Bulletin of the American Physical Society
2009 APS March Meeting
Volume 54, Number 1
Monday–Friday, March 16–20, 2009; Pittsburgh, Pennsylvania
Session W17: Quantum Algorithms, Simulation, and Error Correction |
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Sponsoring Units: GQI Chair: Yaakov Weinstein, Mitre Corporation Room: 318 |
Thursday, March 19, 2009 11:15AM - 11:27AM |
W17.00001: Time dependent DMRG for spectral functions of Heisenberg chains Steven White, Ian Affleck, Rodrigo Pereira Recently developed real-time DMRG techniques allow the calculation of space and time dependent spin-spin correlation functions for spin chain systems. These correlation functions can be Fourier transformed to obtain momentum and frequency dependent spectral functions. The growth of entanglement in the simulation as a function of time prevents extremely long simulation times, limiting the frequency resolution. We have found that the long time behavior can be extrapolated using either of two different techniques, allowing us to obtain very high resolution spectra with very high accuracy. We demonstrate these techniques for the S=1 Heisenberg chain and the XXZ S=1/2 chain. [Preview Abstract] |
Thursday, March 19, 2009 11:27AM - 11:39AM |
W17.00002: High-threshold surface code quantum computing: threshold calculation Peter Groszkowski, Austin Fowler, Ashley M. Stephens Surface codes are topological quantum error correcting codes. In such codes, information is encoded in a collection of physical qubits arranged on a lattice, with only nearest-neighbor interaction required for processing and readout. In this talk we present a detailed account of a numerical threshold calculation for a planar surface code with boundaries (arXiv:0803.0272). In the end we find a threshold value that's approaching 1{\%}. [Preview Abstract] |
Thursday, March 19, 2009 11:39AM - 11:51AM |
W17.00003: Error threshold in topological quantum-computing models with color codes Helmut Katzgraber, Hector Bombin, Miguel A. Martin-Delgado Dealing with errors in quantum computing systems is possibly one of the hardest tasks when attempting to realize physical devices. By encoding the qubits in topological properties of a system, an inherent protection of the quantum states can be achieved. Traditional topologically-protected approaches are based on the braiding of quasiparticles. Recently, a braid-less implementation using brane-net condensates in 3-colexes has been proposed. In 2D it allows the transversal implementation of the whole Clifford group of quantum gates. In this work, we compute the error threshold for this topologically-protected quantum computing system in 2D, by means of mapping its error correction process onto a random 3-body Ising model on a triangular lattice. Errors manifest themselves as random perturbation of the plaquette interaction terms thus introducing frustration. Our results from Monte Carlo simulations suggest that these topological color codes are similarly robust to perturbations as the toric codes. Furthermore, they provide more computational capabilities and the possibility of having more qubits encoded in the quantum memory. [Preview Abstract] |
Thursday, March 19, 2009 11:51AM - 12:03PM |
W17.00004: Performing measurement based quantum computation on ground states Andrew Doherty, Stephen Bartlett One of the most exciting developments in quantum computing in recent years has been the realisation that there exist states of quantum many-body systems that can serve as a universal resource for quantum computing, where computation proceeds solely through single-qubit measurements. Although currently only a few isolated examples of such universal resource states are known, we discuss the possibility that there exist models of interacting spin systems in which an ordered phase is characterized by the ability to perform measurement-based quantum computation (MBQC). To identify such phases, we propose to use nonlocal correlation functions that quantify the fidelity of quantum gates performed between well separated qubits. The quantum computing phase is then characterized by set of order parameters corresponding to a universal set of quantum gates. We investigate a simple spin-lattice system based on the cluster-state model for MBQC by using a series of dualities with better studied models. We demonstrate that the model possesses a zero temperature phase transition between a disordered phase and an ordered ``cluster phase'' in which it is possible to perform a large class of one and two qubit quantum gates. [Preview Abstract] |
Thursday, March 19, 2009 12:03PM - 12:15PM |
W17.00005: Operator Theoretic Quantum Fault Tolerance Gerald Gilbert, Yaakov S. Weinstein, Vaneet Aggarwal, A. Robert Calderbank We outline the advantages of an operator approach to quantum fault tolerance. Operator quantum fault tolerance is based on an explicitly stated halting condition, exact solutions of quantum error correction code dynamics, and as accurate and realistic descriptions as possible of the error models. This allows the proper integration of error correction and concatenation strategies with the system dynamics so as to better allocate quantum computational resources such as qubits, quantum gates, and computation time for quantum circuit design. We demonstrate these characteristics of the operator approach with an example of an asymmetric error model. [Preview Abstract] |
Thursday, March 19, 2009 12:15PM - 12:27PM |
W17.00006: Parallel Environment for Quantum Computing Frank Tabakin, Bruno Julia Diaz To facilitate numerical study of noise and decoherence in QC algorithms,and of the efficacy of error correction schemes, we have developed a Fortran 90 quantum computer simulator with parallel processing capabilities. It permits rapid evaluation of quantum algorithms for a large number of qubits and for various ``noise'' scenarios. State vectors are distributed over many processors, to employ a large number of qubits. Parallel processing is implemented by the Message-Passing Interface protocol. A description of how to spread the wave function components over many processors, along with how to efficiently describe the action of general one- and two-qubit operators on these state vectors will be delineated.Grover's search and Shor's factoring algorithms with noise will be discussed as examples. A major feature of this work is that concurrent versions of the algorithms can be evaluated with each version subject to diverse noise effects, corresponding to solving a stochastic Schrodinger equation. The density matrix for the ensemble of such noise cases is constructed using parallel distribution methods to evaluate its associated entropy. Applications of this powerful tool is made to delineate the stability and correction of QC processes using Hamiltonian based dynamics. [Preview Abstract] |
Thursday, March 19, 2009 12:27PM - 12:39PM |
W17.00007: Relation of operator Schmidt decomposition and CNOT complexity Mark Coffey, Ron Deiotte We consider two-qubit operators and provide a correspondence between their Schmidt number and controlled-NOT (CNOT) complexity, where the CNOT complexity is up to local unitary operations [1]. The results are obtained by complementary means, and a number of examples are given. Additionally, we present results for exact decompositions of two-qubit operators in terms of CNOT [2]. Instances of these results are applicable to superconducting-flux qubit and other systems. [1]M. W. Coffey and R. Deiotte, Quant. Info. Proc. 7, 117 (2008). [2]M. W. Coffey and R. Deiotte, preprint (2008). [Preview Abstract] |
Thursday, March 19, 2009 12:39PM - 12:51PM |
W17.00008: New Method for the Calculation of Qubit Decoherence in the Presence of 1/f Noise Dong Zhou, Robert Joynt We present a new mathematical method for the calculation of qubit decoherence subject to classical noise coming from an ensemble of two-level fluctuators. The time evolution of the qubit density matrix is governed by a non-Hermitian quasi-Hamiltonian, mapping the problem onto a system consisting of a spin-1 particle (the qubit) coupled to spin-1/2 particles (the fluctuators). The method gives non-perturbative results for the energy relaxation, free induction decay (FID) and spin echo pulse measurements. This extends the range of known results to strong coupling, beyond the range of validity of Redfield theory and the commonly-used Gaussian approximation. New functional forms are suggested to explain the recent experiments by Kakuyanagi [PRL 98, 047004 (2007)] and Yoshihara [PRL 97, 167001 (2006)] on qubit decoherence with 1/f noise. [Preview Abstract] |
Thursday, March 19, 2009 12:51PM - 1:03PM |
W17.00009: Quantum Darwinism for mixed-state environment Haitao Quan, Michael Zwolak, Wojciech Zurek We exam quantum darwinism when a system is in the presence of a mixed environment, and we find a general relation between the mutual information for the mixed-state environment and the change of the entropy of the fraction of the environment. We then look at a particular solvable model, and we numerically exam the time evolution of the ``mutual information" for large environment. Finally we discuss about the exact expressions for all entropies and the mutual information at special time. [Preview Abstract] |
Thursday, March 19, 2009 1:03PM - 1:15PM |
W17.00010: Static environments in open quantum systems Ian Durham An open quantum system is one that interacts in some way with another quantum system external to itself, e.g. an environment. In some cases this environment is constrained to be static under unitary transformations. It turns out that there are severe limitations on the types of systems and environments that can interact in such cases. For instance, we find that the system may only be in a pure state under most such transformations. In addition we find that a static environment cannot serve as a sub-system of a Bell state. It \emph{may} serve as a sub-system of a GHZ state in most cases, but its feasibility is dependent upon the unitary transformation that is applied to any part of the system. We note an implication that these results have for recent studies of quantum computation in the presence of closed time-like curves. [Preview Abstract] |
Thursday, March 19, 2009 1:15PM - 1:27PM |
W17.00011: Decoherence in the hypercube quantum walk Frederick Strauch A new model of decoherence in the hypercube quantum walk will be presented, in which dephasing occurs between every vertex of the hypercube. Surprisingly, in this model the hitting probability remains bounded for arbitrarily large hypercubes. This result can be obtained by a simple analytical argument, and has implications for perfect quantum state transfer in qubit networks. This argument, and related numerical and perturbative results, will be discussed. [Preview Abstract] |
Thursday, March 19, 2009 1:27PM - 1:39PM |
W17.00012: Phase transitions in random quantum satisfiability Chris Laumann, Roderich Moessner, Antonello Scardicchio, Shivaji Sondhi The potential power of quantum computers is a subject of great current interest and the raison d'etre for the intense effort and progress to build them. Naturally much theoretical interest has focused on algorithms that outperform their classical counterpart but recent developments in quantum complexity theory suggest that we already know problems, those shown to be QMA-complete, whose worst case instances would take a quantum computer exponentially long to solve. As in classical complexity theory the supposed difficulty of QMA complete problems follows from the existence of polynomial transformations relating any of the large class of QMA problems to instances of QMA-complete questions. This does not directly address the question of why this problem has hard instances and what features they posses. In this work we attempt to investigate the features of hard instances of a QMA complete problem introduced by S. Bravyi: quantum k-SAT. We use techniques of statistical physics of disordered systems in order to study a random ensemble of quantum k-SAT instances parametrized by clause density $\alpha$ in a program that is analogous to recent studies of classical random k-SAT. We establish a phase transition in satisfiability as a function of clause density and show that the problem almost always reduces to identifying a classical graph property. [Preview Abstract] |
Thursday, March 19, 2009 1:39PM - 1:51PM |
W17.00013: Asymptotic convergence rates for statistical moments of pseudorandom quantum circuits Winton Brown, Lorenza Viola We investigate the statistical moments of pseudorandom quantum circuits acting on an n-qubit system. We show that for pseudorandom quantum circuits that are invariant under arbitrary permutations of the qubit labels, there exists a representation of the linear map which describes the evolution of moments of fixed order, t, such that the dimension of the map scales polynomially in the number of qubits. The long time asymptotic convergence rate for low-order moments may be obtained by means of a perturbation expansion, shedding light on the question of how well pseudorandom quantum circuits approximate unitary t-designs. [Preview Abstract] |
Thursday, March 19, 2009 1:51PM - 2:03PM |
W17.00014: Quantum annealing for the ground state problem via exactly solvable models Yohei Saika, Jun-ichi Inoue In this study, in order to clarify the efficiency of quantum annealing for optimization, we study the ground state problem using solvable spin systems, such as the spin 1/2 quantum Ising-XY chain under the Lorentzian field. First, we exactly estimate static properties, such as the ground state energy and the energy gap. We find that the ground state energy depends on the selection of the control field, although the ground state energy is same if the control field vanishes respectively. We also find that the energy gap between the ground state and the first excited state is inversely proportional to the system size and becomes zero in the thermodynamic limit. Also, via the numerical simulation on the Schrodinger equation, we clarify that the quantum annealing using various control fields, such as the transverse field, the ferromagnetic Ising interaction, is available of the ground state problem not also for the spin 1/2 quantum Ising-XY chain and also for the random field Ising model. [Preview Abstract] |
Thursday, March 19, 2009 2:03PM - 2:15PM |
W17.00015: Quantum adiabatic algorithm for the Ising model on a Bethe lattice Erin Handberg, Sergey Knysh, Vadim Smelyanskiy, Andre Petukhov We investigate the average-case complexity of the quantum adiabatic algorithm (QAA) for the Ising model on the Bethe lattice by studying the asymptotic scaling of the minimum gap. For $N < 30$, this is done using direct diagonalization of the quantum Ising Hamiltonian with randomly chosen Gaussian exchange couplings. We use the Mezard-Parisi definition of the Bethe lattice for finite $N$ with the connectivity graph chosen uniformly at random among all $k$-regular graphs. The results of the direct diagonalization are compared to those from the parallel tempering version of quantum Monte Carlo (QMC) algorithm, and we use the QMC to study larger values of $N$. [Preview Abstract] |
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