It was recently shown that continuous-time quantum walks on dynamic graphs, i.e., sequences of static graphs whose edges change at specific times, can implement a universal set of quantum gates. This result treated all isolated vertices as having self-loops, so they all evolved by a phase under the quantum walk. In this paper, we permit isolated vertices to be loopless or looped, and loopless isolated vertices do not evolve at all under the quantum walk. Using this distinction, we construct simpler dynamic graphs that implement the Pauli gates and a set of universal quantum gates consisting of the Hadamard,
T, and controlled-not gates, and these gates are easily extended to multiqubit systems. For example, the
T gate is simplified from a sequence of six graphs to a single graph, and the number of vertices is reduced by a factor of 4. We also construct a generalized phase gate, of which
Z,
S, and
T are specific instances. Finally, we validate our implementations by numerically simulating a quantum circuit consisting of layers of one- and two-qubit gates, similar to those in recent quantum supremacy experiments, using a quantum walk.
Abstract We consider an continuous-time quantum walk on triple graphs with a potential well in one of the terminal sites on the main chain and study its dynamical properties in the open system by introducing a ‘spontaneous damping’ process between the terminal site and its nearest neighbour one. Calculating the stable probability of the terminal site and the time evolution of probabilities of the other sites on the main chain, we show that the system can manifest the factorization and coherence protection phenomena when the length and position of the side chain satisfy some concrete conditions. Furthermore we investigate the effect of disorder on the hopping strength and show that the existence of the disorder can destroy the coherence protection phenomena and increase the stable probability of the terminal site on the main chain.
It is well known that any quantum gate can be decomposed into the universal gate set {
T,H, cnot}, and recent results have shown that each of these gates can be implemented using a dynamic quantum walk, which is a continuous-time quantum walk on a sequence of graphs. This procedure for converting a quantum gate into a dynamic quantum walk, however, can result in long sequences of graphs. To alleviate this, in this paper, we develop a length-3 dynamic quantum walk that implements any single-qubit gate. Furthermore, we extend this result to give length-3 dynamic quantum walks that implement any single-qubit gate controlled by any number of qubits. Using these, we implement Draper's quantum addition circuit, which is based on the quantum Fourier transform, using a dynamic quantum walk.
The continuous-time quantum walk is a quantum version of a random walk that evolves by Schrodinger's equation. With the Hamiltonian equal to the adjacency matrix of a sequence of graphs, called a dynamic graph, continuous-time quantum walks have been shown to implement quantum gates, including the T gate, Hadamard gate, and the Controlled{NOT gate. Since these gates make up a universal set of quantum gates, they can implement any other arbitrary quantum gate up to an arbitrary approximation. This process, however, can be tedious and ine cient. To alleviate this, we have developed a parameterized dynamic graph on which a continuous-time quantum walk can implement any arbitrary single-qubit quantum gate, and we have also extended this result to implement any two-qubit controlled-unitary quantum gate. These dynamic graphs have at most length three. Using these, we implemented Draper's quantum addition circuit, which is based on the quantum Fourier transform, using a continuous-time quantum walk on a dynamic graph.
Quantum walk (QW) on a disordered lattice leads to a multitude of interesting phenomena, such as Anderson localization. While QW has been realized in various optical and atomic systems, its implementation with superconducting qubits still remains pending. The major challenge in simulating QW with superconducting qubits emerges from the fact that on-chip superconducting qubits cannot hop between two adjacent lattice sites. Here we overcome this barrier and develop a gate-based scheme to realize the discrete time QW by placing a pair of qubits on each site of a one-dimensional (1D) lattice and treating an excitation as a walker. It is also shown that various lattice disorders can be introduced and fully controlled by tuning the qubit parameters in our quantum walk circuit. We observe a distinct signature of transition from the ballistic regime to a localized QW with an increasing strength of disorder. Finally, an eight-qubit experiment is proposed where the signatures of such localized and delocalized regimes can be detected with existing superconducting technology. Our proposal opens up the possibility of exploring various quantum transport processes with promising superconducting qubits.