# What can gate teleportation be used for in practice?

I was quite fascinated when I learned about gate teleportation, but I am struggling with understanding what it may be used for.

Sure, one can use ancillae bits to teleport gates into various parts of a circuit... but why? In general would it not be more difficult to add extra qubits to the system as opposed to performing extra gate operations?

Gate teleportation looks great to me on paper (by this I mean drawing the physical circuits... it's so cool!), but when I think about an ion trap I can't seem to convince myself of a time where one would really use it if they have access to the proper lasers to implement gates.

• Quantum gate teleportation is used to create fault tolerant gates by circumventing the Eastin-Knill theorem . Jan 26 at 21:49
• Without gate teleportation, I couldn't start a hypothetical future company selling high quality distilled T states to anyone with a fast quantum internet connection, so they didn't have to make them themselves. (This is a joke. Maybe.) Jan 27 at 2:22
• @CraigGidney haha -- while reading your comment my thoughts were "is this a joke" followed immediately by "maybe not?"...glad to see we were on the same page Jan 27 at 3:10
• @VictoryOmole This should be an answer. Jan 27 at 11:54
• @NorbertSchuch I didn't have time to write a detailed answer. The current accepted answer is pretty much what I would have said if I had time to write it. Jan 27 at 16:59

The key practical value of gate teleportation lies in the fact that it provides a simple and fairly general protocol for reliable and fault-tolerant execution of a wide range of gates.

Depending on the quantum architecture, some gates may be difficult or impossible to execute reliably or fault-tolerantly. The operations that make up the teleportation protocol are often simpler to implement fault-tolerantly than the gate being teleported (see application 1 below). Also, gate teleportation enables us to achieve scalability by performing unreliable operations offline where failures do not restart the entire computation (see application 2 below).

Application 1: Fault-tolerant implementation of non-transversal gates

By the Eastin-Knill theorem, no quantum error correcting code admits transversal implementation of a universal set of gates. While transversality is a simple way to ensure fault-tolerance, it is not the only way. However, most other techniques are ad hoc and do not generalize to many types of gates. Gate teleportation provides fault-tolerant implementation of a large class of gates called the Clifford hierarchy.

The Clifford hierarchy is defined recursively. The first level $$\mathcal{C}_1$$ is the Pauli group and for $$k>1$$

$$\mathcal{C}_k = \{U\,|\,U\mathcal{C}_1U^\dagger \subset \mathcal{C}_{k-1}\}.$$

All levels of the hierarchy contain potentially useful gates. For example, $$\mathcal{C}_2$$ is the Clifford group, $$\mathcal{C}_3$$ contains the $$T$$ gate and the Toffoli gate and each $$\mathcal{C}_k$$ contains the controlled rotations used in Shor's algorithm.

The key fact about gate teleportation and the Clifford hierarchy is that to achieve teleportation of $$U\in\mathcal{C}_k$$ we only need gates in $$\mathcal{C}_{k-1}$$ (together with measurements and classical control). Consequently, recursive application of the gate teleportation scheme allow us to implement any gate in the Clifford hierarchy. The recursion terminates at $$\mathcal{C}_2$$ in codes such as the 7-qubit Steane code that implement the Clifford group transversally. In other stabilizer codes it terminates at $$\mathcal{C}_1$$ or at $$\mathcal{C}_2$$ if other techniques provide fault-tolerant implementation of the Clifford group.

See this paper for details.

Application 2: Reliable implementation of two-qubit gates using linear optics

KLM protocol enables universal quantum computation using only linear optical elements by using post-selection to implement a non-deterministic two-qubit gate. However, if a gate which succeeds with probability $$p$$ is used directly in the execution of a quantum algorithm $$k$$ times then the chance of the overall success of the computation is $$p^k$$.

We can overcome this exponentially small chance of success by replacing gate application with gate teleportation. The advantage of this approach lies in the fact that failures of the non-deterministic gate do not force a computation to restart. Instead, they merely slow down the offline process of collecting the special states for teleportation.