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Efficient quantum algorithms for simulating sparse Hamiltonians

Dominic W. Berry, Graeme Ahokas, Richard Cleve, Barry C. Sanders

TL;DR

This work addresses the quantum simulation of sparse Hamiltonians by leveraging Suzuki's higher-order integrators to reduce the temporal scaling of simulating $e^{-iHt}$ to $t^{1+1/(2k)}$ for an arbitrary order $k$. It introduces an efficient Hamiltonian-decomposition strategy that achieves a $\log^* n$-scaled overhead in the sparse, black-box setting and provides explicit bounds: $N_{\rm exp} \le 2m 5^{2k} (m\tau)^{1+1/2k}/\epsilon^{1/2k}$ for general cases and $N_{\rm bb} \in O((\log^* n) d^2 5^{2k} (d^2 \tau)^{1+1/2k}/\epsilon^{1/2k})$ for sparse cases with $\tau=\|H\|t$. A linear lower bound shows sublinear-in-$\tau$ simulation is impossible in the black-box model. Together, these results yield near-linear-time quantum simulation for sparse Hamiltonians and substantially improved decomposition costs, advancing practical quantum simulations beyond prior $t^2$ or $t^{3/2}$ scaling and polynomials in $n$-dependent resources. The findings have broad impact for simulating quantum systems and computational problems encoded in sparse Hamiltonians.

Abstract

We present an efficient quantum algorithm for simulating the evolution of a sparse Hamiltonian H for a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and |H| is bounded by a constant, we may select any positive integer $k$ such that the simulation requires O((\log^*n)t^{1+1/2k}) accesses to matrix entries of H. We show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible.

Efficient quantum algorithms for simulating sparse Hamiltonians

TL;DR

This work addresses the quantum simulation of sparse Hamiltonians by leveraging Suzuki's higher-order integrators to reduce the temporal scaling of simulating to for an arbitrary order . It introduces an efficient Hamiltonian-decomposition strategy that achieves a -scaled overhead in the sparse, black-box setting and provides explicit bounds: for general cases and for sparse cases with . A linear lower bound shows sublinear-in- simulation is impossible in the black-box model. Together, these results yield near-linear-time quantum simulation for sparse Hamiltonians and substantially improved decomposition costs, advancing practical quantum simulations beyond prior or scaling and polynomials in -dependent resources. The findings have broad impact for simulating quantum systems and computational problems encoded in sparse Hamiltonians.

Abstract

We present an efficient quantum algorithm for simulating the evolution of a sparse Hamiltonian H for a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and |H| is bounded by a constant, we may select any positive integer such that the simulation requires O((\log^*n)t^{1+1/2k}) accesses to matrix entries of H. We show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible.

Paper Structure

This paper contains 6 sections, 6 theorems, 29 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problems and results
  3. Higher order integrators
  4. Linear limit on simulation time
  5. Efficient decomposition of Hamiltonian
  6. Conclusions

Key Result

Theorem 1

When the permissible error is bounded by $\epsilon$, $N_{\rm exp}$ is bounded by for $\epsilon\le 1 \le 2m 5^{k-1}\tau$, where $\tau=\|H\|t$, and $k$ is an arbitrary positive integer.

Figures (2)

  • Figure 1: Graph representing example the Hamiltonian in the proof of Theorem \ref{['th3']}. States are represented by ellipses, and nonzero elements of the Hamiltonian are indicated by lines. The sequence of states $|{k_j,j}\rangle$ with $k_0=0$ is indicated by the solid line.
  • Figure 2: A portion of the graph for the example given in Tables \ref{['xtable']} and \ref{['wtable']}. The vertices $w$, $x$, $y$, etc each have $i=1$ and $j=3$ for the edge labels, so it is necessary for the $\nu$ to differ to ensure that adjoining edges have distinct labels. The $x_l^{(0)}$ and $w_l^{(0)}$ which the vertices correspond to are also given. The numbers in the first columns of Tables \ref{['xtable']} and \ref{['wtable']} are the binary representations of the vertex numbers given here.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • ...and 2 more