which the control pulses can be implemented and that this quickness may have physical limits, we believe that our method is a practical way to implement a small quantum register with solid two-qubit gate operations. Furthermore, our decoupling scheme is robust against eigenenergy discrepancies (diagonal disorder) in the flux qubits, which commonly occur due to engineering deficiencies.

15:00—15:40 James Wootton (Basel) [ pdf | video ]

Error Correction for Non-Abelian Topological Quantum Computation

The possibility of quantum computation using non-Abelian anyons has been considered for over a decade. However the question of how to obtain and process information about what errors have occurred in order to negate their effects has not yet been considered. This is in stark contrast with quantum computation proposals for Abelian anyons, for which decoding algorithms have been tailor-made for many topological error-correcting codes and error models. Here we address this issue by considering general properties of non-Abelian error correction, and also study a concrete anyon model in more detail. The anyons studied are those of the charge submodel of D (S3), which share many properties with Fibonacci anyons. For these we develop a minimum weight perfect matching based decoder which demonstrates that error correction up to a high threshold can be realized.

16:15—16:55 Todd Brun (USC) [ ppt | video ]

Fault-tolerant quantum computation in multi-qubit block codes

A major goal for fault-tolerant quantum computation is to reduce the overhead needed for error correction. One approach is to use block codes that encode multiple qubits, which can achieve significantly higher rates for the same code distance than single-qubit code blocks. We present a scheme for universal quantum computation using multi-qubit CSS block codes, where codes admitting different transversal gates are used to achieve universality, and logical teleportation is used to move qubits between code blocks. All circuits for both computation and error correction are transversal. We present estimates of information lifetime for a few possible codes, which suggest that highly nontrivial quantum computations can be achieved at reasonable error rates, using codes that require significantly less that 100 physical qubits per logical qubit. Open questions regarding ancilla preparation and other issues are also discussed.

17:00—17:40 Graeme Smith (IBM) [ pdf | video ]

Trading Quantum and Classical computational resources

**Thursday**

09:00—09:55 Daniel Gottesman (Perimeter) [ pdf | video ]

Stabilizer codes for prime power qudits

There is a standard generalization of stabilizer codes to work with qudits which have prime dimension, and a slightly less standard generalization for qudits whose dimension is a prime power. However, for prime power dimensions, the usual generalization effectively treats the qudit as multiple prime-dimensional qudits instead of one larger object. There is a finite field GF (q) with size equal to any prime power, and it makes sense to label the qudit basis states with elements of the finite field, but the usual stabilizer codes do not make use of the structure of the finite field. I introduce the true GF (q) stabilizer codes, a subset of the usual prime power stabilizer codes which do make full use of the finite field structure. The true GF (q) stabilizer codes have nicer properties than the usual stabilizer codes over prime power qudits and work with a lifted Pauli group, which has some interesting mathematical aspects to it.

10:00—10:25 Lucy Liuxuan Zhang (Toronto) [ pdf | video ]

Fibre bundle framework for unitary quantum fault tolerance

This work is the beginning of an approach to view general fault tolerance geometrically. Our formu- lations of two important classes of examples of fault tolerance — namely using transversal gates and toric codes — in terms of fibre bundles encourage us to look deeper into the geometric underpinning of fault tolerance. The applicability of the geometric intuition to these two examples led us to make a couple of more general conjectures. For simplicity, at this early stage, our conjectures apply to unitary fault tolerance, namely fault-tolerant protocols involving only unitary gates. The conjecture more or less says: The group of logical operations implementable by a set of unitary fault-tolerant operations on a QECC are given by the projective monodromy representation, obtained from a certain fibre bundle, of the fundamental group of a certain topological space (the base space of the bundle). The key to this conjecture is to define an appropriate fibre bundle given a unitary fault-tolerant protocol. We will need to do it in a physically sensible way, and this will be described in more detail in the revised version. The fault tolerant nature of the protocol corresponds to the existence of a natural flat projective connection on the bundle. In short, the conjecture claims that all unitary fault tolerance is topological in nature. Much of the posted paper is focused on proving the first conjecture in the case of our two examples, transversal gates and toric code. In other words, we proved that “transversal gates are topological”, and “the toric code is topological” (in a somewhat different but related sense). There is also the converse conjecture, which we hope to give some intuitions about. That is, how does the existence of a natural flat projective connection on a physically motivated bundle allow us to read off a fault-tolerant protocol associated to it? Full paper: arXiv:1309.7062.

11:00—11:40 Robin Blume-Kohout (Sandia) [ pdf | video ]

Severe Obstacles to Fault-Tolerant Adiabatic QC

Adiabatic quantum computing (AQC) should be intrinsically robust against some of the failure modes that afflict circuit-model QC, including dissipation and dephasing in the energy eigenbasis. Some other faults can be suppressed by dynamical decoupling and penalty Hamiltonians. But fault tolerance requires robustness against all failure modes stemming from a reasonable error model, and this requires active error correction. In this talk, I introduce the whimsical term “ptheorem”, akin to the well-known “equals-sub-p” notation, to denote an argument that is compelling to a physicist (but not sufficiently rigorous to be a theorem). I will then pprove a simple ptheorem that AQC cannot be error-corrected, without access to either (1) physical Hamiltonians that directly couple O (N) qubits, or (2) perturbative gadgets that simulate such Hamiltonians at cost poly(N).

11:45—12:25 Earl Campbell (Sheffield) [ pdf | video ]

The advantages of qudit fault-tolerance

Qudit quantum computers are built from d-level systems rather than 2-level qubit systems. At first, it might seem these two models of computation would be very similar. However, once made fault tolerant the qudit model acquires both qualitative and quantitative differences to comparable qubit fault tolerance schemes. Thresholds for error correction typically increase with qudit dimension. Furthermore, implementing a universal set of fault tolerant gate becomes much more efficient for prime d>3. The key technical step in implementing these gates is to find codes with a transversal non-Clifford gate, which then provide protocols for either magic state distillation or computation through gauge-fixing. The talk includes results from: arXiv:1406.3055, New J. Phys. 16 063038 (2014), Phys. Rev. X. 2 041021 (2012) .

14:00—14:25 Michael Herold (FU Berlin) [ pdf | video ]

Cellular-automaton decoders for topological quantum memories

Decoders are one of the necessary building blocks for robust local quantum memories, basic primitives that must arguably be part of any functioning topological quantum computer. While surface codes have raised the prospects for scalability and performance of quantum storage devices, the task of decoding resorts to complicated classical algorithms. Such decoding algorithms are typically designed for traditional centralized computing architectures. This questions whether active decoding can be fast enough to outpace the decoherence time in systems with many qubits, giving rise to significant challenges. In this work, we address this question by recasting error recovery as a dynamical process on a field generating cellular automaton. We envisage quantum systems controlled by a classical hardware composed of small local memories, communicating with nearest neighbors only, and repeatedly performing identical simple update rules. This framework for constructing topological quantum memories does not require any global operations or complex decoding algorithms. Furthermore, the local updates rules do not have to be perfect or synchronized, relaxing many of the former requirements on decoding devices. Hence, our work raises prospects for the technical feasibility of scalable and ultra-fast decoding devices. The working principle of our decoders closely resembles fundamental physical laws. In fact, it corresponds to an attractive interaction between anyonic excitations stabilizing a topological state, one reminding of gravity or electrostatics. Thus our work is a first step in connecting decoders with the design of naturally noise resistant quantum memories. In this mindset it is particularly interesting that our 3D-field decoder can also continuously counteract the creation of errors. This transfers the study of decoders from the pure algorithmic viewpoint to the realm of interacting particle systems. Further unpublished numerical findings provide the first evidence for a threshold error rate of ∼ 5 × 10^{−4} in the dynamical setting. The inclusion of measurement errors effects this threshold via a relative shift only.

14:30—14:55 Simon Burton (Sydney) [ pdf | video ]

Error Correction in a Fibonacci Anyon Code

Topologically ordered systems in two dimensions are of great interest for quantum in- formation processing and storage. These systems can be realized as ground states of local Hamiltonians and have natural robustness to local Hamiltonian perturbations. Furthermore, these systems typically have anyonic excitations that may be used to realize fault-tolerant quantum computation by braiding of these particles. The Fibonacci anyon model is one of the simplest abstract anyon models to describe, having only one non-trivial particle type. Nonetheless, it allows for universal quantum computation by braiding and so is of great theoretical interest. Furthermore, the Fibonacci anyons are experimentally motivated as the expected excitations of the ν = 12/5 fractional quantum Hall states, and can also be realized in several spin models. In this work we focus on demonstrating one of the main claims of topological quantum computing: that the effects of local noise can be overcome using error correction techniques. We use monte-carlo simulations of a phenomenological model. This ignores any microscopic structure such as an underlying lattice of spins. The two dimensional manifold of our model has the global geometry of a torus which supports a two-fold degenerate vacuum state. This space is the codespace. The noise model consists of iid pair-creation events for some varying amount of time. Error correction proceeds by first measuring a syndrome: this is the total charge in each tile of some tiling of the surface. The (classical) error correction algorithm performs repeated hierarchical clustering (fusing) of anyons, until there are no more charges remaining . This fails, destroying the encoded state, exactly when a topologically non-trivial operation occurs. Note that the error correction algorithm runs in polynomial time, but the simulation of the anyon system naively takes exponential time and memory. Here we find that a divide and conquer approach, as well as heuristics to minimise braid operations, allow simulation of systems to a reasonably large size (128x128 tiles), and we observe an error correction threshold of around 12.5%.

15:00—15:40 J. M. Taylor (JQI/NIST) [ pdf | video ]

Fault-resistant quantum simulation with fractional quantum Hall states of light

Photons provide high coherence, long-lived quantum excitations with a diverse set of applications in quantum optics, quantum communication, and quantum computation. Considering the viability of photons as a platform for quantum simulation, we describe an approach for generating many-body states of light that may be robust with respect to a variety of experimental imperfections. As an example case, we focus our efforts on the creation of fractional quantum Hall states for light. This example highlights challenges in single-particle physics, such as the creation of synthetic gauge fields for photons; nonlinear optics, including developing strong photon-photon interactions sufficient for fractional quantum Hall regimes; and fundamental questions about stabilizing near equilibrium many-body states of light via artificial chemical potentials. While we do not find any fault-tolerance in such systems generically, the actual question asked of the simulator system -- do excitations of the system have appropriate non-abelian statistics -- provides a robust readout of the performance of the many-body system.

16:15—16:55 Ruben Andrist (Santa Fe Institute) [ pdf | video ]

Error Thresholds for Abelian Quantum Double Models

Current approaches for building quantum computing devices focus on two-level quantum systems which nicely mimic the concept of a classical bit, albeit enhanced with additional quantum properties. However, rather than artificially limiting the number of states to two, the use of d-level quantum systems (qudits) could provide advantages for quantum information processing. Among other merits, it has recently been shown that multi-level quantum systems can offer increased stability to external disturbances — a key problem in current technologies. In this study we demonstrate that topological quantum memories built from qudits, also known as abelian quantum double models, exhibit a substantially increased resilience to noise. That is, even when taking into account the multitude of errors possible for multi-level quantum systems, topological quantum error correction codes employing qudits can sustain a larger error rate than their two-level counterparts. In particular, we find strong numerical evidence that the thresholds of these error-correction codes are given by the hashing bound. Considering the significantly increased error thresholds attained, this might well outweigh the added complexity of engineering and controlling higher dimensional quantum systems.

17:00—17:25 Kristan Temme (Caltech) [ pdf | video ]

Thermalization time bounds on Pauli stabilizer Hamiltonians

We prove a lower bound to the spectral gap of the Davies generator for general N - qubit commuting Pauli Hamiltonians. We expect this bound to provide the correct asymptotic scaling of the gap with the systems size up to a factor of 1/N in the low temperature regime. We derive rigorous thermalization time bounds, also called mixing time bounds, for the Davies generators of these Hamiltonians. Davies generators are given in the form of a Lindblad equation and are known to converge to the Gibbs distribution of the particular Hamiltonian for which they are derived. The bound on the spectral gap essentially depends on a single number ε referred to as the generalized energy barrier. When any local defect can be grown into a logical operator of a stabilizer code S by applying single qubit Pauli operators and in turn any Pauli operator can be decomposed into a product of the clusters of such excitations, ε corresponds to the largest energy barrier of one of the canonical logical operators. The main conclusion that can be drawn from our result is, that the presence of an energy barrier for the logical operators is in fact, although not sufficient, a necessary condition for a thermally stable quantum memory when we assume the full Davies dynamics as noise model.

**Friday**

09:00—09:40 Andrew Ferris (ICFO) [ pdf | video ]

Tensor networks and coding theory: the polar and branching-mera codes

In this talk, I will relate the decoding problem for quantum (and classical) error-correcting codes to tensor network methods used in condensed matter physics. In particular, I will analyze the structure of Arikan’s “polar” code, introduced in 2007 as the first provably efficient and capacity-achieving classical code for arbitrary symmetric (classical) channels. The quantum polar code is understood to protect quantum data at a rate up to the coherent information, but suffers from non-optimal asymptotic behavior. By understanding similarities between the polar code and a recently-discovered tensor network for describing highly-entangled quantum systems called the “branching MERA”, we introduce a brand-new code that has superior error-correction capabilities to the polar code with little increase in computational cost. We also suggest an enhancement to the quantum successive cancelation decoder that reduces significantly the error rate.

09:45—10:10 Olivier Landon-Cardinal (Caltech) [ pdf | video ]

Can long-range interactions stabilize quantum memory at nonzero temperature?

Devising a self-correcting quantum memory, capable of storing qubits for a very long time without active error correction, would be a scientific milestone with far-reaching implications for the scalability of quantum computing and the security of quantum communication protocols. A self-correcting quantum memory is a quantum many-body system which encodes information in its degenerate ground space for a time which grows without bound as the system size increases. Crucially, the degeneracy of the local Hamiltonian must be stable with respect to generic small local perturbations. Topologically ordered systems (such as the toric code) are widely studied candidates since they display this stability. However, topological systems typically have a quantum memory time independent of system size. Indeed, at non-zero temperature, thermal anyons appear; once created, they can propagate without energy penalty, quickly destroying the quantum information. We investigate the recent proposals of introducing effective long-range interactions between anyons by coupling the 2D toric code to an auxiliary bosonic system. The goals of these approaches is two-fold: (1) an attractive long-range interaction would prevent anyons from freely propagating and (2) coupling to a macroscopic number of bosons can introduce a diverging energy barrier, thus forcing the rate for thermal anyon production to approach zero as the system size grows. To attain both goals, effective long-range interaction requires the mediating bosons to be massless. By rephrasing those proposals from a quantum field theory perspective, we emphasize a general issue — generic small local perturbations introduce a mass gap for the bosons. Once the bosons acquire a mass, the effective interactions becomes short-range. As a consequence, anyons propagate freely when sufficiently far away from one another and the (perturbed) energy barrier does not scale with system size. Hence, generic perturbations compromise the thermal stability of such proposals.

11:00—11:25 Michael Beverland (Caltech) [ pdf | video ]

Protected gates for topological quantum field theories

We provide restrictions on the logical gates that can be performed by constant-depth local circuits (CDLC)s in topological quantum error-correcting codes (TQECC)s. In particular, our results apply to codes that can be described as topological quantum field theories on two-dimensional manifolds, without reference to microscopic details of the system. Such codes have a simple mathematical description in terms of anyons, and well-known examples include Kitaev’s toric code and quantum double models, the Levin-Wen models, color codes, as well as fractional quantum Hall systems and topological insulators with topologically nontrivial surfaces. Our main result is that the logical gates implementable by CDLCs generate a finite group in any such code, and hence are not universal for quantum computation. For codes in which the anyons are abelian, we find the group is a proper subgroup of the generalized Clifford group strengthening the result of Bravyi and König. For codes with non-abelian anyons, such gates are even further restricted. For example, for Ising anyons, they must be elements of the Pauli group whereas for Fibonacci anyons there are no non-trivial gates. The motivation for this work is the fact that a fully operational quantum computer should be capable of both (1) storing, and (2) processing quantum information in a fault-tolerant manner. 1. TQECCs are ideal candidates for fault-tolerant quantum storage hardware, since they are naturally protected against typical (local) noise and involve local error correcting operations, making it feasible that such systems could be constructed using a scalable architecture in two-dimensions. 2. CDLCs provide a natural way to process information fault-tolerantly in a TQECC, since CDLCs expand the support of typical (i.e. sufficiently local) errors by a fixed amount — leaving them correctable in the TQECC. Unfortunately, our results exclude the possibility of achieving universal quantum computation through the exclusive use of CDLCs in these types of code. However, by supplementing these gates with additional resources, universality can be recovered. In particular, this may be achieved by relying on non-local processing, be it by braiding of non-abelian anyons or by the (resource intensive) magic state distillation, as well as for cutting-out and re-gluing macroscopic regions on the lattice. The full version of this paper is available at arXiv:1409.3898.

11:30—11:55 Grant Salton (Stanford) [ pdf | video ]

Spacetime replication of continuous variable quantum information

Our work builds on the results of Hayden and May who found necessary and sufficient conditions for information to be replicated and transmitted through spacetime. The only constraints on the system are no-cloning and no-signalling, and the authors found a simple set of conditions based on the causal structure of a summoning problem. In the present work, we developed a set of continuous variable quantum error correcting codes that efficiently satisfy the conditions for summoning. These codes are described and analyzed using a graph theoretic approach but they are special cases of a more general topological framework that will be the subject of a follow-up paper. We then change gears and present a different error correcting code to solve a particularly interesting summoning problem involving four spacetime regions. The code is a CSS code that uses five bosonic modes to encode one. We fully characterize the properties of the code, and we provide experimentally feasible optical implementations of the encoding and decoding operations. We hope to inspire an experimental group to demonstrate our code on an optical bench, since it has interesting fundamental roots in spacetime information replication.

12:00—12:25 Nicholas Delfosse (Sherbrooke) [ pdf | video ]

A combinatorial application of quantum information in percolation theory

Percolation is one of most simple models which exhibits a phase transition. The deter- mination of the threshold for this phase transition is usually quite difficult. It is the central problem of percolation theory. The relation between percolation and quantum error correction has been exploited to bound the performance of topological codes. In the present work, we propose to go in the other direction. We derive bounds on the percolation threshold of a family of hyperbolic graphs based on bounds on the performance of topological codes. Our first bound is a consequence of the no-cloning theorem, then it is refined by a precise study of the performance of topological codes. Finally, we simplify this result by studying the threshold for the appearance of homology in the tiling defining the codes, providing a much more precise bound. This result relies on the remark that uncorrectable erasures for topological codes correspond to subset of edges covering non-trivial homology. To our knowledge, it is the most precise upper bound on the percolation threshold of these graphs. Moreover, the homological threshold determined in arXiv:1408.4031 furnishes also an upper bound on the threshold of hyperbolic codes used over the quantum erasure channel.