论文标题
带有通用量子计算机的BRGC代码的Laplacian的基于门的实现
Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers
论文作者
论文摘要
我们研究基于门反射的灰色代码（BRGC）的基于门的实现和由于在具有周期性边界条件的晶格上离散的Laplacian而导致的单一时间演变操作员的二进制代码。我们发现，通过贝克 - 贝克贝尔 - 霍斯多夫公式的固定晶格间距的固定晶格间距独立于系统尺寸。然后，我们提出用于构建BRGC量子电路的算法。对于此电路，对于绝热的演变时间$ t $，光谱规范错误$ε$，我们发现所需的电路成本（大门）和深度为$ \ mc {o}（t^2 n a d /ε）$，带有$ n-3 $ auxiliary Qubits，用于$ 2^n $ latte $ lattement $ lattension $ d $ d $ d $ $ d $ $ d $ $ d $ $ d $ $ d $ $ d $ $ d $ $ d $ $ d $ a $ a $ a $ a $ a $ a;对二进制位置编码的改进，需要指数级的$ n $ - 本地运营商。此外，在合理的假设是$ [t，v] $界限$Δt$，具有$ t $的动能和$ v $是非平凡的潜力，qft（量子傅立叶变换）的成本（量子傅里叶变换）的实现是将laplacian量表实现为$ \ mc {o} {o} {o} \ weft（n^2 \ right）$ pepth $} $ \ c}作为$ \ mc {o} \ left（n \右）$，为BRGC实现提供了优势。
We study the gate-based implementation of the binary reflected Gray code (BRGC) and binary code of the unitary time evolution operator due to the Laplacian discretized on a lattice with periodic boundary conditions. We find that the resulting Trotter error is independent of system size for a fixed lattice spacing through the Baker-Campbell-Hausdorff formula. We then present our algorithm for building the BRGC quantum circuit. For an adiabatic evolution time $t$ with this circuit, and spectral norm error $ε$, we find the circuit cost (number of gates) and depth required are $\mc{O}(t^2 n A D /ε)$ with $n-3$ auxiliary qubits for a system with $2^n$ lattice points per dimension $D$ and particle number $A$; an improvement over binary position encoding which requires an exponential number of $n$-local operators. Further, under the reasonable assumption that $[T,V]$ bounds $Δt$, with $T$ the kinetic energy and $V$ a non-trivial potential, the cost of QFT (Quantum Fourier Transform ) implementation of the Laplacian scales as $\mc{O}\left(n^2\right)$ with depth $\mc{O}\left(n\right)$ while BRGC scales as $\mc{O}\left(n\right)$, giving an advantage to the BRGC implementation.
