As quantum computing resources remain scarce and error rates high, minimizing the resource consumption of quantum circuits is essential for achieving practical quantum advantage. Here we consider the natural problem of, given a circuit $C$, computing a circuit $C'$ which behaves equivalently on a desired subspace, and that minimizes a quantum resource type, expressed as the count or depth of (i) arbitrary gates, or (ii) non-Clifford gates, or (iii) superposition gates, or (iv) entanglement gates. We show that, when $C$ is expressed over any gate set that can implement the H and TOF gates exactly, each of the above optimization problems is hard for $\text{co-NQP}$, and hence outside the Polynomial Hierarchy, unless the Polynomial Hierarchy collapses. This complements recent results in the literature which established an $\text{NP}$-hardness lower bound when equivalence is over the full state space, and tightens the gap to the corresponding $\text{NP}^{\text{NQP}}$ upper bound known for cases (i)-(iii) over Clifford+T and (i)-(iv) over H+TOF circuits.

翻译：由于量子计算资源仍然稀缺且错误率较高，最小化量子电路的资源消耗对于实现实用量子优势至关重要。本文考虑一个自然问题：给定一个电路$C$，计算一个在期望子空间上行为等效的电路$C'$，并最小化某种量子资源类型，该资源类型可表示为（i）任意门、（ii）非克利福德门、（iii）叠加门或（iv）纠缠门的数量或深度。我们证明，当$C$用任何能精确实现H门和TOF门的门集表示时，上述每个优化问题对于$\text{co-NQP}$都是困难的，因此位于多项式层次结构之外，除非多项式层次结构坍缩。这补充了文献中最近的结果，那些结果建立了在全状态空间上等效时的$\text{NP}$困难下界，并缩小了与已知上界之间的差距：对于Clifford+T电路中的情况（i）-（iii）以及H+TOF电路中的情况（i）-（iv），相应的上界为$\text{NP}^{\text{NQP}}$。