The Dirac equation as a quantum walk: higher dimensions, observational convergence
The Dirac equation as a quantum walk: higher dimensions, observational convergence
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DOI：
10.1088/1751-8113/47/46/465302
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发表时间：
2014-11-21
影响因子：
2.1
通讯作者：
Forets, Marcelo
中科院分区：
文献类型：
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作者：
Arrighi, Pablo;Nesme, Vincent;Forets, Marcelo
狄拉克方程可以被模拟为量子行走（QW），其主要特征是：在时间和空间上离散（即晶格上粒子的波函数的幺正演化）;齐次（即不变性和时间无关）和因果（即信息以有限的速度传播，在严格意义上）。这种联系，这是已经提出的Succi和Benzi，Bialynicki-Birula和迈耶，是持有巴格曼-维格纳方程和对称双曲系统一般。然后，我们分析证明的QW的解决方案的Dirac方程的柯西问题的解决方案的收敛性。我们通过采用标准数值分析中的强大方法来做到这一点，该方法对量子模拟领域具有普遍意义。在实践层面上，这一结果提供了精确的误差界和收敛速度，从而验证了量子阱作为量子模拟方案。在理论层面上，它强化了量子阱作为相对论粒子的简单、离散玩具模型的地位。
The Dirac equation can be modelled as a quantum walk (QW), whose main features are being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and time-independent) and causal (i.e. information propagates at a bounded speed, in a strict sense). This link, which was proposed already by Succi and Benzi, Bialynicki-Birula and Meyer, is shown to hold for Bargmann-Wigner equations and symmetric hyperbolic systems in general. We then analytically prove the convergence of the solution of the QW to the solution of the Cauchy problem for the Dirac equation. We do so by adapting a powerful method from standard numerical analysis, which is of general interest to the field of quantum simulation. At the practical level, this result provides precise error bounds and convergence rates, thereby validating the QW as a quantum simulation scheme. At the theoretical level, it reinforces the status of this QW as a simple, discrete toy model of relativistic particles.