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Black-box Hamiltonian simulation and unitary implementation

Dominic W. Berry, Andrew M. Childs

TL;DR

The paper tackles the problem of simulating black-box Hamiltonians and implementing black-box unitaries using quantum walks. It introduces an oracle-based framework, delivers linear-in-$t$ and sparsity-aware bounds for sparse Hamiltonians, and provides a sharper $D^{2/3}$-scaling bound for general (non-sparse) cases through Hamiltonian breaking and nested Trotter formulas. For unitaries, the authors show sublinear query complexity, achieving $O(N^{2/3} (\\log\\log N)^{4/3} \\delta^{-1/3})$ in general and often near $\\tilde{O}(\\sqrt{N})$ in practice, which can outperform explicit gate decompositions. The results hold promise for efficient quantum simulation and highly quadratically improved unitary implementation in black-box settings, with open questions about universal sqrt-$N$ scaling and further optimization.

Abstract

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.

Black-box Hamiltonian simulation and unitary implementation

TL;DR

The paper tackles the problem of simulating black-box Hamiltonians and implementing black-box unitaries using quantum walks. It introduces an oracle-based framework, delivers linear-in- and sparsity-aware bounds for sparse Hamiltonians, and provides a sharper -scaling bound for general (non-sparse) cases through Hamiltonian breaking and nested Trotter formulas. For unitaries, the authors show sublinear query complexity, achieving in general and often near in practice, which can outperform explicit gate decompositions. The results hold promise for efficient quantum simulation and highly quadratically improved unitary implementation in black-box settings, with open questions about universal sqrt- scaling and further optimization.

Abstract

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.

Paper Structure

This paper contains 15 sections, 13 theorems, 125 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and results
  3. Model
  4. Results
  5. Review of Hamiltonian simulation
  6. Sparse Hamiltonian simulation
  7. Improved state preparation
  8. Non-sparse Hamiltonians
  9. Breaking up the Hamiltonian
  10. Small norms
  11. Norms of components
  12. Large norms
  13. Implementation of unitaries
  14. Examples
  15. Conclusion

Key Result

Theorem 1

For a given Hamiltonian $H$, let $\Lambda\ge\|H\|$ and $\Lambda_{\max}\ge\|H\|_{\max}$. Then the evolution under $H$ for time $t$ can be simulated with error at most $\delta\in(0,1]$ using queries to $O_H$ and $O_F$.

Figures (3)

  • Figure 1: The function ${\rm break}(H)$ for random Hamiltonians. The plusses and squares are the maximum and mean values, respectively, obtained for 100 randomly generated Hermitian matrices. The crosses and circles are the maximum and mean values, respectively, for sets of 100 Hamiltonians composed of random unitaries.
  • Figure 2: The function ${\rm break}(H)$ for a Hamiltonian composed of a matrix that has been produced by perturbing a quantum Fourier transform. The solid line is $\sqrt{M}$ for comparison.
  • Figure 3: The matrix elements of $U=\exp(-i\pi J_x/2)$ in the basis of $J_z$ eigenstates for $J=100$. The separate points are $|\langle{j}|U|{0}\rangle|$, and the solid curve is $|\langle{j}|U|{J}\rangle|$.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['theorem1']}.
  • Lemma 6
  • proof
  • ...and 16 more