User:XFeng256/sandbox

Definition
Before stating the theorem, firstly introduce several definitions. We define the error between two quantum gates $$U$$ and $$U'$$ as $$E(U, U') = \max_{|\psi\rangle} \| U|\psi\rangle - U'|\psi\rangle\|$$, where $$|\psi\rangle$$ is a quantum state.

We say that a quantum gate $$U$$ $$\epsilon$$-approximate another quantum gate $$U'$$ if the error between the two gates is not larger than $$\epsilon$$.

We call a quantum circuit $$\Lambda$$-circuit for some set $$\Lambda$$ of quantum gates if this quantum circuit only consists of gates from the set $$\Lambda$$.

Statement
Let $$\Gamma$$ be a set of single-qubit quantum gates satisfying the conditions: (1) $$\Gamma$$ generates a dense subgroup of SU(2); (2) $$\Gamma$$ is closed under inverse. Then any quantum gate $$U \in SU(2)$$ can be $$\epsilon$$-approximated by a product of at most $$\log^d (1/\epsilon)$$ gates from the set $$\Gamma$$, where $$1 \leq d < 4$$.

Consequences
There are several consequences of the Solovay-Kitaev theorem, one is a corollary stated below.

Corollary: Let $$\Lambda$$ be a gate set such that $$\Lambda = \{ CNOT \} \cup \Gamma$$, where $$\Gamma$$ satisfies the conditions of the Solovay-Kitaev theorem and $$CNOT$$ is a controlled NOT gate. Then for any circuit $$C$$, there exists a $$\Lambda$$-circuit $$C'$$ that $$\epsilon$$-approximates $$C$$ for any $$\epsilon > 0$$. The size of $$C'$$, which is denoted as $$|C'|$$ and means the number of gates in $$C'$$, is not larger than $$|C| \cdot \log^d (|C|/\epsilon)$$.

Basically the overhead $$\log^d (|C|/ \epsilon)$$ is very small. Thus we can $$\epsilon$$-approximate any circuit with a new circuit that consists of only single-qubit quantum gates satisfying the conditions of the Solovay-Kitaev theorem in addition to a CNOT gate. This means that as long as you have a single-qubit gate set that satisfies the conditions of the Solovay-Kitaev theorem, you could $$\epsilon$$-approximate any quantum gate you want efficiently with this gate set plus a CNOT gate.