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Quantum Computing


Qubit and Entanglement
Quantum Logic Processing
Factoring and Encryption
Error Correction
Quantum Computing with Discord

Qubit and Entanglement

Quantum Computing Physics and Computer science have combined to create a new field: quantum computing and quantum information. The spark that ignited world wide interest in this new field sprang forth in 1994 with Peter Shor's discovery of a theoretical way to use quantum mechanical resources to unravel a mathematical problem at the heart of electronic commerce and cryptography.

Basic steps towards the creation of a quantum computer have been taken, with the demonstrations of elementary data storage and manipulation using photons and atoms or trapped ions as the quantum bits, or "qubits". Recently, it has been shown that it is possible to build solid-state qubits made from tiny samples of superconducting material. Figure 01 shows some of the subjects, which are currently being investigated in the field of quantum computing.

Figure 01 Qunatum Computing [view large image]

    There are several requirements for a working quantum computer:

  1. It must be scalable: it needs a set of qubits that can be added to indefinitely.
  2. It must be possible to set all of the qubits to a simple initial state, such as all 0.
  3. The interactions between qubits must be controllable enough to make quantum logic gates.
  4. To perform operations using these gates, the decoherence times must be much longer than the gate-operation time (typically milliseconds to seconds).
  5. There must be some readout capability.
  6. To link up the computer's circuitry, it must be possible to convert memory qubits into processing qubits, and vice versa.
  7. It must be possible to move processing qubits accurately between specified locations.
Qubit Qunatum computing exploits two resources offered by the laws of quantum mechanics: the principle of superposition of states and the concept of entanglement. Superposition is a "one-particle" property; while entanglement is a characteristic of two or more particles.

Consider a particle with spin such as the electron. With reference to a given axis (say along the z axis), the spin of the particle can point in two opposite directions, say "up" or "down", and the spin states can be denoted as |1 and |0. But by the laws of quantum mechanics, the particle can exist in a superposition of these two states (or wave of probability), corresponding to arbitrary orientation as shown in Figure 02.

Figure 02 Qubit [view large image]

Mathematically, the superposition of these two states can be written as:
|f = a |1 + b |0 ------ (1)
where a and b are related to the probability of finding the electron in state |1 and |0 respectively satisfying
|a|2 + |b|2 = 1. This normalization defines the total probability of finding the electron to be 1. In general, the
|1 and |0 states can be represented by any two-states entity such as "on" and "off", horizontal and vertical polarization of a photon, one particle vs no particle, ... etc.
|f is called a qubit. If a photon in state |f passes through a polarizing beamsplitter -- a device that reflects (or transmits) horizontally (or vertically) polarized photons -- it will be found in the reflected (or transmitted) beam with probability |a|2 (or |b|2). Then the general state |f has been projected either onto |1 or onto |0 by the action of the measurement (sometimes it is referred as collapse or decoherence of |f). Thus according to the rule of quantum mechanics, a measurement of the qubit would yield either
|1 or |0 but not |f (See Figure 02).
Entanglement1 Entanglement2 Now, consider a two-particle state: there are four "basis states",
|11|12, |01|02, |11|02 and
|01|12, where the subscript indicates particle 1 and 2. Again, superpositions can be made of these states, including in particular, the four "maximally entangled Bell states":
|11|12 + |01|02 ------ (2)
|11|12 - |01|02   ------ (3)

Figure 03 Entangle-ment
[view large image]

Figure 04 Entanglement Implementation
[view large image]

|11|02 + |01|12  ------ (4)
|11|02 - |01|12   ------ (5)
Such Bell states have the peculiar property that the particles always "know" about each other, even if they are separated by huge distances; this property is commonly associated with the "non-locality" of quantum mechanics. Entanglement such as this is a basic ingredient of quantum computing. Figure 03 shows that a measurement of one entangled member will determine the outcome for the other member -- either in the same state if the Bell state is Eq.(2), Eq.(3) or in the opposite state if the Bell state is Eq.(4), Eq.(5). Figure 04 shows an experiment that implements the entangled states of two photons.
Teleportation In teleportation there are three spin spaces entangled together. Suppose particle 1 which Alice wants to teleport is in the initial state:
|f1 = a |11 + b |01 ------ (6)
and the entangled pair of particles 2 and 3 shared by Alice and Bob is in the state:
|f23 = (|12|03 - |02|13)/21/2 ------ (7)

Figure 05 Teleportation [view large image]

which is produced by an Einstein-Podolsky-Rosen (EPR) source1.

The teleportation scheme proceeds as following. Alice has the particle 1 in the initial state |f1 and particle 2. Particle 2 is entangled with particle 3 in the hands of Bob. The essential point is to perform a joint Bell-state measurement (BSM)2 on particles 1 and 2 which projects them onto the entangled state:
|f12 = (|11|02 - |01|12)/21/2 ------ (8)
This is only one of four possible Bell states into which the two particles can be entangled.

According to the rule of quantum physics once particles 1 and 2 are projected into |f12, particle 3 is instantaneously projected into the initial state of particle 1. (See Figure 05). This is because when we observe particles 1 and 2 in the state |f12 we know that whatever the state of particles 1 is, particle 2 must be in the opposite state. But we had initially prepared particle 2 and 3 in the state |f23, which means particle 2 must be in the opposite state of particle 3. This is only possible if particle 3 is in the same state particle 1 was initially. The final state of particle 3 is therefore:
|f3 = a |13 + b |03 ------ (9)
Note that during the Bell-state measurement particle 1 loses its identity because it becomes entangled with particle 2. Therefore the state |f1 is destroyed on Alice's side during teleportation.

The transfer of quantum information from particle 1 to particle 3 can happen instantly over arbitrary distances, hence the name teleportation. Experimentally, quantum entanglement has been shown to survive over distances of the order of 10 km. In the teleportation scheme it is not necessary for Alice to know where Bob is.

A complete Bell-state measurement not only give the result that the two particles 1 and 2 are in the antisymmetric state in Eq.(8), but with equal probabilities of 25% we could find them in any one the remaining three Bell states. When this happens, the state of particle 3 is determined by one of these three different states. Therefore Alice has to inform Bob, via a classical communication channel, which of the Bell state result was obtained; depending on the message, Bob leaves the particle unaltered or changes it to the opposite state. Either way it ends up a replica of particle 1. It should be emphasized that even if it can be demonstrated for only one of the four Bell states as discussed above, teleportation is successfully achieved, albeit only in a quarter of the cases (see "Entanglement and Teleportation" for further detail).

Table 01 lists the types of qubit that have been tried up till 2010.

Type Qubit Initialization Interaction Data Transmission Detection Coherence
Error Rate(%)
1 or 2 Qubits
Infrared Photon Polarization Stimulated Emission Beam Splitter Waveguide Avalanche Photodiode 0.1 ms 0.016/1
Trapped Ion Energy Levels
Optical Pumping Electric Fields Induced Vibrations Optical Fluorescence 15 s 0.48/0.7
Trapped Atom Energy Levels
Optical Pumping Atomic Interaction Laser Beams Optical Fluorescence 3 s 5
Liquid Molecule Nuclear Spins Spin Orientations Radio-frequency Pulse Molecular Electron Coupling Radio-frequency Pulse Induced Current 2 s 0.01/0.47
e- Spin (GaAs Quantum Dot) Electron Spin States Optical Pumping Electrical or Optical Voltage Variation Spin-to-Charge Conversion 3 s 5
e- Spin (P in Si) Electron Spin or P Nuclear Spin Optical Pumping e--Nuclear Hyperfine Coupling Voltage Variation Optical Pulses QND Measurement 0.6 s 5
29Si Nuclear Spin in 28Si Nuclear Spin of 29Si Optical Pumping e--Nuclear Hyperfine Coupling Voltage Variation Optical Pulses QND Measurement 25 s 5
NV Center in Diamond Spin State of N + C-Vacancy Optical Pumping Resonant Microwave Voltage Variation Optical Microscope 2 ms 2/5
Superconducting Circuit Energy Levels RF Pulse Capacitive or Inductive Coupling resonant cable Magnetometer or Electrometer 4 s 0.7/10
e- Spin (InAs Nanowire) Electron Spin States Electric Field to control orbital motion Electrical     < 3 s  

Table 01 Types of Qubit

Photonic Entanglement Quantum computing requires entanglement between qubits. It is usually produced by sending a single photon through a certain type of crystal lattice (such as the beam splitter). The process is not very reliable until recently in June 2010, a technique has been found to generate an entangled pair of photon from a quantum dot with 82% reliability. The InAs quantum dot is embedded in a GaAs LED. When the LED is supplied with electric current, two electrons hop into two positively charged "holes" in the quantum dot's lattice, releasing a pair of entangled photon (Figure 06c). There are rooms for improvement in producing more reliable quantum dot (that can be used to generate entangled pair) and in raising the operating temperature from 5oK.

Figure 06c Photonic Entanglement

In Figure 06c The "R" and "L" in the Bell state represent right- and left-handed circularly polarization and the first label denotes photon #1 (blue), the second label for photon #2 (green), e.g., |RL> means photon #1 in right polarization, photon #2 in left polarization.
1An EPR-source is used to provide an entangled pair. An example is the decay of the pi meson into an electron-positron pair. Since the spin for the pi meson is 0, the spin for the electron-positron pair must be opposite according to the conservation of angular momentum. Therefore, no matter how far apart are the members of this pair, if the spin is flipped for one of the member, the spin for the other member will also be flipped to the opposite at precisely the same moment. This non-local influence (non-locality) occur instantaneously, as if some form of communication, which Einstein called a "spooky action at a distance", operates not just faster than the speed of light, but infinitely fast. Figure 04 is another method to prepare entangled pair. In this case, it is the entanglement of the horizontal and vertical polarizations of the photon. It has been demonstrated recently in 2004 that entanglement and teleportation is possible using pair of trapped ions such as Ca+ or Be+.

2To achieve projection of photon 1 and 2 into a Bell state the two photons are superposed at a beam splitter.

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