QIC Abstracts

 Vol.14 No.1&2, January 1, 2014

Research Articles:

Gate-efficient discrete simulations of continuous-time quantum query algorithms (pp0001-0030)
Dominic W. Berry, Richard Cleve, and Sevag Gharibian
We show how to efficiently simulate continuous-time quantum query algorithms that run in time T in a manner that preserves the query complexity (within a polylogarithmic factor) while also incurring a small overhead cost in the total number of gates between queries. By small overhead, we mean T within a factor that is polylogarithmic in terms of T and a cost measure that reflects the cost of computing the driving Hamiltonian. This permits any continuous-time quantum algorithm based on an efficiently computable driving Hamiltonian to be converted into a gate-efficient algorithm with similar running time.

Comparisons between quantum state distinguishability measures (pp0031-0038)
Koenraad M.R. Audenaert
We provide a compendium of inequalities between several quantum state distinguishability measures. For each measure these inequalities consist of the sharpest possible upper and lower bounds in terms of another measure. Some of these inequalities are already known, but new or more general proofs are given, whereas other inequalities are new. We also supply cases of equality to show that all inequalities are indeed the sharpest possible.

Effects of quantum error correction on entanglement sudden death (pp0039-0055)
Muhammed Yonac and Joseph H. Eberly
We investigate the effects of error correction on non-local quantum coherence as a function of time, extending the study by Sainz and Bjork. We consider error correction of amplitude damping, pure phase damping and combinations of amplitude and phase damping as they affect both fidelity and quantum entanglement. Initial two-qubit entanglement is encoded in arbitrary real superpositions of both $\Phi$-type and $\Psi$-type Bell states. Our main focus is on the possibility of delay or prevention of ESD (early stage decoherence, or entanglement sudden death). We obtain the onset times for ESD as a function of the state-superposition mixing angle. Error correction affects entanglement and fidelity differently, and we exhibit initial entangled states for which error correction increases fidelity but decreases entanglement, and vice versa.

Quantum algorithms for one-dimensional infrastructures (pp0056-0090)
Pradeep Sarvepalli and Pawel M. Wocjan
Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of infrastructures are the computation of the circumference of the infrastructure and the generalized discrete logarithms. Both these problems are not known to have efficient classical algorithms for an arbitrary infrastructure. Our main contributions are polynomial time quantum algorithms for one-dimensional infrastructures that satisfy certain conditions. For instance, these conditions are always fulfilled for infrastructures obtained from number fields and function fields, both of unit rank one. Since quadratic number fields give rise to such infrastructures, this algorithm can be used to solve Pell's equation and the principal ideal problem. In this sense we generalize Hallgren's quantum algorithms for quadratic number fields, while also providing a polynomial speedup over them. Our more general approach shows that these quantum algorithms can also be applied to infrastructures obtained from complex cubic and totally complex quartic number fields. Our improved way of analyzing the performance makes it possible to show that these algorithms succeed with constant probability independent of the problem size. In contrast, the lower bound on the success probability due to Hallgren decreases as the fourth power of the logarithm of the circumference. Our analysis also shows that fewer qubits are required. We also contribute to the study of infrastructures, and show how to compute efficiently within infrastructures.

Coding-based quantum private database query using entanglement (pp0091-0106)
Fang Yu and Daowen Qiu
We propose an efficient coding-based quantum protocol which uses entanglement to help a user retrieve one out of $N$ items from a database without revealing which one he/she is interested in. The query is accomplished through an encoding-decoding process with the support of a database-specific unitary operation. For each query, $O(logN)$ time is needed for communication and no more than one bit of information about the database can be learnt by even a dishonest user. We prove rigorously that the protocol is secure against a dishonest database which causes little deviation on the original system. A theorem quantitatively describes the tradeoff between information and disturbance in such a setting. Examples are given to illustrate that both coding-based quantum private database query protocols proposed so far, including ours and Olejnik's, are insecure against dishonest databases which are free to cause deviations. Compared to Olejnik's protocol, which utilizes superposed states rather than entanglement, our protocol can prevent a dishonest-but-conscientious database, which tries to provide the user right answers to his/her queries while eavesdropping, from learning significant information when it evaded his/her detection. Finally, for the purpose of enhancing user privacy, two extensions of the protocol are proposed and discussed.

Nonlocal entanglement concentration of separate nitrogen-vacancy centers coupling to microtoroidal resonators (pp0107-0121)
Chuan Wang, Yong Zhang, Ming Lei, Guang-sheng Jin, Hai-qiang Ma, and Ru Zhang
Here we propose two practical protocols to concentrate entanglement between separate nitrogen-vacancy (N-V) centers in less entangled state via coupling to microtoroidal resonators. We construct the parity check gate of the N-V center and microtoroidal resonator systems via the interaction with the ancillary photon input-output process near the microtoroidal resonators. Thus the parity of the N-V center state can be readout by the measurement on the ancillary photon. Then we introduce the parity check operations to entanglement concentration protocols. Considering current techniques, we also discuss the feasibility of our proposal and its experimental challenges.

Many-to-one remote information concentration for qudits and multipartite entanglement (pp0122-0136)
Xin-Wen Wang, Shi-Qing Tang, Li-Jun Xie, Deng-Yu Zhang, and Le-Man Kuang
Telecloning and its reverse process, referred to as remote information concentration (RIC), have attracted considerable interest because of their potential applications in quantum-information processing. We here present a general scheme for RIC in $d$-level systems (qudits), in which the quantum information initially distributed in many spatially separated qudits can be remotely and deterministically concentrated to a single qudit via an entangled channel without performing any global operations. We show that the entangled channel of RIC can be different types of entangled states, including mixed states as well as pure ones. More interestingly, these mixed states include a bound entangled state which has a similar form to the generalized Smolin state but has different characteristics from it. We also show that there exists a multipartite entangled state which can be used to implement both telecloning and RIC in the two-level system. Our many-to-one RIC protocol could be slightly modified to perform some types of many-to-many RIC tasks.

One-step implementation of quantum comtrolled-phase gate via quantum Zeno dynamics (pp0137-0143)
Wen-An Li and Lian-Fu Wei
We propose a scheme to implement a quantum controllable-phase gate via quantum Zeno dynamics. The two qubits are asymmetrically encoded by two four-level atoms coupled via a quantized cavity mode. Under proper conditions, the desirable logic operation can be implemented in one step. Since the qubit is encoded by the ground and the metastable states of the atom and the cavity mode is not really excited, our protocol is robust against the spontaneous decays of the atoms and cavity. Specifically, the feasibility of our generic proposal is demonstrated with two nitrogen-vacancy centers coupled to whispering-gallery microresonator.

Quantum Systems on Non-$k$-Hyperfinite Complexes: a generalization of classical statistical mechanics on expander graphs (pp0144-0180)
Michael H. Freedman and Matthew B. Hastings
We construct families of cell complexes that generalize expander graphs. These families are called non-$k$-hyperfinite, generalizing the idea of a non-hyperfinite (NH) family of graphs. Roughly speaking, such a complex has the property that one cannot remove a small fraction of points and be left with an object that looks $k-1$-dimensional at large scales. We then consider certain quantum systems on these complexes. A future goal is to construct a family of Hamiltonians such that every low energy state has topological order as part of an attempt to prove the quantum PCP conjecture. This goal is approached by constructing a toric code Hamiltonian with the property that every low energy state without vertex defects has topological order, a property that would not hold for any local system in any lattice $Z^d$ or indeed on any $1$-hyperfinite complex. Further, such NH complexes find application in quantum coding theory. The hypergraph product codes\cite{hpc} of Tillich and Z\'{e}mor are generalized using NH complexes.

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