Quantum algorithms for quantum dynamics: A performance study on the spin-boson model

Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter approximation of the time-evolution operator. This approach typically relies on deep circuits and is therefore hampered by the substantial limitations of available noisy and near-term quantum hardware. On the other hand, variational quantum algorithms (VQAs) have become an indispensable alternative, enabling small-scale simulations on present-day hardware. However, despite the recent development of VQAs for quantum dynamics, a detailed assessment of their efficiency and scalability is yet to be presented. To fill this gap, we applied a VQA based on McLachlan's principle to simulate the dynamics of a spin-boson model subject to varying levels of realistic hardware noise as well as in different physical regimes, and discuss the algorithm's accuracy and scaling behavior as a function of system size. We observe a good performance of the variational approach used in combination with a general, physically motivated wave function ansatz, and compare it to the conventional first-order Trotter evolution. Finally, based on this, we make scaling predictions for the simulation of a classically intractable system. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage for the solution of time-dependent problems.

Figure 1

Schematic representation of the qubit-mapped spin-boson model. The spin's state is captured by a single qubit, while under the direct mapping for bosonic modes, each energy level $n_k$ corresponds to a qubit.

Figure 2

Variational simulation results for three qubits ($M=1,n_k^{\max }=1$), Trotter depth $d=1$, i.e., $N_{\theta }=4$ variational parameters, and under varying influence of noise with 8192 shots per circuit evaluation. Using the noise model from one of IBM's devices, ibmq_santiago v1.3.22 [2] (see Appendix pp2, Table 1), $\eta$ denotes the fraction of noise employed. That is, $\eta =1$ indicates results obtained with the full realistic hardware noise together with statistical noise, while $\eta =\infty$ means only statistical and no hardware noise. The top row in (a)–(c) shows the spin-orientation evolving for different system setups (Hamiltonian parameters $\epsilon ,\mathrm{\Delta }$), while the bottom row shows the respective infidelities of the variational state. In panel (d), we plot the mean of the infidelities in (a)–(c), $\mathrm{\Delta }\overline{\mathrm{\Phi }}$, as a function of $\eta$. Results indicate that already a reduction of current hardware noise by one order of magnitude yields an accuracy comparable to that obtained with only statistical noise.

Figure 3

(a) Variational results for five qubits ($M=1,n_k^{\max }=3$) and $g/\omega =0.5$, for Trotter depth $d=1,2$, i.e., $N_{\theta }=8,16$ variational parameters. (b) Variational results for five qubits ($M=2,n_k^{\max }=1$) and varying coupling strength $g/\omega =0.2,0.6,1.0$, for Trotter depth $d=1,2$, i.e., $N_{\theta }=6,12$ variational parameters. Top rows (a1) and (b1) correspond to system parameters $(\epsilon ,\mathrm{\Delta })=(-1,0)$, bottom rows (a1) and (b1) the parameters $(\epsilon ,\mathrm{\Delta })=(0,1)$.

Figure 4

Final depth, i.e., number of Trotter steps, required to achieve an accuracy $\mathrm{\Delta }\mathrm{\Phi }$ below ${\varepsilon }_{\mathrm{thresh}}$, comparing Trotter and variational simulation for different system sizes (data points). Top and bottom plots show the same data with a linear and log scale, respectively. Legend entries denote different values of ${\varepsilon }_{\mathrm{thresh}}$ and variational simulation, respectively. Variational simulation achieves a final infidelity below ${10}^{-4}$ throughout all numerical examples and is therefore to be compared to the ${\varepsilon }_{\mathrm{thresh}}={10}^{-4}$ Trotter curve. The scaling of the depth is estimated with a linear fit (see main text), which is used to extrapolate to system sizes of $N_{\mathrm{qubits}}=120$, the number of qubits necessary for a simulation comparable to state of the art classical spin-boson simulations.

Figure 5

(a) Quantum circuit representation of the variational ansatz $U_{\mathrm{H}}$ in Eq. (8), showing the spin and one bosonic qubit register. The blue box represents the coupling term in $U_{\mathrm{H}}$, while the red two-qubit gates represent bosonic self-interaction terms. (b) Decomposition of the coupling gate from (a), with each cascade of three-qubit gates representing a term in ${\tilde{X}}_k^{\mathrm{e},\mathrm{o}}$. (d) Final decomposition of the three-qubit gate into one- and two-qubit gates, coupling the spin and the respective bosonic mode. (c) Decomposition of the bosonic self-interaction gate. Variational parameters enter through rotational gates $R_z\left(\theta \right)$. The Hadamard gate $H$ and $Y^{†}=R_x(\pi /2)$ rotate qubits from ${\sigma }^z$– into the ${\sigma }^x$– and ${\sigma }^y$–basis, respectively.