We define a map from an arbitrary quantum circuit to a local Hamiltonian whose ground state encodes the quantum computation. All previous maps relied on the Feynman-Kitaev construction, which introduces an ancillary `clock register' to track the computational steps. Our construction, on the other hand, relies on injective tensor networks with associated parent Hamiltonians, avoiding the introduction of a clock register. This comes at the cost of the ground state containing only a noisy version of the quantum computation, with independent stochastic noise. We can remedy this - making our construction robust - by using quantum fault tolerance. In addition to the stochastic noise, we show that any state with energy density exponentially small in the circuit depth encodes a noisy version of the quantum computation with adversarial noise. We also show that any `combinatorial state' with energy density polynomially small in depth encodes the quantum computation with adversarial noise. This serves as evidence that any state with energy density polynomially small in depth has a similar property. As an application, we show that contracting injective tensor networks to additive error is BQP-hard. We also discuss the implication of our construction to the quantum PCP conjecture, combining with an observation that QMA verification can be done in logarithmic depth.

翻译：我们定义了一种从任意量子电路到局域哈密顿量的映射，其基态编码了量子计算过程。以往所有此类映射均依赖于费曼-基塔耶夫构造，该构造引入辅助的"时钟寄存器"来追踪计算步骤。而我们的构造则基于具有关联父哈密顿量的单射张量网络，避免了时钟寄存器的引入。其代价是基态仅包含含有独立随机噪声的含噪量子计算版本。通过量子容错技术，我们可以弥补这一缺陷——使构造具有鲁棒性。除随机噪声外，我们证明任何能量密度随电路深度指数级小的量子态都编码了含对抗噪声的含噪量子计算版本。我们还证明任何能量密度随深度多项式级小的"组合态"也能编码含对抗噪声的量子计算。这为"任何能量密度随深度多项式级小的量子态具有类似性质"提供了证据。作为应用，我们证明在加法误差范围内收缩单射张量网络是BQP-困难的。结合QMA验证可在对数深度完成的观察，我们还讨论了该构造对量子PCP猜想的影响。