Quantum Computing

Contents

Qubit and Entanglement

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.

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).

		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)

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) which is produced by an Einstein-Podolsky-Rosen (EPR) source1. The teleportation scheme works as follows. Alice has the particle 1 in the initial state |f >1 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:

Figure 05 Teleportation [view large image]

|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. The state given in Eq.(8) distinguishes itself from the others by the fact that it changes sign upon interchanging particle 1 and 2. This unique anti- symmetric feature plays an important role in the experiment. 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 top portion of 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. Furthermore, the initial state of particle 1 can be completely unknown not only to Bob but to anyone. It could even be quantum mechanically completely undefined at the time the Bell-state measurement takes place. This is the case when particle 1 itself is a member of an entangled pair and therefore has no well-defined properties on its own. This ultimately leads to entanglement swapping (See lower portion of Figure 05). 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.

Table 01 lists the types of qubit that have been tried up till 2010. Unfamiliar terms are explained briefly in the followings :


• Types of Qubit - There are 4 main types:
  1. Photonic qubits have been implemented with optical and infrared photons. Early development was plagued by the requirement of nonlinear optics, in which the dielectric polarization P responds nonlinearly to the electric field E of the light. This nonlinearity is typically only observed at very high light intensities (values of the electric field comparable to inter-atomic electric fields, typically 108 V/m) such as those provided by pulsed lasers. In nonlinear optics, the superposition principle no longer holds.
  2. Trapped ions or atoms are localized by electric field or laser beams. Scaling (to large number of qubits) becomes difficult when large number of ions participate in collective motion. The central challenges in using optical lattices (array of atoms) are the controlled initialization, interaction and measurement of the qubits.
  3. Electron or nuclear spin states are controlled by radio or optical pulses. They can be submerged in liquid or more commonly in solid in the forms of impurities or dopants which bind one or more electrons or holes. Scaling a system of coupled spins remains a challenge.
  4. Superconductor can be assembled macroscopically to form qubits. Since such qubits involve the collective motion of a large number (~ 10¹⁰) of Cooper-pair electrons, the coherence time is very short in the order of microseconds.
• Optical Pumping - It is a process in which light is used to raise electrons from a lower energy level in an atom or molecule to a higher one.
• Wave Plate - A wave plate is an optical device (usually a birefringent crystal such as the calcite) that alters the polarization state of a light wave traveling through it.
• Waveguide - It is a structure (usually a metal pipe or optical fibres) for transmitting electromagnetic wave. Cavity resonator keeps the wave inside when both ends of the waveguide are closed.
• QND Measurement (Quantum Non-Demolition measurement) - A quantum measurement that does not disturb the system, i.e., the measurement can be repeated. All successive QND measurements on the same system will give exactly the same result as the first one.
• Josephson Junction - It is a non-superconducting thin barrier to separate two supercondutors. It allows electrons to pass through even in the absence of an applied difference (the Josephson Junction current). When a voltage is applied, the current stops flowing and oscillates at a high frequency. It is often used to substitute for the inductance.
• Quantum Computing with Superconductors:
  • A LC-circuit made of superconducting material acts like the harmonic oscillator, the equation of motion for which is :
    m d²x/dt² = - kx,
    where m is the mass, x the displacement of the object, k is the force constant for the vibrating spring, and t is the time. This equation describes exactly the case of superconducting LC-circuit if m is replaced by the inductance L, k by the inverse of the capacitance 1/C, and x by the magnetic flux (the magnetic field through an area A), i.e.,
    L d²/dt² = - /C.
  • In analogy to the harmonic oscillator,
    1. The potential energy is given by the formula: U = ²/2C
    2. The flux is oscillating according to: = _max sin(2ft), where the oscillating frequency f=(1/LC)^1/2.
    3. The quantized energy levels are prescribed by E_n=(n+1/2)hf, where h is the Planck constant. Note that the spacing between adjacent levels is constant, i.e., E=hf.
    4. Since the harmonic energy levels are evenly spaced, resonant electromagnetic excitation leads to multi-photon processes that occupy excited states, and thus do not provide the clean two-level system that is needed for quantum computing. Thus a small nonlinear term must be added to the equation to product anharmonic energy levels. This requirement can be satisfied by adding the Josephson junctions to the LC-circuit (a in Figure 06a).
    5. Depending on the values of the single electron charging energy E_C=e²/2C and the Josephson energy E_J. There are three types of superconducting qubits : 1. Charge qubit with E_J/E_C<<1, inductance is omitted; 2. Flux qubit with E_J/E_C>>1, and 2 currents loop in opposite directions; 3. Phase qubit with E_J/E_C>>1, and the potential is biased at a different point (see b,e, c,g, d,h in Figure 6a).
    6. Superconductive quantum computing has the advantage that it can be supported by current technology to build the components and to have it scaled up to practical size. Figure 06b presents a simplified architecture for a superconductive quantum computer :
       • Input and output can be implemented using existing control logic in classical computers. The program in quantum computing is the sequence of gate operations that act on the qubits. This sequence is supplied by a classical computer.
       • The drivers (e.g., RF pulse) are used to control single qubits and also serve as on and off switch for inter-qubit interactions. Only couplings between adjacent qubits are depicted in the diagram.
       • The qubits are constructed as tiny LC-circuit (as mentioned above) about 0.01 to 0.0001 cm in size.
       • Couplers in the forms of superconductive circuits, electromagnetic coupling with capacitors, inductors, or microwave cavities can be used as long as they can be modulated.
       • Sensor serves as the interface between the quantum world of the qubit and classical control logic. The qubit can be coupled to a magnetometer or electrometer, and then processed by an amplifier and comparator (a device which searches for the larger one between two voltages or currents).

    Figure 06a Superconducting Qubits [view large image]	Figure 06b Superconducting Quantum Computer [view large image]

Type	Qubit	Initialization	Interaction	Data Transmission	Detection	Coherence Time	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 (Occupancy)	Optical Pumping	Electric Fields	Induced Vibrations	Optical Fluorescence	15 s	0.48/0.7
Trapped Atom	Energy Levels (Occupancy)	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

Table 01 Types of Qubit

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.

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

		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
Figure 08a Number of Entangled States	Figure 08b Symphonic Parallelism [view large image]	disturbed in the middle of its run, otherwise the operation will be terminated prematurely by decoherence.

Figure 08c Quantum Computing Example [view large image]

Figure 08c shows an example of quantum computing using 2 qubits:
• 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.

Factoring and Encryption

Factoring large numbers is the basis for a powerful method of information protection. It utilizes a technique called public key cryptography. For example (see Figure 09), Alice sends a public key consisting of a large number (equals to the product of two smaller prime numbers) to Bob. Bob uses this public key to scramble or encode the returning message. Alice then decode the message with the private key consisting of the two original prime numbers. Since it is very difficult to factor the public key, the information is secure, that is, until the arrival of the quantum computing age.

Figure 09 Encry- ption

message can be created in which each letter is substituted with the letter that is two down from it in the alphabet. So "A" becomes "C", "B" becomes "D", ... and so on. The recipient was told that the code (key) is: "Shift by 2". He/she can then decodes the message accordingly. Anyone else will only see a garbled message. Modern encryption employs two keys to provide greater security. The sender selects a public-key such as 1525381, which is the product of two prime numbers: 10667 and 143. This key is used to convert a block of text via an algorithm (a formula for combining the key with the text). The recipent must used a private key such as 143 to decode the encrypted text in the reverse process. The security of public-key cryptography depends on factorization - the fact that it is easy to compute the product of two large numbers but extremely hard to factor it back into the primes. But the

Figure 10 Quantum Encryption

advent of the quantum information era - and, in particular, the capability of quantum computers to rapidly perform monstrously challenging factorizations - may portend the eventual demise of such cryptographic scheme.

Unlike public-key cryptography, quantum cryptography should remain secure when quantum computers arrive on the scene. One way of sending a quantum-cryptographic key between sender and receiver requires that a very low intensity laser transmits single photons that are polarized in one of two modes as shown in Figure 10. Followings illustrate the steps in establishing the key (see Figure 10):


1. Alice sends a photon through either the 0 or 1 slot of the rectilinear or diagonal polarizing filters, while making a record of the various orientations.
2. For each incoming bit, Bob chooses randomly which filter slot he uses for detection and writes down both the polarization and the bit value.
3. After all the photons have reached Bob, he tells Alice over a public channel, perhaps by telephone or an e-mail, the sequence of filters he used for the incoming photons, but not the bit value of the photons.
4. Alice tells Bob during the same conversation which filters he chose correctly. Those instances constitute the bits that Alice and Bob will use to form the key that they will use to encrypt messages.
5. If Eve the eavesdropper tries to spy on the train of photons, she may create errors when wrong filters are used to intercept the photons. The crucial safeguard is in hiding the key established by Alice and Bob.

Searching

Teleportation

the polarizing beam splitters (PBS) for Bell-state measurement (BSM). The logic electronics identify the Bell state and convey the result through the microwave channel (RF unit) to Bob's electro-optic modulator (EOM). Depending on the message, it either leaves the photon state unaltered or changes it to the opposite state. Note that because of the reduced velocity of light within the fibre-based quantum channel, the classical signal arrives about 1.5 microseconds before photon 3. Thus, there is enough time to set the EOM correctly before photon 3 arrives. Polarization rotation (which introduces errors) in the fibres is corrected by polarization controllers (PC) before each run of measurements.

Figure 11 Teleportation over River Danube [view large image]

Polarization stability proved to be better than 10o on the fibre between Alice and Bob, corresponding to an ideal teleportation fidelity of 0.97.

Quantum teleportation has only been done between similar objects - from light to light or matter to matter until 2006, when the first step has been taken to teleport the quantum state between a photon and an atom. This technique is critical in transferring the light qubits into atomic storage. The experiment achieved only for a transmission distance of half a meter. The traveling distance can be extended with improvement on the control of signal degradation. Figure 12 shows the experimental set-up for the experiment. As usual again, Alice is the keeper of particle (1) to be teleported, and the entangled particle (2); while Bob has the entangled particle (3) waiting to receive the teleportation. Here's the protocol:

Atoms are initially optically pumped into the F = 4, mF = 4 state with a 4-ms pulse (see insert). A 2-ms pulse of light is then sent through the atomic sample at Bob's location and becomes entangled with the atoms (3). The pulse travels 0.5 m to Alice's location (2), where it is mixed on a beamsplitter (BS) with the object of teleportation (1) - a few-photon coherent pulse of light - generated by electro-optical modulator (EOM). A Bell measurement is performed, and the results are sent via a classical communication channel to Bob. There they are used to complete the teleportation onto atoms by shifting the atomic collective spin state

Figure 12 Teleportation of Light to Atom [view large image]

with a pulse of radio-frequency (RF) magnetic field of 0.2-ms duration. After a delay of 0.1 ms, a verifying pulse is sent to read out the atomic state, in order to prove the successful teleportation.

Error Correction

Therefore, quantum error correction is performed differently as shown in the followings:
1. Prepare the primary physical qubit such as P = a |1 + b |0, which is to be protected from error.
2. Prepare two auxiliary A1 = |1, A2 = |1, which are then entangled with the primary qubit to form a logical state.
3. Noise (causing random error) is applied to this logical state, which is represented by P, A1, A2 in Figure 13.
4. The primary qubit is decoded from the auxiliary qubits. Now P is separated from A1, A2.
5. Syndrome measurement, which can determine if the error is a bit flip, or a sign (of the phase) flip, or both, is performed on the four possibilities for the auxiliary states A1, A2, e.g.,
   
   1. |1|1 for no error - association of the most often outcome with the most easily distinguishable measurement.
   2. |1|0 for auxiliary state 2 flipped - no correction required.
   3. |0|1 for auxiliary state 1 flipped - no correction required.
   4. |0|0 for primary qubit flipped - e.g., both the bit and sign are flipped: P = -a |0 + b |1.
6. The primary state has been altered in Case 4. Appropriate correction is applied to recover the primary qubit initial state.

Such error-correction protocols have been implemented in 2004 using three beryllium atomic-ion qubits (the qubits comprise the two electronic ground state hyperfine levels, which are equated to the two spin 1/2 states - up and down) confined to a linear, multi-zone trap. The trap acts like a quantum register with the internal state of each ion playing the role of a qubit. It has been demonstrated that fidelity of 0.7 - 0.8 can be achieved in the experiments. However, the method works well only when at most one of the three qubits undergoes a spin-flip error. Figure 13 shows the transportation of the ions in the trap during the error-correction protocol as a function of time. Each experiment requires approximately 4 ms to perform. The ions are kept together by careful tuning of the phases of the optical-dipole force. Refocusing operations are required to counteract qubit dephasing caused by fluctuations in the local magnetic field.

Figure 13 Quantum Error Correction [view large image]

Footnotes