Quantum Fourier Transform

In Quantum Computing, the Quantum Fourier Transform (QFT) is behind the exponential gain in the speed of algorithms [10] such as Shor s factoring algorithm [11,12], The operator QFT can be implemented using only Q(n ) operations, whereas its classical analogue, the Fast Fourier Transform (FFT) requires about Q(n2 ) operations. Therefore, QFT is implemented exponentially faster than the FFT. 

Now, the QFT must be applied. Assuming that r = 11 qubits were used on the first register, the value of s/r will be determined with a precision of 1/4. This means that there are 2 = 2048 allowed states, and after the application of the inverse Quantum Fourier Transform the system will be in superposition of the following states 0), 512), 1024) and 11536) ... 

This last procedure, the application of the inverse Quantum Fourier Transform, is not straightforward, but can be easily calculated numerically, and the outcome of this result depends on the number of qubits in the first register. 

Quantum Fourier Transform (QFT), as explained in Chapter 3, is a key step for quantum algorithms which exhibit exponential speed up. Its main application is in the Shot s factorization algorithm, which uses order finding and period finding [12]. These are in turn variations of the general procedure known as phase estimation [13],... 

Y.S. Weinstein, T.F. Havel, J. Emerson, N. Boulant, M. Saraceno, S. Lloyd, D.G. Cory, Quantum process tomography of the Quantum Fourier Transform, J. Chem. Phys. 121 (2004) 6117. 

Pig. 4. Photo dissociation of ArHCl. Left hand side the number of force field evaluations per unit time. Right hand side the number of Fast-Fourier-transforms per unit time. Dotted line adaptive Verlet with the Chebyshev approximation for the quantum propagation. Dash-dotted line with the Lanczos iteration. Solid line stepsize controlling scheme based on PICKABACK. If the FFTs are the most expensive operations, PiCKABACK-like schemes are competitive, and the Lanczos iteration is significantly cheaper than the Chebyshev approximation. 

One of the teehniques to whieh ehemists are frequently exposed is Fourier transforms. They are used in NMR and IR speetroseopy, quantum meehanies, and elassieal meehanies. 

Besides the intrinsic usefulness of Fourier series and Fourier transforms for chemists (e.g., in FTIR spectroscopy), we have developed these ideas to illustrate a point that is important in quantum chemistry. Much of quantum chemistry is involved with basis sets and expansions. This has nothing in particular to do with quantum mechanics. Any time one is dealing with linear differential equations like those that govern light (e.g. spectroscopy) or matter (e.g. molecules), the solution can be written as linear combinations of complete sets of solutions. 

The isotope has a nuclear spin quantum number I and so is potentially useful in nmr experiments (receptivity to nmr detection 17 X 10 that of the proton). The resonance was first observed in 1951 but the low natural abundance i>i S(0.75%) and the quadrupolar broadening of many of the signals has so far restricted the amount of chemically significant work appearing on this rcsonance, However, more results are expected now that pulsed fourier-transform techniques have become generally available. 

Much of the regularity in classical systems can often be best discerned directly by observing their spatial power spectra (see section 6.3). We recall that in the simplest cases, the spectra consist of few isolated discrete peaks in more complex chaotic evolutions, we might get white noise patterns (such as for elementary additive rules). A discrete fourier transform (/ ) of a typical quantum state is defined in the most straightforward manner ... 

Given equilibrium quantum expectation values, we can calculate moments of the infra-red vibrational lineshape using a procedure originally outlined by Gordon.The infrared vibrational lineshape is given as the Fourier transform of the dipole moment correlation function ... 

Section IIC showed how a scattering wave function could be computed via Fourier transformation of the iterates q k). Related arguments can be applied to detailed formulas for S matrix elements and reaction probabilities [1, 13]. For example, the total reaction probability out of some state consistent with some given set of initial quantum numbers, 1= j2,h), is [13, 17]... 

Figure 1.33 The underlying principle of the Redfield technique. Complex Fourier transformation and single-channel detection gives spectrum (a), which contains both positive and negative frequencies. These are shown separately in (b), corresponding to the positive and negative single-quantum coherences. The overlap disappears when the receiver rotates at a frequency that corresponds to half the sweep width (SW) in the rotating frame, as shown in (c). After a real Fourier transformation (involving folding about n = 0), the spectrum (d) obtained contains only the positive frequencies.

Single-quantum coherence is the type of magnedzadon that induces a voltage in a receiver coil (i.e., Rf signal) when oriented in the xy-plane. This signal is observable, since it can be amplified and Fourier-transformed into a frequency-domain signal. Zero- or multiple-quantum coherences do not obey the normal selection rules and do not... 

Tsoucaris, decided to treat by Fourier transformation, not the Schrodinger equation itself, but one of its most popular approximate forms for electron systems, namely the Hartree-Fock equations. The form of these equations was known before, in connection with electron-scattering problems [13], but their advantage for Quantum Chemistry calculations was not yet recognized. 

QUANTUM CHEMISTRY COMPUTATIONS IN MOMENTUM SPACE Fourier transform of orbital products ... 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

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