The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a Hamiltonian -- using Hamiltonian evolutions of input operators. We show how to maintain this $\textit{Hamiltonian block encoding}$, so that matrix operations can be composed one after another, and the entire quantum computation takes $\leq 2$ ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadmard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain of interest, while the constructed polynomial is upper bounded by the target function over the entire unit interval. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the doubly factorized tensor hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with $1$ ancilla qubit.

翻译：高效实现矩阵算术运算是众多量子算法获得加速的基础。我们开发了一套方法，利用输入算符的哈密顿演化执行矩阵算术运算——其结果编码在哈密顿量的非对角块中。我们展示了如何维持这种$\textit{哈密顿块编码}$，使得矩阵运算能够依次组合，且整个量子计算仅需$\leq 2$个辅助量子比特。我们针对矩阵乘法、矩阵加法、矩阵求逆、厄米共轭、分数缩放、整数缩放、复相位缩放，以及奇偶多项式的奇异值变换均实现了这一目标。我们还提出了一种重叠估计算法，用于提取哈密顿块编码算符的经典性质，其功能类似于著名的Hadamard测试，且无需额外量子比特开销。我们的哈密顿矩阵乘法采用了李群交换子乘积公式及其由Childs和Wiebe提出的高阶推广。我们的哈密顿奇异值变换采用了一种支配多项式逼近方法，该逼近在目标域内成立，同时构造的多项式在整个单位区间上被目标函数上界控制。我们描述了一类平方和型哈密顿量的模拟电路，在步数上实现了交换子标度，并利用矩阵运算能力降低了每步模拟的成本。特别地，我们将此应用于近期量子化学研究中提出的双分解张量超收缩哈密顿量，针对具有固定粒子数的初始态获得了进一步改进。我们仅用$1$个辅助量子比特实现了上述所有目标。