Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States

A long-standing open question about Gaussian continuous-variable cluster states is whether they enable fault-tolerant measurement-based quantum computation. The answer is yes. Initial squeezing in the cluster above a threshold value of 20.5 dB ensures that errors from finite squeezing acting on encoded qubits are below the fault-tolerance threshold of known qubit-based error-correcting codes. By concatenating with one of these codes and using ancilla-based error correction, fault-tolerant measurement-based quantum computation of theoretically indefinite length is possible with finitely squeezed cluster states.Comment: (v3) consistent with published version, more accessible for general audience; (v2) condensed presentation, added references on GKP state generation and a comparison of currently achievable squeezing to the threshold; (v1) 13 pages, a few figure

Schroedinger operators involving singular potentials and measure data

We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data $$\left\{ \begin{alignedat}{2} -\Delta u + Vu & = \mu && \quad \text{in } \Omega,\\ u & = 0 && \quad \text{on } \partial \Omega. \end{alignedat} \right.$$ We characterize the finite measures $\mu$ for which this problem has a solution for every nonnegative potential $V$ in the Lebesgue space $L^p(\Omega)$ with $1 \le p \le N/2$. The full answer can be expressed in terms of the $W^{2,p}$ capacity for $p > 1$, and the $W^{1,2}$ (or Newtonian) capacity for $p = 1$. We then prove the existence of a solution of the problem above when $V$ belongs to the real Hardy space $H^1(\Omega)$ and $\mu$ is diffuse with respect to the $W^{2,1}$ capacity.Comment: Fixed a display problem in arxiv's abstract. Original tex file unchange

Passive interferometric symmetries of multimode Gaussian pure states

As large-scale multimode Gaussian states begin to become accessible in the laboratory, their representation and analysis become a useful topic of research in their own right. The graphical calculus for Gaussian pure states provides powerful tools for their representation, while this work presents a useful tool for their analysis: passive interferometric (i.e., number-conserving) symmetries. Here we show that these symmetries of multimode Gaussian states simplify calculations in measurement-based quantum computing and provide constructive tools for engineering large-scale harmonic systems with specific physical properties, and we provide a general mathematical framework for deriving them. Such symmetries are generated by linear combinations of operators expressed in the Schwinger representation of U(2), called nullifiers because the Gaussian state in question is a zero eigenstate of them. This general framework is shown to have applications in the noise analysis of continuous-various cluster states and is expected to have additional applications in future work with large-scale multimode Gaussian states.Comment: v3: shorter, included additional applications, 11 pages, 7 figures. v2: minor content revisions, additional figures and explanation, 23 pages, 18 figures. v1: 22 pages, 16 figure

Anti-de Sitter branes with Neveu-Schwarz and Ramond-Ramond backgrounds

We review some facts about AdS2xS2 branes in AdS3xS3 with a Neveu-Schwarz background, and consider the case of Ramond-Ramond backgrounds. We compute the spectrum of quadratic fluctuations in the low-energy approximation and discuss the open-string geometry.Comment: 8 pages, uses JHEP3.cl

Sound clocks and sonic relativity

Sound propagation within certain non-relativistic condensed matter models obeys a relativistic wave equation despite such systems admitting entirely non-relativistic descriptions. A natural question that arises upon consideration of this is, "do devices exist that will experience the relativity in these systems?" We describe a thought experiment in which 'acoustic observers' possess devices called sound clocks that can be connected to form chains. Careful investigation shows that appropriately constructed chains of stationary and moving sound clocks are perceived by observers on the other chain as undergoing the relativistic phenomena of length contraction and time dilation by the Lorentz factor, with c the speed of sound. Sound clocks within moving chains actually tick less frequently than stationary ones and must be separated by a shorter distance than when stationary to satisfy simultaneity conditions. Stationary sound clocks appear to be length contracted and time dilated to moving observers due to their misunderstanding of their own state of motion with respect to the laboratory. Observers restricted to using sound clocks describe a universe kinematically consistent with the theory of special relativity, despite the preferred frame of their universe in the laboratory. Such devices show promise in further probing analogue relativity models, for example in investigating phenomena that require careful consideration of the proper time elapsed for observers.Comment: (v2) consistent with published version; (v1) 15 pages, 9 figure

Temporal-mode continuous-variable cluster states using linear optics

I present an extensible experimental design for optical continuous-variable cluster states of arbitrary size using four offline (vacuum) squeezers and six beamsplitters. This method has all the advantages of a temporal-mode encoding [Phys. Rev. Lett. 104, 250503], including finite requirements for coherence and stability even as the computation length increases indefinitely, with none of the difficulty of inline squeezing. The extensibility stems from a construction based on Gaussian projected entangled pair states (GPEPS). The potential for use of this design within a fully fault tolerant model is discussed.Comment: 9 pages, 19 color figure

Flexible quantum circuits using scalable continuous-variable cluster states

We show that measurement-based quantum computation on scalable continuous-variable (CV) cluster states admits more quantum-circuit flexibility and compactness than similar protocols for standard square-lattice CV cluster states. This advantage is a direct result of the macronode structure of these states---that is, a lattice structure in which each graph node actually consists of several physical modes. These extra modes provide additional measurement degrees of freedom at each graph location, which can be used to manipulate the flow and processing of quantum information more robustly and with additional flexibility that is not available on an ordinary lattice.Comment: (v2) consistent with published version; (v1) 11 pages (9 figures