Qubit Coupled Cluster Method: A Systematic Approach to Quantum Chemistry on a Quantum Computer

A unitary coupled cluster (UCC) form
for the wave function in the
variational quantum eigensolver has been suggested as a systematic
way to go beyond the mean-field approximation and include electron
correlation in solving quantum chemistry problems on a quantum computer.
Although being exact in the limit of including all possible coupled
cluster excitations, practically, the accuracy of this approach depends
on the number and type of terms are included in the wave function
parametrization. Another difficulty of UCC is a growth of the number
of simultaneously entangled qubits even at the fixed Fermionic excitation rank. Not all quantum computing architectures
can cope with this growth. To address both problems, we introduce
a qubit coupled cluster (QCC) method that starts directly in the qubit
space and uses energy response estimates for ranking the importance
of individual entanglers for the variational energy minimization.
Also, we provide an exact factorization of a unitary rotation of more
than two qubits to a product of two-qubit unitary rotations. Thus,
the QCC method with the factorization technique can be limited to
only two-qubit entanglement gates and allows for very efficient use
of quantum resources in terms of the number of coupled cluster operators.
The method performance is illustrated by calculating ground-state
potential energy curves of H2 and LiH molecules with chemical
accuracy, ≤1 kcal/mol, and a symmetric water dissociation curve.

在量子计算机上求解量子化学问题时，变分量子本征求解器（variational quantum eigensolver, VQE）中用于波函数的酉耦合簇（unitary coupled cluster, UCC）形式，被提出为一种超越平均场近似、纳入电子关联的系统性方法。尽管在包含所有可能耦合簇激发的极限下该方法严格精确，但实际应用中其精度取决于波函数参数化中所纳入项的数量与类型。酉耦合簇还存在另一项难点：即使在固定费米子激发秩的情况下，同时纠缠的量子比特数目仍会增长，并非所有量子计算架构都能适配这一增长。为解决上述两个问题，我们提出了直接在量子比特空间中构建的量子比特耦合簇（qubit coupled cluster, QCC）方法，该方法利用能量响应估计对各个纠缠算子在变分能量最小化中的重要性进行排序。此外，我们还给出了将多于两个量子比特的酉旋转精确分解为两量子比特酉旋转乘积的方案。因此，结合该分解技术的QCC方法可仅使用两量子比特纠缠门，并能在耦合簇算子数量层面实现量子资源的高效利用。我们通过计算达到化学精度（≤1 kcal/mol）的氢分子（H₂）与氢化锂分子（LiH）的基态势能曲线，以及对称水解离曲线，验证了该方法的性能。