Noisy Control, the Adiabatic Geometric Phase, and Destruction of the Efficiency of Geometric Quantum Computation

We examine the adiabatic dynamics of a quantum system coupled to a noisy classical control field. A stochastic phase shift is shown to arise in the off-diagonal elements of the system's density matrix which can cause decoherence. We derive the condition for onset of decoherence, and identify the noise properties that drive decoherence. We show how this decoherence mechansim causes: (1) a dephasing of the observable consequences of the adiabatic geometric phase; and (2) the loss of computational efficiency of the Shor algorithm when run on a sufficiently noisy geometric quantum computer.Comment: 4 pages, no figures, submitted to Phys. Rev. Let

Simulation of Quantum Adiabatic Search in the Presence of Noise

Results are presented of a large-scale simulation of the quantum adiabatic search (QuAdS) algorithm in the presence of noise. The algorithm is applied to the NP-Complete problem Exact Cover 3 (EC3). The noise is assumed to Zeeman-couple to the qubits and its effects on the algorithm's performance is studied for various levels of noise power, and for 4 different types of noise polarization. We examine the scaling relation between the number of bits N (EC3 problem size) and the algorithm's noise-averaged median run-time . Clear evidence is found of the algorithm's sensitivity to noise. Two fits to the simulation results were done: (1) power-law scaling = aN**b; and (2) exponential scaling = a[exp(bN) - 1]. Both types of scaling relations provided excellent fits. We demonstrate how noise leads to decoherence in QuAdS, estimate the amount of decoherence in our simulations, and derive an upper bound for the noise-averaged QuAdS success probability in the weak noise limit appropriate for our simulations.Comment: 15 pages, 13 figures, 6 table

Noise-Induced Sampling of Alternative Hamiltonian Paths in Quantum Adiabatic Search

We numerically simulate the effects of noise-induced sampling of alternative Hamiltonian paths on the ability of quantum adiabatic search (QuAdS) to solve randomly generated instances of the NP-Complete problem N-bit Exact Cover 3. The noise-averaged median runtime is determined as the noise-power and number of bits N are varied, and power-law and exponential fits are made to the data. Noise is seen to slowdown QuAdS, though a downward shift in the scaling exponent is found for N > 12 over a range of noise-power values. We discuss whether this shift might be connected to arguments in the literature that suggest that altering the Hamiltonian path might benefit QuAdS performance.Comment: 16 pages; 5 figures; 4 tables; to appear in Complexit

Berry's Phase in the Presence of a Non-Adiabatic Environment

We consider a two-level system coupled to an environment that evolves non-adiabatically. We present a non-perturbative method for determining the persistence amplitude whose phase contains all the corrections to Berry's phase produced by the non-adiabatic motion of the environment. Specifically, it includes the effects of transitions between the two energy levels to all orders in the non-adiabatic coupling. The problem of determining all non-adiabatic corrections is reduced to solving an ordinary differential equation to which numerical methods should provide solutions in a variety of situations. We apply our method to a particular example that can be realized as a magnetic resonance experiment, thus raising the possibility of testing our results in the lab.Comment: 21 pages, 1 Postscript figure, submitted to Phys. Rev.

Microscopic Analysis of the Non-Dissipative Force on a Line Vortex in Superconductor

A microscopic analysis of the non-dissipative force F_nd acting on a line vortex in a type-II superconductor at T=0 is given. All work assumes a charged BCS superconductor. We first examine the Berry phase induced in the BCS ground state by movement of the vortex and show how this phase enters into the hydrodynamic action S_hyd of the condensate. Appropriate variation of S_hyd gives F_nd and variation of the Berry phase term is seen to contribute the Magnus force of classical hydrodynamics to F_nd. This analysis confirms in detail the arguments of Ao and Thouless within the context of the BCS model. Our Berry phase, in the limit e -> 0, reproduces the Berry phase they obtain for a neutral superfluid. A second independent determination of F_nd is given through a microscopic derivation of the continuity equation for the condensate linear momentum. This yields the acceleration equation for the superflow. The vortex is seen to act as a momentum sink and the rate of momentum loss yields F_nd. Both calculations yield the same F_nd and show that the Magnus force contribution to F_nd is a consequence of the vortex motion and topology.Comment: abbrev. version of cond-mat/9408025, 4 pages, RevTex, no figure

GL(3,R) gauge theory of gravity coupled with an electromagnetic field

Consistency of $GL(3,R)$ gauge theory of gravity coupled with an external electromagnetic field, is studied. It is shown that possible restrictions on Maxwell field can be avoided through introduction of auxiliary fields.Comment: 5 pages, to appear in CIENCI

High-fidelity universal quantum gates through group-symmetrized rapid passage

Twisted rapid passage is a type of non-adiabatic rapid passage that generates controllable quantum interference effects that were first observed experimentally in 2003. It is shown that twisted rapid passage sweeps can be used to implement a universal set of quantum gates that operate with high-fidelity. The gate set consists of the Hadamard and NOT gates, together with variants of the phase, pi/8, and controlled-phase gates. For each gate g in the universal set, sweep parameter values are provided which numerical simulations indicate will produce a unitary operation that approximates g with error probability less than 10**(-4). Note that all gates in the universal set are implemented using a single family of control-field, and the error probability for each gate falls below the rough-and-ready estimate for the accuracy threshold of 10**(-4).Comment: 11 pages; LaTex; no figures; version to appear in Quantum Information and Computatio

The Distribution of Ramsey Numbers

We prove that the number of integers in the interval [0,x] that are non-trivial Ramsey numbers r(k,n) (3 <= k <= n) has order of magnitude (x ln x)**(1/2).Comment: Published version of manuscript; 5 pages, no figure

Improving quantum gate performance through neighboring optimal control

Successful implementation of a fault-tolerant quantum computation on a system of qubits places severe demands on the hardware used to control the many-qubit state. It is known that an accuracy threshold $P_{a}$ exists for any quantum gate that is to be used in such a computation. Specifically, the error probability $P_{e}$ for such a gate must fall below the accuracy threshold: $P_{e} < P_{a}$. Estimates of $P_{a}$ vary widely, though $P_{a}\sim 10^{-4}$ has emerged as a challenging target for hardware designers. In this paper we present a theoretical framework based on neighboring optimal control that takes as input a good quantum gate and returns a new gate with better performance. We illustrate this approach by applying it to all gates in a universal set of quantum gates produced using non-adiabatic rapid passage that has appeared in the literature. Performance improvements are substantial, both for ideal and non-ideal controls. Under suitable conditions detailed below, all gate error probabilities fall well below the target threshold of $10^{-4}$.Comment: 27 pages; 11 figures; 13 tables; to appear in Phys. Rev.

High-fidelity quantum state preparation using neighboring optimal control

We present an approach to single-shot high-fidelity preparation of an $n$-qubit state based on neighboring optimal control theory. This represents a new application of the neighboring optimal control formalism which was originally developed to produce single-shot high-fidelity quantum gates. To illustrate the approach, and to provide a proof-of-principle, we use it to prepare the two qubit Bell state $|\beta_{01}\rangle = (1/\sqrt{2})\left[\, |01\rangle + |10\rangle\,\right]$ with an error probability $\epsilon\sim 10^{-6}$ ($10^{-5}$) for ideal (non-ideal) control. Using standard methods in the literature, these high-fidelity Bell states can be leveraged to fault-tolerantly prepare the logical state $|\overline{\beta}_{01}\rangle$.Comment: 16 pages, 1 figure, 9 tables, all MATLAB files used in numerical computations include