High-Order Randomized Compiler for Hamiltonian Simulation

Hamiltonian simulation is known to be one of the fundamental building blocks of a variety of quantum algorithms such as its most immediate application, that of simulating many-body systems to extract their physical properties. In this work, we present qSWIFT, a high-order randomized algorithm for Hamiltonian simulation. In qSWIFT, the required number of gates for a given precision is independent of the number of terms in the Hamiltonian, while the systematic error is exponentially reduced with regard to the order parameter. In this respect, our qSWIFT is a higher-order counterpart of the previously proposed quantum stochastic drift protocol (qDRIFT), the number of gates in which scales linearly with the inverse of the precision required. We construct the qSWIFT channel and establish a rigorous bound for the systematic error quantified by the diamond norm. qSWIFT provides an algorithm to estimate given physical quantities by using a system with one ancilla qubit, which is as simple as other product-formula-based approaches such as regular Trotter-Suzuki decompositions and qDRIFT. Our numerical experiment reveals that the required number of gates in qSWIFT is significantly reduced compared to qDRIFT. In particular, the advantage is significant for problems where high precision is required; e.g., to achieve a systematic relative propagation error of ${10}^{-6}$, the required number of gates in third-order qSWIFT is 1000 times smaller than that of qDRIFT.

Popular Summary

In our research, we tackle a key problem in the field of quantum computing: simulating complex systems using a quantum computer. To perform these simulations, we need to use a process called Hamiltonian simulation, which is a fundamental building block of many quantum algorithms. The better we can simulate these systems, the more we can learn about their properties and behavior, which will have broad implications for various scientific disciplines.

Our work introduces a method called qSWIFT, a high-order randomized algorithm for Hamiltonian simulation. This method is a significant improvement over existing methods, such as qDRIFT, as it requires far fewer computational steps (or gates) to achieve the same level of precision. The qSWIFT algorithm is designed in such a way that the number of gates required does not depend on the number of terms in the Hamiltonian, making it more efficient for complex systems. Additionally, the algorithm’s error rate decreases exponentially as we increase its order, that is, the level of complexity of the algorithm.

We rigorously tested qSWIFT by comparing it with the existing qDRIFT method and found that qSWIFT requires significantly fewer gates to achieve the same precision, particularly for problems that demand high precision. In some cases, the number of gates required in qSWIFT was 1000 times smaller than that of qDRIFT.

The development of qSWIFT represents a significant milestone in quantum computing, as it enables more efficient simulations of complex systems. This advancement not only helps push the boundaries of quantum information science but also opens up new possibilities for researchers and practitioners in various fields that rely on accurate simulations of physical systems.

Figure 1

Quantum circuits for implementing the swift operators ${\tilde{\mathcal{S}}}_{\ell }^{(b)}:=S_{\ell }^{(b)}\rho S_{\ell }^{(b)†}(b\in \{0,1\})$. The top line corresponds to an ancilla qubit and the bottom line corresponds to the qubits in the system. The $S$ gate is defined by the operator $e^{\mathrm{i}\pi /2}{e}^{\mathrm{i}{\sigma }_z}$, where ${\sigma }_z$ is the Pauli-$Z$ operator. (a) The quantum circuit for ${\tilde{\mathcal{S}}}_{\ell }^{(0)}(\rho )$. (b) The quantum circuit for ${\tilde{\mathcal{S}}}_{\ell }^{(1)}(\rho )$.

Figure 2

Examples of quantum circuits used for qDRIFT and qSWIFT. (a) A quantum circuit for qDRIFT with $N$ segments. (b) A swift circuit with $N$ segments, where two swift operators are performed in the $(r+1)\mathrm{th}$ segment.

Figure 3

The asymptotic behavior of $N$ for each time for each molecule to achieve the systematic error $\epsilon =0.001$. The three subfigures correspond to the three molecules: (a) propane with the STO-3G basis, (b) ethane with the 6-31G basis, and (c) carbon dioxide with the 6-31G basis. We show the third-order qSWIFT by the pink line (qSWIFT-3), the sixth-order qSWIFT by the pink dotted line (qSWIFT-6), and the qDRIFT by the black line (qDRIFT). For the Trotter-Suzuki decomposition, the best of the deterministic methods among the first, second, and fourth orders is shown by the gray dotted line [TS (Best)], and the best of the randomized methods is shown by the green dotted line [RTS (Best)].

Figure 4

The asymptotic behavior to achieve $\epsilon ={10}^{-6}$. The other settings are the same as in Fig. 3.

Figure 5

The estimation error of $\mathrm{Tr}(Q\mathcal{U}({\rho }_{\mathrm{init}}))$ for each number of gates $N$ for each method with $Q=ZIIIIIII$ and ${\rho }_{\mathrm{init}}=|+{⟩}^{\otimes 8}$. The time evolution is performed by the hydrogen-molecule Hamiltonian with the 6-31G basis transformed by the Bravyi-Kitaev transformation. The evolution time is set to be $t=1$. In plotting each point, we perform six trials and show the mean value and the standard deviation of the mean. For qSWIFT, we show the result of the second order (qSWIFT-2) with the purple line and the third order (qSWIFT-3) with the pink line. The result of the qDRIFT algorithm (qDRIFT) is plotted with the black line. For the deterministic Trotter-Suzuki decomposition, the first-order result (TS-1st) is plotted with the dotted gray line and the second-order result (TS-2nd) is plotted with the dotted green line. For the randomized Trotter-Suzuki decomposition, the first-order result (RTS-1st) is plotted with the dotted yellow line and the second-order result (RST-2nd) is plotted with the dotted blue line.

Figure 6

The quantum circuits for evaluating Eq. (D8) by the LCU-based approach.