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arXiv:quant-ph/0609118 · Physical Review A

Entangling spins by measuring charge: A parity-gate toolbox

Radu Ionicioiu

Published Mar 27, 2007• 47 citations
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Abstract

The parity gate emerged recently as a promising resource for performing universal quantum computation with fermions using only linear interactions. Here we analyze the parity gate ($P$ gate) from a theoretical point of view in the context of quantum networks. We present several schemes for entanglement generation with $P$ gates and show that native networks simplify considerably the resources required for producing multiqubit entanglement, such as $n$--Greenberg-Horne-Zellinger (GHZ) states. Other applications include a Bell-state analyzer and teleportation. We also show that cluster state fusion can be performed deterministically with parity measurements. We then extend this analysis to hybrid quantum networks containing spin and mode qubits. Starting from an easy-to-prepare resource (spin-mode entanglement of single electrons) we show how to produce a spin $n$-GHZ state with linear elements (beam splitters and local spin flips) and charge-parity detectors; this state can be used as a resource in a spin quantum computer or as a precursor for constructing cluster states. Finally, we construct an alternative spin-controlled-$Z$ gate by using the mode degrees of freedom as ancill\ae{}.

DOI
10.1103/physreva.75.032339

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The parity gate emerged recently as a promising resource for performing universal quantum computation with fermions using only linear interactions. Here we analyze the parity gate (PPP gate) from a theoretical point of view in the context of quantum networks. We present several schemes for entanglement generation with PPP gates and show that native networks simplify considerably the resources required for producing multiqubit entanglement, such as nnn--Greenberg-Horne-Zellinger (GHZ) states. Other applications include a Bell-state analyzer and teleportation. We also show that cluster state fusion can be performed deterministically with parity measurements. We then extend this analysis to hybrid quantum networks containing spin and mode qubits. Starting from an easy-to-prepare resource (spin-mode entanglement of single electrons) we show how to produce a spin nnn-GHZ state with linear elements (beam splitters and local spin flips) and charge-parity detectors; this state can be used as a resource in a spin quantum computer or as a precursor for constructing cluster states. Finally, we construct an alternative spin-controlled-ZZZ gate by using the mode degrees of freedom as ancill\ae{}.
It is known that a quantum computer operating on electron-spin qubits with single-electron Hamiltonians and assisted by single-spin measurements can be simulated efficiently on a classical computer. We show that the exponential speedup of quantum algorithms is restored if single-charge measurements are added. These enable the construction of a CNOT (controlled NOT) gate for free fermions, using only beam splitters and spin rotations. The gate is nearly deterministic if the charge detector counts the number of electrons in a mode, and fully deterministic if it only measures the parity of that number.
Measurements play an important role in quantum computing (QC), either by providing the nonlinearity required for two-qubit gates (linear optics QC), or by implementing a quantum algorithm using single-qubit measurements on a highly entangled initial state (cluster state QC). Parity measurements can be used as building blocks for preparing arbitrary stabilizer states, and, together with one-qubit gates, are universal for quantum computing. Here we generalize parity gates by using a higher-dimensional (qudit) ancilla. This enables us to go beyond the stabilizer or graph state formalism and prepare other types of multiparticle entangled states. The generalized parity module introduced here can prepare in one shot, heralded by the outcome of the ancilla, a large class of entangled states, including Greenberger-Home-Zeilinger states GHZn{\mathrm{GHZ}}_{n}GHZn​, Wn{\mathrm{W}}_{n}Wn​, Dicke states Dn,k{\mathcal{D}}_{n,k}Dn,k​, and, more generally, certain sums of Dicke states, like Gn{G}_{n}Gn​ states used in secret sharing. For Wn{\mathrm{W}}_{n}Wn​ states it provides an exponential gain compared to linear-optics-based methods.
This paper proposes a fermionic linear optical scheme for the teleportation and entanglement concentration via entanglement swapping based on charge detection. It also proves that this method is useful in generating entangled states such as GHZ states, W states, and cluster states by using fermionic polarizing beam splitters and single spin rotations assisted by a parity check on the fermionic qubits. This scheme is nearly deterministic (i.e., with 100% successful probability) and does not need the joint Bell state measurement required in the previous schemes.
We propose a realization of a charge parity meter based on two double quantum dots alongside a quantum point contact. Such a device is a specific example of the general class of mesoscopic quadratic quantum measurement detectors previously investigated by Mao et al. [Phys. Rev. Lett. 93, 056803 (2004)]. Our setup accomplishes entangled state preparation by a current measurement alone, and allows the qubits to be effectively decoupled by pinching off the parity meter. Two applications of the parity meter are discussed: the measurement of Bell's inequality in charge qubits and the realization of a controlled-NOT gate.