Bulletin of the American Physical Society
2007 APS March Meeting
Volume 52, Number 1
Monday–Friday, March 5–9, 2007; Denver, Colorado
Session U33: Quantum Algorithms, Simulation, and Error Correction |
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Sponsoring Units: GQI Chair: Ian Durham, Saint Anselm College Room: Colorado Convention Center 403 |
Thursday, March 8, 2007 8:00AM - 8:12AM |
U33.00001: Quantum Simulations of Classical Systems Rolando Somma, Gerardo Ortiz, Cristian Batista, Emanuel Knill Understanding the properties of classical systems on a lattice using numerical methods is, in general, a very hard problem. In this talk I will focus then on the quantum simulations of classical systems. That is, if we had a quantum computer, which properties could be obtained more efficiently on it than on a conventional one?. For this purpose, I will introduce a classical-to-quantum mapping that will allow us to understand classical and quantum annealing procedures as two independent paths on Quantum-Hamiltonian space. I present then the corresponding quantum algorithms to simulate classical systems and give convergence rates, determined by the adiabatic theorem, to assure successful simulation. [Preview Abstract] |
Thursday, March 8, 2007 8:12AM - 8:24AM |
U33.00002: Simple proof of equivalence between adiabatic quantum computation and the circuit model Ari Mizel, Daniel Lidar, Morgan Mitchell We prove the equivalence between adiabatic quantum computation and quantum computation in the standard circuit model. An explicit adiabatic computation procedure is given that generates a ground state from which the answer can be extracted. The amount of time needed is evaluated by computing the gap. We show that the procedure is computationally efficient. [Preview Abstract] |
Thursday, March 8, 2007 8:24AM - 8:36AM |
U33.00003: Quantum cellular automata and quantum simulation Peter Love Quantum information theory can address ground state properties through DMRG-like classical methods, and through proposals for the use of phase estimation-based quantum algorithms. However, the dynamics of quantum systems present greater challenges to both classical and quantum computational methods. Quantum cellular automata provide a simple arena in which to address questions about quantum dynamics. Prior work has yielded decision procedures to determine when the local rule leads to globally unitary dynamics. I will describe a particular class of automata, unitary by construction, and discuss their relevance to the quantum computational complexity of quantum dynamics. [Preview Abstract] |
Thursday, March 8, 2007 8:36AM - 8:48AM |
U33.00004: A Two-dimensional Lattice Ion Trap for Quantum Simulation Tongyan Lin, Kenneth Brown, Xie Chen, Grace Cheung, Christopher Pearson, Isaac Chuang Quantum simulations offer the possibility of answering quantum spin system dynamics questions which may otherwise require unrealistic classical resources to solve. Such simulations may be implemented using well-controlled systems of effective spins, including, as we demonstrate, two-dimensional lattices of locally interacting ions. We present experimental results from a model ion lattice system, realized as a surface electrode rf trap with a square lattice geometry. Using 440 nm diameter charged microspheres, we loaded a 30 $\times$ 36 lattice with a spacing of 1.67 mm. When the trap is driven at 2 kHz and 375 volts, we observe isolated ion secular frequencies of 170 Hz perpendicular to the trap, and Coulomb repulsion between ions at different lattice sites consistent with numerical modeling. These results, when scaled to single-atom ion charge-to-mass ratios, and linewidths achievable using standard microlithography, are promising for quantum simulations with planar ion trap lattices. [Preview Abstract] |
Thursday, March 8, 2007 8:48AM - 9:00AM |
U33.00005: Making Moderately Large Ancillae Bryan Eastin Block coding has long been hailed as a practical method of decreasing the effective error probability. Compared with concatenated coding, large block codes require fewer encoding qubits to achieve the same logical error rate. The size of the quantum code employed, however, is not the only factor in determining the total number of qubits required. Another important aspect is the efficiency of procedures for constructing ancillary logical states used for computation and error correction. In this regard, block codes are at a disadvantage because there is no known method of efficiently preparing logical ancillae for block codes that does not rely on concatenation. For practical applications, however, it is sufficient to be capable of preparing logical ancillae for codes with large finite block size. This presentation describes a method of making moderately large ancillae with benign error properties. The form of the errors will be discussed as well as the method's resource scaling and the range of parameters for which it is effective. [Preview Abstract] |
Thursday, March 8, 2007 9:00AM - 9:12AM |
U33.00006: Quantum Error Correction Beyond Completely Positive Maps Daniel Lidar, Alireza Shabani We present a generalized theory of quantum error correction (QEC) that applies to any linear map, in particular maps that are not completely positive (CP). This theory of ``linear quantum error correction'' is applicable in cases where the standard and restrictive assumption of a factorized initial system-bath state does not apply. For linear maps that preserve positivity and/or Hermiticity, we find that standard QEC based on CP recovery maps still applies. Other linear maps generally require non-CP recovery operations. We illustrate our findings with examples of QEC for non-CP maps. [Preview Abstract] |
Thursday, March 8, 2007 9:12AM - 9:24AM |
U33.00007: Adiabatic Quantum Computation and the Theory of Quantum Phase Transitions William Kaminsky, Seth Lloyd We present a general approach to determining the asymptotic scaling of adiabatic quantum computational resources (space, time, energy, and precision) on random instances of NP-complete graph theory problems. By utilizing the isomorphisms between certain NP-complete graph theory problems and certain frustrated spin models, we demonstrate that the asymptotic scaling of the minimum spectral gap that determines the asymptotic running time of adiabatic algorithms is itself determined by the presence and character of quantum phase transitions in these frustrated models. Most notably, we draw the conclusion that adiabatic quantum computers based on quantum Ising models are much less likely to be efficient than those based on quantum rotor or Heisenberg models. We then exhibit practical rotor and Heisenberg model based architectures using Josephson junction and quantum dot circuits. [Preview Abstract] |
Thursday, March 8, 2007 9:24AM - 9:36AM |
U33.00008: Simulating the time-dependent Schr\"odinger equation with a quantum lattice-gas algorithm Zachary Prezkuta, Mark Coffey Quantum computing algorithms promise remarkable improvements in speed or memory for certain applications. Currently, the Type II (or hybrid) quantum computer is the most feasible to build. This consists of a large number of small Type I (pure) quantum computers that compute with quantum logic, but communicate with nearest neighbors in a classical way. The arrangement thus formed is suitable for computations that execute a quantum lattice gas algorithm (QLGA). We report QLGA simulations for both the linear and nonlinear time-dependent Schr\"odinger equation. These evidence the stable, efficient, and at least second order convergent properties of the algorithm. The simulation capability provides a computational tool for applications in nonlinear optics, superconducting and superfluid materials, Bose-Einstein condensates, and elsewhere. [Preview Abstract] |
Thursday, March 8, 2007 9:36AM - 9:48AM |
U33.00009: Implement quantum search and quantum counting algorithms by geometric quantum computation Qinghua Zhang, Guilu Long, Zidan Wang We derive the matching condition in the SO(3) picture of the generalized Grover's quantum search algorithm. We also give a more concise formula for evaluating the number of the iterations needed in the searching. This will help us a lot in the operation and get the target with certainty easily. We propose a new adiabatic Abelian geometric quantum computation strategy to implement quantum search and quantum counting algorithms based on the non-degenerate energy eigenstates in superconducting phase-qubit systems. [Preview Abstract] |
Thursday, March 8, 2007 9:48AM - 10:00AM |
U33.00010: Multi-dimensional linear diffusion for image enhancement on a type-II quantum computer. Gabriel Colburn, Mark Coffey We describe how multi-dimensional linear diffusion with application to image processing could be carried out on a hybrid classical-quantum computer. By using a hybrid approach, a variety of applications are possible that purely quantum computers are not suited for. Although the speed-up over classical computers is not as high as the latter approach, physical realization may be much sooner. One particular hybrid technique utilizes a lattice of quantum computers with relatively few quantum bits per node. Each node is connected by classical communication channels to its nearest neighbors. Such an architecture is referred to as a Type-II quantum computer. This class of architecture is able to efficiently simulate a variety of partial differential equations and is the platform for which our quantum-lattice-gas-based algorithms have been designed. We present the effective finite difference approximations that yield multi-dimensional linear diffusion, and representative simulation results. Additionally we demonstrate an extension to constrained linear diffusion that provides for nonuniform image smoothing. These methods are particularly relevant to image enhancement tasks. [Preview Abstract] |
Thursday, March 8, 2007 10:00AM - 10:12AM |
U33.00011: Quantum Algorithm for Partial Search Vladimir Korepin Searching and sorting is used as subroutines in many important algorithms. Grover discovered a quantum algorithm that searches faster than any classical algorithm. If we want less information we can do it faster. Partial search used to find an approximate location of the target item. An example: the exact location of the target item is given by a sequence of many bits, but we want to find only some of them. A partial search considers the following problem: a database is separated into several blocks. We want to find a block with the target item, not the target item itself. Efficiency of search algorithms is measured by number of queries to the oracle [total number of iterations]. Partial search can use the same hardware as the full search. Essential references follow:=$>$ 1) K. Grover, Quantum Mechanics helps in searching for a needle in a haystack, Phys. Rev. Letters, 78(2), 325, 1997. 2) L. K. Grover and J.Radhakrishnan, quant-ph/0407122, 3) V. E. Korepin and L. K. Grover, e-print quant-ph/0504157; Quantum Information Processing, vol. 5, issue 1, page 3, 2006, 4) V. E. Korepin, Journal of Physics A: Math. Gen. vol 38, pages L731-L738, 2005 see also quant-ph/0503238 5) V.E. Korepin, J. Liao, quant-ph/0510179 6) B.-S. Choi, T. A. Walker, S. L. Braunstein, e-print quant-ph/0603136 7) B-S Choi, V. E Korepin, quant-ph/0608106 8) V. E. Korepin, B. C. Vallilo, quant-ph/0609205 [Preview Abstract] |
Thursday, March 8, 2007 10:12AM - 10:24AM |
U33.00012: An Index Theorem for the Majorana Zero Modes in Chiral P-Wave Superconductors Sumanta Tewari, Sankar Das Sarma, Dung-Hai Lee We show that the Majorana fermion zero modes in the cores of odd winding number vortices of a 2D $p_x+ip_y$-paired superconductor is due to an index theorem. This theorem is analogous to that proven by Jackiw and Rebbi for the existence of localized Dirac fermion zero modes on the mass domain walls of a 1D Dirac theory. The important difference is that, in our case, the theorem is proven for a two component fermion field theory where the first and second components are related by parity reversal and hermitian conjugation. The vortices with Majorana zero modes can be used, in principle, to build a topological quantum computer. [Preview Abstract] |
Thursday, March 8, 2007 10:24AM - 10:36AM |
U33.00013: Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension Simon Trebst, Philipp Werner, Matthias Troyer, Kirill Shtengel, Chetan Nayak We discuss the stability of topological order against local perturbations by considering the effect of a magnetic field on a spin model -- the toric code -- which is in a topological phase. The model can be mapped onto a quantum loop gas where the perturbation introduces a bare loop tension. When the loop tension is small, the topological order survives. When it is large, it drives a continuous quantum phase transition into a magnetic state. The transition can be understood as the condensation of `magnetic' vortices, leading to confinement of the elementary `charge' excitations. We also show how the topological order breaks down when the system is coupled to an Ohmic heat bath and discuss our results in the context of quantum computation applications. [Preview Abstract] |
Thursday, March 8, 2007 10:36AM - 10:48AM |
U33.00014: D-Colexes: Topological Computation and Brane-Net Condensates Hector Bombin, Miguel A. Martin-Delgado Topological quantum order provides a way to reliably store quantum information. These systems show an energy gap and a topology dependent ground state degeneracy. Then one can use this degenerate subspace to protect quantum information from local interactions with the environment. Topological error correction codes are an interesting constructive approach towards topologically ordered systems. However, when one faces the issue of performing computations, not every topological code is suitable. We introduce a new family of topological codes that have such computational capabilities. They are constructed from certain $D$-dimensional lattices that we call $D$-colexes. For $D=2$, they allow to perform any operation in the Clifford group without individually addressing the physical qubits that form the quantum memories. The point is that Clifford group operations are enough for many tasks in quantum information such as quantum distillation. In the $D=3$ case even universal quantum computation is possible. From the perspective of topological order, the resulting systems give rise to interesting new topological orders: brane-net condensates. [Preview Abstract] |
Thursday, March 8, 2007 10:48AM - 11:00AM |
U33.00015: Quantum search algorithm with many bosons in optical lattices David Feder One approach to implementing quantum algorithms is the quantum walk (QW). In the continuous-time formulation, the QW is equivalent to the time-evolution of a quantum state under the influence of a discrete-space Hamiltonian. Building on a duality between many boson systems and weighted graphs, I will discuss how one can implement a QW search algorithm with ultracold bosons confined in optical lattices. By applying a specified external potential (using an external laser), the atoms in a shallow lattice will evolve from the uniformly populated ground state to all occupying one particular site. The results indicate that quantum algorithms exhibiting polynomial speed-up should be feasible with current experimental technology. [Preview Abstract] |
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