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Physical Review A

Linear optics simulations of the quantum baker’s map

John C. Howell, John A. Yeazell

Published Dec 13, 1999• 9 citations
Publisher

Abstract

The unitary evolution of linear optics can be used to model quantum computational networks. In this paper, a quantum simulation of a classically chaotic map (the baker's map) is developed using linear optics. Two different models are presented. The first model employs only 50-50 beam splitters and phase shifters to simulate universal 2-qubit gates of a quantum computer. The second model uses the discrete Fourier transform generated by symmetric $N\ifmmode\times\else\texttimes\fi{}N$ fiber couplers. If single photons are used as inputs for these linear optics models, the result is a physical realization of the quantum baker's map.

DOI
10.1103/physreva.61.012304

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The unitary evolution of linear optics can be used to model quantum computational networks. In this paper, a quantum simulation of a classically chaotic map (the baker's map) is developed using linear optics. Two different models are presented. The first model employs only 50-50 beam splitters and phase shifters to simulate universal 2-qubit gates of a quantum computer. The second model uses the discrete Fourier transform generated by symmetric N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN fiber couplers. If single photons are used as inputs for these linear optics models, the result is a physical realization of the quantum baker's map.
Integrated optical elements can simplify the linear optics used to simulate quantum circuits. These linear optical simulations of quantum circuits have been developed primarily in terms of the free space optics associated with single-photon interferometry. For an L-bit simulation the number of required free-space optical elements is $\ensuremath{\propto}{2}^{L}$ if 50/50 beam splitters are used. The implementation (construction and alignment) of these circuits with these free-space elements is nontrivial. On the other hand, for the cases presented in this paper in which linear integrated optics (e.g., ${2}^{L}\ifmmode\times\else\texttimes\fi{}{2}^{L}$ fiber couplers) are used, the number of optical devices does not grow exponentially with L. The problem is changed from having an exponential growth of the number of devices to having devices with an exponential growth in the number of ports. In addition to simplifying the construction, the association of an $N\ifmmode\times\else\texttimes\fi{}N$ fiber coupler with the discrete Fourier transform suggests alternative formulations for the circuits. Several examples of circuit reductions are given.
This thesis covers several topics in optical quantum computation. A quantum computer is a computational device which is able to manipulate information by performing unitary operations on some physical system whose state can be described as a vector (or mixture of vectors) in a Hilbert space. The basic unit of information, called the qubit, is considered to be a system with two orthogonal states, which are assigned logical values of 0 and 1. Photons make excellent candidates to serve as qubits. They have little interactions with the environment. Many operations can be performed using very simple linear optical devices such as beam splitters and phase shifters. Photons can easily be processed through circuit-like networks. Operations can be performed in very short times. Photons are ideally suited for the long-distance communication of quantum information. The great difficulty in constructing an optical quantum computer is that photons naturally interact weakly with one another. This thesis first gives a brief review of two early approaches to optical quantum computation. It will describe how any discrete unitary operation can be performed using a single photon and a network of beam splitters, and how the Kerr effect can be used to construct a two photon logic gate. Second, this work provides a thorough introduction to the linear optical quantum computer developed by Knill, Laflamme, and Milburn. It then presents this author's results on the reliability of this scheme when implemented using imperfect photon detectors. This author finds that quantum computers of this sort cannot be built using current technology. Third, this dissertation describes a method for constructing a linear optical quantum computer using nearly orthogonal coherent states of light as the qubits. It shows how a universal set of logic operations can be performed, including calculations of the fidelity with which these operations may be accomplished. It discusses methods for reducing and correcting errors and recovering from failed operations. Lastly it describes an analysis of the long distance transmission of the coherent state qubits and shows how transmission errors can be corrected.
A constructive method for simulating small-scale quantum circuits by use of linear optical devices is presented. It relies on the representation of several quantum bits by a single photon, and on the implementation of universal quantum gates using simple optical components (beam splitters, phase shifters, etc.). This suggests that the optical realization of small quantum networks with present-day quantum optics technology is a reasonable goal. This technique could be useful for demonstrating basic concepts of simple quantum algorithms or error-correction schemes. The optical analog of a nontrivial three-bit quantum circuit is presented as an illustration.
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.
One of the main problems that optical quantum computing has to overcome is the efficient construction of two-photon gates. Theoretically these gates can be realized using Kerr-nonlinearities, but the techniques involved are experimentally very difficult. We therefore employ linear optics with projective measurements to generate these non-linearities. The downside is that the measurement-induced nonlinearities achieved with linear optics are less versatile and the success rate can be quite low. This project is mainly the result of a literature study but also a theoretical work on the physics behind quantum optical multiports which is essential for realizing two-photon gates. By applying different postcorrection techniques we increase the probability of success in a modifed non-linear sign shift gate which is foundational for the two photon controlled-NOT gate. We prove that it's not possible to correct the states by only using a single beam splitter. We show that it might be possible to increase the probability of success using a more complex setup with at least two error-correcting beam splitters.
This work presents a developed Wolfram Mathematica software tool based on the Quantum extension package. This tool is able to simulate the output state for any linear optical circuit based on Beam Splitters (BSs) and Phase Sifters (PSs) by using as input, quantum states of light. The input of the software is described via the use of photon number states capable of emulating Fock states or coherent states, whereas the output provides the probability of detecting a photon at each output port. This photonic circuit may include any number of linear optical elements, including beam splitters with variable splitting ratios, as well as phase shifters. By using the unitary transformation of any photonic circuit (e.g., photonic chip), the output states can be computed, and detection probabilities of the chip can be obtained. The unitary transformation of the Controlled NOT (CNOT) quantum gate through a linear optical layout is presented, validating the functionality of the proposed software tool. In conclusion, the proposed software aims to enable testing of simple photonic circuits in the quantum regime.