Tetsuo Ohmi

Quantum computing : from linear algebra to physical realizations

From linear algebra to quantum computing Basics of Vectors and Matrices Vector Spaces Linear Dependence and Independence of Vectors Dual Vector Spaces Basis, Projection Operator, and Completeness Relation Linear Operators and Matrices Eigenvalue Problems Pauli Matrices Spectral Decomposition Singular Value Decomposition (SVD) Tensor Product (Kronecker Product) Framework of Quantum Mechanics Fundamental Postulates Some Examples Multipartite System, Tensor Product, and Entangled State Mixed States and Density Matrices Qubits and Quantum Key Distribution Qubits Quantum Key Distribution (BB84 Protocol) Quantum Gates, Quantum Circuit, and Quantum Computer Introduction Quantum Gates Correspondence with Classical Logic Gates No-Cloning Theorem Dense Coding and Quantum Teleportation Universal Quantum Gates Quantum Parallelism and Entanglement Simple Quantum Algorithms Deutsch Algorithm Deutsch-Jozsa Algorithm and Bernstein-Vazirani Algorithm Simons Algorithm Quantum Integral Transforms Quantum Integral Transforms Quantum Fourier Transform (QFT) Application of QFT: Period-Finding Implementation of QFT Walsh-Hadamard Transform Selective Phase Rotation Transform Grovers Search Algorithm Searching for a Single File Searching for d Files Shors Factorization Algorithm The RSA Cryptosystem Factorization Algorithm Quantum Part of Shors Algorithm Probability Distribution Continued Fractions and Order-Finding Modular Exponential Function Decoherence Open Quantum System Measurements as Quantum Operations Examples Lindblad Equation Quantum Error-Correcting Codes (QECC) Introduction 3-Qubit Bit-Flip Code and Phase-Flip Code Shors 9-Qubit Code Calderbank-Shor-Steane (CSS) 7-Qubit QECC DiVincenzo-Shor 5-Qubit QECC Physical realizations of quantum computing DiVincenzo Criteria Introduction DiVincenzo Criteria Physical Realizations Beyond DiVincenzo Criteria NMR Quantum Computer Introduction NMR Spectrometer Hamiltonian Implementation of Gates and Algorithms Time-Optimal Control of NMR Quantum Computer Measurements Preparation of Pseudopure State DiVincenzo Criteria Trapped Ions Introduction Electronic States of Ion as Qubit Ions in Paul Trap Ion Qubit Quantum Gates Readout DiVincenzo Criteria Quantum Computing with Neutral Atoms Introduction Trapping Neutral Atoms 1-Qubit Gate Quantum State Engineering of Neutral Atoms Preparation of Entangled Neutral Atoms DiVincenzo Criteria Josephson Junction Qubits Introduction Nanoscale Josephson Junctions and SQUIDs Charge Qubit Flux Qubit Quantronium Current-Biased Qubit Readout Coupled Qubits DiVincenzo Criteria Quantum Computing with Quantum Dots Introduction Mesoscopic Semiconductors Electron Charge Qubit Electron Spin Qubit DiVincenzo Criteria Appendix: Solutions to Selected Exercises Index

Quantum Computing with p-Wave Superfluid Vortices

It is shown that Majorana fermions trapped in three vortices in a p -wave superfluid implement a qubit for universal quantum computing. Several similar ideas have already been proposed: Ivanov [Phys. Rev. Lett. 86 (2001) 268] and Zhang et al. [Phys. Rev. Lett. 99 (2007) 220502] have proposed schemes in which a qubit is implemented with two and four Majorana fermions, respectively, where a qubit operation is performed by exchanging the positions of Majorana fermions. The set of gates thus obtained is a discrete subset of the relevant unitary group. We propose, in this paper, a new scheme, where three Majorana fermions form a qubit. We show that continuous 1-qubit gate operations are possible by exchanging the positions of Majorana fermions complemented with dynamical phase change. Two-qubit gates are realized through the use of the coupling between Majorana fermions belonging to different qubits.

Fast control of topological vortex formation in Bose-Einstein condensates by counterdiabatic driving

Topological vortex formation has been known as the simplest method for vortex formation in BEC of alkali atoms. This scheme requires inversion of the bias magnetic field along the axis of the condensate, which leads to atom loss when the bias field crosses zero. In this Letter, we propose a scheme with which the atom loss is greatly suppressed by adding counter-diabatic magnetic field. A naive counter-diabatic field violates the Maxwell equations and we need to introduce an approximation to make it physically feasible. The resulting field requires an extra currents, which is experimentally challenging. Finally we solve this problem by applying a gauge transformation so that the counter-diabatic field is generated by controlling the original trap field with the additional control of the bias field.