Quantum Computing based on Nuclear Magnetic Resonance (NMR) technologies is one proposal for a potential Quantum Computation device. The method makes use of an ensemble of nuclear spins as the Qubit. The nuclear spins of different atoms in a molecule can be manipulated by Radio Frequencies (RF) and be observed through their Free Induction Decay (FID).

**Introduction**

If a strong, constant magnetic field is applied in one direction (let's set it to the Z-axis) it gives rise to a precession of the nuclear spin around the Z-axis. That frequency is called *Larmor Frequency* and depends only on the external magnetic field and on the nuclei's magnetogyric ratio. Each nuclei has a special Larmor Frequency and can therefore be distinguished in a spectrum. Even atoms of the same species can be distinguished, taken that the molecule doesn't have symmetries. They 'see' a different environment in the molecule and that changes their energy slightly. This phenomena is known as the *Chemical Shift*.

The use of a *Rotating Frame* makes these calculations a lot easier. In a Rotating Frame the spin's sate is a constant in time and doesn't precise around the Z-axis any more. Instead the magnetic field oscillates at Larmor Frequency relative to the spins static Z-axis.

A resonant circuit generates an oscillating weak magnetic field: a RF pulse. That RF has to be in resonance with the Larmor Frequency and gives rise to a change in the spin's state. Usually the spins states are desribed in terms of the Bloch Sphere with each Pauli Matrix on one axis. A RF pulse changes either the population (polarization) and moves the spin from the +Z axis to the -Z axis or changes the coherences (transitions) which are described in the Bloch Sphere as an oscillation in the X-Y plane around the Z-axis.

The spins decohere. As soon as the RF pulse is applied the spins move spirally back to their original position in the static magnetic field at the Z-axis. By doing that they cause a FID and the exponential decay of the oscillating magnetic field can be measured. After a Fourier Transform the different Larmor Frequencies can be determined in a spectrum. The real part gives an Absorbtion Lorentzian Sreal(ω) and the imaginary part a Dispersion Lorentzian Simag(ω). The whole signal is than S(ω)= Simag(ω) + iSreal(ω). The imaginary and the real part are rectangular in the complex plane and can be separated by a phase factor. In a system with 2 qubits we get 4 peaks in the spectrum. FID In a larger system there will be 2^{n} multiplets, where n is the number of qubits. That means the Density matrix is really huge! As huge as the whole Hilbert Space of the system. At the same time it means that there will be a lot of possible outcomes with only a few qubits. Two single transistors give rise to 4 possitions. Two nuclear spins can have more than 4 distinguishable spectrums. Some examples: Spin1Spin2Spin3Spin4 Possible outcomes: IX+XIIX+YIIX+ZIIY+XI IY+YIIY+ZI IZ+XIIZ+YI IZ+ZI

**Theory**

**State and Density Matrix**

The calculation of each individual spin is impracticle and probably not possible. But there's an alternative: the density operator. The density operator makes it possible do describe the quantum state of the entire ensemble.

1. Quantum description: *ρ* = ∑_{i = 0}^{n}|Ψ_{i} > < Ψ_{i} |

2. Classical (statistic) description: in terms of Boltzman statistic the density matrix can also be written as $\rho = \frac{\exp(-\beta E_k)}{Z}$ where $\beta = \frac{1}{k_B T}$ with the Boltzman factor *k*_{B} and T as the temperature. $E_k = \frac{1}{2} h \omega_k$ is the eigenergy of the spin with Planck's constant h and *Z* = ∑_{k = 1}^{n} *E*_{k} as the partition sum. In terms of matrix description the sum in Z has to be replaced by the trace which means nothing else than the sum over all eigenenergies (note that the density matrix has to be diagonal in this case). And *E*_{k} is being replaced by the Hamiltonian H of the system. In the folling the notations *σ*_{1} = *X*, *σ*_{2} = *Y*and *σ*_{3} = *Z* are used. If kT >> ωh it is possible to expand the exponential as a power series (*Taylor expansion*) and to take only the first term: Than $\rho = \frac{1}{2 Z}(1-\beta H) = \frac{1}{2 Z} (1-\sum \, r \sigma)$ The identity matrix 1 cannot be measured and is not visible in the spectrum.

**Hamiltonians**

The whole system Hamilton is described by: *H*_{S} = *H*_{Z} + *H*_{J} + *H*_{E} + *H*_{P}

1. Zeeman Engergies: *H*_{Z} = ∑_{i = 1}^{k} *ω*_{k}*Z*_{k} describes the sum of all Zeeman energies in a static magnetic field in Z direction. *ω*_{k} denotes the Larmor frequency of Spin No. k and to make it easier some constants have been ignored. If there is more than one spin you have to take care of the dimension of your matrix. Tensor the identity matrix to each Z in the correct order. 2. J-coupling term: *H*_{J} = ∑_{i = 1}^{k} *Z*_{j} ⊗ *Z*_{k} describes the interaction energy of two nuclear spins. here the *Ising Model* is used which ignores *X* ⊗ *X* and *Y* ⊗ *Y*couplings and only takes care of the next neighbour interaction. 3. Environment: *H*_{E} In Liquid State NMR the molecules environment averages to zero. In solid state NMR this term usually has to be measured. 4. The Pulse Hamiltonian or RF Hamiltonian: *H*_{R} depends on what you want the spin to do. A pulse is can be written as a unitary matrix $U_R = \exp(\frac{-i}{h}\phi H_S t)$in generally. Some examples: (constants are beeing ingored in the following)

Y_90: rotates Z to X $U_R = \exp(\frac{-1}{2}\alpha \pi Y)$(don't forget to tensor the identity matrix to Y according to the spin you want to adress) The spin will rotate in the X-Y plane now. This movement is called the

free time evolutionexp(2JZ⊗Z). Unfortunatelly the spins decohere in some seconds. X_90: rotates Z to Y $U_R = \exp(\frac{-1}{2}\alpha \pi X)$ Y_180: rotates Z to -Z $U_R = \exp(\frac{}{}\alpha \pi X)$

The (basis)-transformation or the density matrix after the pulse looks like: *ρ̃* = *U*ʹ_{R}*ρ**U*_{R}

In solid State NMR there will be more complicated internal Hamiltonian terms. Usually they are so complicated that it is a lot easier to calculate the Hamiltonian with the help of the spectrum.

**advantages:** - NMR technology is already well used in Chemistry and in Medicine to distinguish elements. - The experimental set up doesn't has to change much and reliable experiments have been already done.

**disadvantages:** - NMR is not scalable up to the recent state of knowledge. Scalable in terms of pulses means that a gate or an algorithm can be decomposed in less than polynomial many operations. The amount of pulses is important because the gate's performance time has to be faster than the spins decoherence time. That's one of the major problems in QC. But people work on finding pulses and I'll keep my fingers crossed.

*This page is still under construction* Category:Physical Realisations Category:NMR