We provide a quantum algorithm for simulating the dynamics of sparse
Hamiltonians with complexity sublogarithmic in the inverse error, an
exponential improvement over previous methods. Specifically, we show that a
$d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$
with precision $\epsilon$ using $O\big(\tau
\frac{\log(\tau/\epsilon)}{\log\log(\tau/\epsilon)}\big)$ queries and
$O\big(\tau \frac{\log^2(\tau/\epsilon)}{\log\log(\tau/\epsilon)}n\big)$
additional 2-qubit gates, where $\tau = d^2 \|{H}\|_{\max} t$. Unlike previous
approaches based on product formulas, the query complexity is independent of
the number of qubits acted on, and for time-varying Hamiltonians, the gate
complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our
algorithm is based on a significantly improved simulation of the continuous-
and fractional-query models using discrete quantum queries, showing that the
former models are not much more powerful than the discrete model even for very
small error. We also simplify the analysis of this conversion, avoiding the
need for a complex fault correction procedure. Our simplification relies on a
new form of "oblivious amplitude amplification" that can be applied even though
the reflection about the input state is unavailable. Finally, we prove new
lower bounds showing that our algorithms are optimal as a function of the
error.

本文提出了一种量子算法，可以在误差的倒数的次对数时间内模拟稀疏哈密顿量的动力学，是之前方法的指数级改进，其查询复杂度不依赖于作用的量子位数，门复杂度对哈密顿量的导数阈值的对数级，同时结果表明离散模型的范数微分变换有很高的效率，证明了算法的最优性。

稀疏哈密顿量模拟精度的指数级提升

Exponential improvement in precision for simulating sparse Hamiltonians

We present general methods for simulating black-box Hamiltonians using
quantum walks. These techniques have two main applications: simulating sparse
Hamiltonians and implementing black-box unitary operations. In particular, we
give the best known simulation of sparse Hamiltonians with constant precision.
Our method has complexity linear in both the sparseness D (the maximum number
of nonzero elements in a column) and the evolution time t, whereas previous
methods had complexity scaling as D^4 and were superlinear in t. We also
consider the task of implementing an arbitrary unitary operation given a
black-box description of its matrix elements. Whereas standard methods for
performing an explicitly specified N x N unitary operation use O(N^2)
elementary gates, we show that a black-box unitary can be performed with
bounded error using O(N^{2/3} (log log N)^{4/3}) queries to its matrix
elements. In fact, except for pathological cases, it appears that most
unitaries can be performed with only O(sqrt{N}) queries, which is optimal.

本文提出了用量子行走模拟黑箱哈密顿量的一般方法。这些技术具有两个主要应用：模拟稀疏哈密顿量和实现黑箱酉操作。我们给出了稀疏 Hamiltonians 常数精度下最好的已知模拟方法，复杂度线性地随着稀疏矩阵的最大非零元素数 D 和演化时间 t 增长，并且我们还研究了给定黑箱描述矩阵元素的任意酉操作实现的任务。