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Chapman & Hall/CRC eBooks

Nondeterministic Logic Gates in Optical Quantum Computing

Federico M. Spedalieri, Jonathan P. Dowling, Hwang Lee

Published Sep 19, 2007• 0 citations
Publisher

Abstract

Federico M. Spedalieri, Jonathan P. Dowling, and Hwang Lee Abstract We present a detailed description of nondeterministic quantum logic gates: the crucial component of linear optical quantum computing. We use the qubits that are encoded in the polarization degrees of freedom, as opposed to the original dual-rail encoding. Employing the polarization-encoding scheme, we can avoid the undetected errors due to the inefficiency of the photodetectors and, therefore, achieve high gate fidelity, independent of the quantum efficiency of the detectors. Furthermore, we show that such a high-fidelity gate operation can be performed with no need of number-resolving detectors.

DOI
10.1201/9781584889007-15

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Federico M. Spedalieri, Jonathan P. Dowling, and Hwang Lee Abstract We present a detailed description of nondeterministic quantum logic gates: the crucial component of linear optical quantum computing. We use the qubits that are encoded in the polarization degrees of freedom, as opposed to the original dual-rail encoding. Employing the polarization-encoding scheme, we can avoid the undetected errors due to the inefficiency of the photodetectors and, therefore, achieve high gate fidelity, independent of the quantum efficiency of the detectors. Furthermore, we show that such a high-fidelity gate operation can be performed with no need of number-resolving detectors.
We show that the KLM scheme [Knill, Laflamme, and Milburn, Nature 409, 46 (2001)] can be implemented using polarization encoding, thus reducing the number of path modes required by half. One of the main advantages of this new implementation is that it naturally incorporates a loss detection mechanism that makes the probability of a gate introducing a non-detected error, when non-ideal detectors are considered, dependent only on the detector dark-count rate and independent of its efficiency. Since very low dark-count rate detectors are currently available, a high-fidelity gate (probability of error of order ${10}^{\ensuremath{-}6}$ conditional on the gate being successful) can be implemented using polarization encoding. The detector efficiency determines the overall success probability of the gate but does not affect its fidelity. This can be applied to the efficient construction of optical cluster states with very high fidelity for quantum computing.
The performance of nondeterministic nonlinear gates in linear optics relies on the photon counting scheme being employed and the efficiencies of the detectors in such schemes. We assess the performance of the nonlinear sign gate, which is a critical component of linear optical quantum computing, for two standard photon counting methods: the double detector array and the visible light photon counter. Our analysis shows that the double detector array is insufficient to provide the photon counting capability for effective nondeterministic nonlinear transformations, and we determine the gate fidelity for both photon counting methods as a function of detector efficiencies.
Implementation of quantum logic gates with linear optical elements plays a prominent role in quantum computing due to the relatively easier manipulation and realization. We present efficient schemes to implement controlled-NOT (CNOT) gate and controlled-swap (Fredkin) gate by solely using linear optics. We encode the control qubits and target qudits in photonic polarization (two-level) and spatial degrees of freedom (ddd-level), respectively. Based on the hybrid encoding, CNOT and Fredkin gates are constructed in a deterministic way without any borrowed ancillary photons or measurement-induced nonlinearities. Remarkably, the number of linear optics required to implement a CNOT gate has been reduced to one polarization beam splitter (PBS), while only ddd PBSs are necessary to implement a generalized Fredkin gate. The optical depths of all schemes are reduced to one and dimension-independent. Besides, the fidelity of our three-qubit Fredkin gate is higher than 99.7\% under realistic conditions, which is higher than the previous schemes.
Ever since Knill, Laflamme and Milburn [Nature (London) 409, 46 (2001)] showed that nondeterministic quantum logic operations could be performed with linear optical elements, additional photons (ancilla) and projective measurements, the idea of linear-optics quantum computation has attracted considerable interest. Our group has recently demonstrated several devices of this kind. We give an overview of recent experimental results, including the quantum parity check, the destructive controlled-NOT, and a cyclical quantum memory. The need for high-efficiency detection of single photons, and for detectors capable of distinguishing photon number will be discussed. Some experimental improvements towards meeting that need will be presented.
We demonstrate a quantum error correction scheme that protects against accidental measurement, using a parity encoding where the logical state of a single qubit is encoded into two physical qubits using a nondeterministic photonic controlled-NOT gate. For the single qubit input states $\ensuremath{\mid}0⟩$, $\ensuremath{\mid}1⟩$, $\ensuremath{\mid}0⟩\ifmmode\pm\else\textpm\fi{}\ensuremath{\mid}1⟩$, and $\ensuremath{\mid}0⟩\ifmmode\pm\else\textpm\fi{}i\ensuremath{\mid}1⟩$ our encoder produces the appropriate two-qubit encoded state with an average fidelity of $0.88\ifmmode\pm\else\textpm\fi{}0.03$ and the single qubit decoded states have an average fidelity of $0.93\ifmmode\pm\else\textpm\fi{}0.05$ with the original state. We are able to decode the two-qubit state (up to a bit flip) by performing a measurement on one of the qubits in the logical basis; we find that the 64 one-qubit decoded states arising from 16 real and imaginary single-qubit superposition inputs have an average fidelity of $0.96\ifmmode\pm\else\textpm\fi{}0.03$.