Between NISQ (noisy intermediate scale quantum) approaches without any proof of robust quantum advantage and fully fault-tolerant quantum computation, we propose a scheme to achieve a provable superpolynomial quantum advantage (under some widely accepted complexity conjectures) that is robust to noise with minimal error correction requirements. We choose a class of sampling problems with commuting gates known as sparse IQP (Instantaneous Quantum Polynomial-time) circuits and we ensure its fault-tolerant implementation by introducing the tetrahelix code. This new code is obtained by merging several tetrahedral codes (3D color codes) and has the following properties: each sparse IQP gate admits a transversal implementation, and the depth of the logical circuit can be traded for its width. Combining those, we obtain a depth-1 implementation of any sparse IQP circuit up to the preparation of encoded states. This comes at the cost of a space overhead which is only polylogarithmic in the width of the original circuit. We furthermore show that the state preparation can also be performed in constant depth with a single step of feed-forward from classical computation. Our construction thus exhibits a robust superpolynomial quantum advantage for a sampling problem implemented on a constant depth circuit with a single round of measurement and feed-forward.

中文翻译：

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恒定深度的鲁棒稀疏 IQP 采样

在没有任何鲁棒量子优势证明的 NISQ（噪声中尺度量子）方法和完全容错量子计算之间，我们提出了一种方案，以实现可证明的超多项式量子优势（在一些广泛接受的复杂性猜想下），该方案对噪声具有鲁棒性，并且具有最小的鲁棒性。纠错要求。我们选择一类带有交换门的采样问题，称为稀疏 IQP（瞬时量子多项式时间）电路，并通过引入四螺旋代码来确保其容错实现。这种新代码是通过合并多个四面体代码（3D颜色代码）获得的，具有以下属性：每个稀疏IQP门都允许横向实现，并且逻辑电路的深度可以换取其宽度。结合这些，我们获得了任何稀疏 IQP 电路的深度 1 实现，直到准备编码状态。这是以空间开销为代价的，该空间开销仅是原始电路宽度的多对数。我们还表明，状态准备也可以通过经典计算的单步前馈以恒定的深度进行。因此，我们的结构对于在具有单轮测量和前馈的恒定深度电路上实现的采样问题表现出强大的超多项式量子优势。