Universal fault-tolerant gates on concatenated stabilizer codes

It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of non-transversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here we demonstrate precisely the existence of such gates. In particular, we show how the limits of non-transversality can be overcome by performing rounds of intermediate error-correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching.Comment: 18 pages + 5 pages appendix, 12 figure

Symposium Review: \u3cem\u3eAmish and Old Order Mennonite Schools: A Concise History\u3c/em\u3e—Joseph Stoll; and The School by the Cornfield—Samuel Coon

Joseph Stoll, in Amish and Old Order Mennonite Schools: A Concise History, and Samuel Coon, in The School by the Cornfield, provide two very different perspectives on the struggle to establish Anabaptist schools. The books contrast primarily in their geographic and chronological scope. However, both write about parochial schools with a voice sympathetic to the vision of Amish and Mennonite school founders. They use similar sources, drawing on newspaper accounts, published Amish schools’ histories, and Amish and Mennonite periodicals, as well as personal recollections from individuals involved in school conflicts. [First paragraph.

Quantum imaging by coherent enhancement

Conventional wisdom dictates that to image the position of fluorescent atoms or molecules, one should stimulate as much emission and collect as many photons as possible. That is, in this classical case, it has always been assumed that the coherence time of the system should be made short, and that the statistical scaling $\sim1/\sqrt{t}$ defines the resolution limit for imaging time $t$. However, here we show in contrast that given the same resources, a long coherence time permits a higher resolution image. In this quantum regime, we give a procedure for determining the position of a single two-level system, and demonstrate that the standard errors of our position estimates scale at the Heisenberg limit as $\sim 1/t$, a quadratic, and notably optimal, improvement over the classical case.Comment: 4 pages, 4 figue

Quantum Inference on Bayesian Networks

Performing exact inference on Bayesian networks is known to be #P-hard. Typically approximate inference techniques are used instead to sample from the distribution on query variables given the values $e$ of evidence variables. Classically, a single unbiased sample is obtained from a Bayesian network on $n$ variables with at most $m$ parents per node in time $\mathcal{O}(nmP(e)^{-1})$, depending critically on $P(e)$, the probability the evidence might occur in the first place. By implementing a quantum version of rejection sampling, we obtain a square-root speedup, taking $\mathcal{O}(n2^mP(e)^{-\frac12})$ time per sample. We exploit the Bayesian network's graph structure to efficiently construct a quantum state, a q-sample, representing the intended classical distribution, and also to efficiently apply amplitude amplification, the source of our speedup. Thus, our speedup is notable as it is unrelativized -- we count primitive operations and require no blackbox oracle queries.Comment: 8 pages, 3 figures. Submitted to PR

Fixed-point quantum search with an optimal number of queries

Grover's quantum search and its generalization, quantum amplitude amplification, provide quadratic advantage over classical algorithms for a diverse set of tasks, but are tricky to use without knowing beforehand what fraction $\lambda$ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction, but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability, and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of $\lambda$.Comment: 4 pages plus references, 2 figure

Optimal arbitrarily accurate composite pulse sequences

Implementing a single qubit unitary is often hampered by imperfect control. Systematic amplitude errors $\epsilon$, caused by incorrect duration or strength of a pulse, are an especially common problem. But a sequence of imperfect pulses can provide a better implementation of a desired operation, as compared to a single primitive pulse. We find optimal pulse sequences consisting of $L$ primitive $\pi$ or $2\pi$ rotations that suppress such errors to arbitrary order $\mathcal{O}(\epsilon^{n})$ on arbitrary initial states. Optimality is demonstrated by proving an $L=\mathcal{O}(n)$ lower bound and saturating it with $L=2n$ solutions. Closed-form solutions for arbitrary rotation angles are given for $n=1,2,3,4$. Perturbative solutions for any $n$ are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to $n=12$. The derivation proceeds by a novel algebraic and non-recursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.Comment: 12 pages, 5 figures, submitted to Physical Review

Low-energy photoelectron transmission through aerosol overlayers

The transmission of low-energy (<1.8eV) photoelectrons through the shell of core-shell aerosol particles is studied for liquid squalane, squalene, and DEHS shells. The photoelectrons are exclusively formed in the core of the particles by two-photon ionization. The total photoelectron yield recorded as a function of shell thickness (1-80nm) shows a bi-exponential attenuation. For all substances, the damping parameter for shell thicknesses below 15nm lies between 8 and 9nm, and is tentatively assigned to the electron attenuation length at electron kinetic energies of ~0.5-1eV. The significantly larger damping parameters for thick shells (> 20nm) are presumably a consequence of distorted core-shell structures. A first comparison of aerosol and traditional thin film overlayer methods is provided