Quantum Computing

Quantum Logic Processing

		Conventional computers process information by breaking it up into its component bits and operating of those bits a few at a time. These computers consist primarily of electronic circuits including bits, wires, and gates. Bits can be implemented by ferrite cores (in memory), magnetic spots (in hard-disk), or the on and off of the voltages. These bits can be sent along wires to the logic gates for processing. It has been shown that any desired logical expression, including complex mathematical calculations, can be built up out of the OR, AND, NOT, and COPY gates (see Figure 06d).
Figure 06d Logic Gates [view large image]	Figure 07 Controlled-NOT Operation

In quantum computing the bits can be implemented by nuclear spins (up or down), polarization of the photons (horizontal or vertical), superconducting loop (clockwise or counter-clockwise of the supercurrent loop), ... etc. Each bit is now associated with states |1

or |0

corresponding to, e.g., spin up or spin down. The states can be superposed to form |1

+ |0

or
|1

- |0

corresponding to rotate the spinning axis 90^o or 270 ^o respectively. The spinning axis can be flipped by radio wave matching the hyperfine structure (the energy difference between the spin up and down states) of the nuclear spin. The axis would be rotated by 90^o if the wave is applied for 1/4 of the time it takes for the spin to precess one cycle, and so on.

Entanglement is achieved by the controlled-NOT logic operation as shown in Figure 07. It transforms the quantum states:
|0

to |0

,
|0

to |0

,
|1

to |1

,
|1

to |1

.
That is, the controlled-NOT operation flips the input state whenever the control state is |1

. Such controlled-NOT logic gate can be constructed by interaction with the kind of radio wave mentioned above. It has been shown that the rotations of individual quantum bits, together with the controlled-NOT operations constitute a universal set of quantum logic operations similar to the classical logic operations in Figure 06d.

The advantage with quantum computing associates mainly with "quantum parallelism", which allows a single quantum processor to

		performs several tasks at once. For example considering the \|1 + \|0 state, each one of the two components can be processed individually at the same time, i.e., a quantum computer can perform two computations simultaneously. The concept can be generalized to more than two input states by superposing many input states into a single entangled state (Figure 08a). It is like the individual instruments in a symphony (Figure 08b), each one plays its own notes. The combination of all the different tones makes the music rich and pleasing. One of the problems with quantum computing is that the processing cannot be disturbed in the
Figure 08a # of Entangled States	Figure 08b Symphonic Parallelism [view large image]	middle of its run, otherwise the operation will be terminated prematurely by decoherence.

Figure 08c Quantum Computing Example [view large image]

a. Initialization - The qubits are in the form of spinning particles with initial state (0, 0). They are enclosed in a copper case to minimize decoherence.
b. Superposition - The computation starts with an operation (shown here as light interacting with the particles) that changes the states to a configuration of uniform superposition. Each state has an amplitude of 0.5 (corresponding to a probability of 1/4).
c. The Black Box - The action in the black box is to mark one state by changing the phase of its amplitude. Quantum parallelism allows the interaction to take place simultaneously on all 4 states (whether changed or not). The problem is to find out which one has its phase switched.
d. Conversion - Since the measured outcomes depend only on the amplitude, an additional step is required to convert the marked phase to marked amplitude. This is done by applying another set of light to produce an interference pattern such that the 0.5 amplitude becomes zero and the -0.5 amplitude turns into 1.
e. Measurement - The state of the qubits is measured by opening the isolating case and checking the orientations of the spins, which turns out to be (1, 0). This quantum computation requires a single application of the black box. Analogous classical computations would need up to three tries by checking for the possible change in (1, 1), (1, 0), (0, 1) - it would certainly be (0, 0) if the others remain unaltered.