Will Quantum Money replace cryptocurrency? (Part 1)
Schrödinger’s cat has some surprising applications to the world of digital currency. This is part 1 in a series of articles on Quantum Money. Read part 2 here.
In recent years, the terms digital currency and quantum computing have appeared countless times in popular science headlines. The general public has started to talk about terms like Blockchain and quantum supremacy, even if they only have a vague idea of what they mean. These are not just empty buzzwords; they refer to specific and fascinating technologies that are currently showing promise for transforming our society.
Although they may seem unrelated, these two concepts have a point of intersection. Quantum Money is a cryptographic protocol that uses quantum computing—computers that leverage the power of quantum mechanics — in order to create a form of digital currency that cannot be forged. Although the original idea dates back to the early 80’s, quantum money has become more relevant with the ongoing quantum computing revolution and with recent research proving its security properties.
The world is now abuzz about electronic money in the form of cryptocurrencies such as Bitcoin, but could quantum money serve as a new and more effective form of digital currency?
In this series of articles I will explain the basics of quantum money from the ground up, assuming only basic knowledge of linear algebra. We will first review the fundamentals of quantum computing, and then we will see how this can be applied to create digital currency with many desirable properties. After reading these articles you will understand how the quantum computing revolution could provide a new type of digital currency, and how it is fundamentally different from today’s cryptocurrency.
In order to discuss quantum money, we first must review the basics of quantum computing. Don’t worry — it doesn’t require a PhD to understand! All that is required is elementary knowledge of linear algebraic concepts such as vectors and matrices.
You might know that classical computers store information using binary — as a sequences of 0’s and 1’s. Digital devices store these in different ways, such as using two different voltage levels or by using a flip-flop circuit. One single unit of data (a single 0 or 1) is called a bit. For example, your favorite cat photo might have size 325 KB which means that it is stored as a collection of 2600000 bits.
Let’s consider what would happen if we try to store information using a more interesting method, by using polarized light.
If you enjoy visiting the beach, you might have used polarized sunglasses to keep the sun’s glare out of your eyes. What does this do exactly?
Classical physics considers light to be a wave which may have different orientations, called polarization. A polarizer is a filter which only lets light with a given polarization through, as seen in the image below:
Just like the systems discussed above, we could use this to store information as bits. For example, we could use horizontally polarized light as the 0 state and vertically polarized light as the 1 state.
Quantum mechanics understands that phenomena like light have properties of both waves and particles (“wave-particle duality”). Let’s consider a single particle of light, known as a photon, and how we could use it in a computer.
Each photon has a property corresponding to polarization (technically this is called a photon’s “spin”). Let’s use the following symbols to represent the two states of our photon: [1]
So far, it seems like we are just reproducing a classical computer. We can store one bit in a photon by setting its state to |0⟩ or |1⟩, so where is the quantum advantage?
Here’s where things start to get weird: We can also have light polarized in a diagonal direction. If we have light polarized at a 45° angle and look at it through a horizontal polarizer, we will only see half of the light go through. But on a quantum scale, the light is made up of individual photons… what happens to each one?
Quantum mechanics says that in this case, each photon randomly collapses to one of the the states |0⟩ or |1⟩. Until we “measure” the photon by passing it through the polarizer, it was in a quantum superposition of the states |0⟩ and|1⟩. Informally it was both states at the same time until we try to measure it. This is the famous Schrödinger’s cat thought experiment, where a cat in a box might be alive or dead based on some quantum phenomenon, until an observer measures its state by looking at it.
Mathematically, our photon could be in any state of the form:
When α=1 and β=0, this is the state |0⟩ (horizontal polarization). When α=0 and β=1, this is the state |1⟩. Intermediate values of α and β represent diagonal polarization, which is a superposition of the states |0⟩ and |1⟩.
We could also write our state as a column vector:
In particular, the two classical (bit) states of our photon are:
The state |ψ⟩=α|0⟩+β|1⟩ above will collapse into |0⟩ or |1⟩ when it is measured. The probability of observing either state depends on the values of α and β.
If α and β are real numbers then the probability of observing |0⟩ is α² and the probability of observing |1⟩ is β². Actually any α and β such that α²+β²=1 define a valid quantum state ψ⟩=α|0⟩+β|1⟩. [2]
For example, light polarized at a 45° angle is represented by the quantum state
This state |+⟩ has probability α²=1/2 of collapsing to state |0⟩ and probability β²=1/2 of collapsing to state |1⟩.
A single particle in a quantum state |ψ⟩=α|0⟩+β|1⟩ is called a qubit. Just like a bit, it can take two basic values, but unlike a bit it can also be in a superposition of these values.
It might seem surprising, but we can use this along with other quantum phenomena to perform calculations that would be very difficult for a classical computer!
Unfortunately, it is difficult to store particles in precise quantum states for long periods of time, so quantum computers today contain very few qubits. Interesting applications like quantum money will require better hardware in order to be feasible.
“When I take a look at the first applications, we’re going to need several thousand, if not 100,000 qubits, to do something useful,” said James Clarke, director of quantum hardware at Intel. “If we’re at 50 to 60 qubits today, it’s going to be a while before we can get to 100,000 qubits. It’s going to be awhile before we can get to 1 million qubits, which would be necessary for cryptography.” — The Great Quantum Computing Race (Lapedus 2021)
This was part 1 in a multi-article series on quantum money. Read part 2 here.
Footnotes
[1] These types of symbols are called “Dirac notation” and are commonly used in quantum mechanics and quantum information theory.
[2] For reasons beyond the scope of this article, α and β can also be complex numbers with |α|²+|β|²=1.
Morris Alper is a data scientist located in Tel Aviv, Israel. He is the Data Science Lead and Lecturer at Israel Tech Challenge and a current MSc student in Computer Science at Tel Aviv University. For more information and contact details, see https://morrisalp.github.io/.
