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Register a new userAdiabatic and Hamiltonian computing on a 2D lattice with simple 2-qubit interactions
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We show how to perform universal Hamiltonian and adiabatic computing using a time-independent Hamiltonian on a 2D grid describing a system of hopping particles which string together and interact to perform the computation. In this construction, the movement of one particle is controlled by the presence or absence of other particles, an effective quantum field effect transistor that allows the construction of controlled-NOT and controlled-rotation gates. The construction translates into a model for universal quantum computation with time-independent 2-qubit ZZ and XX+YY interactions on an (almost) planar grid. The effective Hamiltonian is arrived at by a single use of first-order perturbation theory avoiding the use of perturbation gadgets. The dynamics and spectral properties of the effective Hamiltonian can be fully determined as it corresponds to a particular realization of a mapping between a quantum circuit and a Hamiltonian called the space-time circuit-to-Hamiltonian construction. Because of the simple interactions required, and because no higher-order perturbation gadgets are employed, our construction is potentially realizable using superconducting or other solid-state qubits.
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Incorporating protection against quantum errors into adiabatic quantum computing (AQC) is an important task due to the inevitable presence of decoherence. Here we investigate an error-protected encoding of the AQC Hamiltonian, where qubit ensembles are used in place of qubits. Our Hamiltonian only involves total spin operators of the ensembles, offering a simpler route towards error-corrected quantum computing. Our scheme is particularly suited to neutral atomic gases where it is possible to realize large ensemble sizes and produce ensemble-ensemble entanglement. We identify a critical ensemble size $N_{mathrm{c}}$ where the nature of the first excited state becomes a single particle perturbation of the ground state, and the gap energy is predictable by mean-field theory. For ensemble sizes larger than $N_{mathrm{c}}$, the ground state becomes protected due to the presence of logically equivalent states and the AQC performance improves with $N$, as long as the decoherence rate is sufficiently low.
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 present a Hamiltonian quantum computation scheme universal for quantum computation (BQP). Our Hamiltonian is a sum of a polynomial number (in the number of gates L in the quantum circuit) of time-independent, constant-norm, 2-local qubit-qubit interaction terms. Furthermore, each qubit in the system interacts only with a constant number of other qubits. The computer runs in three steps - starts in a simple initial product-state, evolves it for time of order L^2 (up to logarithmic factors) and wraps up with a two-qubit measurement. Our model differs from the previous universal 2-local Hamiltonian constructions in that it does not use perturbation gadgets, does not need large energy penalties in the Hamiltonian and does not need to run slowly to ensure adiabatic evolution.
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In the Graph Isomorphism problem two N-vertex graphs G and G are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G into G. If yes, then G and G are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and can also determine all automorphisms of a given graph. We show how the GI problem can be converted to a combinatorial optimization problem that can be solved using adiabatic quantum evolution. We numerically simulate the algorithms quantum dynamics and show that it correctly: (i) distinguishes non-isomorphic graphs; (ii) recognizes isomorphic graphs; and (iii) finds all automorphisms of a given graph G. We then discuss the GI quantum algorithms experimental implementation, and close by showing how it can be leveraged to give a quantum algorithm that solves arbitrary instances of the NP-Complete Sub-Graph Isomorphism problem.
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