Bulletin of the American Physical Society
APS March Meeting 2014
Volume 59, Number 1
Monday–Friday, March 3–7, 2014; Denver, Colorado
Session L35: Focus Session: Quantum Computing Architectures and Algorithms: Quantum Error Correction |
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Sponsoring Units: GQI Room: 702 |
Wednesday, March 5, 2014 8:00AM - 8:12AM |
L35.00001: Fibre bundle framework for quantum fault tolerance Lucy Liuxuan Zhang, Daniel Gottesman We introduce a differential geometric framework for describing families of quantum error-correcting codes and for understanding quantum fault tolerance. In particular, we use fibre bundles and a natural projectively flat connection thereon to study the transformation of codewords under unitary fault-tolerant evolutions. We'll explain how the fault-tolerant logical operations are given by the monodromy group for the bundles with projectively flat connection, which is always discrete. We will discuss the construction of the said bundles for two examples of fault-tolerant families of operations, the string operators in the toric code and the qudit transversal gates. This framework unifies topological fault tolerance and fault tolerance based on transversal gates, and is expected to apply for all unitary quantum fault-tolerant protocols. [Preview Abstract] |
Wednesday, March 5, 2014 8:12AM - 8:24AM |
L35.00002: Magic-state encoder and magic teleportation: Efficient fault-tolerant non-Clifford gates with concatenated quantum codes Hayato Goto, Satoshi Nakamura, Mamiko Kujiraoka, Kouichi Ichimura In fault-tolerant quantum computation, Clifford operations, e.g., controlled-NOT gates, can be efficiently implemented in a fault-tolerant manner. However, non-Clifford gates such as the $T$ gate $\pi/8$ rotation) and the Toffoli gate are difficult to implement efficiently. A standard approach to non-Clifford gates is ``magic state distillation,'' which can provide high-fidelity magic states using more low-fidelity magic states. Thus, reliable non-Clifford gates can be performed with the high-fidelity magic states and reliable Clifford operations. However, the resource overhead for magic state distillation is much larger than those for Clifford gates. To solve this problem, here we propose a new approach: magic-state encoder. This can be applied to concatenated quantum codes with a property that Hadamard gates can be implemented transversally. The magic-state encoder encodes (not distills) a high-fidelity level-$(l+1)$ encoded magic state with low-fidelity level-$l$ encoded magic states. As a result, non-Clifford gates (here we focus on the $T$ gate) can be performed with an overhead comparable to Clifford gates. By performing the $T$ gate by teleportation with an entangled state generated with a magic state, which we call ``magic teleportation,'' further improvement is possible. [Preview Abstract] |
Wednesday, March 5, 2014 8:24AM - 8:36AM |
L35.00003: Quantum compiling with low overhead Guillaume Duclos-Cianci, David Poulin I will present a scheme to compile complex quantum gates that uses significantly fewer resources than existing schemes. In standard fault-tolerant protocols, a magic state is distilled from noisy resources, and copies of this magic state are then assembled into produced complex gates using the Solovay-Kitaev theorem or variants thereof. In our approach, we instead directly distill magic states associated to complex gates from noisy resources, leading to a reduction of the compiling overhead of several orders of magnitude. [Preview Abstract] |
Wednesday, March 5, 2014 8:36AM - 9:12AM |
L35.00004: The overhead of fault-tolerant quantum computing Invited Speaker: Daniel Gottesman The 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 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. The result depends on recent progress on quantum low-density parity check codes. [Preview Abstract] |
Wednesday, March 5, 2014 9:12AM - 9:24AM |
L35.00005: Extremal Optimization for estimation of the error threshold in topological subsystem codes at $\mathbf{T=0}$ Jorge E. Mill\'an-Otoya, Stefan Boettcher Quantum decoherence is a problem that arises in implementations of quantum computing proposals. Topological subsystem codes (\emph{TSC}) have been suggested as a way to overcome decoherence. These offer a higher optimal error tolerance when compared to typical error-correcting algorithms. A TSC has been translated into a planar Ising spin-glass with constrained bimodal three-spin couplings. This spin-glass has been considered at finite temperature to determine the phase boundary between the unstable phase and the stable phase, where error recovery is possible.\footnote{R. S. Andrist et al., \emph{Optimal error correction in topological subsystem codes}, Phys. Rev. A., \textbf{85}, 050302(R) (2012)} We approach the study of the error threshold problem by exploring ground states of this spin-glass with the Extremal Optimization algorithm (\emph{EO}).\footnote{S. Boettcher et al., \emph{Optimization with extremal dynamics}, Phys. Rev. Lett., \textbf{86}, 5211 (2001)} EO has proven to be a effective heuristic to explore ground state configurations of glassy spin-systems.\footnote{S. Boettcher, \emph{Stiffness of the Edwards-Anderson model in all dimensions}, Phys. Rev. Lett., \textbf{95}, 197205 (2005)} [Preview Abstract] |
Wednesday, March 5, 2014 9:24AM - 9:36AM |
L35.00006: Error-thresholds for qudit-based topological quantum memories Ruben S. Andrist, James R. Wootton, Helmut G. Katzgraber Extending the quantum computing paradigm from qubits to higher-dimensional quantum systems allows for increased channel capacity and a more efficient implementation of quantum gates. However, to perform reliable computations an efficient error-correction scheme adapted for these multi-level quantum systems is needed. A promising approach is via topological quantum error correction, where stability to external noise is achieved by encoding quantum information in non-local degrees of freedom. A key figure of merit is the error threshold which quantifies the fraction of physical qudits that can be damaged before logical information is lost. Here we analyze the resilience of generalized topological memories built from d-level quantum systems (qudits) to bit-flip errors. The error threshold is determined by mapping the quantum setup to a classical Potts-like model with bond disorder, which is then investigated numerically using large-scale Monte Carlo simulations. Our results show that topological error correction with qutrits exhibits an improved error threshold in comparison to qubit-based systems. [Preview Abstract] |
Wednesday, March 5, 2014 9:36AM - 9:48AM |
L35.00007: The intrinsic error thresholds of the surface code with correlated errors Pejman Jouzdani, Eduardo Mucciolo, Eduardo Novais We study how the resilience of the surface code to decoherence is affected by the presence of a bosonic bath. The surface code experiences an effective dynamics due to the coupling to a bosonic bath that correlates the qubits of the code. The range of the effective induced qubit-qubit interaction depends on parameters related to the bath correlation functions. We show hat different ranges set different intrinsic bounds on the fidelity of the code. These bounds appear to be independent of the stochastic error probabilities frequently studied in the literature and to be merely a consequence of the induced dynamics by the bath. We introduce a new definition of stabilizers based on logical operators that allows us to efficiently implement a Metropolis algorithm to determine the intrinsic upper bounds to the error threshold. [Preview Abstract] |
Wednesday, March 5, 2014 9:48AM - 10:24AM |
L35.00008: Using concatenated quantum codes for universal fault-tolerant quantum gates Invited Speaker: Tomas Jochym-O'Connor Quantum error correction and fault-tolerance are essential for large scale quantum information processing tasks. A standard method for implementing a logical fault-tolerant gate is by applying the gate transversally, that is without coupling qubits within an encoded codeblock. However, Eastin and Knill [Phys. Rev. Lett. 102, 110502 (2009)] proved that it is impossible to have a set of universal transversal gates for a given quantum error correcting code. In this work, we present sufficient conditions to obtain a set of universal fault-tolerant quantum gates by concatenating two quantum error correcting codes. Namely, the concatenation scheme does not require the preparation of special ancillary states in order to obtain universality, unlike schemes such as magic state distillation. The construction exploits the transversality of different sets of gates for the given codes, protecting for the non-transversal gates in one code by implementing these logical gates using transversal gates in the second code. The full distance of the concatenated code is sacrificed to protect against low-weight arbitrary errors, ensuring fault-tolerance. [Preview Abstract] |
Wednesday, March 5, 2014 10:24AM - 10:36AM |
L35.00009: Error correction with machine learning: one man's syndrome measurement is another man's treasure Joshua Combes, Hans Briegel, Carlton Caves, Christopher Cesare, Christopher Ferrie, Gerard Milburn, Markus Tiersch Syndrome measurements that are made in quantum error correction contains more information than is typically used. We show using the data from syndrome measurements (that one has to do anyway) the following: (1) a channel can be dynamically estimated; (2) in some situations the information gathered from the estimation can be used to permanently correct away part of the channel; and (3) can allow us to perform hypothesis testing to determine if the errors are correlated or if the error rate exceeds the ``expected worst case''. The unifying theme to these topics is making use of all of the information in the data collected from syndrome measurements with a machine learning and control algorithms. [Preview Abstract] |
Wednesday, March 5, 2014 10:36AM - 10:48AM |
L35.00010: Finite-size scaling of the decoherence time of the Toric Code in contact with a thermal reservoir C. Daniel Freeman, CM Herdman, Dylan Gorman, Birgitta Whaley We present an analysis of the finite-size scaling of the decoherence time of a topological qubit in contact with a thermal bath. While the relaxation time of the toric code at finite temperature in the thermodynamic limit has a system size independent bound, we find nontrivial finite-size scaling of the decoherence time in the low temperature crossover regime on a finite lattice. Using a continuous-time Monte Carlo method, we explicitly compute the low temperature nonequilibrium dynamics of the toric code on finite lattices. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial 2D classical random walks. As this finite temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of decoherence times of possible physical realizations of topological qubits. [Preview Abstract] |
Wednesday, March 5, 2014 10:48AM - 11:00AM |
L35.00011: Spin glass reflection of quantum error correcting codes Alexey Kovalev, Leonid Pryadko We study the decoding transition for quantum error correcting codes with the help of a mapping to random-bond Wegner spin models. Known families of quantum low density parity check (LDPC) codes lead to unexplored earlier generally non-local Wegner models with rich phase diagrams that include ordered, disordered, and spin glass phases. The decoding transition corresponds to a transition from the ordered phase by proliferation of extended defects which generalize the notion of domain walls to non-local spin models. In recently discovered quantum LDPC code families with finite rates the number of distinct classes of such extended defects is exponentially large, corresponding to extensive ground state entropy of these codes. Here, the transition can be driven by the entropy of the extended defects, a mechanism distinct from that in the local spin models where the number of defect types (domain walls) is always finite. We construct numerically phase diagrams for models corresponding to several families of quantum LDPC codes. We formulate similar mapping to random bond Wegner models for the case of errors in syndrome measurements, and find several examples of code families with highest fault-tolerant thresholds. [Preview Abstract] |
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