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Physical Review A

Reducing the complexity of linear optics quantum circuits

John C. Howell, John A. Yeazell

Published Apr 3, 2000• 26 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.61.052303

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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∝2L\ensuremath{\propto}{2}^{L}\ensuremath∝2L 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., 2L\ifmmode×\else\texttimes\fi2L{2}^{L}\ifmmode\times\else\texttimes\fi{}{2}^{L}2L\ifmmode×\else\texttimes\fi2L 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×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN fiber coupler with the discrete Fourier transform suggests alternative formulations for the circuits. Several examples of circuit reductions are given.
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 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.
This work proposes to carry out a study on integrated photonic circuits. Our goal is to develop a methodology for obtaining the parameters of a photonic circuit that allows us to perform a set of quantum operations necessary for the realization of a given task or quantum processing. More specifically, the quantum tasks involve the implementation of a POVM (Positive Operator Value Measure) through integrated photonic circuits. Thus, we seek to obtain optimized configurations for these devices so that the reduction of their structural complexity simplifies their manufacturing process. In addition, it is also our goal that such objects are robust against errors and losses and that the experimental applications that involve them present a minimization of the time necessary for their realization. We studied the symmetries of the proposed POVMs to, from them, configure the circuit in an optimized way. Initially, we studied the case of quantum state tomography for qubits and qutrits. We obtained planar circuits that had a smaller number of optical elements and an optical depth than those obtained by methods already known in the literature. In addition, these circuits implement all POVM elements at once, making it possible to obtain all their associated probabilities at once, minimizing the time spent on the experiment compared to experiments on an optical table or with reconfigurable circuits, where it is necessary to make changes to the configuration of the experimental apparatus from one POVM element to another. Looking for a way to generalize our protocol to larger systems, we developed a method to obtain three-dimensional photonic circuits that can be used in the experimental realization of quantum state tomography in qudits of any finite dimension. Our protocol generates circuits whose number of beam splitters and optical depth scale with functions less than in other protocols already known, showing a drastic reduction in their complexity. In addition, these circuits were highly robust in relation to the occurrence of losses. The same protocol used to obtain three-dimensional circuits for performing quantum tomography can be used to obtain circuits to implement quantum coherence measurements. The circuits obtained here also have the same advantages as the tomographic case.
This Spotlight aims to provide a general introduction to linear and nonlinear optical components that are frequently used to implement the protocols for quantum computation and communication. The role of each optical element is described briefly, and it is shown that these optical elements can be combined to constitute quantum circuits for performing various quantum computing and communication tasks. Present challenges and future scopes are also mentioned briefly. Specifically, after introducing the qubit, it is stated that a quantum gate is a device that transforms a quantum state into another quantum state. Thus, most of the optical components (e.g., a beamsplitter, a wave plate, a mirror, a polarizing beamsplitter) can be viewed as quantum gates because each of these optical elements can transform a quantum state into another quantum state. This fact is the basis for constructing useful quantum circuits using optical elements. Finally, a few proposals for optical realizations of schemes for quantum communication and computation are described. In classical computation and communication, we use classical gates and bits. Similarly, in quantum computation and communication, we use their quantum analogs, i.e., quantum gates and qubit(s) [quantum bit(s)]. We have already mentioned that a bit can be realized in various ways. In the same way, a qubit can also be realized in various ways, and accordingly, corresponding quantum gates that would operate on qubits can also be realized in various ways. Here, we focus on a specific implementation: optical implementation, where qubits are realized using photons and quantum gates, and circuits are realized using various optical components. Specific merits of this realization lie in the following facts: (i) in quantum communication we need to transmit qubits from one location to the other, and transmission of photons is easier than any other realization of qubits; (ii) the effect of the environment (e.g., external electric and magnetic fields) on photons is less compared to any other implementations of qubits; and (iii) integrated optics are already a well-developed field. Keeping these points in mind, this Spotlight describes basic optical elements and how to use those elements to implement various schemes of quantum computation and communication. We will mostly describe realizations using bulk optics. However, in most cases, it is not difficult to transform these bulk-optics-based realizations into integrated optics. To begin with, in Section 1.1, we formally define a qubit and describe two alternative ways of realizing a qubit using a photon.
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.
We present a constructive method to translate small quantum circuits into their optical analogues, using linear components of present-day quantum optics technology only. These optical circuits perform precisely the computation that the quantum circuits are designed for, and can thus be used to test the performance of quantum algorithms. The method 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.). The optical implementation of Brassard et al.'s teleportation circuit, a non-trivial 3-bit quantum computation, is presented as an illustration.