To implement quantum algorithms on a quantum computer, we must overcome the twin problems of fault-tolerance -- how can we realize a relatively noiseless computation by cleverly combining noisy components? -- and compilation -- how can we realize an arbitrary quantum algorithm given the basic operations available on the quantum device at hand? We show how treating the former problem via error-correcting codes enables greater flexibility in resolving the latter. Specifically, we explicitly leverage the fact that error-correcting codes introduce redundancy which renders physically distinct operators logically indistinguishable. In terms of computation, it suffices to implement any operator logically equivalent to some target, yet from a compilation perspective, certain choices may be preferable to others. Our novel contribution is making this intuition precise in the general setting of the special unitary group. In particular, we describe how to reduce the problem of making a compilation-ideal choice to a least squares problem and provide a closed form solution thereof. Using our framework, it is possible to circumvent inserting costly swaps to adhere to hardware connectivity; instead, we could realize the logical target through a distinct physical Hamiltonian that is natively accessible. We elucidate our approach using the $[[4,2,2]]$ code. We discuss connections to compressed sensing that may pave the way to efficient compilation leveraging physical degrees of freedom.

翻译：要在量子计算机上实现量子算法，我们必须克服两个相互关联的问题：容错性——如何通过巧妙组合噪声组件实现相对无噪声的计算——以及编译——如何利用当前量子设备提供的基本操作来实现任意量子算法。研究表明，通过纠错码处理前者问题能为解决后者问题带来更大灵活性。具体而言，我们明确利用了纠错码引入冗余这一事实，该冗余使得物理上不同的算子在逻辑上不可区分。从计算角度看，只要实现与某个目标逻辑等价的任意算子即可满足要求；但从编译角度看，某些选择可能优于其他选择。我们的创新贡献在于将这一直觉精确地推广到特殊酉群的一般性设定中。特别地，我们描述了如何将编译最佳选择问题转化为最小二乘问题，并给出了其闭式解。利用我们的框架，可以避免插入昂贵的交换门来适配硬件连接约束；相反，我们可以通过原生可访问的不同物理哈密顿量来实现逻辑目标。我们使用$[[4,2,2]]$码阐释了该方法，并讨论了与压缩感知的联系，这或将为利用物理自由度实现高效编译铺平道路。