[论文解读] Quantum approximate optimization is computationally universal

Seth Lloyd|arXiv (Cornell University)|Dec 28, 2018

Quantum Computing Algorithms and Architecture参考文献 8被引用 75

一句话总结

论文表明量子近似优化算法(QAOA)可以通过选择相互作用时间来实现任意量子门，从而可编程地执行通用量子计算。

ABSTRACT

The quantum approximate optimization algorithm (QAOA) applies two Hamiltonians to a quantum system in alternation. The original goal of the algorithm was to drive the system close to the ground state of one of the Hamiltonians. This paper shows that the same alternating procedure can be used to perform universal quantum computation: the times for which the Hamiltonians are applied can be programmed to give a computationally universal dynamics. The Hamiltonians required can be as simple as homogeneous sums of single-qubit Pauli X's and two-local ZZ Hamiltonians on a one-dimensional line of qubits.

研究动机与目标

• 通过探索其计算普遍性，推动理解QAOA不仅是优化工具这一观点。
• 证明交替应用特定哈密顿量可以实现普遍量子动力学。
• 证明在QAOA下，简单的一维最近邻哈密顿量即可实现通用计算。

提出的方法

• 描述QAOA动力学为在 n 个量子比特上对 H_Z 和 H_X 进行交替应用，起始于均匀叠加。
• 展示如何选择时间 t 和 tau 以有效开启特定子哈密顿量并实现所需的幺正变换。
• 证明可以构造形式为 U_AB 和 U_BA 的变换，以在相邻的量子比特对上应用任意对角化在 Z（或在 Y 基底对角化）的两量子比特幺正。
• 通过使用 QAOA 动力学实现广播量子元胞自动机架构来实现通用量子计算。
• 讨论扩展到更高维和长程相互作用的潜力，以及使用全局 X/Hadamard 脉冲实现始终开启 Z 哈密顿量。

实验结果

研究问题

• RQ1 Can QAOA dynamics be programmed to realize universal quantum computation?
• RQ2 What classes of Hamiltonians and architectures suffice to achieve universality under QAOA?
• RQ3 How do incommensurate coupling constants enable selective activation of Hamiltonian terms?
• RQ4 What are the practical implications for near-term quantum devices and quantum learning?
• RQ5 Can this universality extend to scalable fault-tolerant schemes in higher dimensions?

主要发现

• QAOA can be programmed to implement any desired sequence of quantum logic gates.
• With H_X = sum_j X_j and a structured H_Z on a 1D lattice, one can realize U_AB and U_BA as products of single- and two-qubit unitaries on adjacent qubit pairs.
• Transformations like e^{-i φ_A H_A}, e^{-i φ_B H_B}, e^{-i φ_AB H_AB}, e^{-i φ_BA H_BA} enable universal diagonal-unitary constructions on qubit pairs.
• Hadamard conjugation maps the Z-based Hamiltonian to a Y-basis form, preserving universality and enabling parallelizable gate sequences.
• The result implies even the IQP-type dynamics with a fixed Z Hamiltonian can be computationally universal under the QAOA framework.