Can a single qubit store infinite information?

Qubits are to quantum computing what bits are for classical (non-quantum) computing: they form the basic unit of computation and determine how data can be stored and manipulated in a computer. While each classical bit can store a value of 0 or 1, a single qubit can be in one of the infinite superposition states of the two.

Since qubit values can be chosen from a continuous value range, this raises the question of how much information can be stored in a single qubit. Or, more boldly: Can a single qubit be used to store infinite information?

Similar to the question of whether quantum computing is parallel, the answer to this question is a resounding “Yes, but…”.

The Yes-Part:

While a classical bit can be either in a state of 0 or a state of 1, a qubit can be in any state \(|s> = \alpha |0> + \beta |1>\) that fulfils the condition that \(|\alpha|^2 + |\beta|^2 = 1\) where \(\alpha\) and \(\beta\) are complex numbers.

In other words: qubits do not take on discrete values. A qubit can be in a state of \(|0>, |1>, \frac{1}{\sqrt{2}} (|0> + |1>)\), or even \(0.707106781187 |0> – 0.707106781187 i |1>\). As long as the aforementioned condition is satisfied, the qubit state is valid and can be physically realized in a quantum computer.

So much for the qubit part of the question, but how does any of this help us store information?

Let’s take a two-digit number in a decimal basis and call it C.

C can take on any number between 0 and 99. If we limit C to only one of the numbers between 0 and 63, C can be used to store exactly 6 bits of information, anything between 000000 (0) and 111111 (63).

If a two-digit number (in decimal) can store 6 bits, so can any number with a precision of 2 decimal places. The same amount of information that could be encoded in a number ranging from 0 to 63 can also be encoded in the range of 0.00 and 0.63. All we have to do is change the position of the decimal point.

The more decimal places we have at our disposal, the more information we can encode. Using 5 decimal places, we can encode 16 bits and so on. With infinite precision therefore comes infinite information!

Unfortunately, though, there is a catch…

The But-Part…

While a qubit’s value range* bears the potential to encode an infinite amount of information, how to encode it, how to use it, and how to return it, are all factors that limit the utility of that potential.

* Or any continuous value range, really…

For starters, if we wish to encode an arbitrarily high amount of information, we need to be able to manipulate qubits with an arbitrarily high degree of precision. If we wanted to put a qubit in a superposition state of \(|s> = 0.21179691621932634718744437537 |0> + … |1>\), in the case of a trapped-ion quantum computer we would need to be able to produce laser beams with a comparable degree of precision. Even further, infinite information would require infinite precision, an unreachable target.

Secondly, suppose we want to use the potentially infinite amount of precision encoded in a qubit state. We would need to define operations that manipulate it in meaningful ways without measuring it. To the best of my knowledge, there is not much research going on in this regard at the moment.

Even without the ability to manipulate the infinite information we stored in a qubit, it would still be very useful as a datastore. Just imagine you could store whole movies, books, and databases, in a single (qu)bit…

The catch is that while we might be able to store an infinite amount of information in a single qubit, we cannot return this information without loss. As soon as we measure the qubit in question, its superposition will collapse and we will be left with either a state of 0 or 1. To extract the information without any loss, we would have to perform an amount of measurements proportional to the information encoded. This would destroy the advantage we sought to create.

Although all of these catches make the whole endeavor seem rather gloomy, some interesting research directions arise from this. While storing infinite amounts of information might be a bit ambitious, there are multiple ways in which qubits can hold more information than their classical counterparts.

For once, the coefficients of a qubit state are complex numbers, so they already possess an additional dimension compared to classical bits. Furthermore, these coefficients can also be negative, a property already used to distinguish different states in a superposition for example in Grover’s Algorithm.

Finally, Entanglement enables quantum computers to encode certain information in a quantum-native way that would require exponential resources on a classical computer. Through entanglement, a linear increase in qubits can match an exponential growth in classical information, at least in certain cases.

The resulting question in this regard should therefore not be how to store an infinite amount of information in a single qubit, but rather how to use the additional possibilities quantum computing provides us with in a meaningful way.

Summary

Based on the differences between the classical and quantum bits, qubits bear the potential to store an infinite amount of information in the decimal places of their coefficients. This potential, however, is limited by how we can encode, manipulate, and read out this information. Most notably, whenever we measure such a qubit, its state will collapse to either |0> or |1>, losing all the information encoded in the decimal places.