Quantum Theory and Random Numbers

The generation of random numbers has been a subject of interest to human beings since ancient times. Proof of this is the existence of objects such as the dice, whose origin is so distant that it is not exactly known to us.

However, classical algorithms and traditional computing are not capable of generating truly random numbers, but what are known as pseudorandom numbers. To fix ideas and explain this concept, we can think of one of the most common algorithms for generating pseudorandom numbers, the linear congruential generator. This algorithm works by repeatedly implementing the expression shown in Fig. 1:

Figure 1. The linear congruential generator generates a number X_n+1 from a previous number X_n, given values for a, b and N. The first value X₀ from which all others are generated is called the seed.

Thus, for example, if we set a = 1, b = 5 and N = 6, with the seed X_0 = 1, we will obtain the sequence of values shown in Fig. 2.

Figure 2. Simple example of a sequence generated by a linear congruential generator.

As we can see, we achieve our goal of generating 6 numbers with equal probability, just as the dice does. However, the sequence is repeated and for the same seed the sequence is the same. It is in this sense that we say we generate pseudo-random numbers. Although more sophisticated classical techniques can be used to mitigate phenomena of this type, in no case will it be possible to obtain a truly random number.

For quantum nature, however, this task is simple. Think, for example, of the famous example of Schrödinger's cat. Let there be a cat in a box whose state is a superposition of the states |alive} = |0} and |dead} = |1} as shown in Fig.3.

Figure 3. Quantum state of Schrödinger cat.

When we open the box, we will find the cat in the dead or alive state with a certain probability. This probability is determined by the Born rule, one of the fundamental postulates of quantum mechanics, which states that the probability of observing each of the states of the superposition is the coefficient squared with it. Thus, when we open the box, we will have a probability of (1/√2 ^2 = ½ → 50% of observing both dead and alive states.

More generally, we can have a quantum system in superposition of 6 states to build our... quantum dice! As shown in Fig. 4., some companies already offer quantum technology for random number generation [1], with major implications for cybersecurity.

Figure 4. Quantum state of the quantum dice.

This dice will generate truly random numbers between 0 and 5, with 1/6 → 16% probability.

We can generalise further and think of the case of a trick dice, where the possible states are the same but the probabilities of obtaining each of them after measurement are not the same. An example of a quantum state for a trick quantum dice is shown in Fig. 5.

Figure 5. Quantum state of the trick quantum dice.

The possibility of measuring each of the states forms a probability distribution that can be found in Fig. 6.

Figure 6. Probability distribution for the trick quantum dice. On this dice, rolling 2 and 3 is twice as likely as rolling 1 and 4, which is twice as likely as rolling 0 and 5.

There are several proposals for constructing quantum states that represent probability distributions like the one above, among which we can highlight the Born Machine [2] or the Quantum Generative Adversarial Network [3]. All of them are promising tools for the construction of Quantum Generative AI.