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Quantum computing by color-code lattice surgery
Andrew Landahl University of New Mexico
PIRSA:14070006Quantum Error Correction for Ising Anyon Systems
PIRSA:14070005Strong majorization entropic uncentainty relations
Karol Zyczkowski Jagiellonian University
Fault-Tolerant Quantum Computation with Constant Overhead
Daniel Gottesman University of Maryland, College Park
PIRSA:14070004Spin glass reflection of the decoding transition for space-time codes
Alexey Kovalev University of California, Riverside
PIRSA:14070003Maximum likelihood decoding threshold as a phase transition
Leonid Pryadko University of California, Riverside
PIRSA:14070002Overview of the theory of spin glasses and its applications to quantum codes
Hidetoshi Nishimori Tokyo Institute of Technology
PIRSA:14070001
Injectivity radius bounds on the minimum distance of quantum LDPC codes
PIRSA:14070007Only a rare number of constructions of quantum LDPC codes are equipped with an unbounded minimum distance. Most of them are inspired by Kitaev toric codes constructed from the a tiling of the torus such as, color codes which are based on 3-colored tilings of surfaces, hyperbolic codes which are defined from hyperbolic tilings, or codes based on higher dimensional manifolds. These constructions are based on tilings of surfaces or manifolds and their parameters depend on the homology of the tiling.
In the first part of this talk, we recall homological bounds on the parameters of these quantum LDPC codes. In particular, the injectivity radius of the tiling provides a general lower bound on the minimum distance of these quantum LDPC codes.
Then, we extend the injectivity radius method to bound the minimum distance of a family of quantum LDPC codes based on Cayley graphs.
Finally, we improve these results by studying a notion of expansion of these Cayley graphs.
This talk is based on a joint work with Alain Couvreur and Gilles Zémor, and a joint work with Zhentao Li and Stephan Tommassé.Quantum computing by color-code lattice surgery
Andrew Landahl University of New Mexico
PIRSA:14070006In 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.Searching for Other Universes
Matthew Johnson York University
PIRSA:14070025Centuries of astronomy and cosmology have led to an ever-larger picture of our ‘universe’ — everything that we can observe. For just as long, there have been speculations that there are other regions beyond what is currently observable, each with diverse histories and properties, and all inhabiting a ‘Multiverse’. A nexus of ideas from cosmology, quantum gravity, and string theory lead to the prediction that we inhabit one of the most interesting sorts of Multiverses one could imagine: one that arises as a natural consequence of compelling explanations for other physics, and one that at least in principle can be tested with observations. In this talk, I will outline these ideas, and discuss the first observational tests of the Multiverse using data from the Wilkinson Microwave Anisotropy Probe.Quantum Error Correction for Ising Anyon Systems
PIRSA:14070005We consider two-dimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath, and we propose a phenomenological model to describe the resulting short-time dynamics, including pair-creation, hopping, braiding, and fusion of anyons. By explicitly constructing topological quantum error-correcting codes for this class of system, we use our thermalization model to estimate the lifetime of quantum information stored in the code space. To decode and correct errors in these codes, we adapt several existing topological decoders to the non-Abelian setting: one based on Edmond's perfect matching algorithm and one based on the renormalization group. These decoders provably run in polynomial time, and one of them has a provable threshold against a simple iid noise model. Using numerical simulations, we find that the error correction thresholds for these codes/decoders are comparable to similar values for the toric code (an Abelian sub-model consisting of a restricted set of allowed anyons). To our knowledge, these are the first threshold results for quantum codes without explicit Pauli algebraic structure. Joint work with Courtney Brell, Simon Burton, Guillaum Dauphinais, and David Poulin, arXiv:1311.0019.Strong majorization entropic uncentainty relations
Karol Zyczkowski Jagiellonian University
We analyze entropic uncertainty relations in a finite dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained in projective measurements with respect to any two orthogonal bases. We improve the recent bounds by Coles and Piani, which are known to be stronger than the well known result of Maassen and Uffink. Furthermore, we find a novel bound based on majorization techniques, which also happens to be stronger than the recent results involving largest singular values of submatrices of the unitary matrix connecting both bases. The first set of new bounds give better results for unitary matrices close to the Fourier matrix, while the second one provides a significant improvement in the opposite sectors. Some results derived admit generalization to arbitrary mixed states, so that corresponding bounds are increased by the von Neumann entropy of the measured state. The majorization approach is finally extended to the case of several measurements.Fault-Tolerant Quantum Computation with Constant Overhead
Daniel Gottesman University of Maryland, College Park
PIRSA:14070004The threshold theorem for fault tolerance tells us that it is possible to build arbitrarily large reliable quantum computers provided the error rate per physical gate or time step is below some threshold value. Most research on the threshold theorem so far has gone into optimizing the tolerable error rate under various assumptions, with other considerations being secondary. However, for the foreseeable future, the number of qubits may be an even greater restriction than error rates. The overhead, the ratio of physical qubits to logical qubits, determines how expensive (in qubits) a fault-tolerant computation is. Earlier results on fault tolerance used a large overhead which grows (albeit slowly) with the size of the computation. I show that using quantum LDPC codes, it is possible in principle to do fault-tolerant quantum computation with low overhead, and with the overhead constant in the size of the computation.Spin glass reflection of the decoding transition for space-time codes
Alexey Kovalev University of California, Riverside
PIRSA:14070003We introduce space-time quantum code construction which is based on repeating the layers of an arbitrary quantum error correcting code. The error threshold of such space-time construction is shown to be related to the fault tolerant error threshold of the original quantum error correcting code in the presence of errors in syndrome measurements. The decoding transition for space-time codes can be further mapped to random-bond Wegner spin models.
Families of quantum low density parity-check (LDPC) codes with a finite decoding threshold lead to both known models (e.g., random bond Ising and random plaquette Z2 gauge models) as well as unexplored earlier and generally non-local disordered spin models with non-trivial phase diagrams that include the spin glass phase.
We apply this construction to the simplest examples of recently discovered hypergraph-product codes and numerically find the fault tolerant threshold in excess of 5% by employing Monte-Carlo simulations.Maximum likelihood decoding threshold as a phase transition
Leonid Pryadko University of California, Riverside
PIRSA:14070002In maximum likelihood (ML) decoding, we are trying to find the most likely error given the measured syndrome. While this is hardly ever practical, such a decoder is expected to have the highest threshold.
I will discuss the mapping between the ML threshold for an infinite family of stabilizer codes and a phase transition in an associated family of Ising models with bond disorder [1]. This is a generalization of the map between the toric codes and the square lattice Ising model. Quantum LDPC codes produce generally non-local spin models with few-body interactions. A relatively simple Monte Carlo simulation of such a model can give an upper bound on the decoding threshold for the original code family. This can be used to compare code families irrespectively of decoders, and to establish an absolute measure of decoder performance.
[1] A. A. Kovalev and L. P. Pryadko, "Spin glass reflection of the decoding transition for quantum error correcting codes," unpublished,
arXiv:1311.7688 (2013).Overview of the theory of spin glasses and its applications to quantum codes
Hidetoshi Nishimori Tokyo Institute of Technology
PIRSA:14070001I will review the theory of spin glasses with an emphasis on gauge symmetry. A number of exact results will be shown to be derived, some of which are useful to discuss the properties of quantum LDPC codes. Also will be explained is the combination of gauge symmetry, replica method, and duality argument to predict the precise location of a multicritical point, which is equivalent to the error-tolerance limit of toric code.An introduction to quantum LDPC code
David Poulin Université de Sherbrooke
PIRSA:14070000In this talk, I will cover some basic notions of quantum LDPC codes, focusing on the similarities and distinctions with their classical cousins. Topics will include definitions of stabilizer quantum LDPC codes (CSS and general), iterative decoding algorithms, dual spin models, and obstructions caused by error degeneracy. The talk will be informal and a good occasion to ask questions.