We all know about the qubit.
At least, everyone who works in quantum computing.
But there is a new kid in town—the quaterbit.
Unlike qubits, which have 2 states, quaterbits have 4 states.
The idea is very old.
It is based on the concept of quaternions, a mathematical theory developed by the legendary mathematician Hamilton as far back as 1843.
It was first put forth as a workable computing model in 2008; the research paper’s link is provided below.
Quaternionic Computing, by Jose M. Fernandez and William A. Schneeberger
Now I will explain both for you, with the mathematical formalism.
The Qubit
Quantum computing is based on complex numbers, hence two states, and hence a superposition can be achieved through a superposition in probabilities.
The mathematical formulation is given below:
This is the standard formulation of qubits that every quantum computing enthusiast is familiar with.
And of course, to explain how these systems work in practice, we need to explain the linear algebra concepts:
This is the bread and butter of everyone who has ever done serious work in quantum computation.
It is completely different from classical computing and takes some getting used to.
But practice enough times, and eventually, it will feel (at least somewhat) natural.
I suggest the following links for a better introduction:
IBM offers this module:
And D-Wave offers this one:
I hope some curious readers will take the time to go through the above modules in depth!
Now for the quaterbit.
Introduction to Quaternions and Quaterbits
The mathematics:
A quaternion was initially proposed (by Hamilton, as already stated) to handle 3-dimensional systems and particular 3-dimensional rotations.
Now scientists have repurposed them for a completely different application.
Quaterbits are an almost entirely different breed of beast.
For starters, the vector representations are 4 by 1, and the corresponding quantum gates, when expressed in matrices, will be 4 by 4.
The extra dimensions add a pleasing level of computational power that is not available to standard qubits.
By the same logic of superposition, where the quantum parallelism acceleration of qubits is 2^N, where N is the number of processing units, the corresponding parallelism power of qubits is, by the same logic, 4^N.
Many other rules are common, but one thing is clear: we have a vastly more powerful mathematical toolkit.
Quaterbits: A Quantum Revolution in the Making
Quaterbits offer compelling advantages over qubits.
There are many reasons to ditch qubits and go for quaterbits as the basic building block of quantum computing.
The advantages of quaterbits are given below:
- Superior information capacity: Quaterbits can represent four distinct states at the same time! This increased capacity leads to efficient encoding and faster compute processing. A quaterbit doubles the amount of information that a qubit can carry. It also unlocks the door to more powerful algorithms and more powerful quantum computers.
- Increased computational complexity: The computational power of systems using quaterbits grows super-exponentially with the number of quaterbits. For n quaterbits, the number of possible states is 4^n, compared to 2^n for qubits. That possibly indicates that 2048-bit RSA encryption could be broken with 1024-quaterbit computers. Which, alone, makes it worth exploring in more detail.
- Improved Error Correction: Quaterbits could allow better error correction frameworks that make large-scale quantum computing practical at last. The higher dimensionality could be utilized to reduce decoherence and noise at the quantum level.
- Mathematically Rich Quaternion Algebra Framework: The enhanced algebra around quaternions could allow greater complexity of quantum algorithms, produce richer and more powerful variations of existing quantum algorithms, and even help to create completely new quantum algorithms. This enhanced functionality is one of the biggest advantages of quaterbits.
- Potential for New Applications: Quaterbits are a fertile breeding ground for new algorithms. In particular, the power of 4^N over 2^N could lead to exponentially faster algorithms in optimization, machine learning, quantum chemistry, post-quantum cryptography, and molecular simulation. This single fact should have every quantum computing scientist running towards their research to learn about quaterbits in detail and start applying them to existing algorithms!
China Is Already A Leader in this Field
There is compelling evidence that China is already a world leader in this field.
China is, as usual, secretive, but there have been some very telltale links published already.
The first link we examine here is:
This scientist has created a memory with no less than 25 different states. To say that quaterbits are a likely foundation would not be an exaggeration.
The next scientist referred to has published some very advanced research in spinors based upon quaternions—the fundamental idea behind quaterbits.
In case you don’t know what a spinor is, check out the Wiki page below. Warning: some very advanced mathematics here:
To even try to understand the page above, you may need to google at least 20 times (not an exaggeration).
But the information will be invaluable for quantum computing researchers, so those of you who qualify (or want to) try it!
And of course, we cannot leave out the 500-qubit experimental quantum computer:
Conclusion: And Addressing Some Critics
Quaterbits are an intriguing technology, but they have had their critics.
One of the first papers I read states that N quaterbits can be simulated by N+1 qubits, and some interpret that to mean that there is no real speed-up.
I respectfully choose to provide a different viewpoint.
Complex quantum computing could be possible with fewer quaterbits than qubits in far less complex circuits.
In this NISQ age where quantum scalability is a huge issue and control and calibration of a large number of quantum gates are so difficult for quantum computers:
A less complex circuit involving fewer gates and few quaterbits could do far more work in a far simpler manner than the corresponding qubit circuit, which could be exactly what the quantum computing world needs.
Give quaterbits a try.
You could find that you can accomplish far more important results with fewer gates and units than the corresponding qubit circuit.
Give it a shot!
And see what breakthroughs you can accomplish.
Agree? Disagree? Comment below to let me know!
References
Epilogue: Octonionic Computing
You might ask me that if quaternions are good for quantum computing, what about higher hypercomplex numbers?
What about octonions? (Numbers with one real part, 7 imaginary values)
Surely 8^N is faster than 4^N?
I’m sorry to disappoint you.
This same question came to my head, and I searched until I found this answer:
Octonions are not associative in multiplication.
To put it simply, association means:
a(bc) = (ab)c.
Changing the order of multiplication of the multiple components gives you the same answer if your system is associative.
This is immensely important for today’s quantum systems, where most equations involve linear algebra and eigenvalues.
Quantum computing is full of such matrix multiplication where order should not be a problem (I won’t bore you with equations).
And choosing octonions for our basic computing units violates the associativity rule.
While octonions hold promise in areas such as:
- Machine Learning
- Quantum AI
- Cognition Science
- Advanced Quantum Physics
- String Theory
Unless someone tries to rewrite current quantum systems from the bottom up, they will never be applied to quantum computing.
They are a very elegant theory, however!
Congratulations on your insight, and all the best for your future!
Enjoy quantum!
Epilogue References
- On Octonions violating associativity (and other things):https://arxiv.org/pdf/1208.4999
- A very elegant read on Octonions and cognition:https://www.linkedin.com/pulse/qubits-consciousness-daemon-husk-knbme/
- What are octonions?https://en.wikipedia.org/wiki/Octonion
- Applications of Octonions in Mechanicshttps://www.researchgate.net/publication/253342563_Quaternions_and_Octonions_in_Mechanics
All images were AI-generated by Microsoft Designer. Not as good as it used to be once, but still gets the job done!
