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Journal of Physics B Atomic Molecular and Optical Physics

Resource-efficient generation of linear cluster states by linear optics with postselection

Dmitry B. Uskov, Paul M. Alsing, Michael L. Fanto, Lev Kaplan, R Kim, Attila A. Szep, A. Matthew Smith

Published Jan 30, 2015• 6 citations
Publisher

Abstract

Abstract We report on theoretical research in photonic cluster-state computing. Finding optimal schemes of generating non-classical photonic states is of critical importance for this field as physically implementable photon–photon entangling operations are currently limited to measurement-assisted stochastic transformations. A critical parameter for assessing the efficiency of such transformations is the success probability of a desired measurement outcome. At present there are several experimental groups that are capable of generating multi-photon cluster states carrying more than eight qubits. Separate photonic qubits or small clusters can be fused into a single cluster state by a probabilistic optical CZ gate conditioned on simultaneous detection of all photons with 1/9 success probability for each gate. This design mechanically follows the original theoretical scheme of cluster state generation proposed more than a decade ago by Raussendorf, Browne and Briegel. The optimality of the destructive CZ gate in application to linear optical cluster state generation has not been analyzed previously. Our results reveal that this method is far from the optimal one. Employing numerical optimization we have identified that the maximal success probability of fusing n unentangled dual-rail optical qubits into a linear cluster state is equal to an m -tuple of photonic Bell pair states, commonly generated via spontaneous parametric down-conversion, can be fused into a single cluster with the maximal success probability of (1/4) m −1 .

DOI
10.1088/0953-4075/48/4/045502

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Abstract We report on theoretical research in photonic cluster-state computing. Finding optimal schemes of generating non-classical photonic states is of critical importance for this field as physically implementable photon–photon entangling operations are currently limited to measurement-assisted stochastic transformations. A critical parameter for assessing the efficiency of such transformations is the success probability of a desired measurement outcome. At present there are several experimental groups that are capable of generating multi-photon cluster states carrying more than eight qubits. Separate photonic qubits or small clusters can be fused into a single cluster state by a probabilistic optical CZ gate conditioned on simultaneous detection of all photons with 1/9 success probability for each gate. This design mechanically follows the original theoretical scheme of cluster state generation proposed more than a decade ago by Raussendorf, Browne and Briegel. The optimality of the destructive CZ gate in application to linear optical cluster state generation has not been analyzed previously. Our results reveal that this method is far from the optimal one. Employing numerical optimization we have identified that the maximal success probability of fusing n unentangled dual-rail optical qubits into a linear cluster state is equal to an m -tuple of photonic Bell pair states, commonly generated via spontaneous parametric down-conversion, can be fused into a single cluster with the maximal success probability of (1/4) m −1 .
We report on theoretical research in photonic cluster-state computing. Finding optimal schemes of generating non-classical photonic states is of critical importance for this field as physically implementable photon-photon entangling operations are currently limited to measurement-assisted stochastic transformations. A critical parameter for assessing the efficiency of such transformations is the success probability of a desired measurement outcome. At present there are several experimental groups which are capable of generating multi-photon cluster states carrying more than eight qubits. Separate photonic qubits or small clusters can be fused into a single cluster state by a probabilistic optical CZ gate conditioned on simultaneous detection of all photons with 1/9 success probability of each gate. This design mechanically follows the original theoretical scheme of cluster state generation proposed more than a decade ago by Raussendorf, Browne and Briegel. The optimality of the destructive CZ gate in application to linear optical cluster state generation has not been analyzed previously. Our results reveal that this method is far from the optimal one. Employing numerical optimization we have identified that maximal success probability of fusing n unentangled dual-rail optical qubits into a linear cluster state is equal to 1/2^(n-1); m-tuple of photonic Bell pair states, commonly generated via spontaneous parametric down-conversion, can be fused into a single cluster with the maximal success probability of 1/4^(m-1).
We analyze the generation of linear-optical cluster states (LOCSs) via sequential addition of one and two qubits. Existing approaches employ the stochastic linear-optical two-qubit controlled-Z (cz) gate with success rate of 19\frac{1}{9}91​ per operation. The question of optimality of the cz gate with respect to LOCS generation has remained open. We report that there are alternative schemes to the cz gate that are exponentially more efficient and show that sequential LOCS growth is indeed globally optimal. We find that the optimal cluster growth operation is a state transformation on a subspace of the full Hilbert space. We show that the maximal success rate of postselected entangling nnn photonic qubits or mmm Bell pairs into a cluster is ${(\frac{1}{2})}^{n\ensuremath{-}1}$ and ${(\frac{1}{4})}^{m\ensuremath{-}1}$, respectively, with no ancilla photons, and we give an explicit optical description of the optimal mode transformations.
We analyze the generation of linear optical cluster states (LOCS) via addition of one and two qubits. Existing approaches employ the stochastic linear optical two-qubit CZ gate with success rate of 1/9 per fusion operation. The question of optimality of the CZ gate with respect to LOCS generation remains open. We report that there are alternative schemes to the CZ gate that are exponentially more efficient and show that sequential LOCS growth is globally optimal. We find that the optimal cluster growth operation is a state transformation on a subspace of the full Hilbert space. We show that the maximal success rate of fusing n photonic qubits or m Bell pairs is 1/2^n-1 and 1/4^m-1 respectively and give an explicit optical design.
Cluster state plays a crucial role in one-way quantum computation. Here, we propose and experimentally demonstrate a scheme to prepare an ultrahigh-fidelity four-photon linear cluster state via a spontaneous parametric down-conversion process. The state fidelity is measured to be $0.9517\ifmmode\pm\else\textpm\fi{}0.0027$. Our scheme can be directly extended to more photons to generate an NNN-qubit linear cluster state. Furthermore, our scheme is optimal for generating photonic linear cluster states in the sense of achieving the maximal success probability and having the simplest strategy. The key idea is that the photon pairs are prepared in some special nonmaximally entangled states instead of the normal Bell states. To generate a 2N2N2N-qubit linear cluster state from NNN pairs of entangled photons, only $(N\ensuremath{-}1)$ Hong-Ou-Mandel interferences are needed and a success probability of ${(\frac{1}{4})}^{N\ensuremath{-}1}$ is achieved.
We introduce a scheme for linear optics quantum computation, that makes no use of teleported gates, and requires stable interferometry over only the coherence length of the photons. We achieve a much greater degree of efficiency and a simpler implementation than previous proposals. We follow the "cluster state" measurement based quantum computational approach, and show how cluster states may be efficiently generated from pairs of maximally polarization entangled photons using linear optical elements. We demonstrate the universality and usefulness of generic parity measurements, as well as introducing the use of redundant encoding of qubits to enable utilization of destructive measurements--both features of use in a more general context.
Entangled dual-rail photonic qubit states are important resource states for measurement-based and fusion-based quantum computing. However, their generation is highly challenging due to the probabilistic nature of entangling operations. Here, the authors introduce a formalism using ZX diagrams to design linear optical systems that can generate entangled multiqubit states of dual-rail photonic qubits. This formalism makes it easier to compare methods for creating a desired photonic state and to find the most optimal one.