Spectral implementation of some quantum algorithms by one- and two-dimensional nuclear magnetic resonance

Quantum information processing has been effectively demonstrated on a small number of qubits by nuclear magnetic resonance. An important subroutine in any computing is the readout of the output. ``Spectral implementation'' originally suggested by Z.L. Madi, R. Bruschweiler and R.R. Ernst, [J. Chem. Phys. 109, 10603 (1999)], provides an elegant method of readout with the use of an extra `observer' qubit. At the end of computation, detection of the observer qubit provides the output via the multiplet structure of its spectrum. In "spectral implementation" by two-dimensional experiment the observer qubit retains the memory of input state during computation, thereby providing correlated information on input and output, in the same spectrum. "Spectral implementation" of Grover's search algorithm, approximate quantum counting, a modified version of Berstein-Vazirani problem, and Hogg's algorithm is demonstrated here in three and four-qubit systems.Comment: 39 pages,11 figure

Boson Sampling from Gaussian States

We pose a generalized Boson Sampling problem. Strong evidence exists that such a problem becomes intractable on a classical computer as a function of the number of Bosons. We describe a quantum optical processor that can solve this problem efficiently based on Gaussian input states, a linear optical network and non-adaptive photon counting measurements. All the elements required to build such a processor currently exist. The demonstration of such a device would provide the first empirical evidence that quantum computers can indeed outperform classical computers and could lead to applications

Non-Gaussian states for continuous variable quantum computation via Gaussian maps

We investigate non-Gaussian states of light as ancillary inputs for generating nonlinear transformations required for quantum computing with continuous variables. We consider a recent proposal for preparing a cubic phase state, find the exact form of the prepared state and perform a detailed comparison to the ideal cubic phase state. We thereby identify the main challenges to preparing an ideal cubic phase state and describe the gates implemented with the non-ideal prepared state. We also find the general form of operations that can be implemented with ancilla Fock states, together with Gaussian input states, linear optics and squeezing transformations, and homodyne detection with feed forward, and discuss the feasibility of continuous variable quantum computing using ancilla Fock states.Comment: 8 pages, 6 figure

On the Complexity of Random Quantum Computations and the Jones Polynomial

There is a natural relationship between Jones polynomials and quantum computation. We use this relationship to show that the complexity of evaluating relative-error approximations of Jones polynomials can be used to bound the classical complexity of approximately simulating random quantum computations. We prove that random quantum computations cannot be classically simulated up to a constant total variation distance, under the assumption that (1) the Polynomial Hierarchy does not collapse and (2) the average-case complexity of relative-error approximations of the Jones polynomial matches the worst-case complexity over a constant fraction of random links. Our results provide a straightforward relationship between the approximation of Jones polynomials and the complexity of random quantum computations.Comment: 8 pages, 4 figure

On Quantum Algorithms

Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. In effect, they follow the same logical paradigm as (multi-particle) interferometers. We show how most known quantum algorithms, including quantum algorithms for factorising and counting, may be cast in this manner. Quantum searching is described as inducing a desired relative phase between two eigenvectors to yield constructive interference on the sought elements and destructive interference on the remaining terms.Comment: 15 pages, 8 figure