Optimal Hamiltonian Simulation by Quantum Signal Processing (1606.02685v2)

Abstract: The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a $d$-sparse Hamiltonian $\hat{H}$ for time-interval $t$ with error $\epsilon$ is $\mathcal{O}(td|\hat{H}|_{\text{max}}+\frac{\log{(1/\epsilon)}}{\log{\log{(1/\epsilon)}}})$, which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (1) transducing eigenvalues of $\hat{H}$ into a single ancilla qubit, (2) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (3) projecting this ancilla with near unity success probability.

• The paper introduces quantum signal processing to achieve optimal Hamiltonian simulation with query complexity that meets known lower bounds.
• The paper’s method simplifies implementation by using single-qubit rotations, significantly reducing the need for additional ancilla qubits.
• The algorithm achieves a high success probability while lowering query costs by up to a square-root compared to previous state-of-the-art techniques.

Optimal Hamiltonian Simulation by Quantum Signal Processing: An Expert Overview

In the field of quantum computation, Hamiltonian simulation is fundamental, representing a problem encapsulating the full expanse of quantum computing capabilities. The paper "Optimal Hamiltonian Simulation by Quantum Signal Processing," authored by Guang Hao Low and Isaac L. Chuang, addresses this issue by demonstrating an algorithm for Hamiltonian simulation that is both optimal in terms of query complexity and rooted in a conceptually straightforward approach.

Key Contributions

1. Quantum Signal Processing Methodology: The paper introduces a three-step method known as quantum signal processing. This approach revolves around manipulating the eigenvalues of a Hamiltonian via a single ancilla qubit and a sequence of single-qubit rotations. The methodology consists of signal transduction, transformation, and projection, effectively reducing the task of Hamiltonian simulation to one of optimal quantum control.
2. Query Complexity: The authors establish that the query complexity for simulating a $d$-sparse Hamiltonian $\hat{H}$ over a time-interval $t$ with error $\epsilon$ is $$\mathcal{O}(td\|\hat{H}\|_{\text{max}} + \frac{\log(1/\epsilon)}{\log\log(1/\epsilon)})$$. This optimal complexity aligns with known lower bounds, showcasing that the query cost is additive with respect to the simulation length and the target error.
3. Implementation Simplicity: Through the use of single-qubit rotations, the proposed algorithm requires significantly fewer ancilla qubits compared to earlier methods. This simplification in the algorithmic implementation not only reduces the complexity but also approaches the problem with a distinct physical intuition that was previously elusive in Hamiltonian simulation problems.

Numerical Results and Complexity

The crux of the methodology is achieving error bounds through Fourier series approximations of trigonometric polynomials, leveraging the Jacobi-Anger expansion. The result is a rigorous yet intuitive procedure, achieving the optimal transformation for a given Hamiltonian, mathematically expressed with the error $\epsilon$ being efficiently bounded as per Thm. 2 in the paper.

The paper asserts improvements in query complexity, achieving up to a square-root reduction compared to previous state-of-the-art techniques. Furthermore, the algorithm maintains a high success probability (close to unity) without the decay typically seen in previous methods that rely heavily on ancilla and controlled operations.

Theoretical and Practical Implications

The presented algorithm substantiates that a Hamiltonian simulation can be achieved with optimal resource usage by embedding physical quantum dynamics directly into computation theory. This approach can feasibly enhance the performance and understanding of quantum simulations on future quantum computers.

The insight provided by leveraging single-qubit rotations for general quantum transformations suggests a broader applicability. The potential for quantum algorithms, beyond Hamiltonian simulation, to benefit from this approach is significant, hinting at possibilities for simplification and optimization in other quantum processes, such as quantum walks and Grover search.

Future Perspectives

The results advocate for a deeper exploration into the structured dynamics of quantum systems that can be abstracted into computational frameworks. The methodologies discussed here may inspire additional algorithmic development, yielding further insights into the intersection of physics and computation. As quantum computing resources continue to evolve, adapting these physically motivated and mathematically rigorous techniques could unveil new pathways for efficient quantum algorithm design.

In conclusion, Low and Chuang's work exemplifies an intersection of quantum control and computation, heralding a noteworthy advance in Hamiltonian simulation. The simplicity and efficiency of their approach provide a compelling vision for the future of quantum algorithms through leveraging intrinsic physical operations.