By Killian Robinson
In traditional computers, data is represented as an extraordinary array of 0’s and 1’s, these 0’s and 1’s are known as bits. Data can be stored as a combination of 0’s and 1’s, 8 of them for example is a ‘byte’, with megabytes being 8 million bits or 1 million bytes. These ‘bits’ (binary digits) can only hold a certain amount of information and this is limited by having the bits as definitively being either a 0 or a 1. This may seem like a strange comment, how can there be any other way of bits existing? Enter Qubits. Quantum bits (aka Qubits) allow superposition between a 0 and a 1, thus allowing the ‘qubit’ to be a 0 and a 1 at the same time. Furthermore, as qubits are entangled, if a qubit happens to be a 0, the other qubit it is entangled with is also a 0. These qualities allow qubits to vastly improve the power and efficiency of computers and it seems that we can model the brain in the exact same way by treating phosphates as qubits, storing information in an entangled state.
Before we go on to impacts in the brain, we shall see how particles can become entangled and how we can represent the superposition of binary states of a qubit. In quantum physics, the symbol psi ( ψ ) represents wave function, which is like any other function but is used to model the wave like nature of particles and light. It can also be expressed as a vector, however in quantum physics a vector is denoted by the presence of ket notation instead of an arrow on the top. For example, |ψ> is a wave function vector (as seen in figure 1.) (Sabine Hossenfelder, 2020)
V= a1 a2 a3 =a11 0 0 +a20 1 0 +a30 0 1 = a1 a2 a3 (where an is a real number)
Instead of a column vector, we use ket of x, y, and z for |ψ> :
> = i1X>+ i2Y>+ i3Z> =i1 i2 i3 (where in is an imaginery number)
Figure 1 – Vector V is just a normal vector in 3-dimensional space, the wave function is similar to this with 3 main differences. The first is that the coefficients are imaginary numbers (i, not a), the second is that the column vectors are replaced by ket notation and lastly, that the wave function is not limited to 3D space like the normal vector is. (Sabine Hossenfelder, 2020)
Qubits can be represented as the wave function vector, |ψ>, as a point on a so called ‘Bloch’ sphere. Using Euler’s formula, we can derive an equation for its position on the sphere.
|ψ> =cos(2)|0> +eisin(2)
We can reduce this formula further to:
ψ> = 0>+ |1> , where 2+ 2=1
Figure 2 – A Bloch sphere and its equation, where the wave function vector can be defined in terms of φ (the angle between the x and y axis) and θ (the angle between the z and y axis). Taken from (Sebastiano et al., June 18, 2017)
If for example, the wave function vector is found directly on the x axis, then φ=0 and θ=2 , substituting these into the equation above gives us:
|ψ> =cos(4)|0> +eisin(4) = 12|0>+12|1>
We know that the squares of the coefficients (alpha and beta) are equal to 1 as shown in figure 2, but the squares themselves represent the probability of the qubit taking on the state of a 0 or a 1, as seen in figure 3.
Figure 3 – A diagram showing how a measurement of a wave function allows us to calculate the probability of getting a 0 or a 1. With 2 and 2 representing the probabilities of getting a 0 and a 1, respectively. (Brilliant, 2018)
Using figure 3 and applying it to figure 2 we know that on the x-axis the wavefunction yields the equation:
|ψ> =12|0>+12|1> which is equivalent to |ψ> =|0>+ |1>
Since 12 2=12 we now know that there is an equal probability of getting either a 0 or a 1 in our qubit, and these probabilities vary depending on its point on the sphere.
To explain how we can hold information using these qubits, we can investigate how two qubits can become entangled in a two-qubit based system using this formula:
|+> =12 |0A0B>+12 |1A1B>
Figure 4 – The bell state (where |+> is equivalent to |ψ>, where phi is just another symbol to denote that this is the simplest quantum entangled state for two qubits, that is they have an equal superposition (1/2 chance of happening). Taken from (Beil, n.d)
Imagine two people each holding a qubit, lets call them Alice and Bob. Alice possesses qubit A (subscript A in figure 4) which as we know has the potential to be a 0 or a 1. Bob also has a qubit the same as this (subscript B), however his qubit is qubit B. Let us say, Alice is at one end of the universe with a computer, with an infinitely long cable connecting to another computer at the other end of the universe, where bob is sitting. If Alice makes a measurement of her qubit, and it is a 0, because of quantum entanglement Bob will get the exact same state, a 0. Regardless of distance, if one of them makes the measurement, the other will get the exact same, and this is quantum entanglement. (Quantiki, 2015).
Implications for the brain
Physicists have been predicting a quantum level-based explanation for many biological processes for years now, and maybe the properties of the brain may just be explained by what we’ve just discussed, albeit more complex.
The theory of quantum memory proposed by Michael. E. Fisher suggests that phosphates in the pyrophosphate molecules from the hydrolysis of ATP can become entangled (Ball, 2017) like our qubits exemplified by Alice and Bob. The pyrophosphates can be held in clusters known as ‘Posner clusters’, and even if a phosphate from the pyrophosphate couples manages to leak out into another posner cluster, they can remain entangled regardless of distance (Ouellette, 2016).
Figure 5 – How phosphate nuclei can become entangled in the brain in Posner clusters and how this facilitates a quantum memory.
In this way, we can model the entangled phosphates as qubits, each representing a superposition of information instead of a 0 or a 1. This information may be utilized in the process of memory retention and even make it more efficient rather than dedicating particular phosphates to specific information. Perhaps, by using phosphates as quantum objects it facilitates the ability of the brain to retain much more information, and therefore explains it’s extraordinary ability to keep memories. This theory is supported by the fact that the other molecules incorporated into a posner cluster, namely oxygen and calcium do not possess a ‘nuclear spin’ which allows the entangled phosphates to remain coherent and retain information important for memory.
In conclusion, qubits are a great tool for us in quantum computing, they allow us to store much more information than we could ever have hoped for and have significant advantages over binary digits. There are many other processes in biology though, such as photosynthesis and even magnetoreception that have been proposed to be driven by the effects of quantum physics and therefore their role in the brain may not be such an unreasonable one. Perhaps the reason we have not discovered exactly how the brain retains memories is because we have not been looking at it at the right level, the quantum level. Using a quantum physics-based approach then, may reveal hidden secrets about the brain that will be essential to us in the future.
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Beil, R, G. (n.d) Correspondence of Bell state and One-particle State Transformations. Available from: https://arxiv.org/ftp/quant-ph/papers/0605/0605028.pdf [Accessed Feb 4, 2021]
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