Embedding classical dynamics in a quantum computer
Embedding classical dynamics in a quantum computer
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DOI：
10.1103/physreva.105.052404
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发表时间：
2022-05-03
影响因子：
2.9
通讯作者：
Slawinska, Joanna
中科院分区：
文献类型：
--
作者：
Giannakis, Dimitrios;Ourmazd, Abbas;Slawinska, Joanna
我们开发了一个框架，用于在量子计算机上模拟测量保持，遍历动力系统。我们的方法通过将遍历理论与量子信息科学相结合，提供了经典动力学的算子理论表示。由此产生的经典动力学量子嵌入（QECD）能够使用二次量子门数有效地模拟具有指数大维度的经典可观测空间。QECD框架基于一个量子特征映射，我们引入该映射用于在再现核Hilbert空间H上用密度算子表示经典态。此外，建立了将经典可观测值嵌入到H上的自伴随算子中，使得量子力学期望值与点向函数求值一致。在该方案中，量子态和观测值在经典系统的库普曼演化算子的提升作用下统一演化。此外，由于H的再现特性，量子系统与底层经典动力学是点向一致的。为了实现量子计算优势，我们将量子系统的状态投射到与n个量子位相关的2n维张量积希尔伯特空间上的有限秩密度算子上。通过对谱函数进行离散傅里叶-沃尔什变换，将有限维量子系统的演化算子分解为张量积形式，使其能够通过大小为O(n)的n通道量子电路实现，无需通道间通信。此外，该电路还具有一个大小为O(n)的状态准备阶段和一个大小为O（n(2)）的量子傅立叶变换阶段，这使得通过在标准计算基础上的测量来预测可观测值成为可能。我们在大量子位极限n ->无穷大下证明了这些预测的理论收敛结果。鉴于这些性质，QECD提供了一个通过投影量子测量实现的经典可观测物演化的一致模拟器，它能够使用大小为O（n(2)）的电路模拟2n维的经典可观测物空间。我们证明了该格式在环面上包含周期和准周期振子的原型动力系统中的一致性。这些例子包括在Qiskit Aer中的模拟量子电路实验，以及在IBM量子系统一号上的实际实验。
We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides an 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 that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, H. Furthermore, an embedding of classical observables into self-adjoint operators on H is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. 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 H, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a 2n-dimensional tensor product Hilbert space associated with n qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an n-channel quantum circuit of size O(n) and no interchannel communication. 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 in the large-qubit limit, n -> infinity In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension 2n using circuits of size O(n(2)). We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.