Quantum gates represent the operations we can perform on qubits. A universal gate set is a set of these gates that enable universal quantum computation, which means that all possible operations are enabled. The gates that go into this set must include both Clifford gates and non-Clifford gates.
Clifford gates can be efficiently simulated with classical computers and they are transversal, which means they can be efficiently applied to logical qubits. They can be applied by quantum error correction codes (QECC), which means they are useful for fault-tolerant quantum computing (FTQC). Non-Clifford gates cannot be efficiently simulated with classical computers, nor can they be efficiently applied to logical qubits. To achieve universal FTQC, there must be some way to efficiently implement non-Clifford logical gates.
Magic state quantum computing is a possible solution, and it works much like a magic trick. Most magic tricks require some kind of preparation in advance, after which they are ready to be performed at the appropriate times during the show. Because non-Clifford gates are inefficient to implement, their effects are encoded into quantum states in advance. At the appropriate times during computation, these pre-prepared “magic states” are used, leaving the rest of the computation to the Clifford gates, which can be implemented efficiently.
For more information, the Stack Exchange Quantum Computing question “What are magic states?” has some upvoted answers with mathematics, a visualization, and links to internal and external resources. Furthermore, the Earl T. Campbell article “Magic States” is written in plain English and includes analogies, a visualization, and links to external resources.
A magic state, simply put, is a quantum state that is prepared in advance because it otherwise cannot be implemented efficiently. Magic state injection is the incorporation of this pre-prepared state into fault-tolerant quantum computation. However, this process can be noisy and may not be implemented perfectly.
The solution for this is magic state distillation. Analogous to how distillation is used to separate the components of liquids to achieve higher concentrations of certain components, magic state distillation takes many noisy quantum states and returns more accurate quantum states, possibly fewer in number. The resultant magic state based effects, in conjunction with Clifford gates, implement the desired non-Clifford operations.
Magic states have the potential to have critically important applications. First of all, they apply to FTQC, which means that they are of long-term interest. Second, they may be essential to universal quantum computation during the FTQC era. A lack of a universal logical gate set would greatly restrict quantum algorithm development. And third, the pre-preparation of magic states may increase the speed and efficiency of complex calculations.
Magic state preparation, distillation, and injection have their ongoing research challenges. However, it is theorized that QECC and logical qubits will be essential to achieving fault tolerance, which is essential to realizing useful quantum computing. But fault tolerance isn’t enough; a transversal universal gate set is required. Magic states might be the key to unlocking such a gate set.
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