The ZX-calculus is a graphical language for reasoning about quantum computation using ZX-diagrams, a certain flexible generalisation of quantum circuits that can be used to represent linear maps from $m$ to $n$ qubits for any $m,n \geq 0$. Some applications for the ZX-calculus, such as quantum circuit optimisation and synthesis, rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. While several sufficient conditions are known for describing families of ZX-diagrams that can be efficiently transformed back into circuits, it has previously been conjectured that the general problem of circuit extraction is hard. That is, that it should not be possible to efficiently convert an arbitrary ZX-diagram describing a unitary linear map into an equivalent quantum circuit. In this paper we prove this conjecture by showing that the circuit extraction problem is #P-hard, and so is itself at least as hard as strong simulation of quantum circuits. In addition to our main hardness result, which relies specifically on the circuit representation, we give a representation-agnostic hardness result. Namely, we show that any oracle that takes as input a ZX-diagram description of a unitary and produces samples of the output of the associated quantum computation enables efficient probabilistic solutions to NP-complete problems.

翻译：ZX 计算仪是一种图形语言,用于推算使用ZX- diagrams(ZX- diagrams) 进行量子计算的方法的推理。ZX- diacululus 是用来代表从美元到美元的任何美元,n\geq 0美元的线性地图的量子电路。ZX- calculs的一些应用程序,例如量子电路优化和合成,依靠能够将ZX- diagraph 有效转换回一个类似大小的量子电路。虽然在描述能够有效转换回回电路的ZX- diagrams家庭时已经知道若干充分的条件,但以前曾推测电路提取的一般问题很难。也就是说,将描述单线性线性图的任意的ZX- diagraph转换成一个等量电路。在本文中,我们通过显示电路提取问题是硬性的,因此至少是硬的量电路路路路的模拟。除了我们主要的硬性结果之外,还具体以直径的电路运描述来显示一个硬性结果。