Linear Optical Quantum Computing Anna Dardia I. INTRODUCTION There are a wide range of proposed systems with which quantum computing could be physically realized, including atom and ion-trap quantum computing, nuclear magnetic resonance, and quantum dots. Quantum computing with linear optics uses photons as qubits. Photons have the advantage of minimal decoherence. However, photons do not natrually interact with one another, so in order to make two-qubit gates an effective interaction needs to be induced. In this paper, we will discuss the construction of quantum gates using linear optics, how multi-qubit gates can be constructed, as well as the Knill-Laflamme-Millburn (KLM) protocol for linear optical quantum computing (LOQM). Some of the error correction methodology will be discussed, as well as sources of error arising from the physical linear optical components and resources. Finally, we briefly discuss some developing improvements on the KLM regime. II. A BRIEF BACKGROUND ON OPTICS A. Quantum Optics The energy of a classical electromagnetic field is Z 1 X h!k H = (E2 + B2)dr = a a∗ 2 2 k k V k ∗ The electromagnetic field is quantized by turning the coefficients ak and ak into operators with commutation relations y [a ^i ; a^j ] = δij y y [a ^i; a^j] = [a ^i ; a^j ] = 0 Letting 1 rm! 1 a^ = p x^ + i p ^p 2 ¯h m¯h! The energy can now be expressed in the form of a harmonic oscillator p^2 m!2x^2 H^ = + 2m 2 The eigenstates of H^ are called Fock states, labelled jni :a^ anda ^y are creation and annihilation operators acting on the Fock states as follows p p a^ jni = n jn − 1i ; a^y jni = n + 1 jn + 1i B. Linear Optics Optical components are linear if their output modes are linear combinations of their input modes. ^y X y bj = Mjka^j k Basic linear optical components are beamsplitters, half and quarter-waveplates and phase-shifters. These components can be described mathematically by unitary operations. 2 A single-mode phase shift changes the phase of the electromagnetic field in a particular mode. It's interaction Hamiltonian is given by ^ y Hφ = φa^ina^in using the convention ¯h = 1. This Hamiltonian commutes with the number operator, and so the total photon number is a conserved quantity. FIG. 1. Diagrams of a beamsplitter (a) and a polarizing beamsplitter (b) with incoming and outgoing modes depicted [1] In general, the interaction Hamiltonion of a beamsplitter can be written iφ y ^ −iφ ^y HBS = θe a^inbin + θe a^inbin This operator also commutes with the number operator, and the number of photons is again a conserved quantity. This description also applies to polarization evolution due to the action of a waveplate. Rather than spatial modes, incoming modes have different polarization, and the angles θ and φ refer to angles of rotation. We may also construct polarizing beamsplitters, often cut to separate horizontal and vertical polarization. In this case, the transformation is as follows ^ a^in;H ! a^out;H ; a^in;V ! bout;V ^ ^ ^ bin;H ! bout;H ; bin;V ! a^outV Where the input and output modes are as depicted in Fig 1. III. QUBITS IN LINEAR OPTICS Th e qubits are quantum systems with SU(2) symmetry. In the case of LOQC, the typical qubit is a photon that can be in one of two modes. Qubits are encoded with dual rail logic, in this case with respect to spatial modes. We define logical zero and one: j0iL ≡ j1ia ⊗ j0ib j1iL ≡ j0ia ⊗ j1ib Where a denotes one spatial mode and b denotes another. It is mathematically equivalent to use polarization instead of spatial modes, defining j0iL = jHi ; j1iL = jV i. While the two are equivalent, going forward this paper will be concerning mostly dual rail logic with spatial modes. IV. SINGLE-QUBIT GATES In order for a physical system to be capable of quantum computation, we require a universal set of gates [2]. We will first discuss single-qubit operations, which are generated by the Pauli operators σx; σy; σz, as any single-qubit 3 unitary can be decomposed into rotations in the Bloch sphere. As such, all single-qubit gates can be implemented using only beamsplitters and phase shifters. Rotations about the Z axis can be accomplished using a phase shifter. Consider the application of a phase shifter on the first mode of the qubit α j0iL + β j1iL: α j0iL + β j1iL = α j01i + β j10i ! α j01i + βeiφ j10i iφ/2 −iφ/2 iφ/2 = e (e α j0iL + e β j1iL) iφ/2 = e RZ (φ)(α j0iL + β j1iL) Up to an arbitrary phase factor, a rotation about the Z axis of the Bloch sphere has been performed. Rotations about the Y axis can be performed by a beamsplitter. A rotation of 2θ requires a beamsplitter with angle θ, and φ = 0, as the action of a beamsplitter on a photon in modes a and b is a^y ! (cos(θ)^ay + sin(θ)^by) ^by ! (−sin(θ)^ay + cos(θ)^by) Let us consider the action of the beamsplitter on an arbitrary qubit α j0iL + β j1iL: α j0iL + β j1iL = α j01i + β j10i ! α(cos(θ) j01i − sin(θ) j10i) + β(cos(θ) j10i + sin(θ) j01i) = cos(θ)(α j01i + β j10i) − sin(θ)(α j10i − β j01i) iθYq = e (α j0iL + β j1iL) = RY (−2θ)(α j0iL + β j1iL) We can see that a rotation of −2θ about Y has been accomplished. From here we may construct any deterministic single-qubit gate. V. THE KLM PROTOCOL Multi-qubit gates are difficult to construct with photons, as photons do not interact with one another- actually we cannot construct deterministic two-qubit gates using linear optics. In 2001, Knill, LaFlamme, and Millburn constructed a protocol to create universal quantum computers using linear optics. This protocol addressed the difficulty of constructing multi-qubit gates by performing probabilistic operations offline, and then boosting the probability of success using teleportation [3]. The gate we will focus on is the controlled-phase sign gate (CZ). Control Target CZ j0i j0i j00i j0i j1i j01i j1i j0i j10i j1i j1i -j11i q1q2 This transformation can be expressed as jq1q2i ! (−1) jq1q2i and can be constructed using two nonlinear-sign (NS) gates [3]. The NS gate transforms the three lowest Fock states as α j0i + β j1i + γ j2i ! α j0i + β j1i − γ j2i Consider Fig 2 with input jφ1i ⊗ jφ2i = (α j01i + β j10i)(γ j01i + δ j10i), which is separable. The NS gates impart a phase shift π, and after the second beamsplitter the output is αγ j00i + αδ j01i + βγ j10i − βδ j11i, which is entangled. While an NS gate cannot be constructed with linear optics, the gate can be constructed probabilistically with the 4 FIG. 2. CZ gate constructed from two NS gates and two beamsplitters [1] 1 use of projective measurements [1]. The maximum probability of success for an NS gate is 4 , which means that the 1 probability of success of a CZ gate constructed from two NS gates will be at most 16 . This, of course, is insufficient for scalable LOQM, as it is very likely that the gate will fail, overwhelmingly likely as the number of probabilistic gates increase. The solution is then to remove the probabilistic gate from the circuit, prepare it as an off-line resource, and then "teleport" the gate into the circuit. That is, the CZ gate is applied offline (probabilistically), and once we know that the gate was successful, the qubits are teleported into the circuit and the information remains uncorrupted. FIG. 3. Teleportation using a QFT[1] We are interested then in teleporting an arbitrary state α j0i + β j1i, as linearity ensures that if this is possible, any superposition of the arbitrary state is also possible. This is accomplished using an n+1 point discrete quantum Fourier transform (QFT)(Which can also be seen as a generalized 50:50 beamsplitter [4]). We note that the QFT erases path information of incoming modes. In the quantum channel we have the general state n 1 X j n−j j n−j jtni = p j1i j0i j0i j1i n + 1 j=0 We can write the QFT in matrix notation as 1 (i − 1)(j − 1 (F ) = p exp 2πi n ij n n The QFT is applied to the input and to the first n modes of jtni. The number of photons m in the output mode are counted, and the state is teleported to the n+m mode of the quantum channel. Measuring m photons means either the input mode did not have a photon, and the m photons came from jtni, or the input mode did have a photon, and the other photons originated from jtni. But by construction, the m mode of the remaining jtni modes must have the same number of photons as the input mode. The erasure of incoming mode path information means that these two possibilities are added. Thus, the qubit has been teleported to the n+m mode. There is still a chance of failure- if 0 5 photons are counted in the output, the state collapses to j0i, and if n+1 photons are counted, the state collapses to n 2 j1i. But this chance is far smaller than that of the probabilistic gate alone; the probability of success is then ( n+1 ) for the teleportation of two qubits. This chance of failure may be further minimized through error correction. We should note the complexity of the required auxiliary states.