Point of this paper and small introduction into Quantum Mechanics and Computing telling the reader about Qubits quantum states quantum phenomena quantum hardware quantum algorithms real world applications quantum states measurements observables quantum programming classical vs quantum history of quantum computing uses of quantum computing
I will be outlining the philosophy of quantum computing mainly the quantum mechanical theory behind it, and then I will go on to show you the mathematical representation as well as hope to show you how a quantum computer strives to operate as well as how it is different from a classical computer
Difference between a quantum bit and a probabilistic bit
Around the 1980s Physicists sought to combine two fields of study, information theory and quantum mechanics, information theory being the study of how to code information in certain forms such as sequences symbols and impulses as well has how rapidly this information could be transmitted through circuits or tele communication channels; Quantum Mechanics being a fundamental theory in physics which describes how things behave at a subatomic level as opposed to classical physics which describe things in an everyday sense classical physics could be seen as an approximation to quantum mechanics . Most people have heard of quantum mechanics through a famous thought experiment by Physicist Erwin Schrödinger in 1935 the simple version involves a cat in a box along with poison a Geiger counter a hammer and radioactive substance, if the radioactive substance decays the Geiger counter will active and trigger the hammer to release the poison which will kill the cat. Radioactive decay is a random process there is no way to predict when it will happen so physicists say the atom exists in a state know as Superposition both decayed and not decayed at the same time. So until the box is opened you don’t know if the cat is dead or alive it is in a combined state. But now when you open the box you can find out if the cat is dead or alive this action of opening the box disturbs the state of the system observing the system itself gives you an answer but also gives you the only answer say you had 100 boxes with 100 cats and so on in some cases the cat would be dead in some cases the cat would be alive and as soon as you open the box you get a final answer a perfect copy of the state could never be created. This is a huge part of quantum computing – the fact that observing a state disrupts it and after this you can gain no more information from said state. Double split experiment more relevant than Schrödinger (this thought experiment actually critiqued quantum mechanics saying that a cat could not exist in a dead/alive state so why should a sub atomic particle also be able to do the same.) a qubit is represented as a vector in a two dimensional vector space with inner product. We use Dirac notation to denote the vector psi as a mix between two states 0 and 1 combined with coefficients a and b the sum of magnitudes of these two is equal to 1 a and b are part of the complex sets of numbers the probability that the qubit is in state 0 is the magnitude of a and the probability it is in state 1 is the magnitude of 1 the portability is related to the orthonormal basis given we start with 0 and 1 but there are others which will be seen in the rest of the paper. The quantum state of n cubits is represented as a vector in dimensional space 2N (talk about how we can choose a orthonormal basis for the vectors labelled in binary tiring which relates to each number it represents the qubit (I believe?)
The joining of Information Theory and Quantum Mechanics birthed a new field dubbed as Quantum computing, this field is the use of quantum mechanical phenomena such as superposition which we have already explained and entanglement (EXPLAIN QUANTUM ENTANGLEMENT IN DETAIL) now although a quantum computer uses these phenomena it cannot do anything that a classical computer can’t do it just does these things differently. Classical computers compute using bits quantum computers compute using quantum bits or as I will refer to them from now qubits having computation on a quantum mechanical foundation led us to discover faster algorithms and novel cryptographic mechanisms as well as improved communication protocols. Superposition of qubits allows it to store much more information that bits as it can exist in a lot more forms than 1 and 0 it can exist in any superposition of the values in the form of (Dirac Notation)
Quantum computing can be described as assembling N qubits apply a transformation of Unitary nature after u is applied we measure all qubits along the basis by projecting them onto the {0 1} basis the measurement outcome is the output of the computation.
What is a quantum computer?
What does it do differently from a classical computer?
A classical computer sometimes emulates a quantum computer to solve some problems but it does not do things as efficiently which is fair as a quantum computer should be able to operate more efficiently in doing the thing it was built for operating specifically using qubits and such in 1994 a physicist named Shor found that large numbers could be factored efficiently factoring a number is an example of an intractable problem with the properties that the solution could be easily verified and that the solution is hard to find, now a classical computer can find the factor of large numbers as well but it takes longer in fact there is a formula for how long it takes the formula is equal to time is approximately equal to e[ c(ln N)1/3(ln ln n)2/3] where c = (64/9)1/3~ 1.9 sing this it was estimated that factoring a 400 digit number would take 1010 years which is laughable and seemingly impossible using quantum computing it was found that the formula changes into o[(ln n)3] from this it was found we could factor a 400 digit number in 3 years the difference between 1010 and 3 is immense.
Why is this important
Knowing this is important as we now know we can tackle problems differently things that seemed impossible for classical computers such as cracking encryptions are an easier task for quantum computers much easier
- quantum errors can be corrected
- quantum hardware can be constructed
-Who is important in the founding of this?
Overview
It seems imperative to educate the reader about quantum states so you can full understand how a quantum computer hopes to use the fact that matter operates very strangely at an atomic scale to its benefit.
A quantum mechanical systems state is described by a wave function(exit) with x being co-ordinate and time t due to wave particle duality. This is also represented in Dirac notation as stated above as a ray in Hilbert space.
Observables can normally be describes as any property in a physical system that can in effect be measured, specifically in quantum mechanics an observable is a ‘self adjoint operator’ an operator in linear algebra is a linear map which takes one vector to another an operator is self adjoint if it is Hermitian which means it is equal to its on conjugate transpose. Epr paradox
There are several axioms of quantum mechanics which I will go through to help you understand what a quantum state is.
A state is a description of a physical system, represented as a complex wave function dependant on variables (x,t )co-ordinates and time
A measurement in quantum mechanics is a process in which information about the state of a system is found by an observer, it is a one way process once you measure a system there can be no other result found even if you measure it again. In quantum mechanics a measurement of an observable gives you an eigenstate and the said observer will then learn the corresponding eigenvalue if the quantum state is the vector (psi) we will learn the probability of it being an as the ||En(psi)||2
There is a phenomena called spooky action at a distance or as we call it today quantum entanglement, this is when particles are derived from a common source this phenomena involved atomic particles such as electrons, but I will first explain it in a simple sense let us say we have a two boxes and inside are two identical objects with opposite colours- black and white- with box a having the black object and box b having the white object in a classical sense entanglement is simply stating that given these two boxes if you open box b and see the colour to be white you will know before opening box a that the colour of the object inside is black. In a quantum sense we treat the box as a quantum system which means the object inside exists as a superposition of both states black and white this means there are two possible outcomes when you open the first box which is in effect ‘measuring’ the system and collapsing it meaning no further measurement can be made the outcomes are box A is white or black meaning box b is black or white respectively this is very interesting as it means collapsing one quantum system transfers information to the other about what colour the other object should be , as due to quantum superposition the object was not any particular colour because before the measurement it was in a superposition state and also collapses that state. This seems okay until we place each box at extremely large distances from each other let’s say at different ends of the universe the laws of quantum entanglement and superposition still hold no matter how far this implies that measuring the quantum system known as box a send information instantaneously across the universe to box b at speeds which are faster than light. Einstein did not like this at all because seemingly because of a famous theory of his. The theory of relativity which stated nothing can move faster than light but because of this quantum weirdness here we have quantum system transferring information and speeds much faster than light depending on how far the objects are.
(Heisenberg Uncertainty principle) states that the hidden variables are not just unobservable they simply don’t exist outside the context of an observation. EPR said either there are hidden variables or the attributed of a particle are not real and defined until they are observed he then said that a particle must have a separate reality independent of the measurements it has all its attributes even when not being measured he compared this to knowing the moon existed without looking at it
Let us now abandon these boxes and instead use an electron and a positron born from pair production made by energy as they are from a common source they are entangled, let us now give the electron to a lab run by cito and the positron to a lab run by kash. We then ask cito to measure the spin of the electron if the spin is found to be spin up, we know without even measuring the positron the spin will be the opposite spin down this has been experimentally verified. The two particles are in some way linked. We know because of wave particle duality electrons act as a wave so we can describe them as a wave function psi as they are born from the same source they share the same wave function and exist in a superposition of state so the two particles contribute to one wave function which is the reason they are entangled and gives us certainty that if one has an up spin the other will have down spin. Now let us gain a different electron and a different positron given again to cito and kash, cito will measure the electron in the y direction (up down) and again finds the spin to be up this time kash will measure in the x direction (left right) not the y direction and he finds the spin to be pointing to the right he then learns from cito that the particle also has up spin this gies him co-ordinates in the x and y direction but this impossible due to a thing called the Heisenberg uncertainty principle. And kash cannot measure the positron in the x direction if cito has measured the electron in the y direction. How is this possible? The particles are communicating instantaneously as previously stated above which breaks the rules of special relativity. There is another possibility that when the electron and positron are born they have a set of hidden variables which will govern their behaviour such that they always act in a way which is always complimentary to have so much information about any such thing that could happen to a particle which seems highly unlikely. This gives us the Einstein Podolsk Paradox or EPR as I will call it from now which focuses on how these particles seem to know information about their entangled partners seemingly instantaneously. The main points of Epr focused on the interpretation on a quantum state ‘wave functions’ and uses the standard system collapsing method. It also says that the measurement corresponding to a collapsed system is determined by the real physical state of the system as well as the fact the spatially separated systems have real physical states it also states that if systems are spatially separated the measurement of one system does not affect the reality of the other states directly. If two different states have always seemingly opposite values the values are definite and were written at the birth of the state i.e the ‘hidden variables’ if the description of quantum systems by wave functions were complete, then the definite values of position etc. could be inferred from a wave function for the state itself. Altogether separated systems as described by EPR have definite position and momentum values simultaneously since this cannot be inferred from any wave function the quantum mechanics descriptions of quantum systems by the means of a wave function is incomplete. ‘if an observed particle could be predicted with 100% without disturbing the system then it must correspond with an element of reality
EPR gave strong doubt towards the completeness of Quantum mechanics and describing a system through a wave function, until John S. Bell he stated that the paradox of Einstein, Podolsky and Rosen was an argument that quantum mechanics is incomplete but could be supplemented by hidden variables to restore the theories of realism [1] and Locality [2]. He then went on to show mathematically that that was incompatible with the statistical predictions of quantum mechanics giving us the ‘Bell Inequalities’ First I will talk about photons and polarization and then I will give you an example of one of the experiments performed by Bell.
Photons are simply particles of light which can also be described as a wave the majority of photons are linearly polarized, this can be observed using a filter for polarization when using one filter most of the time 50% of the light will be absorbed the filter which means 50% will go through this is a general rule no matter what the orientation is. And after a photon passed through a filter its polarization will be aligned with the filter, polarization simply means the direction the wave is vibrating in.
Okay now let us consider a single photon of light and ask ourselves a question. Will the photon have a definite polarization at 3 separate angles: 0 22.5 and 45 degrees? Which I will denote angle a,b and c respectively. According to EPR the photons should contain hidden variables telling them whether it should pass through the filter at said angles this can be confirmed through a simple experiment simply pass an electron with a known polarization through a filter of the same angle and see if the photon has the same polarization after passing through the filter the prediction would be of 100% certainty. This will hold up to be true after an experiment. From this we can infer that the 3 angles must correspond to individual elements of reality (if you have forgotten what this means se EPR)
This bring us on to bells experiment do the 3 angles correspond to 3 Simultaneous elements of reality as EPR stated an element of reality existed independent of the act of observation in simpler word the hidden variables exist always so there will be definite values depending on the photon and not the filter lens itself. Or in other words is the expected outcome equal to the statistical outcome gained from an experiment.
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Let us have an arbitrary large number of entangled pairs of photons as well as polarizing filters set at different angles, because these photons are entangled if you were to pass them through filters oriented at the same angle either both pass through or both are blocked, in essence the photons behave the same way when measured along the same axis and as always with entanglement this behaviour persists no matter how far in space the photons are separated the aim of this experiment was to see the percentage of photons let through and blocked by the different filters set at different angles and see if statistically it could shed some doubt on the hidden variable theory. The experiment was simply sending entangled pairs of photons through the filters separated by a large amount of space and simultaneously at each one of those locations change the orientation of the filter so at lab A it could be at 90 degrees and at lab B it could be at 45 degrees doing this many times we collect a lot of information about how often the different pairs of photons pass through the different orientations, the results that were gotten seemed to show the hidden variable was incompatible with quantum mechanics statistical predictions. How? It’s simple the maths did not quite add up here we have a table with different outcomes AA 100 AB 85//15 BC 85//15 AC 50//50 this does not hold up with the predictions as we would expect the
Quantum states and observables
To give the reader a short look at typical quantum states to explain what an observable is to brush over spin states and entangled particles as an example of two and 3 level quantum states.
Two level quantum state
A two level quantum state can simply be described as a quantum state with one qubit I can give you an example of a two level quantum state, this example exists in the form of a photon, a Photon is a particle of light which can also act as a wave it is important to note two things when considering photons, photons are massless and photons have a spin of 1. Photons can have two values of polarization in the x direction or the y direction for horizontal and vertical polarization, the polarization of a photon simply means the direction the electromagnetic field vibrates, for a photno this is usually transverse to the direction of propagation. (Denote photon using x = cosax + sinay and y = -sinax + cosay) this is how the photon is represented in two dimensions
Classical Vs Quantum Computing
Quantum computers are not a replacement for classical computers. They are simply better when tackling certain problems because they use different means to go about tackling these problems, a classical computer uses bits with distinct values 0,1 and quantum computer uses qubits which can exist in a superposition of 0 and 1. In a classical state the logical state of a register is determinded by all the bits it contains they can be changed localy and independent from one another measuring one does not affect any other bit in the register.
On the other hand it is impossible to describe a quantum registerby listing the states of each qubit given a state of two qubits it can be described by 22 complex amplitudes (show dirac notation) but manipulations on a single qubit of the register will effectthe complex amplitudes of the overall state due to the entanglement of the state they have a global character. To describe the combined state of n qubit 22 complex numbers are needed
In a classical computer the measurement of the value is the same as the binary value of the concerned bits, i.e the output is the binary value of the given bits in addition the state is not effected by the measurement it is ‘non-destructive’ yet when measuring a quantum state according to kopenhagner the outcome must be formulated in classical terms i.e binary bits. The quantum state therefore collapses and a value is given this means that measuring the state changes is it.
Calculations are done by Unitary Transformations on the qubits a long with principles of superposition this allows for a lot more possibilities that are not available in classical computation this gives more efficient algorithms for factoring numbers (which is very important in cryptography) and simulation of quantum mechanical systems.
In a classical computer calculations are done the same way as by hand meaning the type of things that can be solved efficiently are the type of things we can solved efficiently by hand
No reversible quantum computing.
These are just some basic differences between classical and quantum computing to put this into perspective it just means that they operate differently and because of this they serve different purposes quantum computers are not for every day sue such as surfing the web and such, they could probably still achieve this but the speed would not be necessarily faster than a classical computer so building a quantum computer for such activities would be a waste of time, quantum computing is tailor made for things such as performing simulations of the quantum mechanical process in physics chemistry and biology it is important to note that a classical computer can emulate a quantum computer but not to the same degree of efficiency
- Electron spin
- Photon polarization
- Energy level of particle
As we are talking about quantum computing it seems suffice to talk about the hardware, the physical devices we create to make these computations and take advantage of how atomic particles operate at a quantum.
The Spin of an Electron
When a charged particle moves it creates an electromagnetic field this is a law in classical physics, which has been demonstrated many times, to introduce you to the term ‘Spin’ I will first start with a experiment conducted. In the experiment we have a Stern Gerlach apparatus, the apparatus works by measuring the orientation of the magnet that passes through it(from an up and down perspective). We then take electrons (charged particles) which are static and unmoving and put them in the apparatus. When using actual magnets the results were that there was an even spread of orientation over several different values. When using the electrons the expectation was that they would be randomly orientated but there was only two possible results, completely up or completely down from a quantum mechanics perspective this makes a lot of sense as we describe single qubit systems as a combination superposition of two states and when the qubit is measured it gives us either one or the other nothing in-between. After testing the electrons this way to make sure there was no human or systematic error taking place we turned the machine on its side, this allows us to measure the orientation from a right left perspective. The expected results was that nothing would happen to the electrons as they were already seen to be orientated in the up down direction but again half of the projected electrons went left and half went right. In essence The up/down orientation of the particle is an observable the eigenstates are simply vectors in the up and down direction and a have function to describe it is simply a(up)+b(down). We can convert this to a different basis for left and tight. In 1922 a similar experiment to this was conducted by Otto Stern and Walter Gerlach and the results concluded that individual electrons were in possession of an intrinsic angular momentum and a magnetic moment. This is possible if the electron is thought to be a spinning ball of charge. This property is called spin.
Quantum Hardware, Theory and Circuits
Things required to be classified as a quantum computer in terms of the mathematics operations it must be able to carry out on qubits -these are called gates (think logic gates)- later I will show that set of qubits that include all one qubit operations and a controlled not gate will be universal meaning that I can construct all other gates with said gate. To achieve this a quantum computer must be able to perform arbitrary rotations on a single qubit and be able to alter the state of one qubit when the second qubit is in it.
In addition there are physical demands that a quantum computer must meet to be able to implement these operations.
A quantum computer must be able to apply the transformations to the qubits without inspecting their state. If the state were inspected the whole system would become entangled and the quantum aspects of the system would be negated.
The computer must be able to hold a distinct number of qubits.
It must be able to operate individually on arbitrary qubits without affecting any of the other qubits.
The qubits must be able to maintain their state when they are unobserved and the transformations must not differ to much from what was theoretical specified. This is a big problem with quantum computers because tradition error correction techniques are not applicable to them.
During operation the computer must not inspect their state. But otherwise it must be able to inspect the state in the form of measurement to any arbitrary amount of qubits in the system either directly or indirectly.
Of all the requirements stability is the chief obstacle towards building a quantum computer. (Expand on this point)
I will now talk about the theory behind a quantum computer the mathematical operations they hope to perform when talking about this in the terms of a quantm system with a state space we can denote S we hope to find the unitary transformation T : S S (linear transformation) that can be computed on the quantum hardware models that exist right now.
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