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Harnessing the Exchange Interaction for Quantum Information Processing

Mrittunjoy Guha Majumdar

Published Sep 5, 2018• 1 citation
Publisher

Abstract

In this paper, I propose new models of quantum information processing using the exchange interaction in physical systems. The partial SWAP operator that can be realized using the exchange interaction is used as the underlying resource for defining models of quantum computation, quantum communication, quantum memory and decoherence-free subspaces. Given the non-commutativity of these operators (for adjacent operators operating on a common qubit), a number of quantum states and entanglement patters can be obtained. This zoo of states can be classified, due to the parity constraints and permutation symmetry of the states, into invariant subspaces that are used for the definition of some of the applications in this paper.

DOI
10.20944/preprints201809.0079.v1

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In this paper, I propose new models of quantum information processing using the exchange interaction in physical systems. The partial SWAP operator that can be realized using the exchange interaction is used as the underlying resource for defining models of quantum computation, quantum communication, quantum memory and decoherence-free subspaces. Given the non-commutativity of these operators (for adjacent operators operating on a common qubit), a number of quantum states and entanglement patters can be obtained. This zoo of states can be classified, due to the parity constraints and permutation symmetry of the states, into invariant subspaces that are used for the definition of some of the applications in this paper.
In this paper, I propose new models of quantum information processing using the exchange interaction in physical systems. The partial SWAP operator that can be realized using the exchange interaction is used as the underlying resource for defining models of quantum computation, quantum communication, quantum memory and decoherence-free subspaces. Given the non-commutativity of these operators (for adjacent operators operating on a common qubit), a number of quantum states and entanglement patters can be obtained. This zoo of states can be classified, due to the parity constraints and permutation symmetry of the states, into invariant subspaces that are used for the definition of some of the applications in this paper.
This project is a comprehensive investigation into the application of the exchange interaction, particularly with the realization of the SWAP^1/n quantum operator, in quantum information processing. We study the generation, characterization and application of entanglement in such systems. Given the non-commutativity of neighbouring SWAP^1/n gates, the mathematical study of combinations of these gates is an interesting avenue of research that we have explored, though due to the exponential scaling of the complexity of the problem with the number of qubits in the system, numerical techniques, though good for few-qubit systems, are found to be inefficient for this research problem when we look at systems with higher number of qubits. Since the group of SWAP^1/n operators is found to be isomorphic to the symmetric group Sn, we employ group-theoretic methods to find the relevant invariant subspaces and associated vector-states. Some interesting patterns of states are found including onedimensional invariant subspaces spanned by W-states and the Hamming-weight preserving symmetry of the vectors spanning the various invariant subspaces. We also devise new ways of characterizing entanglement and approach the separability problem by looking at permutation symmetries of subsystems of quantum states. This idea is found to form a bridge with the entanglement characterization tool of Peres-Horodecki’s Partial Positive Transpose (PPT), for mixed quantum states. We also look at quantum information taskoriented ‘distance’ measures of entanglement, besides devising a new entanglement witness in the ‘engle’. In terms of applications, we define five different formalisms for quantum computing: the circuit-based model, the encoded qubit model, the cluster-state model, functional quantum computation and the qudit-based model. Later in the thesis, we explore the idea of quantum computing based on decoherence-free subspaces. We also investigate ways of applying the SWAP^1/n in entanglement swapping for quantum repeaters, quantum communication protocols and quantum memory.
The states of the physical algebra, namely the algebra generated by the operators involved in encoding and processing qubits, are considered instead of those of the whole-system algebra. If the physical algebra commutes with the interaction Hamiltonian, and the system Hamiltonian is the sum of arbitrary terms either commuting with or belonging to the physical algebra, then its states are decoherence free. One of the examples considered shows that, for a uniform collective coupling to the environment, the smallest number of physical qubits encoding a decoherence-free logical qubit is reduced from four to three.
During the last decade the field of quantum information has seen considerable progress both theoretically and experimentally. The building block of this theory is the so-called qubit, the quantum counterpart of the classical bit. It turns out that spin degrees of freedom of quantum particles are among the most promising candidates for quantum information processing and computation. Thus, a deep understanding of the different processes that govern the dynamics of these objects is of fundamental importance. For instance, the decoherence process, caused the coupling of the qubits to their surrounding environment, is the main obstacle to quantum information processing: it leads them to lose their quantum coherence and thus behave classically. In addition, quantum systems exhibit correlations that have no classical counterpart. This phenomenon is called entanglement and is the vital resource for quantum teleportation and quantum computing. Decoherence and entanglement dynamics were extensively studied within the framework of the Markovian approximation using the master equation approach. However, due to the non-commuting character of quantum observables, only few models are known to be exactly solvable. Moreover, many spin systems display a strongly non-Markovian behavior. This thesis is devoted to the development of new techniques for deriving the exact dynamics of spin qubits, coupled through Heisenberg and/or Ising interactions to spin environments that have internal dynamics. The basic idea behind these techniques is to use the underlying symmetries exhibited by the Hamitonian operators of the composite systems. This allows for the study of problems related to decoherence and entanglement.
Decoherence-free subspaces protect quantum information from the effects of noise that is correlated across the physical qubits used to implement them. Given the ability to impose suitable Hamiltonians upon such a multi-qubit system, one can also implement a set of logical gates which enables universal computation on this information without compromising this protection. Real physical systems, however, seldom come with the correct Hamiltonians built-in, let alone the ability to turn them off and on at will. In the course of our development of quantum information processing devices based on liquid-state NMR, we have found the task of operating on quantum information encoded in decoherence-free subspaces rather more challenging than is commonly assumed. This contribution presents an overview of these challenges and the methods we have developed for overcoming them in practice. These methods promise to be broadly applicable to many of the physical systems proposed for the implementation of quantum information processing devices.