Quantum Fourier Transforms

Quantum Fourier transform (QFT) plays a key role in many quantum algorithms, but the existing circuits of QFT are incomplete and lacking the proof of correctness. Furthermore, it is difficult to apply QFT to the concrete field of information processing. Thus, this chapter firstly investigates quantum vision representation (QVR) and develops a model of QVR (MQVR). Then, four complete circuits of QFT and inverse QFT (IQFT) are designed. Meanwhile, this chapter proves the correctness of the four complete circuits using formula derivation. Next, 2D QFT and 3D QFT based on QVR are proposed. Experimental results with simulation show the proposed QFTs are valid and useful in processing quantum images and videos. In conclusion, this chapter develops a complete framework of QFT based on QVR and provides a feasible scheme for QFT to be applied in quantum vision information processing.

Programmable Integrated Photonics for Quantum Systems

Programmable photonics can find applications in myriad areas including the quantum information field, which encompasses communications, computing, sensing and tomography. Large-scale bulk optics setups previously prevented the development of more complex and scalable quantum optics configurations. Linear optic systems with the required fidelity require a strict control of interference through demanding phase stability mechanisms. Integrating a considerable number of photonic elements on a chip in order to implement multi-port interferometers has become the only viable technological path towards quantum information systems. This chapter introduces the applications of programmable photonics to quantum information systems. After introducing the general framework of a programmable quantum photonic system integrated on a chip and briefly describing the role of more external components such as sources and detectors, it covers the relationship between reconfigurable integrated optic circuits and linear optical quantum gates, quantum transport simulation, boson sampling and complex Hadamard and quantum Fourier transforms.

Efficient classical simulations of quantum Fourier transforms and Normalizer circuits over Abelian groups

The quantum Fourier transform (QFT) is an important ingredient in various quantum algorithms which achieve superpolynomial speed-ups over classical computers. In this paper we study under which conditions the QFT can be simulated efficiently classically. We introduce a class of quantum circuits, called \emph{normalizer circuits}: a normalizer circuit over a finite Abelian group is any quantum circuit comprising the QFT over the group, gates which compute automorphisms and gates which realize quadratic functions on the group. In our main result we prove that all normalizer circuits have polynomial-time classical simulations. The proof uses algorithms for linear diophantine equation solving and the monomial matrix formalism introduced in our earlier work. Our result generalizes the Gottesman-Knill theorem: in particular, Clifford circuits for $d$-level qudits arise as normalizer circuits over the group ${\mathbf Z}_d^m$. We also highlight connections between normalizer circuits and Shor's factoring algorithm, and the Abelian hidden subgroup problem in general. Finally we prove that quantum factoring cannot be realized as a normalizer circuit owing to its modular exponentiation subroutine.

EXACT QUANTUM FOURIER TRANSFORMS AND DISCRETE LOGARITHM ALGORITHMS

We show how the Quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes). Most quantum algorithms only need a good approximation of the quantum Fourier transform of order 2n to succeed with high probability, and this QFFT can in fact be done exactly. Kitaev1 showed how to approximate the Fourier transform for any order. Here we show how his construction can be made exact by using the technique known as "amplitude amplification". Although unlikely to be of any practical use, this construction allows one to make Shor's discrete logarithm quantum algorithm exact. Thus we have the first example of an exact non black box fast quantum algorithm, thereby giving more evidence that "quantum" need not be probabilistic. We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely, the parameters of the gates can be approximated efficiently.