Avoiding symmetry roadblocks and minimizing the measurement overhead of adaptive variational quantum eigensolvers
Avoiding symmetry roadblocks and minimizing the measurement overhead of adaptive variational quantum eigensolvers
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DOI：
10.22331/q-2023-06-12-1040
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发表时间：
2021-09
期刊：
影响因子：
6.4
通讯作者：
V. O. Shkolnikov;N. Mayhall;S. Economou;J. Dyke;George S. Barron;Edwin Barnes;Ho Lun Tang;Bryan T. G
中科院分区：
文献类型：
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作者：
V. O. Shkolnikov;N. Mayhall;S. Economou;J. Dyke;George S. Barron;Edwin Barnes;Ho Lun Tang;Bryan T. G
强相关系统的量子模拟可能是近期量子计算机最可行的有用应用。最大限度地减少量子计算资源对于实现这一目标至关重要。用于此目的的一类有前途的算法包括变分量子本征求解器（VQE）。其中，针对问题定制的版本（例如 ADAPT-VQE）从预定义的算子池逐步构建变分分析，在电路深度和变分参数计数方面表现得特别好。然而，与标准 VQE 相比，这种性能的提高是以额外的测量开销为代价的。在这里，我们证明这种开销可以减少到仅随量子位数量 n 线性增长的量，而不是像原始 ADAPT-VQE 中那样呈四倍增长。我们通过证明如果选择适当的话，大小为 2n−2 的算子池可以表示希尔伯特空间中的任何状态来实现这一点。我们证明这是这种“完整”池的最小尺寸，讨论它们的代数性质，并为它们的完整性提供必要和充分的条件，使我们能够有效地找到这样的池。我们进一步表明，如果模拟问题具有对称性，那么完整的池可能无法产生收敛结果，除非选择池遵守某些对称规则。我们通过在几个强相关分子的 ADAPT-VQE 经典模拟中使用这种对称适应完整池来展示它们的性能。我们的发现与任何使用基于泡利弦的 ansatz 的 VQE 相关。
Quantum simulation of strongly correlated systems is potentially the most feasible useful application of near-term quantum computers. Minimizing quantum computational resources is crucial to achieving this goal. A promising class of algorithms for this purpose consists of variational quantum eigensolvers (VQEs). Among these, problem-tailored versions such as ADAPT-VQE that build variational ansätze step by step from a predefined operator pool perform particularly well in terms of circuit depths and variational parameter counts. However, this improved performance comes at the expense of an additional measurement overhead compared to standard VQEs. Here, we show that this overhead can be reduced to an amount that grows only linearly with the number n of qubits, instead of quartically as in the original ADAPT-VQE. We do this by proving that operator pools of size 2n−2 can represent any state in Hilbert space if chosen appropriately. We prove that this is the minimal size of such "complete" pools, discuss their algebraic properties, and present necessary and sufficient conditions for their completeness that allow us to find such pools efficiently. We further show that, if the simulated problem possesses symmetries, then complete pools can fail to yield convergent results, unless the pool is chosen to obey certain symmetry rules. We demonstrate the performance of such symmetry-adapted complete pools by using them in classical simulations of ADAPT-VQE for several strongly correlated molecules. Our findings are relevant for any VQE that uses an ansatz based on Pauli strings.