QIC Abstracts

 Vol.16 No.15&16, November 1, 2016

Research Articles:

Noise thresholds for the [4,2,2]-concatenated toric code (pp1261-1281)
Ben Criger and Barbara Terhal
We analyze the properties of a 2D topological code derived by concatenating the \cfour code with the toric/surface code, or alternatively by removing check operators from the 2D square-octagon or 4.8.8 color code. We show that the resulting code has a circuit-based noise threshold of $\sim 0.41\%$ (compared to $\sim 0.6\%$ for the toric code in a similar scenario), which is higher than any known 2D color code. We believe that the construction may be of interest for hardware in which one wants to use both long-range two-qubit gates as well as short-range gates between small clusters of qubits.

A note on coherence power of n-dimensional unitary operators (pp1282-1294)
Maria Garcia-Diaz, Dario Egloff, and Martin B. Plenio
The coherence power of a quantum channel, that is, its maximum ability to increase the coherence of input states, is a fundamental concept within the framework of the resource theory of coherence. In this note we discuss various possible definitions of coherence power and coherence rate and their basic properties. Then we prove that the coherence power of a unitary operator acting on a qubit, computed with respect to the $l_1$-coherence measure, can be calculated by maximizing its coherence gain over pure incoherent states. We proceed to show that this result fails in the general case, that is, the maximal coherence gain is found when acting on a state with non-vanishing coherence in the case of the l1-coherence and dimension $N>2$, the relative entropy of coherence and the geometric measure of coherence.

Corrected quantum walk for optimal Hamiltonian simulation (pp1295-1317)
Dominic Berry and Leonardo Novo
We describe a method to simulate Hamiltonian evolution on a quantum computer by repeatedly using a superposition of steps of a quantum walk, then applying a correction to the weightings for the numbers of steps of the quantum walk. This correction enables us to obtain complexity which is the same as the lower bound up to double-logarithmic factors for all parameter regimes. The scaling of the query complexity is $O\left( \tau \frac{\log\log\tau}{\log\log\log\tau} + \log(1/\epsilon) \right)$ where $\tau := t\|H\|_{\max}d$, for $\epsilon$ the allowable error, $t$ the time, $\|H\|_{\max}$ the max-norm of the Hamiltonian, and $d$ the sparseness. This technique should also be useful for improving the scaling of the Taylor series approach to simulation, which is relevant to applications such as quantum chemistry.

Quantum Latin squares and unitary error bases (pp1318-1332)
Benjamin Musto and Jamie Vicary
In this paper we introduce \textit{quantum Latin squares}, combinatorial quantum objects which generalize classical Latin squares, and investigate their applications in quantum computer science. Our main results are on applications to \textit{unitary error bases} (UEBs), basic structures in quantum information which lie at the heart of procedures such as teleportation, dense coding and error correction. We present a new method for constructing a UEB from a quantum Latin square equipped with extra data. Developing construction techniques for UEBs has been a major activity in quantum computation, with three primary methods proposed: \textit{shift-and-multiply}, \textit{Hadamard}, and \textit{group-theoretic}. We show that our new approach simultaneously generalizes the shift-and-multiply and Hadamard methods. Furthermore, we explicitly construct a UEB using our technique which we prove cannot be obtained from any of these existing methods.

Generalized Dicke states (pp1333-1348)
Stephan Hartmann
Quantum master equations are an important tool in quantum optics and quantum information theory. For systems comprising a small to medium number of atoms (or qubits), the non-truncated equations are usually solved numerically. In this paper, we present a group-theoretical superoperator method that helps solving these equations. To do so, we exploit the $SU(4)$-symmetry of the respective Lindblad operator and construct basis states that generalize the well-known Dicke states. This allows us to solve various problems analytically and to considerably reduce the complexity of problems that can only be solved numerically. Finally, we present three examples that illustrate the proposed method.

Extension of remotely creatable region via local unitary transformation on receiver side (pp1349-1364)
G. Bochkin and A. Zenchuk
We show that the length of the effective remote state creation via the homogeneous spin-1/2 chain can be increased more than three times using the local unitary transformation of the so-called extended receiver (i.e., receiver joined with the nearest node(s)). This transformation is most effective in the models with all-node interactions. We consider an example of communication lines with the two-qubit sender, one-qubit receiver and two-qubit extended receiver.

Stability of two interacting entangled spins interacting with a thermal environment (pp1365-1378)
S. Dehdashti, M. B. Harouni, Z. Harsij, J. Shen, H. Wang, Z. Xu, B. Mirza, and H. Chen
We study the entanglement dynamics of two entangled spins coupled with a common environment consisting of a large number of harmonic oscillators. Specifically, we study the impacts of both interaction and temperature of the environment on the dynamic quantum correlation, namely, entanglement and quantum discord of two spins via concurrence and global quantum discord criteria. In the present system, we show that the interaction between the spin sub-systems and the common environment causes environmental states to approach a composition of even and odd coherent states, which have different phases, and which are entangled with the spin states. Moreover, using the thermofield approach, we demonstrate quantum correlation stabilization as a result of increasing environmental interaction as well as increasing temperature.

The Clifford group forms a unitary 3-design (pp1379-1400)
Zak Webb
Unitary $k$-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery. We also show that the Clifford group does not form a 4-design, thus characterizing how well random Clifford elements approximate Haar-random unitaries. Additionally, we show that the generalized Clifford group for qudits is not a 3-design unless the dimension of the qudit is a power of 2.

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