An efficient high dimensional quantum Schur transform

The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an $n$ fold tensor product $V^{\otimes n}$ of a vector space $V$ of dimension $d$. Bacon, Chuang and Harrow \cite{BCH07} gave a quantum algorithm for this transform that is polynomial in $n$, $d$ and $\log\epsilon^{-1}$, where $\epsilon$ is the precision. In a footnote in Harrow's thesis \cite{H05}, a brief description of how to make the algorithm of \cite{BCH07} polynomial in $\log d$ is given using the unitary group representation theory (however, this has not been explained in detail anywhere. In this article, we present a quantum algorithm for the Schur transform that is polynomial in $n$, $\log d$ and $\log\epsilon^{-1}$ using a different approach. Specifically, we build this transform using the representation theory of the symmetric group and in this sense our technique can be considered a "dual" algorithm to \cite{BCH07}. A novel feature of our algorithm is that we construct the quantum Fourier transform over the so called \emph{permutation modules}, which could have other applications.Comment: 21 page

Symmetry in quantum walks

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. We derive an expression for hitting time using superoperators, and numerically evaluate it for the walk on the hypercube for various coins and decoherence models. We show that, by contrast to classical walks, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition for their existence. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Symmetries of a graph, given by its automorphism group, can be inherited by the evolution operator. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. Symmetry can also cause the walk to be confined to a subspace of the original Hilbert space for certain initial states. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph and we give an explicit construction of the quotient graph. We conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up. Finally, we use symmetry and the theory of decoherence-free subspaces to determine when the subspace of the quotient graph is a decoherence-free subspace of the dynamics.Comment: 136 pages, Ph.D. thesis, University of Southern California, 200

Anderson localization casts clouds over adiabatic quantum optimization

Understanding NP-complete problems is a central topic in computer science. This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps between the ground and excited states of the quantum computer's Hamiltonian. We show that the statistics of the gaps can be analyzed in a novel way, borrowed from the study of quantum disordered systems in statistical mechanics. It turns out that due to a phenomenon similar to Anderson localization, exponentially small gaps appear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. This implies that unfortunately, adiabatic quantum optimization fails: the system gets trapped in one of the numerous local minima.Comment: 14 pages, 4 figure

Continuous-variable entanglement distillation over a pure loss channel with multiple quantum scissors

Entanglement distillation is a key primitive for distributing high-quality entanglement between remote locations. Probabilistic noiseless linear amplification based on the quantum scissors is a candidate for entanglement distillation from noisy continuous-variable (CV) entangled states. Being a non-Gaussian operation, quantum scissors is challenging to analyze. We present a derivation of the non-Gaussian state heralded by multiple quantum scissors in a pure loss channel with two-mode squeezed vacuum input. We choose the reverse coherent information (RCI)---a proven lower bound on the distillable entanglement of a quantum state under one-way local operations and classical communication (LOCC), as our figure of merit. We evaluate a Gaussian lower bound on the RCI of the heralded state. We show that it can exceed the unlimited two-way LOCCassisted direct transmission entanglement distillation capacity of the pure loss channel. The optimal heralded Gaussian RCI with two quantum scissors is found to be significantly more than that with a single quantum scissors, albeit at the cost of decreased success probability. Our results fortify the possibility of a quantum repeater scheme for CV quantum states using the quantum scissors.Comment: accepted for publication in Physical Review

Coherent Communication with Continuous Quantum Variables

The coherent bit (cobit) channel is a resource intermediate between classical and quantum communication. It produces coherent versions of teleportation and superdense coding. We extend the cobit channel to continuous variables by providing a definition of the coherent nat (conat) channel. We construct several coherent protocols that use both a position-quadrature and a momentum-quadrature conat channel with finite squeezing. Finally, we show that the quality of squeezing diminishes through successive compositions of coherent teleportation and superdense coding.Comment: 4 pages, 3 figure