We introduce a quantum algorithm to efficiently prepare states with a small energy variance at the target energy. We achieve it by filtering a product state at the given energy with a Lorentzian filter of width $\delta$. Given a local Hamiltonian on $N$ qubits, we construct a parent Hamiltonian whose ground state corresponds to the filtered product state with variable energy variance proportional to $\delta\sqrt{N}$. We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in $\mathrm{poly}(N,1/\delta)$ time via adiabatic evolution. We numerically benchmark the algorithm for a particular non-integrable model and find that the adiabatic evolution time to prepare the filtered state with a width $\delta$ is independent of the system size $N$. Furthermore, the adiabatic evolution can be implemented with circuit depth $\mathcal{O}(N^2\delta^{-4})$. Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators by directly preparing a finite energy state, providing access to an approximation of the microcanonical properties at an arbitrary energy.

中文翻译：

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一种用于过滤产品状态的高效量子算法


我们引入了一种量子算法来有效地准备在目标能量处具有较小能量方差的状态。我们通过使用宽度 $\delta$ 的洛伦兹滤波器过滤给定能量下的乘积状态来实现它。给定 $N$ 量子位上的局部哈密顿量，我们构造一个父哈密顿量，其基态对应于滤波后的乘积状态，其可变能量方差与 $\delta\sqrt{N}$ 成比例。我们证明了父哈密顿量是有间隙的，并且它的基态可以通过绝热演化在 $\mathrm{poly}(N,1/\delta)$ 时间内有效地实现。我们对特定不可积模型的算法进行数值基准测试，发现准备宽度 $\delta$ 的滤波状态的绝热演化时间与系统大小 $N$ 无关。此外，绝热演化可以通过电路深度$\mathcal{O}(N^2\delta^{-4})$来实现。我们的算法提供了一种通过直接准备有限能态来研究量子模拟器中许多物体系统的有限能量状态的方法，从而提供对任意能量下的微正则特性的近似。