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arXiv:0806.0982 · Physical Review A

Generalized parity measurements

Radu Ionicioiu, Anca E. Popescu, William J. Munro, Timothy P. Spiller

Published Nov 14, 2008• 31 citations
arXivHTMLPDF

Abstract

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 ${\mathrm{GHZ}}_{n}$, ${\mathrm{W}}_{n}$, Dicke states ${\mathcal{D}}_{n,k}$, and, more generally, certain sums of Dicke states, like ${G}_{n}$ states used in secret sharing. For ${\mathrm{W}}_{n}$ states it provides an exponential gain compared to linear-optics-based methods.

DOI
10.1103/physreva.78.052326

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Quantum entanglement plays an important role in quantum information processes, such as quantum computation and quantum communication. Experiments in laboratories are unquestionably crucial to increase our understanding of quantum systems and inspire new insights into future applications. However, there are no general recipes for the creation of arbitrary quantum states with many particles entangled in high dimensions. Here we exploit a recent connection between quantum experiments and graph theory and answer this question for a plethora of classes of entangled states. We find experimental setups for Greenberger-Horne-Zeilinger states, WWW states, general Dicke states, and asymmetrically high-dimensional multipartite entangled states. This result sheds light on the producibility of arbitrary quantum states using photonic technology with probabilistic pair sources and allows us to understand the underlying technological and fundamental properties of entanglement.
Parity measurements have been explored as building blocks for preparing and discriminating entangled states, as well as for implementing quantum computation. We first develop two alternative high-dimensional generalized parity modules, and then propose a procedure for constructing high-dimensional generalized module-based controlled-NOT gate. The construction of module-based quantum computing introduced here is deterministic, heralded, insensitive to the dimensionality of the computing basis, and postselection technique is not required. The result shows that out of (d−1)!(d-1)!(d−1)! generalized parity modules, only modules $ \mathcal{P}=(j\ominus i)\bmod d$ and $ \mathcal{P}=(j \oplus i)\bmod d$ can be used as building blocks for high-dimensional quantum computing. Furthermore, we proposed an optical nondestructive scheme for implementing generalized parity module through quantum nondemolition measurements, and the success of the parity module is heralded by photon-number-resolving detectors and single-photon detectors.
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{}.
Parity measurements have been explored as building blocks for preparing and discriminating entangled states, as well as for implementing quantum computation. We first develop two alternative high-dimensional generalized parity modules, and then propose a procedure for constructing a high-dimensional generalized module-based controlled-NOT gate. The construction of module-based quantum computing introduced here is deterministic, heralded, and insensitive to the dimensionality of the computing basis, and a postselection technique is not required. The result shows that out of ( d − 1)! generalized parity modules, only modules P =( j ⊖ i )mod d and P =( j ⊕ i )mod d can be used as building blocks for high-dimensional quantum computing. Furthermore, we proposed an optical nondestructive scheme for implementing a generalized parity module through quantum nondemolition measurements, and the success of the parity module is heralded by photon-number-resolving detectors and single-photon detectors.
A model of universal quantum computation with qudits (ddd-level systems) is presented in which the gates are implemented on a register entirely via fixed-form sequential interactions with ancillary qudits followed by adaptive, variable basis measurements of these ancillas. The model maintains determinism via classical feed-forward of measurement outcomes in a similar fashion to the qudit one-way model. An embedding of the qudit one-way model is given which demonstrates that the hybrid quantum-classical advantages of one-way quantum computation are also inherent in the model presented here, including the ability to implement any Clifford gate in constant depth using multiple ancillas in parallel. A model which requires only fixed basis measurements is then presented, but due to the lack of adaptive controls it can only implement universal quantum computation in a stochastic sense. Interestingly, this scheme may then be altered to recover determinism by replacing ancilla measurements with ancilla preparation. These models provide an intuitive setting for understanding the interplay between the benefits gained from classical controls, measurements and state preparation in the context of multi-valued logic ancilla-based quantum computation.
Quantum computers have the potential to solve problems that are believed to be classically intractable. However, building such a device is proving to be very challenging. In this thesis, two physically promising settings for quantum computation are investigated: the one-way quantum computer and ancilla-based quantum gates. The majority of both the theoretical and experimental focus in the field of quantum computation has been on computation using 2-level quantum systems, known as qubits. In contrast to this, in this thesis I consider the relatively less well-understood setting of quantum computation using continuous variables or d-level quantum systems, called qudits. I develop a simple notation that encompasses each different encoding, and is applicable to a `general quantum variable'. These ideas are then used to investigate computational depth (a proxy for time) in quantum circuits and one-way quantum computations in this general quantum variable setting. In doing so, the parallelism inherent in the one-way quantum computer is made precise. In the second half of this thesis, a range of techniques are proposed for implementing entangling gates on a well-isolated computational register via interactions with `ancillary' systems. In particular, ancilla-based quantum gates for general quantum variables are investigated - including the interesting case of hybrid quantum computation, whereby more than one encoding is used in tandem. The methods proposed herein each have their own unique advantages, such as: reducing gate-counts in certain circuits, allowing for inherently parallel computation, or minimising the physical requirements for universal quantum computation. In particular, the final gate techniques that are proposed in this thesis may implement any quantum computation using only a single fixed ancilla-register interaction gate and ancillas prepared in simple states. This then allows the computational register to consist of well-isolated `memory' quantum variables and the ancillas need only be optimised for a single high-quality fixed interaction gate. Hence, this provides a simple and highly promising setting for physically implementing a quantum computer.