We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from [DPMRF22]; (ii) injective matrix product states; (iii) output states of dense Hamiltonian evolution, i.e. states of the form $e^{\iota H^{(p)}} \cdots e^{\iota H^{(1)}} |\psi_0\rangle$ for any $n$-qubit product state $|\psi_0\rangle$, where each $H^{(i)}$ can be any local commuting Hamiltonian satisfying a norm constraint, including dense Hamiltonians with interactions between any qubits. Our proofs use polynomial approximations to show that these states are close to local operators. This implies that the distribution of the Hamming weight of a computational basis measurement (and of other related observables) concentrates. An example of (iii) are the states produced by the quantum approximate optimisation algorithm (QAOA). Using our concentration results for these states, we show that for a random spin model, the QAOA can only succeed with negligible probability even at super-constant level $p = o(\log \log n)$, assuming a strengthened version of the so-called overlap gap property. This gives the first limitations on the QAOA on dense instances at super-constant level, improving upon the recent result [BGMZ22].

翻译：我们证明以下类别量子态的浓度边界：(i) 浅层量子电路的输出态，回答了[DPMRF22]中提出的开放问题；(ii) 内射矩阵乘积态；(iii) 稠密哈密顿量演化的输出态，即具有形式$e^{\iota H^{(p)}} \cdots e^{\iota H^{(1)}} |\psi_0\rangle$的任意$n$量子比特乘积态$|\psi_0\rangle$，其中每个$H^{(i)}$可以是满足范数约束的任意局域交换哈密顿量，包括任意量子比特间存在相互作用的稠密哈密顿量。我们的证明采用多项式逼近方法，证明这些态与局域算符相近。这意味着计算基测量（及其他相关可观测量的）汉明权重的分布具有集中性。第(iii)类的典型例子是量子近似优化算法(QAOA)产生的量子态。基于这些态的浓度结果，我们证明：对于随机自旋模型，即使采用超常数层级$p = o(\log \log n)$，在所谓的重叠间隙性质的强化版本假设下，QAOA仅能以可忽略概率成功。这给出了超常数层级稠密实例上QAOA的首个限制，改进了近期成果[BGMZ22]。