We introduce the first minimal and complete equational theory for quantum circuits. Hence, we show that any true equation on quantum circuits can be derived from simple rules, all of them being standard except a novel but intuitive one which states that a multi-control $2\pi$ rotation is nothing but the identity. Our work improves on the recent complete equational theories for quantum circuits, by getting rid of several rules including a fairly unpractical one. One of our main contributions is to prove the minimality of the equational theory, i.e. none of the rules can be derived from the other ones. More generally, we demonstrate that any complete equational theory on quantum circuits (when all gates are unitary) requires rules acting on an unbounded number of qubits. Finally, we also simplify the complete equational theories for quantum circuits with ancillary qubits and/or qubit discarding.

翻译：我们首次提出了量子电路的最小且完备的等式理论。由此证明，量子电路上的所有真等式均可由简单规则推导得出——除了一条新颖却直观的规则（即多控制$2\pi$旋转等价于恒等操作）外，其余均为标准规则。本研究改进了近期提出的量子电路完备等式理论，消除了包括一条相当不实用规则在内的若干规则。我们的主要贡献之一在于证明了该等式理论的最小性，即没有任何规则可由其他规则推导得出。更一般地，我们论证了任何关于量子电路（所有门均为酉算子时）的完备等式理论，都必须包含作用于无界数量子比特上的规则。最后，我们还简化了带辅助量子比特和/或量子比特丢弃的量子电路的完备等式理论。