Classification of topologically protected gates for local stabilizer codes

Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault-tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological protection must be "turned off" for at least some steps in the computation in order to achieve universality. For the 3D geometry we show that an encoded gate U is implementable by a constant-depth circuit only if the image of any Pauli operator under conjugation by U belongs to the Clifford group. This class of gates includes some non-Clifford gates such as the \pi/8 rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance.Comment: 6 pages, 2 figure

Progressor: Personalized visual access to programming problems

This paper presents Progressor, a visualization of open student models intended to increase the student's motivation to progress on educational content. The system visualizes not only the user's own model, but also the peers' models. It allows sorting the peers' models using a number of criteria, including the overall progress and the progress on a specific topic. Also, in this paper we present results of a classroom study confirming our hypothesis that by showing a student the peers' models and ranking them by progress it is possible to increase the student's motivation to compete and progress in e-learning systems. © 2011 IEEE

Strong Tunneling in Double-Island Structures

We study the electron transport through a system of two low-capacitance metal islands connected in series between two electrodes. The work is motivated in part by experiments on semiconducting double-dots, which show intriguing effects arising from coherent tunneling of electrons and mixing of the single-electron states across tunneling barriers. In this article, we show how coherent tunneling affects metallic systems and leads to a mixing of the macroscopic charge states across the barriers. We apply a recently formulated RG approach to examine the linear response of the system with high tunnel conductances (up to 8e^2/h). In addition we calculate the (second order) cotunneling contributions to the non-linear conductance. Our main results are that the peaks in the linear and nonlinear conductance as a function of the gate voltage are reduced and broadened in an asymmetric way, as well as shifted in their positions. In the limit where the two islands are coupled weakly to the electrodes, we compare to theoretical results obtained by Golden and Halperin and Matveev et al. In the opposite case when the two islands are coupled more strongly to the leads than to each other, the peaks are found to shift, in qualitative agreement with the recent prediction of Andrei et al. for a similar double-dot system which exhibits a phase transition.Comment: 12 page

Tunneling resonances in quantum dots: Coulomb interaction modifies the width

Single-electron tunneling through a zero-dimensional state in an asymmetric double-barrier resonant-tunneling structure is studied. The broadening of steps in the $I$--$V$ characteristics is found to strongly depend on the polarity of the applied bias voltage. Based on a qualitative picture for the finite-life-time broadening of the quantum dot states and a quantitative comparison of the experimental data with a non-equilibrium transport theory, we identify this polarity dependence as a clear signature of Coulomb interaction.Comment: 4 pages, 4 figure

Topological insulator and the Dirac equation

We present a general description of topological insulators from the point of view of Dirac equations. The Z_{2} index for the Dirac equation is always zero, and thus the Dirac equation is topologically trivial. After the quadratic B term in momentum is introduced to correct the mass term m or the band gap of the Dirac equation, the Z_{2} index is modified as 1 for mB>0 and 0 for mB<0. For a fixed B there exists a topological quantum phase transition from a topologically trivial system to a non-trivial one system when the sign of mass m changes. A series of solutions near the boundary in the modified Dirac equation are obtained, which is characteristic of topological insulator. From the solutions of the bound states and the Z_{2} index we establish a relation between the Dirac equation and topological insulators.Comment: 9 pages, published versio

A de Finetti representation for finite symmetric quantum states

Consider a symmetric quantum state on an n-fold product space, that is, the state is invariant under permutations of the n subsystems. We show that, conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems, the state in the remaining subsystems is close to having product form. This immediately generalizes the so-called de Finetti representation to the case of finite symmetric quantum states.Comment: 22 pages, LaTe