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# Technical challenges of quantum computers

## Requirements for physical implementations

DiVincenzo has classified the main ingredients that a physical system should possess in order to be a candidate for a quantum computer. They are:

• Quantum register: A scalable physical system with well characterized representing qubits that in turn compose the quantum register.
• Initialization: The ability to prepare the state of the register in a initial state.
• Universal set of gates: The ability to implement a universal set of logic gates.
• Low error and decoherence rate: High fidelity of gate operation, with probability per gate < 10−3 and qubit decoherence times that are longer than the gate operation time.
• Read-out: The ability to reliably measure the state of individual qubits the computational basis

With the knowledge of the basic requirements, here is an actual scheme for a physical implementation of quantum computation.

## Nuclear magnetic resonance

Nuclear magnetic resonance is a spectroscopic tool that is used to get the structural information of molecule due to the chemical shift and Zeeman effect on the resonant frequencies of the nuclei. With a spin of S=1/2, the distance between the two energy levels is proportional to the magnetic field strength. With the help of a coil, which generates a radio frequency (RF) magnetic field, it is possible to detect transitions between the different spin states.

Qubits

The qubits can be realized by using the spins of freely floating molecules. A quantum register now needs several distinguishable qubits. We can realize that by the fact that different molecules have different Larmor frequencies. So spins whose Larmor frequency differs from the frequency of the radio frequency pulse are not affected by the pulse.

Pseudo effective pure states

Because detecting individual spins is extremely difficult in most cases, signals can be detected only from an ensemble of spins, containing in the order of $10^\left\{20\right\}$ spins. For the description of the mixed states a density operator is used. The corresponding average density operator corresponds to the sum of the unit operator (the totally mixed state) and a pseudo pure state. One big disadvantage of this process is that one loses signal by destroying polarization. This loss of polarization, which increases exponentially with the number of spins in a register, restricts the usefulness of NMR quantum computing.

NMR signals

NMR signals are obtained in the time domain as a response of the system to an RF pulse. The system is virtually unaffected by the measurement. Of course, this is so because the system consists of an ensemble of many spins and not just a single particle. The observation of the spins is achieved through the Faraday effect. The polarized spin ensemble is a macroscopic magnetization. The generated signal is known as free induction decay (FID). This signal is analyzed in the frequency-domain after an Fourier transform. The frequency-domain contains the same information as the time-domain, but it makes it possible to distinguish between different transitions. Two distinct transitions usually have different Larmor frequencies. Their corresponding resonance lines are therefore separated in the frequency space.

Gates

One qubit gates are easily implemented using resonant RF pulses, which excite the corresponding spins. By using pulses with narrow excitation bandwidths it is possible to select single spins, and this perform operations on individual qubits. The implementation of a NOT, for example, can be applied by a 180 degree pulse, which inverts the two states $|0rangle$ and $|1rangle$. Two qubit gates can be implemented by combining one qubit gates with the spin-spin coupling. That way, for example, a Cnot gate can be implemented.

Measurement

Because all spins have a different Larmor frequency to allow addressability for logical operations, their precision frequencies during detection also will be different. A Fourier transformation of the FID from such a system, therefore separates the contributions from different qubits in frequency space.

## Problems

Practical implementations of the many-qubit systems are still in their infancy. NMR based quantum computers, which are rather successful in terms of the number of qubits realized so far because they are not scalable. It is still very difficult to resolve the NMR frequencies of individual qubits and the measurement signal, as the number of qubits becomes larger than a dozen. A more complete list of Quantum Computer implementations can be found at Quantum Computing.

## References

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