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A Radix-4 Chrestenson Gate for Optical Quantum Computation

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A Radix-4 Chrestenson Gate for Optical Quantum Computation A Radix-4 Chrestenson Gate for Optical Quantum Computation Kaitlin N. Smith, Tim P. LaFave Jr., Duncan L. MacFarlane, and Mitchell A. Thornton Quantum Informatics Research Group Southern Methodist University Dallas, TX, USA fknsmith, tlafave, dmacfarlane, mitchg @smu.edu Abstract—A recently developed four-port directional coupler is described in Section III followed by the realization of the used in optical signal processing applications is shown to be coupler with optical elements including its fabrication and equivalent to a radix-4 Chrestenson operator, or gate, in quantum characterization in Section IV. Demonstration of the four-port information processing (QIP) applications. The radix-4 qudit is implemented as a location-encoded photon incident on one of coupler as a radix-4 Chrestenson gate is presented in Section V the four ports of the coupler. The quantum informatics transfer and a summary with conclusions is found in Section VI. matrix is derived for the device based upon the conservation of energy equations when the coupler is employed in a classical II. QUANTUM THEORY BACKGROUND sense in an optical communications environment. The resulting transfer matrix is the radix-4 Chrestenson transform. This result A. The Qubit vs. Qudit indicates that a new practical device is available for use in the The quantum bit, or qubit, is the standard unit of information implementation of radix-4 QIP applications or in the construction for radix-2, or base-2, quantum computing. The qubit models of a radix-4 quantum computer. Index Terms—quantum information processing; quantum pho- information as a linear combination of two orthonormal basis tonics; qudit; states such as the states j0i and j1i. j0i and j1i are Dirac T notation representations where j0i = 1 0 and j1i = T I. INTRODUCTION 0 1 , respectively. The qubit differs from the classical bit In the field of quantum computing, theory has been well by its ability to be in a state of superposition, or a state of developed over the last half-century. There is still room for linear combination, of all basis states. There are theoretically the discovery of new operators and algorithms, but current an infinite number of states for a qubit while in a state of work is enough to show the potential that quantum methods superposition have for solving some of the most difficult scientific problems. T Unfortunately, the physical implementations for QIP have not jΨi = x j0i + y j1i = x y (1) advanced as rapidly as the theory. Since a standard platform has not been chosen for the quantum computer (QC), efforts where x and y are complex values, c 2 C, such that c = a+ib where i is an imaginary number, i2 = −1. For the qubit jΨi, have been divided among many competing technologies with ∗ 2 quantum optics being one of the more promising physical the probability that jΨi = j0i is equal to x x = jxj and the probability that jΨi = j1i is equal to y∗y = jyj2 where the realizations. This work hopes to contribute a new component ∗ to the already well-established optical quantum computing symbol indicates a complex conjugate. The total probability library. of occupying either one basis state or the other must total to 100%, so the inner product, or dot product, of jΨi with itself The four-port directional coupler is an optical component ∗ ∗ shown theoretically to act as a quantum gate that can place must equal 1. In other words, x x + y y = 1. Once a qubit a radix-4 photon encoded qudit into a state of equal super- is measured, it collapses into a basis state as defined by the position. The gate realized optically is known as the radix-4 eigenvectors of the measurement operator. The measurement Chrestenson gate, and it is derived from the generalized radix- operation causes a qubit’s state of superposition to be lost. p Chrestenson transformation. This operator is significant to Qubits are the current standard for encoding data in QIP, the field of QIP due to the need to put quantum information but it is possible to have a quantum system of higher order. in a state of superposition for quantum algorithm execution. A A quantum unit of dimension, or radix, p > 2 is referred description of the four-port coupler as well as a demonstration to as a qudit. In this paper, the radix-4 qudit using four orthonormal basis states is of interest. The set of basis states of its capabilities as a quantum optics operator will be shown T mathematically in this work. used for this qudit includes the vectors j0i = 1 0 0 0 , T T This paper will proceed as follows. A brief summary of j1i = 0 1 0 0 , j2i = 0 0 1 0 , and j3i = T important QIP concepts, details of the Chrestenson gate, and 0 0 0 1 . Just like the radix-2 qubit, the radix-4 qudit information about quantum optics is provided in Section II. is not limited to having the value of only one of its four The four-port directional coupler, the component of interest, possible basis states. The qudit is capable of existing in a linear combination, or a state of superposition, of all four basis groups is outlined in [6] as well as in [7]. The multiple- states, as demonstrated by valued generalization of the radix-2 quantum Hadamard gate and its transfer matrix is composed of discretized versions of T the orthogonal Chrestenson basis function set [7]. This gate is jΦi = v j0i + x j1i + y j2i + z j3i = v x y z (2) generally referred to as the Chrestenson gate [8]. The general- where v, x, y, and z are complex values. These coefficients ized radix-p Chrestenson transfer function is represented with can be multiplied by their respective complex conjugates in a matrix in the form of order to derive the probability that the radix-4 qudit is in a 2 3 particular basis state. The basis state probabilities of jΦi must w0(0) w1(0) : : : w(p−1)0 6 7 sum to 100%, so v∗v + x∗x + y∗y + z∗z = 1. 6 7 6 . 7 For a radix-4 quantum system to be physically realized, 6 w0(1) w1(1) ::: . 7 1 6 7 a methodology must exist for encoding four distinct qudit p 6 7 : (4) p 6 . 7 basis states. In [5], Rabi oscillations are utilized to create 6 . .. 7 6 7 radix-p quantum systems. Orbital angular momentum (OAM) 6 7 4 5 states of light could also be used to encode the qudit [16]. w0(p−1) : : : : : : w(p−1)(p−1) In this paper, a radix-4 quantum state will be created using k the location of light as the information carrier. This technique Eq. 4 contains elements w based on the p-th roots of unity builds on the concept of photonic dual-rail representation of found on the unit circle in the real and imaginary plane accord- the qubit in order to physically realize qudit with a quad-rail ing to w = exp(i2π=p) [7], [8]. The resulting values satisfy k p implementation. (w ) = 1 as roots of unity. Using the fourth roots of unity, exp[(i2π=4)∗1] = i; exp[(i2π=4)∗2] = −1; exp[(i2π=4)∗3] = B. Quantum Operations −i; and exp[(i2π=4) ∗ 4] = 1, in the general form in Eq. 4, A quantum state must undergo purposeful transformations the radix-4 Chrestenson gate transfer matrix of is derived as through quantum operations in order to develop meaningful 2 3 information from QIP. If a quantum algorithm is modeled 1 1 1 1 6 7 as a circuit, quantum operations can be viewed as quantum 6 7 1 61 i −1 −i7 logic gates. Each of these gates is represented by a unique, C4 = p 6 7 : (5) 4 6 7 unitary transfer function matrix, U, that is characterized by 61 −1 1 −17 4 5 the following properties: 1 −i −1 i y y • U U = UU = Ip −1 y The radix-4 Chrestenson gate (C4), allows a radix-4 qudit • U = U originally in a basis state to evolve into a state of equal • Rank(U) = p superposition. The example below shows how the radix-4 qudit • jUj = 1 1 1 1 1 jai = j0i evolves to jbi = 2 j0i + 2 j1i + 2 j2i + 2 j3i after When considering radix-p quantum operations, or gates, the passing through the C4 transform transfer function matrices will always be square matrices each of a dimension that is a power of p. Therefore, radix-4 qudit C4 jai = jbi ; operations will have a dimension that is a power of four, 4k, where the power, k, indicates the number of qudits transformed 2 3 2 3 2 3 by an operation. 1 1 1 1 1 1 6 7 6 7 6 7 6 7 6 7 6 7 C. The Chrestenson Gate 1 61 i −1 −i7 607 1 617 C4 j0i = p 6 7 6 7 = 6 7 ; The power of QIP lies in the ability for a quantum state 4 6 7 6 7 2 6 7 61 −1 1 −17 607 617 to be in superposition. In radix-2 quantum computation, the 4 5 4 5 4 5 Hadamard gate 1 −i −1 i 0 1 1 2 3 C4 j0i = [j0i + j1i + j2i + j3i]: 1 1 1 2 p 4 5 (3) 1 2 1 −1 If jai = j3i, the radix-4 qudit would evolve to jbi = 2 j0i − 1 1 1 2 i j1i − 2 j2i + 2 i j3i is an important operator used to put a qubit in a state of superposition so that it has equal probability of being observed, 2 3 2 3 2 3 1 1 1 1 0 1 or measured, as either j0i or j1i.
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