Quantum computing in future may enables new possibilities for a variety of use cases e.g. autonomous driving, encryption. Why?
Exponential computation complexity (e.g. 2^n) is calculated in polynomial time (n^2) – get more as you investment 🤑 e.g. with n = 10 the
- complexity is 1024 operations and
- runtime requires 100 operations.
The saving is 924 operations. For a moment just let us use the term “operation” as a unit of required work.
Doing calculations in the 2 dimensional space, 2^n states can be described with n quantum-bit, e.g. 32 quantum bits keeps 4294967296 states (simultaneously).
In one dimensional space it is always n times one state e.g. 32 bit take 32 states.
But, a good IEEE paper on the use RSA decryption (here) says:
“100-million-qubit system would be needed to factor a 2,000-bit number— a not-uncommon public key length – in one day”
What is the quantum model?
Ok but what is a “quantum“? It is the name of a model that describes
- What: the quantitative 📏 measurement of energy packets and,
- How: the mechanics of this energy, in sense of
- cause and effect behavior as well as
- resulting state after impact.
What is a quantum?
The describing energy-packages (aka every-elements) are quantum-bits (aka q-bits).
Due to Wikipedia “According to Max Planck, quantities of energy could be … divided into “elements” whose size (E) would be proportional to their frequency (ν) and mass-frequency proportion (Planck constant h).”
E = v x h
Einstein 🧐 in comparison defined the proportion between mass and energy, the C in E=m x C^2 equation.
The icon for energy packets is the Blocksphere. Due to Wikipedia:
What is the quantum behavior?
´The most interesting form of energy is light 🤨, because the related mechanics has its own history:
- In the 17th century Newton physically assigned light a particular behavior. This implies it only can be at one specific place at a time ⚽️.
- In the 18th century the double-slit experiment identified a wave like behavior, spread through the space ⛵️, being at multiple places at a time and weaken or strengthen through its overlapping.
- In the 19th century the particle-wave-duality proofed both behavior via the double-slit experiment.
A nice analogy is the Schrödinger’s cat, sitting in a closed box. The cat is simultaneously, with a certain probability, alive and dead. As soon you look into the box, the state becomes one clear fact (aka state).
Or in other words the state is like a postion. Being “here” and “there”. At positions at the same time – it is a wave. As soon as we measure the state it is clearly “here” or “there”.
What are the quantum states
Now it is becoming interesting 🤯
Like Schrödinger’s cat, the quantum bit is in multiple states at the same time. Two states are mathematically described by 2 dimensions, the x-axis and a y-axis of the probability wave 🌊 (aka amplitude).
More precise:
Note: Vector space has been reduced and simplified in comparison to the Blochsphere afore. We use the 2 dimensional reduced Hilbert-space understanding as the circle plane between |0> and |1>.
Important take away: Independend from the dimensionality, the sum of all dimensions is always 1 or even 0. Due to that only a few e.g. 2 dimensional positions are allowed. In other words, the relationship between all dimensions is always equal (aka sum is 1). Otherwiese it is an entangled state.
Why wave? As soon we observe the wave, it is a 1 dimensional fact horizon (not state). Like a surfer hitting a wave with a certain energy power.
Why probability? The surfer hits the wave with a different energy proportion. The power itself is between 0 and 100%. To map it into a boolean state a threshold of 50:50 is assumed; like hit the wave or even not (aka the factual state). ´
The state before measurement is notated (probability) in the
- ket form: column-vector,
- bra form: row-vector.
The combined bra-ket (Dirac notation) notation only counts the amount of quantum states e.g., 2 qbits with
- |20> with 2 bits in state A and 0 bit in state B
- |11> with 1 bit in state A and 1 bit in state B
- |02> with 0 bit in state A and 2 bits in state B
The state after the measurement is the so called density operator |ψ> (aka wave function, psi) as the state with its position in the n-dimensional space.
For operations the control bit (aka c-bit) / row-vector notation is used.
- |ψ> for a 2D example state |0> = (1,0) as c-bit notation equals 1 time “x” and (+) 0 times “y”
- |ψ> for a 2D example state |1> = (0,1) as c-bit notation equals 0 times “x” and (+) 1 time “y”
- |ψ> for a 2D example state |00> = (1,0,0,0) for the probability of 4 states in parallel:
- 00 with 1
- 01 with 0
- 10 with 0
- 11 with 0
The factual position of the wave in the 2 dimensional space, can be expressed in different forms. e.g.
- As coordinate it is e.g. 2 times “x” and 3 times “y” = 2x + 3i.
- In the Euler-form it is described by the sinus and con-sinus relationship of the angle between x-axis root and the vector itself.
The quantum mechanics and computing
The quantum mechanics is about changing the state probability through respective operations. Applying operations is equal to perform computations.
In other words. The state positon “here” and “there” changing its relationships. If it was “here” to 40% it may afterswards is 60%. The combination with another view on “here” and “there” – the operation.
As wave it is like inferences (constructive- or destructive-inference). Inference can be shown via the double slit experiment:
An operation (aka inference) is also called a gate. Compare the list of available gates here. A gate (c, d) aka ket in the product operation with the given state (a, b) aka bra or vise versa. Mathematically two different operations are applied:
- scalar product or inner product of bra <a | and ket |b> , short <a | b>, e.g. (a,b) · (c,d) = (a · c + b · d) = real number = quantify the distance between the two states (aka fidelity). The Euclidean inner product – hey data scientists – is replaced by the Hilbert-Schmidt product. Usually used for test gates e.g, measurement.
- tensor product or outer product of ket |b> and bra <a | , short |b><a |, e.g. (a,b) ⊗ (c,d) = (a x c, a x d, b x c, b x d) = complex number = new position in space. In words “each energy element of vector an interacts with the energy elements of vector b“. Usually used for operations, e.g., calculations.
Per definition all operations need to be invertible e.g. the circle state machine shows 16 invertible states. If the resulting state cannot be expressed as a tensor product (aka not invertible, aka not de-composable) it is called an entangled state – if one changes the state, the other turns it into the “opposite” position. Entanglement makes it possible create complete 2^n dimensional complex vector space.
May follow this discussion thread on stackexchange, stating e.g.,
- “The AND gate is not reversible as you can see from the way the number of outputs is smaller than the number of inputs“
- “Beware the swap gate! It cannot be written in the form A⊗B, but it is also not entangling. “
Example: A unary adder using the Quirk simulator
Pattern of quantum algorithm application
Quantum algorithm can be separated into three phases
- Pre-processing on common computer, with necessary encoding
- Quantum calculation
- Operations incl. the initial state preparation, applying quantum gates
- Measurement, reading the result
- Post-processing on common computer, required for necessary decoding
Quantum computing shows the important of encoding and the resulting information density better as most e.g, machine learning technique denotes. In the example afore we just used the binary representation of integer.
Sure, also the binary representation (blue balls) is an encoding, but it not allows to use the advantages of quantum computing. A classical computer does the job much better.
In comparison the tensor product encoding uses the full 2D circle to represent the information, e.g. (cos x, sin x), with x as qbit.
Where and why I failed a quantum computer 😜
Quantum programs are hard to program. It is like playing music 🎼 ; you can once start and stop a song.
What I learned from trying to play songs:
- Live without
- stack – e.g. no function arguments, no variables
- heap and the ability to copy values between variables – the “no cloning”-theorem doesn’t allow this.
- memory, data storage 💾, files or I/O of data
- Turing machine formalization for
- superposition and
- entanglement.
- Only the abort of the calculation allows to measure the factual result – consider the cat in the box.
- A calculation setup is hard to construct
- build a physical setup which can only play “one music song”
- for generic calculation machine we need to use electro magnetic fields in “noise-less” environments, e.g. requiring temperatures around the absolute zero of minus 273 degree Celsius.
Good to know, the electromagnetic-effect was the reason why Einstein 🧐 earned its Nobel-price.
A well written paper about the quantum computing issues is the “The Bitter Truth About Quantum Algorithms in the NISQ Era“.
What is needed for a quantum computer
A quantum computer machine requires the ability
- to put energy in superpositions (act in a space > 1 dimensional) and,
- to perform operations on energy packets (implicitly or explicitly).
It is your choice:
Available kinds of “real” quantum computer
- Electromagnetic: Ion trap, somehow the most stable option “not” requiring specific surrounding condition.
- Electrical: Transmon used by the hyperscaler such as Google and Microsoft, but require temperature close to absolutions zero
- Thermal combined with one of the two afore.
Nice over can be found here