Linear Optical Quantum Computing

CSE682 End-term report 1 Linear Optical Quantum Computing RISHABH SAHU* *Roll number: 13817575, email: [email protected] Compiled November 12, 2017 Optical quantum computers provide a variety of advantages over other types of quantum computers including seamless integration of computational and communication part of the quantum technology. Single-qubit unitary operations on photons have been shown to be rather trivial, however, two-qubit gates become a challenge due to non-interacting nature of photons. Although several theoretical two-qubit gates were proposed using non-linear optical elements, any physical realisation of these gates was not seen due to unavailability of required highly nonlinear materials for the scheme. In this report, we discuss implementation of optical quantum computer, especially two-qubit gates, using only linear optical elements. 1. INTRODUCTION 2. PHOTONS AS QUBITS There are various properties of photons that allow them to be Various physical entities can be used as a qubit including ion used as qubits. Major ones include they travel in straight line, traps, superconducting charges, spin of atomic nucleus etc. How- they travel with constant speed in a given media and their polar- ever, each of them have some or the other disadvantage that ization, that is, the field of vibration of electric field. Using these stems realisation of a fully scalable quantum computer using properties we can construct two main types of photonic qubits, that entity. In this race of quantum computers, photons also provide several advantages compared to other entities as qubits. Dual-rail qubits Photons travel in straight line. Hence, if we choose two paths of travel and put photon in either one of them, The biggest advantage that an optical quantum computer depending in which path do we detect the photon, we can call that either a 0 or 1 . Figure1 shows a dual-rail bit. can offer over other computing systems would be that they pro- j i j i vide a seamless integration of quantum communication, that is imagined to be done using photons, and quantum computation. Apart from that photons don’t decohere easily as compared to many other systems. Also, unlike other contenders, they need 0 1 not be operated at extremely low temperatures to keep the ther- | i | i mal noise at minimum. Fig. 1. A dual-rail photonic qubit. Depending on the channel in which the photon is put, the qubit is in either in state 0 or j i However, these advantages come with one big disadvantage. 1 . j i Photons don’t interact among themselves. This is a desired property to have when building single-bit gates but it creates a A photon state that has a specific number of photons is called challenge when designing two-qubit gates. a fock-state. A single photon fock-state is represented by 1 , a two-photon fock state by 2 and so on. A fock-state withj i no In this report, we will describe the approaches used to make photons is called a vacuumj statei is represented by 0 . Raising, two-qubit gates. We start with introducing photons as qubits aˆ†, and lowering operator, aˆ, can be used receptivelyj i to switch and some properties of their properties. We, then, move on to from one fock-state to other. They function as, describe common optical elements that can be used to make single-bit unitary gates. Thereafter, we address the problem of aˆ n = pn n 1 j i j − i two-qubit gates. We start with some earlier approaches that aˆ† n = pn + 1 n + 1 tried to implement two-qubit gates but were either not realisable j i j i or scalable. We, then, describe a protocol originally given by Hence, we can write 1 as aˆ† 0 and so on. These operators will Knill, Laflamme and Milburn in 2001[1]. The protocol describes be used extensivelyj fori analyticalj i treatment of further optical implementation of scalable two-qubit gates. elements that we discuss in next section. CSE682 End-term report 2 Polarization qubit It turns out that the plane of vibration of We also note that a rotation about X axis can be written in electric field can be at any angle independent of direction of terms of rotation about Y and Z axis as, propagation of light (as long it is perpendicular to direction of p p propagation in 3 dimensions). The electric field can hence take Rx(q) = Rz( )Ry( q)Rz( ) 2 − − 2 any direction in 2-dimensional space. This two dimensional space can be represented by a basis consisting of horizontal We can, hence, use phase shifters and beam splitters to perform polarization, H , and vertical polarization, V . Any other di- any rotation of the qubit in Bloch sphere. rection can thenh j be written as a superpositionh ofj these two. Finally, it can be observed that any single-bit unitary gate can be expanded as, 3. SINGLE-QUBIT UNITARY OPERATIONS ia U = e Rz(a)Ry(b)Rx(g) (3) Single-bit unitary operations become rather trivial on the pho- This shows that we can implement any single-bit unitary opera- tonic qubits. They can be implemented using a few simple tion on a dual-rail qubit using a combination of beam splitters optical elements. Here, we discuss the optical elements along and phase shifters. However, to include polarization qubit also, with their operation on qubits. we use another optical element, Phase Shifter These are simplest optical elements that intro- Polarising Beam Splitter This is another kind of beam splitter duce a phase shift to any photons passing through it. The with a slight difference. Instead of splitting the beam in some amount of phase introduced, f, can be controlled by chang- specific ratio, it splits the path of incoming photon according ing the refractive index or length of glass used. Analytically, we to its polarization. Horizontally polarised components pass write this in terms of operators introduced in previous section, without reflection and vertically polarised components suffer a reflection. It is trivial to see how polarising beam splitter can aˆ† = eifaˆ† out in be used to switch between a dual-rail qubit and a polarization Note that in a dual-rail qubit, if we put the phase shifter only in qubit. the second path, we can perform a rotation of qubit about Z axis in Bloch sphere. aôut,V† iφ aˆ† aˆ† = e aˆ† in φ out in Fig. 2. A phase-shifter is just an optical element with different aîn† aôut,H† refractive index. It introduces a pre-determined phase to all incoming modes. Fig. 4. A polarising beam splitter. Horizontally polarized part passes without reflection while the vertically polarised part gets reflected. Beam Splitter The second important element in linear optics is beam splitter. It mixes two incoming modes aˆ and bˆ . Working in in There are other optical elements such a polarization rotators of a generalised beam splitter can be given as follows, and wave-plates that can be used to modify a polarization qubit. † † if † However, in this report we mostly deal with dual-rail qubits and aˆ = cosqaˆ + ie− sinqbˆ (1) out in in would hence skip any details about these elements. ˆ† if † ˆ† bout = ie sinqaîn + cosqbin (2) 4. EARLIER ATTEMPTS AT TWO-QUBIT GATES where q and f are parameters of a general beam-splitter. Re- flection and transmission coefficients are sin2q and cos2q respec- The above section shows how to do single qubit unitary opera- tively which add upto 1. It is worth noting here that if we tion with photonic qubits. Use of simple linear optical elements substitute f = 0 here, a beam splitter with q parameter will make the task rather trivial. However, realisation of optical perform a rotation of dual-rail qubit about Y axis by an amount quantum computer is stemmed by two-qubit gates whose im- of 2q. plementation with photonic qubits becomes extremely difficult − because of non-interacting nature of photons. ˆ bout† Many attempts have been made to implement two-qubit gates but all of them failed to provide a scalable and physically realisable solution. We discuss some of these attempts here. A. Cerf, Adami and Kwait protocol (1998) aîn† aôut† In 1998, Cerf et al.[2] came up with a technique to implement θ,φ CNOT gate using linear optical elements such as beam splitters and phase shifters only. In this protocol, n-qubits were represented by a single photon in 2n different paths. For example, a ˆ bin† two qubit system would be represented by a single photon in 4 different paths representing 00 , 01 , 10 and 11 . Fig. 3. A generalised beam splitter with q and f as characteris- As all the representationj werei j comfortablyi j i separated,j i they tic parameters. The action is given by equations1 and2. can be easily manipulated separately. A CNOT implementation, CSE682 End-term report 3 it. In any other case, it won’t introduce any phase shift in the 90o photons. After this interaction with nonlinear medium, the vertical and horizontal part of both the qubits is then combined back. The only change that occurs, as a result, is 1, 1 part gets a phase shift of p while the rest remains the samej as requiredi by the CZ gate. However, this approach suffers from one big problem. Even the most nonlinear materials available are extremely weak and Fig. 5. CNOT as implemented by Cerf et al. The control bit can introduce a phase shift of only about t = 10 18 while for is represented by spatial degree of freedom while the target − this scheme to work, required phase shift is t = p.

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