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Nanosystems Physics Chemistry Mathematics

On construction of evolutionary operator for rectangular linear optical multiport

M.M. Lipovich, I. S. Lobanov

Published Aug 28, 2015• 0 citations

Abstract

The work of Knill et. al. (2001) established the possibility of nondeterministic realization of certain quantum logic operations using linear optical elements, ancilla photons and postselection techniques.It was also shown that any discrete unitary operator acting on N optical modes can be implemented by a triangular multiport device constructed from a series of beam splitters and phase shifters (see work of Reck, Zeilinger et. al., 1994).Here, we consider the rectangular linear optical multiport that is used for the probabilistic realization of unitary transformations on n qubits.This kind of linear optical scheme is suitable for probabilistic realization of unitary operators using ancilla photons and projective measurements.Qubits are encoded into the bosonic states of optical modes in two possible polarizations, and a number of ancilla photons and photodetectors are used for postselection of the qubits' state, based on the output of the detectors.We derive a procedure of evolutionary operator calculation for schemes of the considered type and present algorithms for their efficient computation on symmetric state space.We also provide complexities for different algorithms for the computation of evolutionary operator and estimate demands of resources in each case.A destructive Toffoli gate, acting on three qubits, using one ancilla photon and a photodetector, is implemented using schemes of the presented type.

DOI: 10.17586/2220-8054-2015-6-4-537-546

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Knill, Laflamme, and Milburn [1] have shown that probabilistic quantum logic operations can be implemented using linear optical elements, such as beam splitters and phase shifters, additional photons (ancilla) and post-selection based on the results of measurements on the ancilla. This may seem somewhat surprising, since logic operations are inherently nonlinear. Here the nonlinearity is produced by the quantum measurement process used in the post-selection. Nonlinearities of this kind are inherently quantum-mechanical and have no classical analogy.

Knill, Laflamme, and Milburn [Nature (London) 409, 46 (2001)] recently showed that nondeterministic quantum logic operations could be performed using linear optical elements, additional photons (ancilla), and postselection based on the output of single-photon detectors. Here we report the experimental demonstration of two logic devices of this kind, a destructive controlled-NOT (CNOT) gate and a quantum parity check. These two devices can be combined with a pair of entangled photons to implement a conventional (nondestructive) CNOT that succeeds with a probability of 1/4.

We design linear-optics multiqubit quantum logic gates. We assume the traditional encoding of a qubit onto the state of a single photon in two modes (e.g., spatial or polarization). We suggest schemes allowing direct probabilistic realization of the fundamental Toffoli and Fredkin gates without resorting to a sequence of single- and two-qubit gates. This yields more compact schemes and potentially reduces the number of ancilla photons. The proposed setups involve passive linear optics, sources of auxiliary single photons or maximally entangled pairs of photons, and single-photon detectors. In particular, we propose an interferometric implementation of the Toffoli gate in the coincidence basis, which does not require any ancilla photons and is experimentally feasible with current technology.

The first section of the work investigates light modes as a means of implementing optical qubits and qudits. The modes considered are polarization mode, path mode, transverse spatial mode, frequency mode, temporal bin-mode and temporal mode. Subsequently, mathematical model of linear optical elements like beam splitters (BS) and phase shifters are deduced and their capability of representing any single qubit optical gate is exposed. Finally Knill, Laflamme and Milburn (KLM) method of using linear optical elements to promote nonlinear operations based on nonlinear (nondeterministic) sign-flip gate (NS) is explained, and designs of two qubits conditional sign flip gate (c-z1/16) and CNOT gate, both based on KLM method, are demonstrated. So, universal set of quantum gates based on linear optics is possible.

Universal multiport interferometers to implement any linear transformation are emerging as a powerful tool for quantum optics. These experimental devices are usually made from an array of beam splitters and phase shifters. Reck et al [3], provided the first algorithmic proof for experimentally realizing any arbitrary discrete unitary operator. Thus, it was made possible to measure any observable corresponding to any discrete Hermitian operator. Our work extends this idea and tries to find the best optimal experimental design for a given unitary operator. We have developed a numerical algorithm which gives an experimental design with the least number of beam splitters for a given unitary operator.