[code]Note that whenever I do ket notation of something, I start with the character | but the arrow doesn't show up, sorry.[/code]
The basic unit of data is a qubit, and a group of qubits together is called a quantum register.
A classical bit is anything you can distinguish two distinct amounts or states of (like North/South poles directions of magnets, or positive/negative charge).
So we need two quantum states to be the zero and one states. An example could be 2 different standing waves.
Remember that these states need to be orthogonal, i.e 90 deg to one another - their eigenstates are perpendicular to one another, where eigenstates are a vector that correspond to the eigenvalue of a wave function.
An eigenvalue is a characteristic of a wave function, as in it can be used to obtain many values.
Photons, electrons and other phenomena can be used as qubits.
To distinguish quantum bits from classical bits, you can write states using "Dirac's ket notation".
The 0 state can be both written, and drawn using 2 dials, it is pronounced "0 ket".
Note the full vector is represented on the upper dial, and nothing on the lower dial.
And then the 1 state is pronounced "1 ket", and written in the 2 dial way as, with the full vector on the lower, nothing on the upper.
These quantum 0/1 states, are waves, they have amplitude and a phase, and also be put into superposition, so be made to produc inteference.
The vector inside the upper circle will represent the amount of the 0 state that is present, and the vector in the lower circle to represent the amount of the 1 state that is present.
This represents a 50% 0, and 50% 1 state.
The 1 state is defined relative to the 0 state, so the phase of the 0 state, is always 0.
However, this means that 1 state can have differing phase, which can be written/shown like this:
Where the phase itself is in radians (remember pi = 180deg, and the angle of the vector is relative to 0, which is a normal kinda - the angle is clockwise from it).
Anyhow, there will normally only be 1 dial with 0 phase in a problem such as this, with the dials.
The quantum waves are quantum probability amplitudes, so they are the probability that we will find a particular state.
The probability in the quantum sense doesn't necessarily always add up to 1 for all possibilties (like how clasically all probabilties in a problem sum to 1).
The probability of states can be written as:
Where alpha and beta represent the quantum probability amplitude of 0 and 1.
In quantum mechanics, the probability of a state is the absolute value of the square of the amplitude, rather than just amplitude:
And as such, they would add to 1:
The process of ensuring they add to 1, is called normalisation.
So far the example have been 50% 0, 50% 1, they can be like this:
Which is 25% probability of 0, 75% probability of 1.
(If you extract alpha and beta, and put it through the equation directly above this, you get 1).
As you can see, the lengths of the vector lines in the dials represents the probabilities (1 is lower, 0 is upper).
So remember, that the phase is always inside normal brackets, and the probability (alpha/beta) is not. The phase determines the angle of the vector, the probabiility determines the length.
The fact that there are 2 states, and you can have differing probabilities of them, makes it a qubit.
Another better way of representing a single Qubit is by use of a Bloch sphere. The length of the vector is always 1, and so lies on the surface of the sphere. The point on the surface can easily define both amplitude
and phase of the qubit.
North pole represents 0, south pole represents 1, and the equator is the 50/50 probability. The lattitude is basically the probability amplitude, the longitude is the phase of the wave.
(East to west is phase, north to south is probability).
The sphere also conveniently shows which states are orthogonal (what I mentioned earlier). If you have two points 180deg from one another, they are considered orthogonal points (not 90deg like in normal maths and
It is useful for representing operations and measurements on a single qubit, however it is not really useful for multiple qubits, so multiple dial representation is preferable in those cases.
A single qubit can be in a superposition state, that is by having multiple probabilities at once.
For example let's say there's a qubit with probability 50% 0, 50% 1, it would be the square root of the 50% sqrt(0.5), that would be the probability amplitude, as a^2 + b^2 = 1, as mentioned before, where a and b are
alpha and beta respectively. So can be written and shown in dial form as the following:
Remember the sum of square of both states must be 100%.
[bold]When you measure the qubits, the wave function collapses, and you observe either 0, or 1.
After you measure that the qubit is of one state, there is no amplitude of the other state left.
It is 100% one state or the other state upon observation.[/bold]
This also means that there is no way to find the original probability amplitudes, and as such the only way to infer the orignal amplitudes is to rerun the experiment multiple times and then observe what the probability
of each state is.
So if you get 49 readings of 1, and 51 readings of 0, you can infer that the probability is about 50:50.
[h2]Three Ways to measure one qubit[/h2]
The Bloch sphere is useful for thinking about measurement.
If the unit vector of the qubit points to the northern hemisphere, it is probably |0 state, and |1 in the southern hemisphere.
The state at the closest hemisphere is then projected onto the one we measured, leaving only |0 or |1.
([i]Projected[/i] means that the unit vector is moved to one of the poles, the states).
The |0 to |1 axis is the [b]Z[/b] axis in the Bloch sphere, and as such [i]when you measure the qubit, it is projected onto the Z axis, so it is more towards the |0 or |1 state, and that is the state of the qubit[/i].
When we measure on the X axis, we will project our qubit to one end of the X axis, which we called our |+ and | states.
The x-axis is:
Measuring along the Y axis is less common, so unlike the Z and X axes, the two ends dont have such simple nicknames. The states at the two ends are
State space is simply how many possible values are representable with the amount of digits.
In denary it is 10^n, where n is the number of digits, in binary it is 2^n, where n is the number of digits.
To measure only one qubit in a multi-qubit state, calculate the probability of each state, and sum up all of the states with zero in the appropriate position to get the probability of finding zero.
To measure only one qubit in a multi-qubit state, calculate the probability of each state, and sum up all of the states with zero in the appropriate position to get the probability of finding zero.
[b]Remember that 1111 bits, is 2^3 state vector, so is represented on 2^3 quantum amplitude dials describing each state, many of them in most cases are likely to be 0, so have no need to be represented.[/b]
The square root of the absolute value of the amplitude of the amplitude is the probability that you will measure that value if you measure a whole register (a register is a number of quantum bits).
When there are 2 or more qubits, their states can be correlated in manners not replicable in classical systems.
As an example, consider the following:
- Directions (up and down) denote two qubits (made up), where up is 0, down is 1.
- Then you can have two quantum states made by those qubits:
- Up down, and down up, or |01 and |10.
However you could also use up/down arrows to represent the states, or whatever, just 2 symbols:
If you take the 2 states, and put them in superposition, then you end up with:
They are in a 50/50 superposition, or 1/2, which is normalised to sqrt(2).
In this superposition, the state of the first qubit is 50% 0, and 50% 1, totally random.
The same with the second one.
But despite this, they are not independent - they may be random, but there is a linking factor.
The system will collapse into a state which has a probability that is greater than 0.
That is, it will never collapse into the |11 and |00 states, as the probabilties are already decided, so 0%.
The will collapse into |01 or |10, that is in the case of 2 qubits, 4 states (more qubits, more states, can be extrapolated).
Furthermore, when you measure them, the probabilties of value 0 (|11 and |00) are never found, just the 2 non-zero ones, despite the fact that the individual probabilties of each qubit would suggest that the other
|00 and |11 states could be found - [i]this is what we mean by saying that they are random, but they are not independent, there is some linking mechanism.[/i]
The following is an extract from the course directly, really useful:
One of the most important uses of entanglement is quantum teleportation. In teleportation, the state of one qubit is destroyed in one place and resurrected in another, accomplished by using entanglement. Teleportation
is a fundamental primitive in quantum networking, and in executing certain types of quantum computing.
Moving quantum states from one place to another is difficult: if we want to go any long distance, we want to use photons (light), which are pretty good at keeping a quantum state intact, but have the habit of getting
lost along the way. Back in 1993, Bennett, Brassard, Crepeau, Jozsa, Peres and Wootters realized that its possible use two qubits entangled into a Bell pair as a tool for moving a qubit from one place to another.
To begin, create a Bell pair shared between two people, always called Alice and Bob in quantum discussions. Doing this over a distance using photons is hard, but has been done in many laboratories around the world.
In the figure at the top of this Article, the entanglement is represented by the squiggly green line between the qubits A and B, but of course theres nothing to see thats just to show you which two qubits are
Alice also creates (or receives from someone else) the qubit she wishes to teleport to Bob, marked D in the figure. (If the procedure for creating Alices data qubit is easy, it might be easier to just send the recipe to
Bob; we are assuming here that the procedure is complicated, or the qubit is only part of a large quantum state, or that Alice was asked to relay the qubit for someone else.)
Next, Alice measures her two qubits (D and A) together, in a special way known as Bell state analysis. This measurement destroys the entanglement in the Bell pair, and also forces the collapse of any superposition in
the data qubit. It leaves Bobs half of the Bell pair in an ambiguous state.
This Bell state analysis gives Alice two classical bits of information. One of those classical bits tells Alice the parity of A and D: if they are both 0 or both 1, then the parity is even; if one of them is 0 and the other
is 1, the parity is odd. Similarly, The other classical bit tells her the relative phase: is their phase the same or opposite?
If Alice sends these two classical bits to Bob, he can use those to turn his qubit into the state of the original qubit, resurrecting an apparently dead qubit. The two bits serve as corrections to the qubit; the first one
tells Bob if he should flip the value of the qubit, the other tells him if he should flip the phase of the qubit. After he applies these corrections, the data has been teleported from Alice to Bob!
We saw in an earlier video Step that the entanglement behaves like there is some communication between the qubits over a distance, but it cant be used to transmit data faster than the speed of light. The need for
the transmission of the classical data and its use in order to recreate the data qubit is why this is so. If Bob measures the qubit without waiting for the data from Alice, he will get only random numbers that are of no
use. It is only once the bits arrive, and tell him how to interpret that data almost like an encryption key that his data becomes useful.
Note that teleportation destroys the entanglement in the Bell pair, so if we want to repeat the teleportation operation (which we almost certainly do), we will need to invest a lot of effort in creating a stream of Bell
Teleportation will be the foundation of wide-area quantum networking. It is also used to execute gates in a fashion known as gate teleportation, and serves as the first key to a specialized type of quantum computing
called measurement-based quantum computation, which operates on entangled states consisting of many qubits. In this course, we wont have any further need for teleportation, but its importance cant be overstated.
[h2]More on Quantum Entanglement[/h2]
So basically, Quantum Entanglement does indeed lead to some faster than light speed correlation between the states of two qubits, however this doesn't mean any information is actually transferred as classically
transferred data is still needed on the other end to interpret the correlation, and actually get useful information from it.
So information isn't communicated faster than light speed, just very very fast, at a max, as fast as the classical data is transmitted and recieved, then used to intepret the other qubit.
[h2]Entropy and Information[/h2]
The destruction of information increases the overall entropy of the universe and generates waste heat.
The harder it is to describe your data, the higher the [i]entropy of information[/i], in that case.
As in if you have 700 0's or 700 1's, you can just describe it like that, but if they're all evenly distributed, with different random combinations of the 700 states, it's harder to describe and higher entropy.
Does nothing to the state, the output is the same as the input, so it is obviously reversible (part of reversible computing I noted in Week 1), you only need to use this gate once to get your input.
Executing twice in a row results in the exact same output as input.
[h2]CNOT gate (controlled not gate)[/h2]
That weird circular kinda symbol is XOR.
Basically, so long as both A and B aren't 1, then C won't be flipped.
[b]The last two gates are powerful enough to make [i]any[/i] classical logic circuit.[/b]
[h2]The No-Cloning Theorem[/h2]
Basically saying it is impossible to make a perfect, not entangled copy of a qubit with an unknown state in [i]all[/i] cases.
2 types of copies, dependent and independent.
Dependent ones are entangled with the original.
Decoherence is any sort of unintended factors affecting the qubits, which may result in damages to the quantum state, so it's incredibly difficult to build a machine doing this stuff properly.
The technology to control and measure individual electrons atoms or photons must be precise and sensitive, if not they may result effects on an entire register of qubits, and negative or unintended effects may
Qubits must be properly isolated from the outside environment: given that we use light, or microwaves, or magnetic fields to control qubits, any stray signals can damage our state. The basic way to isolate something
from radio waves is to put it inside a Faraday cage, a shell of conductive metal that absorbs the signal. Even with our best effort, though, some radio waves can leak through, and some quantum systems are so sensitive
that interference caused by electric trains running a kilometer away from the laboratory can be detected!
Without proper isolation from the environment, it is also possible information leaks out of the system, causing our qubits to be measured inadvertently by the environment, or become entangled with the environment. In
the worst case, this will destroy the superposition or entanglement of our qubits, leaving us with nothing but noisy, random classical data.
[h3]Mixed and Pure states[/h3]
Without errors, all of the states we have presented are pure states. A state that might have errors, on the other hand, is called a mixed state. Note that its very tempting to call a superposition or an entangled state
mixed, but thats incorrect use of the term.
Mixed states are described, in part, in terms of their fidelity. The fidelity of a quantum state is the probability that the state we have created is identical to the state we think we have created, regardless of whether
that state is a simple state such as |0 or a superposition such as (|0+|1)/2 .
The fidelity F is a number between 0.0 and 1.0. F=1.0 implies that our quantum computer works perfectly. For a single qubit, F=0.5 means that the state has a 50% probability of being the state we expect, and
a 50% probability of being the wrong state; its state is completely random. Such a state is referred to as completely mixed.
For more than two qubits, F is the probability that all of them are behaving the way we expect, based on the algorithm weve run so far. For two qubits, the completely mixed state is F=0.25 , for three qubits, its
F=0.125 , and so on.
[h2]Quantum Computer Architecture[/h2]
- Consists of both classical and quantum components
Most proposed quantum computer architectures do not make a distinction between the register(s) and a more passive memory equivalent to a classical computers RAM.
[h2]Basic Algorithmic Flow[/h2]
A quantum algorithm generally has five parts. The first part is the classical pre-processing. Next, we initialize the processor or qubit register to zero and create a superposition of all possible states. After initialization,
quantum algorithm progress using an appropriate combination of one- and two-qubit gates and measurement operations. To obtain the result of the computation, we measure the qubits. Through these measurements, each
qubits state becomes a classical 0 or 1 and the entanglement disappears. The classical post processing not only involves some additional calculations, it involves confirming that the answer is correct. If not, we probably
have to go back to step one and repeat.
An example is factoring, it's hard to find the factors, but easy to get back to the original number.
[h3]Working with larger quantum registers[/h3]
Working with multi-qubit registers allows the number of possible states to grow exponentially. It requires us to keep track of the weight (or amplitude) and phase of each possible state.
[h3]Weights and measures[/h3]
Lets say we have two quantum registers, which we will call A and B , and we add them together and put the result in register C , so that C=A+B . (These registers would each have more than one qubit, but for
our purposes here we can treat them as a single numeric value.)
Lets assume we begin with A in a superposition of all of the numbers 0 to 3. We can write this using our dial representation, but in this case a simple table will do.
Assume A is 0,1,2,3 and B is 0,1,2,3.
When you add them together, C would be the superposition of the numbers 0 to 6 (all the possible values C can be considering A and B).
The 'weight' of C would be greater at 3 than 6, as there are more possible combinations of A and B to get to 3 than there are to 6.
A can be 4 states, B can be 4 states and C can be 7 states, and as all of the registers are entangled during the calculation, 112 dial vectors are needded to represent them.
16 are non-zero however, so only they are needed.
When inspecting it, C = 3 has weight 4x that of C = 6.
When C is measured, and say C does indeed equal 3, the superposition collapses, and C becomes unentangled with A and B, as it's state is known (3), however A's and B's are not known, they are still entangled, and
can be in any of 4 possible combinations.
Before quantum computers existed, quantum algorithms were defined either mathematically, in terms of the transitions on quantum states, or using a graphical circuit notation for individual gates. Nowadays, quantum
computers can be programmed using a GUI for the graphical notation, an assembly language-like text notation for the equivalent gates, or richer programming languages that may also provide low-level access.
When we construct quantum algorithms, we will use gates. Each gate will operate on one, two or three qubits, and we will apply a sequence of gates to our register to form a complete circuit, an implementation of an
algorithm. In most quantum computer implementations, the qubits are stationary, and gates are executed on the qubits where they sit, allowing us to treat them something like low-level computer instructions.
Each of the gates discussed back in Step 2.15 has a symbol, shown in the figure at the top of this article. Circuits are composed to run left to right, with a horizontal line for each qubit. The visual resemblance to
music has lead IBM to referring to such a diagram as a score. IBMs web-based interface uses drag-and-drop programming with colorful, stylized gate symbols.
This approach can show conditional operation, basing the choice to execute individual gates on the outcome of prior measurements. It is also common to draw circuits in terms of blocks of primitive gates. However, it is
inherently difficult to create a loop or complex control structures.
[h2]Photons and Qubits[/h2]
Photon can be used as state variables, and as such can be part of Qubits.
Photons can act as qubits in two ways:
- the first is the way in which they are called [i]Polarisation Photons[/i], where there are 2 of these photons, one is polarised vertically, the other orthogonal to it, horizontally.
- the light waves vibrate orthogonal to one another, where one axis (say the vertical) can act as the one state, and the other as the zero state
- the second way is as [i]Time-bin Qubits[/i], in which a photon can arrive in 2 potential pulses of light, either in the earlier one or latter
- again, as there are 2 distinct states, one can be considered the zero state, the other the one state
Electrons have a characteristic known as spin. Spin has a particular direction, and it creates a magnetic field; if a large number of electrons have their spins pointing in the same direction, they create a large magnetic
field and you have a macroscopic magnet. In this video, Professor Kohei Itoh of Keio University explains how the spin of individual electrons can be used as qubits.
For a single electron, the spin can be either aligned with the surrounding magnetic field, which we call spin up, or anti-aligned with it, which we call spin down. We can write these states using the ket notation we
introduced when we talked about entanglement and Bell pairs, | and | . We then assign one of these to be our |0 state and the other to be our |1 state, so that we can use them as variables.
[h2]Charge and magnetic flux[/h2]
[b]There are three types of state variables using superconductors.[/b]
In a superconductor, there is no electrical resistance. If you start an electrical current flowing, it will run forever. It can also maintain the state of individual quanta.
The first state variable was charge. In a superconductor, electrons form Cooper pairs, in which two electrons, which normally repel each other, become loosely bound to each other, and behave together. A charge qubit
uses the presence of a Cooper pair in a small, isolated island as the |1 state, and the absence of the pair as the |0 state.
The second choice of state variable is magnetic flux. Electrical current flowing in a loop creates a magnetic field, the basis for all electromagnets. If we have a microscopic loop of superconductor, the current can flow
either clockwise or counterclockwise around the loop, so we can use clockwise as our |0 state and counterclockwise as our |1 (or vice versa).
The third type of state variable is an intermediate between the two, known as a transmon.
The key to all three of these state variables is being able to control the presence or absence of Cooper pairs very precisely. A Josephson junction is a tiny gap (perhaps only a few atoms across) in the metal conductor.
It might seem that such a gap would prevent current from flowing (unless the voltage is high enough to make spark across the gap, but the voltages and energies here are much, much too small for that). However,
because of the way that the quantum probability amplitude waves work, there is a small probability that our Cooper pair will tunnel through this barrier. Used appropriately, at close to absolute zero, this gives us the
ability to control very precisely the number of Cooper pairs, giving us the states we can use as our state variable.
A machine called the Cryostat contains the qubit in the University of Tokyo, with temperature of as low as 10 milikelvin, and contains obviously superconducting devices.
Microwave pulses are sent to the device, and the response is the data that they require, through the expensive elctronics they have in and around the Cryostat.
Each electron has a spin, and the spin can be either aligned with the surrounding magnetic field, which we call spin up, or anti-aligned with it, which we call spin down. The nucleus of some types of atoms also has a
spin. You probably know that the nucleus is composed of protons and neutrons. Besides charge and mass, these can give a nucleus a spin. For a given type of atom (say, carbon), the number of protons is fixed, but
there may be several options for the number of neutrons. We call each of these variants an isotope. The rules for nuclei are complicated, but for certain isotopes of different elements, we have a simple, clean system
that we can also use as a qubit.
Nuclear spins were actually one of the first types of qubits used experimentally. Lieven Vandersypen, then a graduate student at Stanford University, performed experiments using a liquid containing molecules with
ordinary hydrogen ( 1 H) and isotopes of flourine ( 19 F) and carbon ( 13 C) as qubits. For solid-state systems, a lot of work has been done on phosphorus ( 31 P), silicon ( 29 Si), and carbon in diamond.
[h2]Control versus Isolation[/h2]
For our quantum state variable, we want something that has two characteristics: one, we want it to be easy to control; and two, we want it to be well isolated from the environment, so that different kinds of noise,
such as the radiation from wireless LANs and microwave ovens, doesnt affect the state very much. When we build a device, we build a lot of shielding around it to protect it from those kinds of things.
Nuclear spins are actually already naturally protected. The electrons around the nucleus serve as a kind of shield, keeping the radio waves away from the nucleus. This makes it very hard to control, and very slow to act
and measure. But the advantage is that we can keep the state exactly the way we want it for a long time, so nuclear spins can make good memories, for example, for quantum networks or long computations where some
of the data isnt used for a long time.
[h2]Atomic energy level[/h2]
The simplest quantum system is the hydrogen atom, with a single proton for a nucleus and a single electron orbiting it. In an undergraduate quantum mechanics course, students are often asked to derive its size and
behavior using Schroedingers equation. The position of the electron isnt fixed, like a satellite in an orbit, but instead is a quantum probability wave in a standing wave of the kind we studied in the first week, known as
an orbital. For an isolated hydrogen atom, both the ground and excited states derive from the simplest standing wave we demonstrated, and are spherical. For most atoms, the shapes are more complex, as they represent
standing waves with a larger number of waves.
[h2]Simple energy levels[/h2]
An atom has a minimum energy level, which we call the ground state. If it has absorbed energy from the environment, usually by absorbing a photon, we say that the atom is in an excited state. It will eventually release
that energy by emitting a photon whose wavelength is determined by the amount of energy released. The atom can only be in one of a fixed set of energy levels, so it always emits a photon of a particular wavelength as
it moves from one state to another. Starting in the ground state and absorbing smallest possible amount
We will discuss the strengths and weaknesses further when we discuss ion trap hardware. of energy, we say it moves to the first excited state. Second and higher excited states are possible, too.
We can create a qubit using the ground state and the first excited state. Just like we used the up and down arrows when talking about spin, we can write g and e inside our kets to describe the physical
phenomenon, then map those states to data values for our computations. We can use the ground state |g as our |0 state, and |e as our |1 state.
Ground and excited states have the advantage of being conceptually simple, but a second approach is also common. In the presence of magnetic fields, the behavior of an atom is more complex than we just described,
especially when the atom has more protons in its nucleus and more electrons than a hydrogen atom. The energy level is primarily defined by the distance between the nucleus and the electron, but magnetic fields distort
the shape of the nucleus and the electrons orbital. Our single, well-defined energy level splits into two or more slightly different energy levels.
Because the energy levels of the hyperfine states are very close together, it takes less energy to move the qubit from one state to the other. We can use microwaves to control hyperfine states, rather than laser light.
We will discuss the strengths and weaknesses further when we discuss ion trap hardware.