Implementing one-photon three-qubit quantum gates using spatial light modulators

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Implementing one-photon three-qubit quantum gates using spatial light modulators

A. F. Abouraddy, G. Di Giuseppe, T. M. Yarnall, M. C. Teich, and B. E. A. Saleh

Phys. Rev. A 86, 050303(R) – Published 8 November 2012

Abstract

Increasing the information-carrying capacity of a single photon may be achieved by utilizing multiple degrees of freedom. We describe here an approach that utilizes two degrees of freedom to encode three qubits per photon: one in polarization and two in the spatial-parity symmetry of the transverse field. In this conception, a polarization-sensitive spatial light modulator corresponds to a three-qubit controlled-unitary gate with one control qubit (polarization) and two target (spatial-parity-symmetry) qubits. We describe the construction of controlled-not (cnot), $\sqrt[n]{c n o t}$ , controlled-phase, and Fredkin gates, and the preparation of one-photon, three-qubit Greenberger-Horne-Zeilinger (GHZ) and $W$ states. This approach enables simple optical implementations of few-qubit tasks in quantum information processing.

Received 20 July 2012

DOI:https://doi.org/10.1103/PhysRevA.86.050303

Authors & Affiliations

A. F. Abouraddy^1,*, G. Di Giuseppe^1,2, T. M. Yarnall^3,†, M. C. Teich^1,4, and B. E. A. Saleh¹

¹CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, Florida 32816, USA
²School of Science and Technology, Physics Division, University of Camerino, 62032 Camerino, Italy
³19 Chestnut Circle, Merrimack, New Hampshire 03054, USA
⁴Departments of Electrical & Computer Engineering and Physics, Boston University, Boston, Massachusetts 02215, USA

^*raddy@creol.ucf.edu
^†Currently with Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, MA 02420, USA.

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Vol. 86, Iss. 5 — November 2012

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Images

Figure 1
(a)–(c) $x$ -parity-symmetry operators. (a) A SLM imparting a phase distribution $\frac{θ}{2} h (x)$ corresponds to a parity-symmetry qubit rotation by an angle $θ$ ; (b) a dove prism that flips the field distribution along $x$ corresponds to the Pauli ${\hat{Z}}_{x}$ operator; and (c) a balanced MZI with ${\hat{Z}}_{x}$ placed in one arm corresponds to a projection operator in the even-odd basis. (d)–(f) The $y$ -parity-symmetry operators corresponding to (a)–(c). (g)–(k) Phase distributions in the $x$ - $y$ plane imparted by a SLM (black-bordered squares) to the transverse optical field in order to implement (g), (h) one-qubit and (i)–(k) two-qubit parity-symmetry rotation operators, and the corresponding quantum-circuit representations. Black lines correspond to 1D parity-symmetry qubits, and black boxes to rotation operators.Reuse & Permissions
Figure 2
Phase distributions on a PS-SLM (with dashed bottom and top borders) to produce controlled one- and two-qubit rotation operators. (a) Controlled- ${\hat{R}}_{x} (θ)$ or ${\hat{C}}_{x} (θ)$ , (b) ${\hat{C}}_{y} (θ)$ , (c) ${\hat{C}}_{x} (θ) \cdot {\hat{C}}_{y} (θ)$ , (d) ${\hat{C}}_{x} (θ_{1}) \cdot {\hat{C}}_{y} (θ_{2})$ , and (e) ${\hat{C}}_{x y} (θ)$ . In the corresponding quantum-circuit representations, the red (black) line corresponds to the polarization control (parity-symmetry target) qubit.Reuse & Permissions
Figure 3
Three-qubit quantum gates. (a) cnot $_{x}$ , (b) cnot $_{y}$ , (c) the cascade ${cnot}_{x} \cdot {cnot}_{y}$ , and (d) $\sqrt{{cnot}_{x}} \cdot \sqrt{{cnot}_{y}}$ gates, each implemented using a PS-SLM with the shown phase distribution. (e)–(g) Controlled-phase (CP) gates. PBS: polarizing beam splitter. The upper and lower paths in the MZI correspond to $| H 〉$ and $| V 〉$ polarization components, respectively. $Z$ refers to the appropriate Pauli- $Z$ operator acting on $x$ or $y$ parity. (h) A controlled-swap gate, or a quantum Fredkin gate.Reuse & Permissions
Figure 4
Preparing one-photon three-qubit entangled states. HWP: half-wave plate rotating $| H 〉$ by $ϕ$ : ${\hat{R}}_{p} (ϕ) | H 〉 = \cos ϕ | H 〉 + \sin ϕ | V 〉$ ; SLM: spatial light modulator (square with black boundary); PS-SLM: polarization-sensitive SLM (square with dashed bottom and top black boundary). The top row shows schematics of the optical arrangements to prepare the desired states starting from $| Ψ_{1} 〉 = | Hee 〉$ . The second row depicts the corresponding quantum circuit; the red line represents the polarization qubit and the black lines represent the $x$ - and $y$ -parity-symmetry qubits. The third row lists the prepared quantum state and the last row shows a pictorial representation of the state. Each qubit is represented by a circle (red for polarization, black for parity) and the blue lines represent entanglement between the connected qubits.Reuse & Permissions
Figure 5
(a) Schematic of an optical arrangement to implement a cnot gate with a control $x$ -parity-symmetry qubit and a target polarization qubit and (b) the equivalent quantum circuit. (c), (d) Corresponding implementation and circuit for $y$ -parity-symmetry control qubit.Reuse & Permissions

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