Single Qubit Arbitrary Unitary Synthesis using Photonic Spectral Encoding

Introduction: In recent years, the photonic implementation of an arbitrary unitary transformation has been a subject of interest because of the attractive idea of building all-photonic quantum integrated devices [1, 2]. In... [ view full abstract ]

Introduction: In recent years, the photonic implementation of an arbitrary unitary transformation has been a subject of interest because of the attractive idea of building all-photonic quantum integrated devices [1, 2]. In [2], the authors propose the concept of spectral linear optical quantum computation (spectral LOQC), where they use a dual-rail encoding scheme to encode a photonic qubit over two frequency modes as a resource for scalable quantum information processing. Furthermore, recent efforts have demonstrated theoretically [3] and experimentally [4] that it is possible to perform non-diagonal gates using electro-optic phase modulation.

Methods: In this work, we improve on the results developed in [2], exploiting the fact that any unitary can be synthesized using just two types of optical components: pulse-shapers (PS, for phase-manipulation of individual spectral modes) and electro-optic phase modulators (EOPM, for cross modal-couplings). By considering encodings more general than dual-rail, we develop a mathematical framework for single-qubit unitary synthesis.

Results: We first exhibit an encoding, within a 4-dimensional spectral mode space, allowing the exact synthesis of the Hadamard gate using just 3 components (2 PS and 1 EOPM, in the sequence PS₂ - EOPM - PS₁) over 4 modes. This improves over previous results from [2], where 4 components (2 PS and 2 EOPMs, in the sequence EOPM₂ - PS₂ - EOPM₁ - PS₁) and a minimum of 8 modes are required to reach > 99% fidelity.  We then show that the same 3 components are sufficient to exactly (100% fidelity) synthesize an arbitrary single-qubit unitary in the following form:

                                           U   =   PS₂ - EOPM - PS₁                                                 

Discussion: As most applications of quantum communications and cryptography can be achieved with single-qubit unitary operations, our technique offers a complete framework to implement and develop such technologies. At the same time, a judicious choice of encoding also enables realizing such unitaries with a minimum number of physical components.

References:

[1] J.-F. Morizur et al, J. Opt. Soc. Am. A 27, 2524–2531 (2010).

[2] J. M. Lukens and P. Lougovski, Optica 4, 8–16 (2017).

[3] J. Capmany and C. R. Fernandez-Pousa, Laser Photon. Rev., vol. 5, Nov. 2011.

 [4] L. Olislager et al, Phys. Rev. A 89, 052323 (2014).