Circuit Extraction for ZX-diagrams can be #P-hard

TL;DR

This paper proves that converting ZX-diagrams into quantum circuits is computationally intractable (#P-hard), establishing fundamental limits on the efficiency of circuit extraction in quantum computing representations.

Contribution

It demonstrates that the general problem of ZX-diagram circuit extraction is #P-hard, confirming a long-standing conjecture and providing a representation-agnostic hardness result.

Findings

Circuit extraction is #P-hard, making it computationally intractable.

Efficient probabilistic solutions to NP-complete problems can be achieved using ZX-diagram oracles.

The hardness result applies broadly beyond specific circuit representations.

Abstract

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.