Quantum Computers operate on the fundamental concept of “qubits” (quantum bits), usually made of sub-atomic particles governed under the laws of Quantum Mechanics. This enables them to achieve mathematical performance advancements in another league, urging researchers to find new and interesting applications for this technology.

But have you ever heard of “qudits” before? With a replacement of just one letter, what the heck are they, and why should you care?

The Basics 🧱

Qudits are a generalization of qubits denoted as d^n. While a qubit is limited to two states (0 and 1), a qudit can exist in d states, where d can be any integer greater than 2. For instance:

• Qubit: 2 states (0, 1)
• Qutrit: 3 states (0, 1, 2)
• Ququart: 4 states (0, 1, 2, 3)

Mathematically, qudits are represented as vectors in a d-dimensional Hilbert space. For a qubit, the states |0⟩ and |1⟩ can be represented as:

For a qutrit (d=3), the states |0⟩, |1⟩, and |2⟩ can be represented as:

A qutrit in superposition might be described by:

where ⍺, β and 𝛄 are complex numbers, and |⍺|² + |β|² + |𝛄|² = 1

Since qudits can have n dimensions instead of just two, traditional quantum computing gates must be modified to work with n dimensions.

The generalization of Pauli X Gates for Qudits is given by:

For Pauli Y Gates, it is given by:

where ⊕ denotes addition modulo d and ω = exp(i2π/d).

The states of a qubit can be visualized using a Quantum Harmonic Oscillator. It is a way of representing the vibrations of a molecule under the effects of quantum mechanics between two “quantized” energy levels.

However, implementing this physically is another story. When expanding this model to higher energy levels, various physical apparatuses such as qubit-oscillator coupling methods are required, significantly increasing the complexity of the physical system.

Companies like IBM and Xanadu are working on technologies like this. But what other ways can we use to model qudits physically?

Photons! 💡

Linear optics, the same technology used to carry your Internet, TV, and Telephone signals are the cornerstone to implementing qudits in a physical system. It involves manipulating light using components that do not change the frequency of the light, only its path, phase, or intensity. 

One of the simplest ways to encode a qudit in a photon-based quantum system is to map its energy levels to d different “optical modes”, aka the different paths that photons can travel through.

For example, if we have a ququart (4-level qudit), we can use four different optical modes labeled 0, 1, 2, and 3. The state |0⟩ means the photon is in the first path, |1⟩ means the photon is in the second path, and so on.

This type of encoding is called “multi-rail encoding”, as you can encode each state into multiple “rails”, modes, or paths (too many names, right?). It has the advantage that any single-qudit transformation can be efficiently implemented using something called Clements Decomposition.

The idea of decomposition in quantum computing isn’t particularly new — it is the act of simplifying complex quantum operations into a few of the most basic, non-simplifyable quantum operations.

Clements Decomposition involves simplifying your optical system into just Beam Splitters and Phase Shifters, some of the simplest apparatus you can use. The former is used to divide a beam of light into two modes, while the latter is used to change the phase of light in a single mode.

This reduces the footprint of the circuit, hence improving its fidelity against optical losses.

Why Qudits 🤔

With qubits being able to solve a very niche set of mathematically complex problems, qudits essentially offer you the ability to solve an even more niche set of problems better than qubits can.

Here’s a challenge. Assign k colors to the vertices of the following graph, such that no two adjacent vertices share the same color.

Pretty simple right? Shouldn’t have taken you more than a couple of minutes.

Here is a possible solution:

These graphs were processed using a classical computer using a greedy algorithm. However, in a graph with hundreds of nodes with many more interconnections, this problem isn’t so easy.

That’s where qubits come in! The problem can be solved more efficiently at scale through the following method:

• Encoding the problem into a binary optimization problem.
• Using the Quantum Approximate Optimization Algorithm (QAOA) to find the optimal coloring by minimizing a cost function that penalizes invalid colorings and adjacent vertices sharing the same color.
• Having the Cost Hamiltonian and Mixing Hamiltonian guide the optimization process.

However, qudits introduce a whole-nother level of efficiency to this — problem:

• Encoding the problem by naturally representing each qudit as one of the k colors.
• Using the QAOA like with the qubits but with d-dimensional operators instead.
• Having the Cost Hamiltonian and Mixing Hamiltonian guide the optimization process, but be expressed in terms of the qudit representation of the Pauli-Z gates

All in all, the qudits can encode more information per computational unit, decrease the number of photons and optical modes required, and simplify the cost function (no penalty terms for invalid colorings!), hence reducing the complexity of the quantum circuit.

Challenges — lots of challenges 😬

Let’s start with the theoretical issues. Not all algorithms can adapt as well as QAOA to the modified gates required to make running algorithms on qudits feasible. And even if they do, the higher number of states and their unique interactions make error mitigation MUCH more complicated.

But how about developing new algorithms? Well just like with qubit-based quantum computation, researchers are still looking for algorithms that will justify the usage of these machines to solve problems of ever-increasing complexity. Taking full advantage of the state space and minimizing physical resource overheads in practical implementations​ are just some of the things that have to be kept in mind when designing new algorithms for qudits.

It’s not hard as in “computationally difficult”, we simply haven’t found a solution to these problems yet. There is not enough incentive to spend the resources to solve these issues, slowing down potential progress even further.

Conclusion

The reality is that qudits are an interesting technology that has very few, niche use cases today like the k-coloring problem.

Much like with qubits, we have found the solution before the problem! At least you have a cool new factoid to bring up next time you’re at the lunch tables with your fellow colleagues. Who knows, you might just inspire the next big breakthrough in quantum computing!