The quantum computer uses quantum physics principles to explore simultaneously an exponential number of computation paths, and is probabilistically queried for those paths leading to a solution.
In this way, the quantum computer is a realization of a non-deterministic turing machine, where a classical computer is the realization of a deterministic turing machine.
Quantum physics principles no more implies how a quantum computer will operate than do the principles of electricity imply how a classical computer will operate. Just as there is a subtle and fascinating history to how our sort of computer came to be, there is this history for the quantum computer.
The approach here appreciates that this history exists, but upon it, remains mostly silent; in the same way other computer science courses are silent on such things as analog computers or Field Programmable Gate Arrays.
Systems and particles in the quantum domain exhibit behaviors that are impossible to replicated in the classical world. They can seem to be in many places at once, and in many states at once. Communication within the system seems to transcend the constraints of time and space to coordinate magically entangled parts of the system.
Such strange behavior collapses into common behavior of a non-quantum system as soon as an observation is made of the system. The multiplicity of possibilities in the quantum state becomes a probability distribution over the outcomes of the observation.
However, this is not to be misunderstod asthe quantum system being in some state, but that state is unknown until an observation confirms it. The quantum system is in a state inconceivable to the non-quantum mind until the observation forces the system to choose an outcome, to fit into the limitations of the non-quantum world.
This choice is made a random, with a pure randomness that cannot be found in classical physics. Although where the ball lands on the Roulette wheel is random, many try to run the complex dynamics through a simulation, and in principle, if the initial data is all known, the final result will calculate out in accordance with the observed result in reality. While this is not possible, that impossibility is more a theoretical impracticality.
In quantum, the randomness of the choice of outcome of an observation is pure and axiomatic.
Imagine that a photon as a sphere a defined axis running through it. The photon travels in the z direction, and the axis of the sphere lies in the x-y plane. The axis still has a degree of freedom to rotate in the x-y plane. We will measure this angle θ from the "up" direction. Hence a photon with linear polarization at angle θ equal 0 is in the "up" direction.
The polaroid filters of this experiment have a direction of polarization as well. Photons whose angle of polarization align with that of the filter will pass through the filter. Photons at complete right angles to the filter's polarization will not pass through, they will be absorbed.
What happens to photons that are in between θ = 0 (pass completely) and θ = 90 (absorbed completely) is a demonstration of how quantum particles act.
One cannot know at quantum state in the classical world. When making an observation of a quantum state, the quantum state will leap into the quantum state that is pure for the observation outcome. In this case, if the photon is at some angle, and passes the filter, it leaps to the state θ = 0. If said photon, at some angle not necessarily 0 or 90, is absorbed, it leaps to the state θ = 90, the classical state for photons that are absorbed.
One can know only what one observes, and one cannot fully observe a quantum state. It is ineffable with words and concepts of the classical world.
The likelihood of the photon aligning its state with the direction of the filter or orthogonal to the direction of the filter is determined by the angle of the photon. If the photon is θ = 45, halfway between the direction of passing and the direction of absorbing, then half the photons will pass and half will be absorbed. Which half is determined by a coin flip of whose origin we know nothing.
If the photon is θ = 30, it is more likely to adopt the θ = 0 state then the θ = 90 state. In fact it will do so with probability 3/4 — three quarters of the photons will pass and one quarter will be absorbed.
P.A.M. Dirac introduced a notation that helps focus on the "stateness" of a state, but allowing them names. We choose a sense of which direction will be a classical 0, and then the ortogonal direction is a classical 1. That is because cos 90 = 0, so a photon right angle to the direction of polarization will never be observed as passing through the filter.
The photon state associated with the classical 0 state is denoted ∣0❭, and associated with a classical 1 is ∣1❭.
We have three polaroid filters. (Linearly polarized. There is also circular polarization, which would be an entirely different experiment.)
We allow a wild source of photons (the light through the window) to pass through a polaroid filter placed upright. According to our theory, each photon enters at some unknown state and either adopts ∣0❭ or ∣1❭ according to a probability the favors the more similar orientation.
Arguing entirely intuitively, by symmetry, about half the photons pass through, and the all emerge from the filter in the ∣0❭ state.
If a second filter is placed next in line,and orientated 90 degrees, so that now ∣0❭ state photons are absorbed, indeed no light passes. If that filter is rotated, more and more photons pass until the polaroid is oriented equal to the first filter, and now all photons pass, as they were all in the ∣0❭ state, so their observation as passing through is a certainly.
At 45 degrees, the light dims, as one half of the photons jump to 45 degree polarization and pass, and have to −45 degres polarization and are absorbed. The name for the 45 degree state is ∣+❭ and for the −45 degree state is ∣−❭
Because we live in a classical, non-quantum world, our knowledge of a quantum state needs to be measured to be known. But a measurement provides information about an unknowable quantum state so the quantum state will be changed by the measure measure to agree with one of the possible observable outcomes.
In the case of our filters, the measurement is the photon either passes or it does not. It passes if it is state ∣0❭ and does not pass if it is in state ∣1❭. Hence the flood of wild photons entering the filter with unknown quantum states will adopt one of these two states and accordingly be absorbed into or transmitted through the filter.
As the second filter is rotated, the probability becomes more evenly divided between the two possible observations we can make with the second filter. At 45 degrees, half of the photons are observed to pass and therefore have adopted the ∣+❭ state, and half are observed to not pass, and have adopted the ∣−❭ state.
Continuing to rotate the filter to φ = 90 degrees, the direction of the filter is perfectly orthogonal to the ∣0❭ state of the photon, so that the probability it will adopt this direction and pass through the second filter is 0. None of the light gets through.
Consider now introducing a a third filter between the two, that filter at 45 degrees. With this filter in between the other two, the passage of photons is restored. That is remarkable and hard to explain with classical models. Quantum it is simple enough.
+---+ +---+ +---+
| ⇑ | | ⇗ | | ⇒ |
all orientations ⟿ | ⇑ | ⟿ ↑↑↑ ⟿ | ⇗ | ⟿ ↗↗↗ ⟿ | ⇒ | ⟿ → → →
| ⇑ | | ⇗ | | ⇒ |
+---+ +---+ +---+
½ pass ½ pass ½ pass
The three polarizer experiement
What is happening is that the first filter provides us with state ∣0❭ photons. Half of which when measure by the second filter take quantum state ∣+❭ and are passed through the second filter. These photons are now only at 45 degrees from the measurement basis of the third filter, not 90 degrees as they were when the middle filter was not inserted. And therefore 1/2 of these photons can adopt quantum state ∣1❭ that pass through the third filter.
❖ Watch it on YouTube!
author: burton rosenberg
created: 26 apr 2020
update: 4 may 2020