Efficient quantum walk on a quantum processor

Abstract

The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise, quantum walks have shown much potential as a framework for developing new quantum algorithms. Here we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to sample from the output probability distributions of quantum walks on circulant graphs efficiently. We also show that solving the same sampling problem for arbitrary circulant quantum circuits is intractable for a classical computer, assuming conjectures from computational complexity theory. This is a new link between continuous-time quantum walks and computational complexity theory and it indicates a family of tasks that could ultimately demonstrate quantum supremacy over classical computers. As a proof of principle, we experimentally implement the proposed quantum circuit on an example circulant graph using a two-qubit photonics quantum processor.

Quantum Computation using Arrays of N Polar Molecules in Pendular States

“…For instance,r ecently it was shown by Qiang et al [62] that one could gain an exponentials peedup in simulating continuous time quantum walk on circulant graphs( graphs whose adjacency matrices satisfy the property where the row j + 1c an be obtained by rotatingr ow j by one element) compared with the best classical algorithm. It is known that any Hamiltonian H for ac ontinuous time quantumw alko na ny circulant graph can be diagonalized by the unitary Fourier Transform: [63] H ¼ Q y LQ where L is diagonal.…”

“…The evolution under H then becomes e ÀiHt ¼ Q y DQ where D ¼ e ÀiLt simulates ad iagonal Hamiltonian. (17)]: [62] p 00ÁÁÁ0 ¼jh00 ÁÁÁ0jQe ÀiLt Q y j00 ÁÁÁ0ij 2 ¼jh00 ÁÁÁ0jH n e ÀiLt H n j00 ÁÁÁ0ij 2 [60,61] Hence aq uantum walk on ac irculant graph of 2 n nodes is efficiently realizable in the gate model with n qubits.M ore interesting is the prospecto fr eplacing the n-qubitq uantum Fouriert ransform Q with easierto-implement Hadamard gates H n when it comes to computing the probability of obtaining j00 ÁÁÁ0i in the final state [Eq.…”

“…[62] for the difficulty of computing p 00ÁÁÁ0 on ac lassical computer is also based on the notion of instantaneous quantumpolynomial time (IQP). [62] for the difficulty of computing p 00ÁÁÁ0 on ac lassical computer is also based on the notion of instantaneous quantumpolynomial time (IQP).…”

“…[60,61] Hence aq uantum walk on ac irculant graph of 2 n nodes is efficiently realizable in the gate model with n qubits.M ore interesting is the prospecto fr eplacing the n-qubitq uantum Fouriert ransform Q with easierto-implement Hadamard gates H n when it comes to computing the probability of obtaining j00 ÁÁÁ0i in the final state [Eq. (17)]: [62] p 00ÁÁÁ0 ¼jh00 ÁÁÁ0jQe ÀiLt Q y j00 ÁÁÁ0ij 2 ¼jh00 ÁÁÁ0jH n e ÀiLt H n j00 ÁÁÁ0ij 2…”

Reconfigurable optical implementation of quantum complex networks

“…We want also to stress here that although the efficient implementation of quantum networks has been achieved in other platforms, e.g. superconducting qubit [36], atoms in optical lattices [37] and single photons [38], the topologies which have been explored are mainly regular ones (lattice, circulant graphs, triangular graphs) and they mainly concern logical encoding rather than physical interactions between the quantum systems. On the contrary, our platform allows for: the deterministic implementation of the network, as the mapping is based on continuous variables, and the implementation of arbitrary complex topology within the limits of the experimentally achievable size.…”