AI 中文总结
该研究在复 Clifford 代数中开发\(n\)量子比特变分量子算法，区分不同旋转并推导规则，通过对临界开放横向场伊辛链实验，给出误差数据，还测试了有限次选择方法，贡献了代数公式、池修剪规则及可重复研究。
AI 中文摘要
我们在复 Clifford 代数\(\mathrm{Cl}(2n,\mathbb{C}) \cong M(2^n,\mathbb{C})\)中开发了一种以稀疏算子为中心的\(n\)量子比特变分量子算法实现。密度算子、门、可观测量、通道、费米子模式和自适应选择可观测量都在一个泡利词代数中表示，约旦 - 维格纳映射提供了与反对易 Clifford 生成元的精确桥梁。我们区分了一般泡利词旋转和自旋群转子，并推导了一个精确的转置奇偶规则。对于临界开放横向场伊辛链，深度为三的哈密顿变分假设在\(n = 4,5,6\)时给出的相对能量误差分别为\(4.84\times10^{-5}\)、\(2.19\times10^{-3}\)和\(3.67\times10^{-3}\)。一个紧凑的局部 ADAPT 池在\(n = 4\)时是精确的，但在更大规模时会留下残余误差；一个系统的连续三局部奇数\(Y\)池在\(n\leq6\)时相对误差低于\(1.3\times10^{-12}\)。在\(n = 4\)的 100 次种子有限次测试中，固定次选择在\(0/100\)次运行中成功，而均匀递增和置信界竞赛分别在\(84/100\)次运行中成功；竞赛将中位数次数降低了\(34\%\)。我们声称没有比矩阵方法更快的渐近加速。贡献在于一个修正的代数公式、一个精确的池修剪规则以及对测量受限自适应选择的可重复研究。
英文摘要
We develop a sparse operator-centric realization of $n$-qubit variational quantum algorithms in the complex Clifford algebra $\mathrm{Cl}(2n,\mathbb{C}) \cong M(2^n,\mathbb{C})$. Density operators, gates, observables, channels, fermionic modes, and adaptive-selection observables are represented in one Pauli-word algebra, with the Jordan--Wigner map providing the exact bridge to anticommuting Clifford generators. We distinguish general Pauli-word rotations from Spin-group rotors and formulate the familiar odd-$Y$ restriction for real-state adaptive ansatzes as an exact transpose-parity statement: for real Hamiltonians and real states, every candidate Pauli word containing an even number of $Y$ factors has zero ADAPT gradient, while odd-$Y$ rotations preserve the real sector. For the critical open transverse-field Ising chain, a depth-three Hamiltonian variational ansatz gives relative energy errors $4.84\times10^{-5}$, $2.19\times10^{-3}$, and $3.67\times10^{-3}$ for $n=4,5,6$. A compact local ADAPT pool is exact at $n=4$ but leaves residual errors at larger sizes; a systematic contiguous three-local odd-$Y$ pool reaches relative errors below $1.3\times10^{-12}$ for $n\leq6$. In 100-seed finite-shot tests at $n=4$, fixed-shot selection succeeds in $0/100$ runs, whereas uniform escalation and confidence-bound racing each succeed in $84/100$ runs; racing lowers median shots by $34\%$. We claim no asymptotic speedup over matrix methods. The contribution is a corrected algebraic formulation, a density-operator derivation and implementation of the real-sector pool filter, and a reproducible study of measurement-limited adaptive selection.
Comments9 pages plus 3 pages Supplemental Material; 6 figures
