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Physical Review A

Generalized flying-qudit scheme in arbitrary dimensions

Thomas Durt, L. C. Kwek, Dagomir Kaszlikowski

Published Apr 22, 2008• 3 citations
Publisher

Abstract

We generalize in higher dimensions the so-called ``flying-qubit scheme'' that was described in the paper by Lim, Beige, and Kwek [Phys. Rev. Lett. 95, 030505 (2005)]. In that paper, the authors proposed a scheme according to which distant atoms get entangled during a measurement, in the Bell basis, of photons (flying qubits) emitted by them. We show that although in principle a generalization of this scheme to arbitrary dimensions is possible, this theoretical proposal is not presently feasible in all dimensions because only qubit Bell states have been successively measured until now. Nevertheless we show that a many-qubits generalization of the flying-qubit scheme factorizes and reduces to the realization, in parallel, of many individual single-qubit schemes, for which it is known that they are realizable experimentally with the techniques that are available today. In other words, our approach shows that when $d$ is an even prime power $(d={2}^{m})$, the flying-qudit scheme reduces to $m$ flying-qubit schemes. For $d={2}^{m}$ and arbitrary $m$ the implementation of a generalized, maximally entangling, conditional qudit phase gate ``with insurance'' or ``repeat-until-success'' is thus shown to be feasible in practice by coupling $m$ (pairs of) two-level atoms to $m$ (pairs of) two-level (polarized) photons. Moreover, due to the parallelism of the task, the time necessary for completing successfully the task scales logarithmically in a function of $m$ while at the same time the dimension of the Hilbert space scales exponentially which presents promising perspectives regarding quantum informational realizations such as the quantum computer.

DOI
10.1103/physreva.77.042318

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We generalize in higher dimensions the so-called flying-qubit scheme'' that was described in the paper by Lim, Beige, and Kwek [Phys. Rev. Lett. 95, 030505 (2005)]. In that paper, the authors proposed a scheme according to which distant atoms get entangled during a measurement, in the Bell basis, of photons (flying qubits) emitted by them. We show that although in principle a generalization of this scheme to arbitrary dimensions is possible, this theoretical proposal is not presently feasible in all dimensions because only qubit Bell states have been successively measured until now. Nevertheless we show that a many-qubits generalization of the flying-qubit scheme factorizes and reduces to the realization, in parallel, of many individual single-qubit schemes, for which it is known that they are realizable experimentally with the techniques that are available today. In other words, our approach shows that when $d$ is an even prime power $(d={2}^{m})$, the flying-qudit scheme reduces to $m$ flying-qubit schemes. For $d={2}^{m}$ and arbitrary $m$ the implementation of a generalized, maximally entangling, conditional qudit phase gate with insurance'' or ``repeat-until-success'' is thus shown to be feasible in practice by coupling mmm (pairs of) two-level atoms to mmm (pairs of) two-level (polarized) photons. Moreover, due to the parallelism of the task, the time necessary for completing successfully the task scales logarithmically in a function of mmm while at the same time the dimension of the Hilbert space scales exponentially which presents promising perspectives regarding quantum informational realizations such as the quantum computer.
Robust quantum computation with ddd-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high-fidelity two-qudit gates. We first describe how to implement parallel single-qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth (time) of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the alkali-metal atoms 87Rb^{87}\mathrm{Rb}87Rb and 133Cs^{133}\mathrm{Cs}133Cs. Second, we provide a protocol for implementing parallelized nonlocal two-qudit gates using the assistance of entangled qubit pairs. Using known protocols for qubit entanglement purification, this offers the possibility of high-fidelity two-qudit gates.
Qudits are the natural generalization of qubits, that is, the basic unit of information in quantum computation. This generalization allows for quantum states to be represented with the same cyclic nature present in qubits, while also permitting the basic unit of information to be stored in d-level systems. In this document, we are concerned with studying universality over qudits on arbitrary dimension d. The question of single-qudit universality of a set of operators can be reduced to studying the adjoint representation of the set and determining whether or not its generated group is finite. Thus, we set out to find universal sets of operators in quantum computation over qudits, for any possible value of d. Moreover, we focus on finding “Clifford + T-like” universal sets of operators in arbitrary dimensions, in the hopes of maintaining a strong analog to the qubits case and its most studied universal set. We accomplish this goal by studying how the Clifford group behaves on arbitrary dimensions, and how this behavior relates to the typical Clifford group in prime dimensions. Furthermore, we find that the Clifford group on arbitrary dimension splits as the typical Clifford group on each of the coprime factors of d, which leads to a fundamental difference between the study of universality on prime power dimensions and arbitrary dimensions containing co-prime factors. Then, we focus on studying the structure of universal sets in dimension d, for any d containing at least two co-prime factors. Thus, after studying which T-like gates drive the single-qudit Clifford group to universality on prime power dimensions; that is, guaranteeing the irreducibility of the adjoint representation and an infinite generated group, we combine these two approaches to obtain “Clifford + T-like” universal sets on all possible dimensions.
We introduce an architecture for robust and scalable quantum computation using both stationary qubits (e.g., single photon sources made out of trapped atoms, molecules, ions, quantum dots, or defect centers in solids) and flying qubits (e.g., photons). Our scheme solves some of the most pressing problems in existing nonhybrid proposals, which include the difficulty of scaling conventional stationary qubit approaches, and the lack of practical means for storing single photons in linear optics setups. We combine elements of two previous proposals for distributed quantum computing, namely the efficient photon-loss tolerant build up of cluster states by Barrett and Kok [Phys. Rev. A 71, 060310(R) (2005)] with the idea of repeat-until-success (RUS) quantum computing by Lim et al. [Phys. Rev. Lett. 95, 030505 (2005)]. This idea can be used to perform eventually deterministic two qubit logic gates on spatially separated stationary qubits via photon pair measurements. Under nonideal conditions, where photon loss is a possibility, the resulting gates can still be used to build graph states for one-way quantum computing. In this paper, we describe the RUS method, present possible experimental realizations, and analyze the generation of graph states.
Here we present an experimentally feasible scheme to entangle flying qubit (individual photon with polarization modes) and stationary qubit (atomic ensembles with long-lived collective excitations). This entanglement integrating two different species can act as a critical element for the coherent transfer of quantum information between flying and stationary qubits. The entanglement degree can be also adjusted expediently with linear optics. Furthermore, the present scheme can be modified to generate this entanglement in a way event-ready, with the employment of a pair of entangled photons. And then successful preparation can be unambiguously heralded by coincident between two single-photon detectors. Its application for individual photons quantum memory is also analyzed. The physical requirements of all those preparation and applications processing are moderate, and well fit the present technique.
Recently, Chau [Phys. Rev. A 92, 062324 (2015)] introduced an experimentally feasible qudit-based quantum-key-distribution (QKD) scheme. In that scheme, one bit of information is phase encoded in the prepared state in a 2n{2}^{n}2n-dimensional Hilbert space in the form $(|i\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|j\ensuremath{\rangle})/\sqrt{2}$ with $n\ensuremath{\ge}2$. For each qudit prepared and measured in the same two-dimensional Hilbert subspace, one bit of raw secret key is obtained in the absence of transmission error. Here we show that by modifying the basis announcement procedure, the same experimental setup can generate nnn bits of raw key for each qudit prepared and measured in the same basis in the noiseless situation. The reason is that in addition to the phase information, each qudit also carries information on the Hilbert subspace used. The additional $(n\ensuremath{-}1)$ bits of raw key comes from a clever utilization of this extra piece of information. We prove the unconditional security of this modified protocol and compare its performance with other existing provably secure qubit- and qudit-based protocols on market in the one-way classical communication setting. Interestingly, we find that for the case of n=2n=2n=2, the secret key rate of this modified protocol using nondegenerate random quantum code to perform one-way entanglement distillation is equal to that of the six-state scheme.
Recently we proposed a hybrid architecture for quantum computing based on stationary and flying qubits: the repeat-until-success (RUS) quantum computing scheme. The scheme is largely implementation independent. Despite the incompleteness theorem for optical Bell-state measurements in any linear optics set-up, it allows for the implementation of a deterministic entangling gate between distant qubits. Here we review this distributed quantum computation scheme, which is ideally suited for integrated quantum computation and communication purposes.