a topic that i have "some" understanding in and it will be a great time to contribute. I will break it into few parts and i hope i do not go on for multiple pages as i tend to do but will keep this short and simple.
(and for those who might be tempted to learn more about this topic i will write some obvious things (for you but not others looking to expand their knowledge)
- Gaussian Boson Sampling (GBS)
GBS is a model of quantum computation that utilizes squeezed states of light as input. These squeezed states are injected into a linear interferometer, where the photons undergo transformations due to interference, and the output is detected using photon number-resolving detectors. GBS operates within the Gaussian regime of quantum optics, meaning the initial quantum states (squeezed vacuum states) and the operations (passive linear optics) are all Gaussian. The term "Gaussian" comes from the fact that the state's Wigner function, which describes the quantum state, has a Gaussian profile.
After sending the squeezed states through the interferometer and applying photon detection, you end up with a measurement distribution, which relates to sampling from a complex output photon distribution. The computational task here is to sample the distribution of photons across the detectors, which is extremely hard for classical computers as the system scales. Thus, GBS demonstrates quantum supremacy in a specialized task.
- Cluster States
In contrast, cluster states (a type of graph states) are fundamental resources for measurement-based quantum computation (MBQC). A cluster state is an entangled multi-qubit state, usually described as a graph where each vertex represents a qubit, and edges represent entanglement between them. Unlike GBS, which operates within the continuous-variable regime (Gaussian states), cluster states can be constructed using discrete qubits in various physical systems, such as photonic qubits, superconducting qubits, or trapped ions.
Cluster states provide a universal resource for quantum computation when combined with appropriate measurements. The qubits in a cluster state are entangled in such a way that measurement in specific bases can guide the computation, eliminating the need for unitary gates as in circuit-based quantum computing.
- Can Cluster States Be Generated from GBS?
Yes, it's possible to generate continuous-variable (CV) cluster states from Gaussian states, which is likely the connection you're referring to. A GBS system produces Gaussian states, and by applying appropriate operations (such as quadrature measurements), one can generate CV cluster states. These cluster states can be thought of as graph states where each node represents a mode of the Gaussian state, and the edges represent the entanglement between them.
The challenge, however, is that Gaussian cluster states are not universal for quantum computation on their own. They require additional non-Gaussian elements (e.g., photon-number measurements or non-Gaussian operations) to enable universal quantum computation.
- Can Cluster States Be Generated Without a Resonant State?
As you pointed out, cluster states can also be generated using discrete qubits, such as superconducting transmon qubits, trapped ions, or single-photon emitters. In this context, you do not need a resonant state. For example, in superconducting qubit systems, qubits are entangled via quantum gates like the controlled-Z gate, which directly creates entanglement necessary for the cluster state. This is fundamentally different from GBS, which operates in the continuous-variable regime.
So to address your question, you don't always need a resonant (Gaussian) state to generate cluster states. Discrete qubits can be used to directly generate cluster states through gate-based entanglement.
- n-Dimensional Cluster States
A 2D cluster state refers to a graph where qubits are arranged in a 2D lattice, with each qubit entangled with its neighbors. Extending this to n-dimensions is possible in theory, but it depends on the physical platform and the kind of qubits or modes you're using. In the case of photonic systems, a resonant state (such as a squeezed state) might be required, but in the case of superconducting qubits, you could directly generate an n-dimensional cluster state by creating multi-qubit entanglement through appropriate quantum gates.
Your approach to understanding is on the right track, and the key distinction lies in the discrete-variable vs. continuous-variable nature of these quantum states and how they’re generated across different physical systems.
I will be more than glad to read different understandings of the topic.
