Hamiltonian Simulation by Qubitization

We present the problem of approximating the time-evolution operator $e^{-i\hat{H}t}$ to error $\epsilon$, where the Hamiltonian $\hat{H}=(\langle G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}})$ is the projection of a unitary oracle $\hat{U}$ onto the state $|G\rangle$ created by another unitary oracle. Our algorithm solves this with a query complexity $\mathcal{O}\big(t+\log({1/\epsilon})\big)$ to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are $d$-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where $\hat{H}$ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large class of operator functions of $\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\hat{H}t}$ is a special case.Comment: 23 pages, 1 figure; v2: updated notation; v3: accepted versio

myCopter: Enabling Technologies for Personal Aerial Transportation Systems: Project status after 2.5 years

Current means of transportation for daily commuting are reaching their limits during peak travel times, which results in waste of fuel and loss of time and money. A recent study commissioned by the European Union considers a personal aerial transportation system (PATS) as a viable alternative for transportation to and from work. It also acknowledges that developing such a transportation system should not focus on designing a new flying vehicle for personal use, but instead on investigating issues surrounding the implementation of the transportation system itself. This is the aim of European project myCopter: to determine the social and technological aspects needed to set up a transportation system based on personal aerial vehicles (PAVs). The project focuses on three research areas: human-machine interfaces and training, automation technologies, and social acceptance. Our extended abstract for inclusion in the conference proceedings and our presentation will focus on the achievements during the first 2.5 years of the 4-year project. These include the development of an augmented dynamic model of a PAV with excellent handling qualities that are suitable for training purposes. The training requirements for novice pilots are currently under development. Experimental evaluations on haptic guidance and human-in-the-loop control tasks have allowed us to start implementing a haptic Highway-in-the-Sky display to support novice pilots and to investigate metrics for objectively determining workload using psychophysiological measurements. Within the project, developments for automation technologies have focused on vision-based algorithms. We have integrated such algorithms in the control and navigation architecture of unmanned aerial vehicles (UAVs). Detecting suitable landing spots from monocular camera images recorded in flight has proven to reliably work off-line, but further work is required to be able to use this approach in real time. Furthermore, we have built multiple low-cost UAVs and equipped them with radar sensors to test collision avoidance strategies in real flight. Such algorithms are currently under development and will take inspiration from crowd simulations. Finally, using technology assessment methodologies, we have assessed potential markets for PAVs and challenges for its integration into the current transportation system. This will lead to structured discussions on expectations and requirements of potential PAV users

Controlling qubit transitions during non-adiabatic rapid passage through quantum interference

In adiabatic rapid passage, the Bloch vector of a qubit is inverted by slowly inverting an external field to which it is coupled, and along which it is initially aligned. In non-adiabatic twisted rapid passage, the external field is allowed to twist around its initial direction with azimuthal angle \phi(t) at the same time that it is non-adiabatically inverted. For polynomial twist, \phi(t) \sim Bt^{n}. We show that for n \ge 3, multiple qubit resonances can occur during a single inversion of the external field, producing strong interference effects in the qubit transition probability. The character of the interference is controllable through variation of the twist strength B. Constructive and destructive interference are possible, greatly enhancing or suppressing qubit transitions. Experimental confirmation of these controllable interference effects has already occurred. Application of this interference mechanism to the construction of fast fault-tolerant quantum CNOT and NOT gates is discussed.Comment: 8 pages, 7 figures, 2 tables; submitted to J. Mod. Op

Deutsch-Jozsa algorithm as a test of quantum computation

A redundancy in the existing Deutsch-Jozsa quantum algorithm is removed and a refined algorithm, which reduces the size of the register and simplifies the function evaluation, is proposed. The refined version allows a simpler analysis of the use of entanglement between the qubits in the algorithm and provides criteria for deciding when the Deutsch-Jozsa algorithm constitutes a meaningful test of quantum computation.Comment: 10 pages, 2 figures, RevTex, Approved for publication in Phys Rev

Quantum Computers, Factoring, and Decoherence

In a quantum computer any superposition of inputs evolves unitarily into the corresponding superposition of outputs. It has been recently demonstrated that such computers can dramatically speed up the task of finding factors of large numbers -- a problem of great practical significance because of its cryptographic applications. Instead of the nearly exponential ($\sim \exp L^{1/3}$, for a number with $L$ digits) time required by the fastest classical algorithm, the quantum algorithm gives factors in a time polynomial in $L$ ($\sim L^2$). This enormous speed-up is possible in principle because quantum computation can simultaneously follow all of the paths corresponding to the distinct classical inputs, obtaining the solution as a result of coherent quantum interference between the alternatives. Hence, a quantum computer is sophisticated interference device, and it is essential for its quantum state to remain coherent in the course of the operation. In this report we investigate the effect of decoherence on the quantum factorization algorithm and establish an upper bound on a ``quantum factorizable'' $L$ based on the decoherence suffered per operational step.Comment: 7 pages,LaTex + 2 postcript figures in a uuencoded fil