近对称优化问题上浅层QAOA的容错代价
Fault-tolerant cost of shallow QAOA on near-symmetric optimization problems
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- IQM Quantum Computers（IQM量子计算机）
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中文总结 AI 辅助
本文研究浅层QAOA在近对称优化问题上的容错代价，发现小角度旋转合成可将non-Clifford代价降至Õ(n^2)，并指出代价的傅里叶系数泄漏不一定暴露植入解。
中文摘要 AI 辅助
Montanaro和Zhou证明了深度为一的量子近似优化算法（QAOA）能够以Ω(1)的概率找到某些近对称约束满足问题的植入解，并观察到通用经典求解器在显式实例上呈指数级运行时间，为低深度QAOA的经验性指数加速提供了有力证据。我们研究这样的电路在容错条件下的代价。尽管代价哈密顿量包含Θ(n^ℓ)个子句，但QAOA成功所需的旋转角度随n^{1-ℓ}缩小，我们证明小角度Clifford+T旋转合成将每个电路的non-Clifford代价降低至Õ(n^2)，适用于任意子句局部性ℓ。然而，编译电路需要显式的子句列表。我们发现，QAOA恒定成功背后的机制固定了代价的傅里叶系数的一阶项，其符号揭示了植入解。然而，这种泄漏不一定暴露解：我们设计了二次non-Clifford缩放与精确优化困难共存的实例族。
英文摘要
Montanaro and Zhou proved that depth-one Quantum Approximate Optimization Algorithm (QAOA) finds the planted solution of certain near-symmetric constraint satisfaction problems with probability $Ω(1)$, and observed exponential runtimes for general-purpose classical solvers on explicit realizations, providing strong evidence of an empirical exponential speedup of low-depth QAOA. We ask what such a circuit costs fault-tolerantly. Although the cost Hamiltonian carries $Θ(n^\ell)$ clauses, the rotation angle at which QAOA succeeds shrinks as $n^{1-\ell}$, and we show that small-angle Clifford$+T$ rotation synthesis reduces the non-Clifford cost per circuit to $\widetilde O(n^2)$ for every clause locality $\ell$. Compiling the circuit, however, requires the explicit clause list. We find that the mechanism underlying constant QAOA success fixes the degree-one Fourier coefficients of the cost, whose signs reveal the planted solution. However, this leakage need not reveal the solution: we design families in which the quadratic non-Clifford scaling and hard exact optimization coexist.
