Black-box Hamiltonian simulation and unitary implementation (0910.4157v4)

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.

• The paper demonstrates that black-box Hamiltonian simulation achieves linear scaling in both evolution time and sparseness parameter D.
• It introduces quantum walk-based methods that significantly reduce computational complexity compared to traditional sparse Hamiltonian simulation techniques.
• The work shows that arbitrary unitary operations can be implemented using O(N(log log N)^(4/3)) queries, paving the way for more efficient quantum algorithms.

Overview of the Paper: Black-box Hamiltonian Simulation and Unitary Implementation

This paper by Dominic W. Berry and Andrew M. Childs presents advancements in the simulation of black-box Hamiltonians using quantum walks, offering robust techniques for simulating sparse Hamiltonians and executing black-box unitary operations. The primary contribution of their work lies in improving the computational complexity of simulating sparse Hamiltonians and implementing arbitrary black-box unitary operations.

Core Contributions and Numerical Analysis

The paper introduces methods that lead to an improved simulation of sparse Hamiltonians. Traditionally, such simulations have complexities that scale superlinearly with evolution time $t$ and are inefficient when confronted with increased sparseness, often growing with the fourth power of the sparseness parameter $D$. Berry and Childs propose a new simulation approach, which improves upon this by delivering computational complexity strictly linear in $t$ and proportional to $D$, a significant optimization in contrast to previous methods. This is reminiscent of the optimal scaling known in the domain.

Furthermore, the paper examines the task of implementing an arbitrary unitary defined by a black-box description of its matrix elements. Conventional methods demand $O(N)$ or more gates to perform a unitary operation, as stated for $N \times N$ matrices. However, utilizing their simulation techniques, the authors show that such unitary operations can be realized with $O(N(\log\log N)^{4/3})$ queries, and often optimally around $O(\sqrt{N})$.

Methodology Innovations

Key to these improvements is the use of quantum walks for both Hamiltonian simulation and unitary implementation. The paper progresses beyond traditional methods such as the Lie-Trotter-Suzuki formulas, offering a different mechanism better suited for error management and short time steps. This strategy comprises several innovative techniques, including:

1. Lazy Quantum Walks and Phase Estimation: The methodologies include steps that modify the lazy quantum walk using phase estimation, enhancing simulation efficiency.
2. State Preparation via Amplitude Amplification: This offers a more efficient simulation even in non-sparse settings.
3. Decomposition and Recomposing Using Lie-Trotter-Suzuki Formulae: This technique breaks down the Hamiltonian into parts based on matrix element magnitudes, leveraging an orthogonal approach to the traditional Lie-Trotter-Suzuki application.

Implications and Speculation on Future Developments

The implications of this work are manifold, touching on both theory and practice. Practically, this paper provides tools that facilitate more efficient execution of quantum algorithms which leverage Hamiltonian dynamics as a computational process. Theoretically, it contributes to a deeper understanding of the trade-offs inherent in Hamiltonian simulation, offering potential pathways to more universally optimized quantum simulations.

The work also invites further exploration into whether these or similar techniques can universally achieve the theoretically efficient bound of $O(\sqrt{N})$ for unitary implementations. Such achievements could dramatically impact the scalability of quantum algorithms across a broader range of applications, potentially reshaping the computational landscape of quantum mechanics simulations and unitary transformations.

Conclusion

Berry and Childs' research provides substantial advancements in quantum computing methodologies, especially in the field of Hamiltonian simulation and unitary implementation. Their findings mark a significant step forward in simulating quantum systems more efficiently, ultimately underlining the versatility and potential of quantum walks as a foundational tool in quantum computing. Future research building on these principles may well unlock new efficiencies and capabilities in the field.