Hello, habrozhiteli! The purpose of this book is to introduce quantum computing to anyone who is familiar with a high school math course and who is willing to work a little. In this book, we will get acquainted with qubits, entanglement (quantum states), quantum teleportation and quantum algorithms, as well as with other topics related to quantum computers. The task is not to give a vague idea of these concepts, but to make them crystal clear.
Quantum computing is often mentioned in the news: China teleported a qubit from Earth to a satellite; Shore’s algorithm jeopardized the current encryption methods; quantum key distribution will again make encryption a reliable means of protection; Grover's algorithm will increase the speed of data retrieval. But what does all this really mean? How does it all work? Chris Bernhard is about to tell about this.
Excerpt. Einstein and local realism
A good example for explaining local realism is gravity. Newton's law of gravity gives the formula of the force of attraction between two masses. If we substitute the dimensions of the masses, the distance between them and the gravitational constant, we can obtain the magnitude of the force of attraction. Newton's law changed physics. Using it, for example, you can prove that the planet rotates around the star in an elliptical orbit. However, despite the fact that the law describes the magnitude of the force, it does not tell us anything about the nature of this force.
Newton's law of gravity can be used for calculations, but it does not explain how gravity works. Newton himself was also worried about this. Everyone thought that there should be some deeper theory explaining the effect of gravity. Many different assumptions were made, often involving the “ether", which should be an integral part of the universe. And although there was no consensus on the mechanism of gravity, no one considered gravity a supernatural action at a distance, and everyone believed that some kind of natural explanation could be found. There was a belief in what we now call local realism.
Newton's law of gravity was replaced by Einstein's theory of gravity. She not only improved Newton's theory in terms of the accuracy of predicting astronomical observations that cannot be derived using Newton's theory, but also explained how gravity works. She described the distortion of space-time. According to her, the planet moves in accordance with the form of space-time in which it is located. No supernatural action at a distance. Einstein's theory was not only more accurate, but also described how gravity works, and this description was local. The planet moves in accordance with the shape of space in its vicinity.
The Copenhagen interpretation in quantum mechanics reintroduced the idea of supernatural action at a distance. When measuring a pair of entangled qubits, their state changes immediately, even if they are physically removed from each other. Einstein's reasoning seems quite natural. He had just ruled out a supernatural action from the theory of gravity and is now facing it again. Unlike him, Bohr did not believe in the existence of a deeper theory capable of explaining the mechanism of this action. Einstein did not agree with him.
Einstein believed that he could prove the fallacy of Bohr's position. In collaboration with Boris Podolsky and Nathan Rosen, he wrote an article in which he pointed out that his special theory of relativity implies the impossibility of disseminating information faster than the speed of light, but instantaneous action at a distance means that the information from Alice to Bob can be delivered instantly. This problem is called the EPR paradox, that is, the Einstein – Podolsky – Rosen paradox.
In our time, the EPR paradox is usually described in terms of spin, and this is exactly what we will do, although Einstein et al described the problem differently. They examined the location and momentum of two entangled particles. And the formulation from the position of the back was suggested by David Bohm. It is Bohm’s wording that is currently in use, and that it was used by John Sewart Bell to calculate his important inequality. Although Bom played an important role in describing and formulating the paradox, his name is usually omitted.
In the previous chapter, it was pointed out that the Copenhagen interpretation does not allow the possibility of transmitting information faster than the speed of light, and therefore, although the EPR paradox is not really a paradox, there is still the question of whether there is an explanation that eliminates the supernatural action.
Einstein and the hidden variables
From a classical point of view, physics is deterministic - if the initial conditions are known with infinite accuracy, you can predict the exact result. Of course, the initial conditions can be known only with some finite accuracy, in the sense that the measurements always have some error - a small difference between the measured and the true value. Over time, this error may increase to a value that will no longer allow an adequate forecast. This idea underlies the so-called sensitive dependence on initial conditions. She explains why the weather forecast for more than a week is extremely unreliable. However, it is important to remember that the underlying theory is determined. The weather looks unpredictable, but this is not due to any inherent accident, just we can not take measurements with sufficiently high accuracy.
Another area where probability invades classical physics is the laws regarding gases, that is, the laws of thermodynamics, but the theory itself is again deterministic. If you know exactly the speed and mass of each molecule in a gas, theoretically you can accurately predict what will happen to each molecule in the future. In practice, there are too many molecules to be able to take into account each of them, so we take the average values and consider the gas from a statistical point of view.
It was to this classical deterministic view that Einstein referred when he very coolly stated that God does not play dice with the Universe. He felt that using probability in quantum mechanics demonstrates the incompleteness of a theory. There must be a deeper theory, possibly including new variables, that is deterministic but looks probabilistic if all of these so far unknown variables are not taken into account. These unknown variables began to be called hidden variables.
The classic explanation of entanglement
Let's start with our quantum clocks in state
Alice and Bob ask the question: does the arrow point to twelve? The quantum model claims that they will both receive the same answer: “yes, the arrow points to twelve” or “no, the arrow points to six”. Both answers are equally probable. In fact, we can experiment with the spins of entangled electrons. The results of these experiments will exactly match what the quantum model predicts. But how does the classical model explain these results?
The classical interpretation of the described situation looks quite simple. Electrons have a definite spin in any direction. Entangled electrons become entangled as a result of some local exposure. And again, we turn to hidden variables and a deeper theory. We do not know exactly what is happening, but there is some local process that transfers electrons to the same spin state. When they are entangled, the spin direction is chosen immediately for both electrons.
This can be compared with the situation when we have a deck of cards, which we first mix, then without looking we remove one card, cut it into two halves and put it in two envelopes, all this time not knowing which card was removed from the deck. Then we send the envelopes to Bob and Alice, living at the opposite ends of the universe. Neither Alice nor Bob suspect which card they received. It can be any card from fifty-two, but as soon as Alice opens her envelope and sees a jack of diamonds, she will know for sure that Bob also received half of the jack of diamonds card. There is no action at a distance and nothing supernatural.
In order to arrive at the results obtained by Bell, we must measure our intricate qubits in three different directions. Now, back to the analogy of the confused clock, we will ask three questions: does the hand point to twelve, four, and eight. The theoretical quantum model claims that each question will be answered either “yes, indicates” or “no, it indicates in the opposite direction.” Both answers to each question are equally probable. But when Alice and Bob ask the same question, they will get the same answer. This can be described from the classical point of view in exactly the same way as before.
There is some local process that confuses the clock. We are not trying to describe exactly how this is done, but simply referring to hidden variables - there is some deeper theory that explains all this. But when the clock is confused, quite definite answers are chosen to three questions. This can be compared with the situation when there are three decks of cards with shirts of different colors. We take one card from the deck with a blue, red and green shirt. We cut each in half and send three halves to Alice and three halves to Bob. If Alice sees a half of a jack of diamonds card with a green shirt, she will know for sure that Bob will receive a half of a card with a green shirt, which is a jack of diamonds.
Regarding our quantum clocks, the classical theory says that for each question there is a definite answer, which is predetermined even before the question is asked. Quantum theory, by contrast, states that the answer to a question is not defined until it is asked.
Bell inequality
Imagine that we generated a stream of qubit pairs and sent them to Alice and Bob. Each pair of qubits is confused.
Alice randomly chooses the direction of 0 °, 120 ° or 240 ° to measure her qubit. Each of these directions is chosen randomly, with a probability of 1/3. Alice does not remember the selected directions, but writes the result, 0 or 1. (I recall that 0 corresponds to the first base vector, and 1 to the second.) After Alice measures her qubit, Bob randomly chooses one with a probability of 1/3 from the same three directions and measures its qubit. Like Alice, he does not remember the direction of measurement, but records the result, 0 or 1.
As a result, Alice and Bob get a long line of 0 and 1. Then they compare their lines, character by character. If the first characters match, they write the letter A, if they do not match, the letter D. Then go to the second character and also write A or D, depending on the coincidence or mismatch. So they compare all the characters in their lines.
The result is a new line consisting of the letters A and D. What proportion of the line will be in the character A? Bell noted that the model of quantum mechanics and the classical model give different answers.
The answer to the model of quantum mechanics
Qubits are confused
We have already seen that if Alice and Bob both choose the same direction for measurement, they will receive the same answer. Now let's see what happens if they choose different bases.
Let's start with the case when Alice chooses
and Bob chooses 
Confused state
can be written using Alice's basis as
When Alice performs her measurement, a transition to the state
or 
each of which is equally probable. If a state transition occurs , it will write 0. If there is a transition to the state she will record 1.
Bob should now take the measurement. Suppose that after Alice’s measurement, the qubits are in a state
that is, Bob's qubit is in a state 
To calculate the measurement result by Bob, you need to rewrite this state using Bob's basis. (We already did similar calculations in the Alice, Bob, and Eve section of Chapter 3.)
Having written down the solution using two-dimensional ketos, we get:
Multiply
to a matrix with rows corresponding to the sconces from Bob's basis.
As a result, we get
Having completed the measurement, Bob will receive 0 with a probability of 1/4 and 1 with a probability of 3/4. That is, when Alice gets 0, Bob will get 0 with a probability of 1/4. It is easy to verify another case. If Alice gets 1, Bob will also get 1 with a probability of 1/4.
Other cases give similar results: if Bob and Alice take measurements in different directions, their results will coincide in 1/4 of the cases and not coincide in 3/4 of the cases.
As a result, in 1/3 of the cases they take measurements in one direction and always get matches; in 2/3 of the cases, they take measurements in different directions and get matches in 1/4 of the cases. Accordingly, the proportion of characters A in a string of A and D is
Thus, according to the model of quantum mechanics, with a sufficiently large number of tests, the fraction of symbols A should be half.
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