魔法态的费米子高斯秩和Fock态的玻色子相干态秩的指数下界
Exponential lower bounds on the fermionic Gaussian rank of magic states and the bosonic coherent state rank of Fock states
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中文总结 AI 辅助
本文证明魔法态张量积的费米子高斯分解项数呈指数下界，并给出Fock态相干态秩的精确公式，为经典模拟量子计算提供理论支撑。
中文摘要 AI 辅助
最近用于经典模拟量子力学的算法，其运行时间的超多项式部分由线性依赖决定，该依赖涉及将某些“魔法”态的大张量积写成“自由”态（可以是稳定子态、费米子高斯态或其他态）的叠加所需的项数。关于此类分解中的项数（称为秩，例如稳定子秩、费米子-高斯秩等）人们知之甚少。出于复杂性理论的原因，这些秩预计会随张量因子数量呈指数增长，然而，虽然魔法态的稳定子秩和费米子-高斯秩的指数上界已知，但稳定子秩的最佳已知下界是二次的，而费米子-高斯秩除了固定常数外没有已知下界。在这项工作中，我们证明任何$\lvert M\rangle^{\otimes k}$的费米子高斯分解包含$\Omega(1.4^k)$项。这里$\lvert M\rangle$是费米子线性光学中最标准的魔法态：可被消耗以实现交换门的4量子比特态。我们还证明了同一态的$\delta$-近似秩的基本匹配的界，其下界由限制精确秩的同一量乘以$1-\delta^2$因子给出。我们关于精确费米子高斯秩的结果直接适用于任意$k$个固定宇称非高斯态的乘积，尽管近似秩并非如此。最后，我们证明具有$m_j$个玻色子在模式$j$中的$n$模玻色子Fock态的相干态边界秩恰好为$\prod_{j}(1+m_j)$，回答了参考文献[1]中的猜想，并获得了近似相干态秩的下界，该下界由相同量乘以近似保真度的函数给出，强调了我们所采用方法的广泛适用性。
英文摘要
Recent algorithms for classical simulation of quantum mechanics have runtime whose superpolynomial component is given by a linear dependence on the number of terms required to write large tensor products of certain "magic" states as superpositions of "free" states (which may be stabilizer states, fermionic Gaussian states or others). Surprisingly little is known about the number of terms in such decompositions, called ranks (e.g. the stabilizer rank, fermionic-Gaussian rank etc.). For complexity theoretic reasons they are expected to grow exponentially in the number of tensor factors but, while exponential upper bounds are known for both the stabilizer and fermionic-Gaussian rank of magic states; the best known lower bounds on the stabiliser rank are quadratic, and no bounds on the fermionic-Gaussian rank are known beyond fixed constants. In this work we prove that any fermionic Gaussian decomposition of $\lvert M\rangle^{\otimes k}$ consists of $Ω(1.4^k)$ terms. Here $\lvert M\rangle$ is the most standard magic state for fermionic linear optics: the $4$ qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the $δ$-approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of $1-δ^2$. Our results on exact fermionic Gaussian rank apply directly to any product of $k$ fixed parity non-Gaussian states, although the same is not true of the approximate rank. Finally, we prove that the coherent state border rank of an $n$-mode bosonic Fock state with $m_j$ bosons in mode $j$ is exactly $\prod_{j}(1+m_j)$, answering a conjecture of Ref. [1] and obtain lower bounds on the approximate coherent state rank given by the same quantity multiplied by a function of the fidelity of the approximation, emphasizing the broad applicability of the method we employ.
发表机构
- Centrum Fizyki Teoretycznej Polskiej Akademii Nauk（波兰科学院理论物理中心）
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