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Journal of Physics A Mathematical and General

Quantum gates and degenerations

Juan Mateo, José M. Cerveró

Published Jan 28, 2004• 1 citation
Publisher

Abstract

A three state Hamiltonian with a degenerate ground state is shown to exhibit the property of generating quantum gates for fixed values of the energy parameters just by using the ability of this quantum system to evolve in time yielding suitable sets of quantum probabilities. A method is presented, based on the conventional rules of quantum mechanics, for obtaining several gates without changing the system configuration, i.e. just by adjusting the times in which the interaction is switched on and off. The mathematical rules for generating such a set of gates are completely defined and described in this work. The corresponding physical implementation should involve fast Rabi pulses and very precise switches of the kind proposed in quantum geometric computation.

DOI
10.1088/0305-4470/37/6/035

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A three state Hamiltonian with a degenerate ground state is shown to exhibit the property of generating quantum gates for fixed values of the energy parameters just by using the ability of this quantum system to evolve in time yielding suitable sets of quantum probabilities. A method is presented, based on the conventional rules of quantum mechanics, for obtaining several gates without changing the system configuration, i.e. just by adjusting the times in which the interaction is switched on and off. The mathematical rules for generating such a set of gates are completely defined and described in this work. The corresponding physical implementation should involve fast Rabi pulses and very precise switches of the kind proposed in quantum geometric computation.
A qubit, or quantum bit, is conventionally defined as "a physical system for storing information that is capable of existing in either of two quantum states or in a superposition of both". In this paper, we examine the simple question of whether two distinct levels, each consisting of multiply degenerate sub-states, could serve as a practical quantum bit. We explore this idea using a well-characterized atomic system of the kind employed in several quantum computing implementations. We approximate the atom as a two-level system without degeneracy lifting in the magnetic quantum number while using the angular momentum addition rules to select the desired state transition. We find that, in the continuous presence of the field, the atom still undergoes Rabi oscillations, which are suitable for quantum gate construction. In addition, we compute the average fidelity in quantum gate performance for a single degenerate atom and postulate the required form of two-atom interaction to construct a controlled Z gate.
We present the analytical generators for arbitrary multi-qubit controlled gates. Closed forms for the generating Hamiltonians are given for gates with both multiple control and target qubits, as well as for arbitrary control conditions. This allows us to go beyond gate-based simulations of quantum circuits and incorporate decoherence and other noise in simulations of quantum computers. We exemplify this by simulating the impact of a harmonic oscillator interacting with two qubits during the application of a controlled NOT gate.
The hydrodynamic re-grounding of quantum mechanics proposes that measurement outcomes arise from a deterministic relaxation of a probability fluid into eigenstate basins. A prior adjudication of this program found its central weakness to be the absence of any concrete mechanism. This paper reports a pre-registered computational test of the strongest available form of that proposal: a two-level system evolved unitarily to a measurement time, then subjected to three candidate relaxation dynamics toward the eigenbasis, with outcome statistics collected over 1e5 shots per state across a fixed test set. The simulation code, parameter values, and analysis rules were fixed and checksummed before any result was produced. Every configuration failed: the maximum deviation from Born probabilities was 0.5, fifty times the pre-registered tolerance of 1e-2, because a deterministic map from a fixed initial state yields a degenerate outcome channel - measured probabilities take only the values 0 or 1. The reader should care because this converts a conceded gap into a demonstrated one: within the tested family, deterministic measurement-triggered relaxation cannot reproduce the Born rule; reproducing it requires an ensemble over initial states, stochasticity, or measurement-contextual variables. Where the premises end: the falsification is scoped to deterministic, fixed-initial-state, threshold-terminated dynamics on two-level systems.
We study the low energy states of one-dimensional spin chain clusters consisting of three spin-1/2 Heisenberg particles. When applying an external magnetic field on the spin chain cluster, the ground state doublet will generate Zeeman splitting. As a two-level quantum system, the split ground state doublet can be defined as a new 'spin cluster qubit'. The cluster quibit is shown to be more robust with a smaller decoherence rate. We realize universal single-qubit gates with the algebraic dynamical method, by applying a time-dependent magnetic field on the spin cluster. We also investigate a two-qubit system made up of two such Heisenberg spin chain clusters which interact through a tunable Heisenberg coupling. In the two-qubit product basis, the controlled quantum phase gates can be generated by applying a magnetic field along the z-axis on the system of the two clusters. One can realize the controlled-NOT gate through a three-step time evolution by adjusting the corresponding physical parameters. Our scheme involves two kinds of dynamically distinct operations, which is much simpler and more feasible.
Geometric phases arise naturally in a variety of quantum systems with observable consequences. They also arise in quantum computations when dressed states are used in gating operations. Here we show how they arise in these gating operations and how one may take advantage of the dressed states producing them. Specifically, we show that for a given, but arbitrary Hamiltonian, and at an arbitrary time $\ensuremath{\tau}$, there always exists a set of dressed states such that a given gate operation can be performed by the Hamiltonian up to a phase $\ensuremath{\varphi}$. The phase is a sum of a dynamical phase and a geometric phase. We illustrate the dressed phase for several systems.