Quantum phase estimation

There are several concepts for the description of phase in quantum theory at present. Some of them are accenting the theoretical aspects, other the experimental ones. Quantization based on the correspondence principle leads to the formulation of operational quantum phase concepts. For example, the well-known operational approach formulated by Noh et al. [63,64] is motivated by the correspondence principle in classical wave theory. Further generalization may be given in the framework of quantum estimation theory. The prediction may be improved using the maximum-likelihood estimation. The optimization of phase inference will be pursued in the following. 

The performance of the Gaussian and Poissonian phase estimators have been determined in a series of experiments utilizing two principal sources of particles laser light [71] and a beam of thermal neutrons [70], The main goal of the experiments was to compare the optimum phase prediction with the NFM semiclassical theory in the (quantum) regime of only a few input particles. As a side result, the theoretical asymptotic uncertainties given in Table V were tested in a real experiment. 

A precise analysis of the noise is not the only merit of the phase estimation. Since phase detection represents an indirect measurement, it could serve as the simplest example of quantum tomography. Similar treatment inspired by the ML estimation may be applied to the reconstruction of a generic quantum state, namely that of entangled qubits. 

A phase diagram based on measurements of viscosity, electrical conductivity, and the temperature coefficient of electrical conductivity has been constructed for the system CF3 COaH-HSOsF the following equilibria are considered to be established HSO3F + CFa COzH HS03F,CF3 -C02H [S03F]-[CF8-C0aH2]+ Quantum chemical estimates of rotational barriers and dipole moments have been made for the compounds CFsX-COY (X = HorF Y = For OH) in order to facilitate interpretation of their microwave spectra. 

The factorization algorithm has 4 stages, but only the last one is quantum in nature. In fact, it turns out that the factorization problem can be reduced to an order finding problem, which can be implemented using basically the same quantum routine for phase estimation. Thus, phase estimation and order finding are subroutines to Shor algorithm, and they will be discussed in the next subsections. 

Figure 3.12 Quantum circuit to implement the routine of phase estimation. Adapted with permission from [1],...

Finding the order of a number is a difficult task to classical computers, particularly if the number is large. Further below, it will be shown why this routine can be used to find the non-trivial factors of a number. Here we will be focused on the necessary quantum procedures for implementing this routine, that is illustrated on Figure 3.13. As one can see, this is just the phase estimation routine, with a different input, at the second register. 

With the eigenket of the operator U at hand, we can use the phase estimation routine for finding the order, r, such as = 1 (mod N). For that, the system must be prepared in the initial state iri i) = 0)/ 1), and a Hadamard gate applied on every t qubit of the first register, in order to create a uniform superposition. The quantum state of the system is then... 

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],... 

Another interesting implementation of the QFT in a three-qubit system (the three of alanine) was reported by Weinstein et al. [15]. With the technique, the authors measured the periodicity of an input state, which was followed by quantum state tomography. Their result is shown in Figure 5.6. Other interesting NMR implementation of QFT can be found in Lee et al. [16] and Weinstein et al. [17]. The first applied QFT to phase estimation and quantum counting, and the second performed the quantum process tomography of QFT. 

The accuracy of the CSP approximation is, as test calculations for model. systems show, typically very similar to that of the TDSCF. The reason for this is that for atomic scale masses, the classical mean potentials are very similar to the quantum mechanical ones. CSP may deviate significantly from TDSCF in cases where, e.g., the dynamics is strongly influenced by classically forbidden regions of phase space. However, for simple tunneling cases it seems not hard to fix CSP, by running the classical trajectories slightly above the barrier. In any case, for typical systems the classical estimate for the mean potential functions works extremely well. 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

FIGURE 2.7. (a) Three active pz orbitals that are used in the quantum treatment of the X + CH3-Y— X-CH3 + Y Sw2 reaction, (b) Valence-bond diagrams for the six possible valence-bond states for four electrons in three active orbitals, (c) Relative approximate energy levels of the valence-bond states in the gas phase (see Table 2.4 for the estimation of these energies). 

With the estimates of Fig. 6.8 we are ready to determine the off-diagonal elements. These elements can be obtained by fitting our four-states gas-phase potential surface to the more rigorous six-states EVB surface given in ref. 6 (or to other gas-phase quantum mechanical surfaces) using the expression given in eq. (6.4). 

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