arXiv 2012.060972020-12quant-phmath.DS

在量子计算机中嵌入经典动力学

Embedding classical dynamics in a quantum computer

Dimitrios Giannakis, Abbas Ourmazd, Philipp Pfeffer, Joerg Schumacher, Joanna Slawinska

备注：: 42 pages, 9 figures

摘要：

我们开发了一个框架，用于在量子计算机上模拟保持测量的遍历动力系统。我们的方法提供了一个新的经典动力学的算子理论表示的遍历理论与量子信息科学相结合。由此产生的量子嵌入的经典动力学（QECD），使有效的模拟空间的经典可观的指数大尺寸使用二次数量的量子门。QECD框架是基于一个量子特征映射的再生核希尔伯特空间，$\mathcal H $上的密度算子，并嵌入到$\mathcal H$上的自伴算子的经典可观的表示经典状态。在该方案中，量子态和观测量在经典系统Koopman演化算符的提升作用下幺正演化。此外，由于$\mathcal H$的再生性质，量子系统与潜在的经典动力学是逐点一致的。为了实现指数量子计算的优势，我们将量子系统的状态投影到与$n$量子位相关联的$2^n$维张量积希尔伯特空间上的密度矩阵。通过采用离散傅立叶-沃尔什变换，有限维量子系统的演化算子被分解为张量积形式，使得能够通过大小为$O(n)$的量子线路来实现。此外，该电路还具有一个同样大小为$O(n)$的状态准备阶段和一个大小为$O(n^2)$的量子傅里叶变换阶段，这使得通过标准计算基础中的测量来预测可观测量成为可能。我们证明这些预测的理论收敛结果为$n\to\infty$。我们目前在Qiskit Aer模拟量子线路实验，以及IBM量子系统一号的实际实验。

英文摘要：

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides a new operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal H $, and an embedding of classical observables into self-adjoint operators on $\mathcal H$. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal H$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve an exponential quantum computational advantage, we project the state of the quantum system to a density matrix on a $2^n$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through a quantum circuit of size $O(n)$. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O(n^2)$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions as $n\to\infty$. We present simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.

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