# Quantum Computing, Qubit Creation/Entanglement

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## Solution 1

Take a proton (the nucleus of Hydrogen - everywhere in water) which has a spin, and since it's charged, it has North and South poles. If you measure it, the North pole points either up or down in your instrument. If you embed it in a magnetic field, it will want to line up with that field, but it can't easily because it's spinning like a little gyroscope, so it precesses like a top. The rate at which it precesses depends on the strength of the field, and that can be detected, and so you have Nuclear Magnetic Resonance, used everywhere in MRI machines.

By manipulating the field, you can put the proton into a state where it's "in-between" up and down. If you measure it, it will be either one or the other, but before you measure it, it's in a mixture of states, called a "superposition".

If you have some number of them, like for example four, by manipulating the field, you can put them all in a mixture state. But it's not like four independent mixtures. Rather it's one mixture of 16 possible states. If you measure all of them at once, you could get any one of the 16 possible answers.

Each one of those states in the superposition is a fully-specified combination of bits, so it's like having 16 different 4-bit computers running in parallel, but they're all running the same program at the same time. The "program" consists of magnetic pulse trains that affect all the states at the same time. That's called "quantum parallelism", and you can see that if you can put enough qubits into this superposition where every one of the 2^N combinations is equally likely, you can carry on 2^N computations in parallel.

Then, suppose one of those computations reaches a result that you want to know. You have to get the result by measuring, but that's complicated to explain and may be a bit much for this answer.

P.S. One of the interesting aspects of quantum computation is that it has to be reversible. So if you have an algorithm that you want to execute on a quantum computer, you have to make sure the algorithm can be run in reverse just as well as forward. So for example, if you have a state machine where either state A or state B can transition to state C, it won't work in a quantum computer unless there is some memory of how C was entered, so the state transition can be "un-done".

P.P.S. Let me take another stab at how you get the results out of a quantum computer. The method I'm familiar with is Lov Grover's Search Algorithm, for doing search in an unsorted table. If the table contains M entries, you create a superposition with M states, one of which will "succeed". Since the only way you can get information out is by measuring, what you need to do is adjust the probability amplitudes of the states so that the successful one has a high probability, so when you measure, it is the one you will most likely see. That is done by a manipulation that transfers some of the probability from the unsuccessful states to the successful one. Then the computation is run in reverse back to the beginning, then run forward again, and the probability-transfer operation is done again. This is done several times, until the successful state has nearly all of the probability. It's important not to do it too many times, because it will start having the opposite effect.

## Solution 2

There are many ways to make a qubit, the most favoured method for lab based work is by using an ion trap - an ion is contained in a mixture of oscillating and static electric fields (wikipedia 'Paul trap'). This gives two qubits - the electric state of the ion and its vibrational mode - and has the advantage of being well separated from the environment (so it takes a long time to decohere), however it is not very practical for a quantum computer as combining the ion traps is very difficult. There is also the additional work of having to split pulses up, so a single operation is split into roughly 3 operations, to prevent the ion from being excited outside of the computational basis (i.e. if you're using the ground state and first excited state, it is possible to excite it to the second excited state by using a large pulse).

One of the most successful quantum computers has been made using liquid NMR, using the same synthetic molecule suspended in a liquid. This has been used to run Shor's algorithm (integer factorisation) on the number 15, using 7 qubits. However, the use of synthetic molecules for a quantum computer is again not very practical as longer chains decohere quicker and the nuclei frequencies become closer together, making them harder (and eventually practically impossible) to address individually.

One of the current pushes is to make a solid state quantum computer, either using electron spin or nuclear spin as the qubits, as the experience with solid state classical computing is very high. But there are some drawbacks, electron spin in GaAs (a common material used for quantum dots) typically decoheres in around 250 us, which doesn't give very long for the calculations. So the use of phosphorus has been suggested, as its nuclear spin could potentially last in the orders of hours, placed on to a silicon lattice. Whilst its similar size would mean it could fit into the silicon lattice very nicely, it is very difficult to place so that each P atom is in a regular array - it uses the doping process, but unlike the manufacture of p and n material, the randomness of the doping is a severe drawback, as you can get multiple P atoms in a cell and then none in others. Cambridge university (and others) are currently doing some interesting work on a different method - using surface acoustic waves to confine a moving electron (see http://www.sp.phy.cam.ac.uk/SPWeb/research/sawqc.html) - this looks like it could be used for a potential quantum computer, but it is again limited by the spin decoherence time.

The biggest quantum 'computer' made is the D-Wave One (http://www.dwavesys.com/en/products-services.html) but it doesn't work in the same way as a quantum computer should - it doesn't implement any gates, instead it uses a network of double well potentials to find the lowest energy state of the system (what they refer to as the programmed energy model). It uses superconducting rings to implement the qubits, with clockwise being spin down and anti-clockwise being spin up.

Qubits can also be made by using laser pulses, with the qubit being the polarisation of the photon and gates being implemented using optics - but that is about all I know of them.

To sum up, things you could use as a qubit

• electron spin
• nuclear spin
• electron occupancy (i.e. in a quantum dot, where the presence of an electron is the $|1\rangle$ state and the dot being empty is the $|0\rangle$ state)
• current direction (i.e. around superconducting loop)
• photon polarisation

(For a rough guide, using Shor's algorithm (without error correction) to factorise a 512 bit number requires ~2500 qubits and 10$^{10}$ gates, the largest quantum computer I am aware of is 7 qubits (excluding the D-Wave), so this should give you an idea of how far we are away from quantum computers.)

Entanglement requires two qubits interacting and that their state is not separable. Two qubits can be written as having the following state, $$| \psi \rangle = \alpha | 00 \rangle + \beta | 01 \rangle + \gamma | 10 \rangle + \delta | 11 \rangle$$ where $| \alpha |^2 + | \beta |^2 + | \gamma |^2 + | \delta |^2 = 1$. This is separable if it can be written as a tensor product, eg $$| \psi \rangle = \frac{1}{\sqrt{2}} \left( |0\rangle + |1\rangle\right) \otimes |1\rangle$$ which is the state $\frac{1}{\sqrt{2}} \left( |01\rangle + |11\rangle \right)$.

$$\frac{1}{\sqrt{2}} \left( |01\rangle + |10\rangle \right)$$ is not separable and is therefore entangled. Entanglement generation can be achieved by using a two qubit gate, such as the CNOT (controlled not), which only flips the target bit if the control bit is in the $|1\rangle$ state, i.e. $\text{CNOT}\left(|x\rangle|y\rangle\right) = |x\rangle|x \oplus y\rangle$, where $x$ is the control bit and $y$ is the target bit.

N.B. hopefully this isn't too high a level for you, there are some nice review articles in a few journals, but I'm not sure if you will have access to them. A good book for a general background read is 'Principles of Quantum Computation and Information: Vol 1: Basic Concepts' by Benenti, Casatio & Strini.

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### 11D Reality Hacker

Updated on December 31, 2020