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arXiv:quant-ph/0511131

Adiabatic Quantum Computing in systems with constant inter-qubit couplings

Sergey Knysh, Vadim Smelyanskiy

Published Nov 14, 2005• 1 citation
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Abstract

We propose an approach suitable for solving NP-complete problems via adiabatic quantum computation with an architecture based on a lattice of interacting spins (qubits) driven by locally adjustable effective magnetic fields. Interactions between qubits are assumed constant and instance-independent, programming is done only by changing local magnetic fields. Implementations using qubits coupled by magnetic-, electric-dipole and exchange interactions are discussed.

DOI
10.48550/arxiv.quant-ph/0511131

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We propose an approach suitable for solving NP-complete problems via adiabatic quantum computation with an architecture based on a lattice of interacting spins (qubits) driven by locally adjustable effective magnetic fields. Interactions between qubits are assumed constant and instance-independent, programming is done only by changing local magnetic fields. Implementations using qubits coupled by magnetic-, electric-dipole and exchange interactions are discussed.
We propose a method that implements a universal set of one- and two-quantum-bit gates for quantum computation in a system of coupled electron pairs with constant non-diagonal exchange interaction. In our proposal, suppression of the exchange interaction is performed by the continual repetition of single-spin rotations. A small g-factor difference between the electrons allows for addressing qubits and avoiding strong magnetic field pulses. Numerical experiments were performed to show that, to implement the one- and two-qubit operations, it is sufficient to change the strength of the magnetic field by a few Gauss. This introduces one and then the other electron in a resonance. To determine the evolution of the two-qubit system, we use the algorithms of optimal control theory.
Quantum computers allow to solve efficiently certain problems that are intractable for classical computers. For the realization of a quantum computer, a qubit design as the basic building block is a nontrivial starting point. Within a nanoscale ferromagnetic domain wall stabilized by achiral energy, two degenerate chirality forms exist which can be regarded as the two qubit states. Our numerical demonstration shows that the manipulation of spin configurations of the ferromagnetic domain walls is governed by magnetic and electric fields for single-qubit quantum gates, while the Ising exchange coupling facilitates the two-qubit gates. The incorporation of these quantum gates permits universal quantum computation. Furthermore, we discuss the estimation of the coherence time, as well as the initialization and readout of the qubits. Our findings show a practical implementation of quantum computing architectures based on the domain-wall qubits in ferromagnetic materials.
Controllable adiabatic evolution of a multi-qubit system can be used for adiabatic quantum computation (AQC). This evolution ends at a configuration where the Hamiltonian of the system encodes the solution of the problem to be solved. As a first steps towards realization of AQC we have investigated two, three and four flux qubit systems. These systems were characterized by making use of a radio-frequency method. We designed two-qubit systems with coupling energies up to several kelvins. For the three-flux-qubit systems we determined the complete ground-state flux diagram in the three dimensional flux space around the qubits common degeneracy point. We show that the system's Hamiltonian can be completely reconstructed from our measurements. Our concept for the implementation of AQC, by making use of flux qubits, is discussed.
The problem 2-LOCAL HAMILTONIAN has been shown to be complete for the quantum computational class QMA. In this paper we show that this important problem remains QMA-complete when the interactions of the 2-local Hamiltonian are between qubits on a two-dimensional (2-D) square lattice. Our results are partially derived with novel perturbation gadgets that employ mediator qubits which allow us to manipulate k-local interactions. As a side result, we obtain that quantum adiabatic computation using 2-local interactions restricted to a 2-D square lattice is equivalent to the circuit model of quantum computation. Our perturbation method also shows how any stabilizer space associated with a k-local stabilizer (for constant k) can be generated as an approximate ground-space of a 2-local Hamiltonian.
A computational method called the local-field response method is proposed, where spins evolve by responding to an effective field consisting of gradually decreasing external fields and spin–spin interactions, similarly to what is carried out in adiabatic quantum computing (AQC). This method is partly quantum–mechanical. That is, spins are treated as classical variables, but the response function of the spins to the effective field is determined a priori by referring to a quantum–mechanical calculation that was carried out for similar problems. This novel response function improves the performance of the ground state being maintained in the time evolution compared with the case without a priori information. The performance is numerically checked in an eight-qubit system by solving random-interaction problems of finding their ground states. The false probability decreases by about half as a result of using a priori information. The operation of this method is classical, but it has a quantum–mechanical advantage through a priori information. This method is practically useful because obtaining a complete quantum system is difficult as it stands.