QIC Abstracts

 Vol.16 No.9&10, July 1, 2016

Research Articles:

Near-linear constructions of exact unitary 2-designs (pp0721-0756)
Richard Cleve, Debbie Leung, Li Liu, and Chunhao Wang
A unitary 2-design can be viewed as a quantum analogue of a 2-universal hash function: it is indistinguishable from a truly random unitary by any procedure that queries it twice. We show that exact unitary 2-designs on $n$ qubits can be implemented by quantum circuits consisting of $\widetilde{O}(n)$ elementary gates in logarithmic depth. This is essentially a quadratic improvement in size (and in width times depth) over all previous implementations that are exact or approximate (for sufficiently strong approximations).

The Unruh effect interpreted as a quantum noise channel (pp0757-0770)
S. Omkar, R. Srikanth, Subhashish Banerjee, and Ashutosh Kumar Alok
We make use of the tools of quantum information theory to shed light on the Unruh effect. A modal qubit appears as if subjected to quantum noise that degrades quantum information, as observed in the accelerated reference frame. The Unruh effect experienced by a mode of a free Dirac field, as seen by a relativistically accelerated observer, is treated as a noise channel, which we term the Unruh channel. We characterize this channel by providing its operator-sum representation, and study various facets of quantum correlations, such as, Bell inequality violations, entanglement, teleportation and measurement-induced decoherence under the effect. We compare and contrast this channel from conventional noise due to environmental decoherence. We show that the Unruh effect produces an amplitude-damping-like channel, associated with zero temperature, even though the Unruh effect is associated with a non-zero temperature. Asymptotically, the Bloch sphere subjected to the channel does not converge to a point, as would be expected by fluctuation-dissipation arguments, but contracts by a finite factor. We construct for the Unruh effect the inverse channel, a non-completely-positive map, that formally reverses the effect, and offer some physical interpretation.

Classification of transversal gates in qubit stabilizer codes (pp0771-0802)
Jonas T. Anderson and Tomas Jochym-O'Connor
This work classifies the set of diagonal gates that can implement a single or two-qubit transversal logical gate for qubit stabilizer codes. We show that individual physical diagonal gates on the underlying qubits that compose the code are restricted to have entries of the form~$e^{i \pi c/2^k}$ along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner. As such, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy. Moreover, we show that for a given stabilizer code, the two-qubit diagonal transversal gates must belong to the same level of Clifford hierarchy as the single-qubit diagonal transversal gates available for the given code. We use this result to prove a conjecture about arbitrary transversal gates made by Zeng et al in 2007.

Phase estimation using an approximate eigenstate (pp0803-0812)
Avatar Tulsi
A basic building block of many quantum algorithms is the Phase Estimation algorithm (PEA). It finds an eigenphase $\phi$ of a unitary operator using a copy of the corresponding eigenstate $|\phi\rangle$. Suppose, in place of $|\phi\rangle$, we have a copy of an approximate eigenstate $|\psi\rangle$ whose component in $|\phi\rangle$ is at least $\sqrt{2/3}$. Then the PEA fails with a constant probability. Using multiple copies of $|\psi\rangle$, this probability can be made to decrease exponentially with the number of copies. Here we show that a single copy is sufficient to find $\phi$ if we can selectively invert the $|\psi\rangle$ state. As an application, we consider the eigenpath traversal problem (ETP) where the goal is to travel a path of non-degenerate eigenstates of $n$ different operators. The fastest algorithm for ETP is due to Boixo, Knill and Somma (BKS) which needs $\Theta(\ln n)$ copies of the eigenstates. Using our method, the BKS algorithm can work with just a single copy but its running time $\mathcal{Q}$ increases to $O(\mathcal{Q}\ln^{2}\mathcal{Q})$. This tradeoff is beneficial if the spatial resources are more constrained than the temporal resources.

An improved asymptotic key rate bound for a mediated semi-quantum key distribution protocol (pp0813-0834)
Walter O. Krawec
Semi-quantum key distribution (SQKD) protocols allow for the establishment of a secret key between two users Alice and Bob, when one of the two users (typically Bob) is limited or ``classical'' in nature. Recently it was shown that protocols exists when both parties are limited/classical in nature if they utilize the services of a quantum server. These protocols are called mediated SQKD protocols. This server, however, is untrusted and, in fact, adversarial. In this paper, we reconsider a mediated SQKD protocol and derive a new proof of unconditional security for it. In particular, we derive a new lower bound on its key rate in the asymptotic scenario. Furthermore, we show this new lower bound is an improvement over prior work, thus showing that the protocol in question can tolerate higher rates of error than previously thought.

Renormalization of quantum deficit and monogamy relation in the Heisenberg XXZ model (pp0835-0844)
Meng Qin, Xin Zhang, and Zhong-Zhou Ren
In this study, the dynamical behavior of quantum deficit and monogamy relation in the Heisenberg XXZ model is investigated by implementing quantum renormalization group theory. The results demonstrate that the quantum deficit can be used to capture the quantum phase transitions point and show scaling behavior with the spin chain size increasing. It was also found that the critical exponent has no change when varying measure from entanglement to quantum correlation. The monogamy relation is influenced by the steps of quantum renormalization group and the ways of splitting the block states. Furthermore, the monogamy relation of generalized $W$ state also is given by means of quantum deficit.

Optimal bounds on functions of quantum states under quantum channels (pp0845-0861)
Chi-Kwong Li, Diane Christine Pelejo, and Kuo-Zhong Wang
Let $\rho_1, \rho_2$ be quantum states and $(\rho_1,\rho_2) \mapsto D(\rho_1, \rho_2)$ be a scalar function such as the trace distance, the fidelity, and the relative entropy, etc. We determine optimal bounds for $D(\rho_1, \Phi(\rho_2))$ for $\Phi \in \mathcal{S}$ for different class of functions $D(\cdot, \cdot)$, where $\mathcal{S}$ is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels.

Improved quantum ternary arithmetic (pp0862-0884)
Alex Bocharov, Shawn X. Cui, Martin Roetteler, and Krysta M. Svore
Qutrit (or ternary) structures arise naturally in many quantum systems, notably in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders. The main distinction from the binary adders is a richer ternary carry which leads potentially to higher resource counts in universal ternary bases. Our ternary ripple adder circuit has a circuit depth of $O(n)$ and uses only $1$ ancilla, making it more efficient in both, circuit depth and width, when compared with previous constructions. Our ternary carry lookahead circuit has a circuit depth of only $O(\log\,n)$, while using $O(n)$ ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate $C(X): \ket{i,j}\mapsto \ket{i, j + \delta_{i,2} \; {\rm mod} \; 3}$ or from the two-qutrit Clifford gates and the non-Clifford gate $P_9=\mbox{diag}(e^{-2 \pi \, i/9},1,e^{2 \pi \, i/9})$. The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomials over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.

Phase diagrams of one-, two-, and three-dimensional quantum spin systems (pp0885-0899)
Briiissuurs Braiorr-Orrs, Michael Weyrauch, and Mykhailo V. Rakov
We study the bipartite entanglement per bond to determine characteristic features of the phase diagram of various quantum spin models in different spatial dimensions. The bipartite entanglement is obtained from a tensor network representation of the ground state wave-function. Three spin-1/2 models (Ising, XY, XXZ, all in a transverse field) are investigated. Infinite imaginary-time evolution (iTEBD in 1D, `simple update' in 2D and 3D) is used to determine the ground states of these models. The phase structure of the models is discussed for all three dimensions.

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