A random compiler for fast Hamiltonian simulation (1811.08017v2)

Abstract: The dynamics of a quantum system can be simulated using a quantum computer by breaking down the unitary into a quantum circuit of one and two qubit gates. The most established methods are the Trotter-Suzuki decompositions, for which rigorous bounds on the circuit size depend on the number of terms $L$ in the system Hamiltonian and the size of the largest term in the Hamiltonian $\Lambda$. Consequently, Trotter-Suzuki is only practical for sparse Hamiltonians. Trotter-Suzuki is a deterministic compiler but it was recently shown that randomised compiling offers lower overheads. Here we present and analyse a randomised compiler for Hamiltonian simulation where gate probabilities are proportional to the strength of a corresponding term in the Hamiltonian. This approach requires a circuit size independent of $L$ and $\Lambda$, but instead depending on $\lambda$ the absolute sum of Hamiltonian strengths (the $\ell_1$ norm). Therefore, it is especially suited to electronic structure Hamiltonians relevant to quantum chemistry. Considering propane, carbon dioxide and ethane, we observe speed-ups compared to standard Trotter-Suzuki of between $306\times$ and $1591\times$ for physically significant simulation times at precision $10^{-3}$. Performing phase estimation at chemical accuracy, we report that the savings are similar.

• The paper introduces qDRIFT, a randomized compiling strategy that reduces gate counts by probabilistically sampling Hamiltonian terms based on their strengths.
• It demonstrates impressive speedups ranging from 306× to 1591× compared to traditional Trotter-Suzuki decompositions in simulating complex quantum systems.
• The approach broadens simulation applicability to dense Hamiltonians and mitigates coherent errors, offering significant theoretical and practical advancements in quantum computing.

Analyzing Randomized Compiling for Efficient Hamiltonian Simulation

The paper "A random compiler for fast Hamiltonian simulation" by Earl Campbell presents a novel approach to quantum system simulation using quantum computers, centering around the concept of a random compiler to improve the efficiency of Hamiltonian simulation.

Overview of Hamiltonian Simulation

Quantum simulation of Hamiltonians is pivotal in quantum computing, enabling researchers to explore quantum systems beyond the reach of classical computation techniques. Traditional methods, specifically Trotter-Suzuki decompositions, simulate quantum dynamics by decomposing a unitary evolution operator into a sequence of quantum gates. However, these methods are limited by the sparsity of Hamiltonians, where the gate count depends heavily on the number of terms, $L$, and the magnitude of the largest term, $\Lambda$.

Introduction of Randomized Compilation

Campbell introduces a randomized compiling strategy that contrasts with the traditional deterministic compilers. This method offers an innovative way to address the scalability issues faced by the Trotter-Suzuki approach. In particular, the proposed method, termed qDRIFT, utilizes randomized sampling of Hamiltonian terms, where the probability of selecting each term is weighted by its absolute strength. This probabilistic treatment leads to a gate count that is independent of $L$ and $\Lambda$, instead relying on $\lambda$, defined as the $\ell_1$ norm of Hamiltonian strengths.

Numerical Insights and Contradictory Claims

The paper provides empirical evidence through the simulation of electronic structure Hamiltonians relevant to quantum chemistry, such as those for propane, carbon dioxide, and ethane. The results highlight substantial speedups between $306\times$ and $1591\times$ over the Trotter-Suzuki method at a precision of $10^{-3}$, for simulation times deemed physically significant. This is noteworthy, considering conventional wisdom suggests that higher-order Trotter methods offer superior efficiency.

Theoretical and Practical Implications

Theoretically, the introduction of qDRIFT adds a stochastic dimension to Hamiltonian simulation that potentially mitigates the impact of coherent errors, a well-known challenge in quantum computations. Practically, the independence of gate counts from $L$ and $\Lambda$ broadens the applicability of the method to more complex and dense Hamiltonians, such as those encountered in quantum chemistry, without a corresponding increase in resource requirements.

Future Directions and Speculations

Given the promising results of the qDRIFT compiler, further investigations could explore extensions and adaptations to different classes of Hamiltonians or assess the practical yields of this compiling methodology in fault-tolerant quantum computing scenarios. There is also notable scope for optimizing phase estimation techniques, leveraging qDRIFT's reduction in gate count, especially in the context of achieving chemical accuracy in energy spectrum calculations.

Conclusion

Campbell's proposal for a random compiler signifies a valuable addition to the toolkit for quantum Hamiltonian simulation, offering both a conceptual shift and a practical advantage over traditional methods. The implication of these findings not only challenges conventional deterministic approaches but also seeks to harmonize computational feasibility with the aspiration to solve quantum problems of increasing complexity. As research continues, the true potential of stochastic approaches in other domains of quantum computing may become more evident.