Variational dynamics as a ground-state problem on a quantum computer

We propose a variational quantum algorithm to study the real-time dynamics of quantum systems as a ground-state problem. The method is based on the original proposal of Feynman and Kitaev to encode time into a register of auxiliary qubits. We prepare the Feynman-Kitaev Hamiltonian acting on the composed system as a qubit operator and find an approximate ground state using the variational quantum eigensolver. We apply the algorithm to the study of the dynamics of a transverse-field Ising chain with an increasing number of spins and time steps, proving a favorable scaling in terms of the number of two-qubit gates. Through numerical experiments, we investigate its robustness against noise, showing that the method can be used to evaluate dynamical properties of quantum systems and detect the presence of dynamical quantum phase transitions by measuring Loschmidt echoes.

Figure 1

Sketch of the VFK method. Coupling the physical system to an auxiliary clock system, we can encode an entire time evolution into a time-independent state. We prepare a variational approximation of the history state by measuring the expectation value of the Feynman-Kitaev Hamiltonian and its derivatives on a quantum computer.

Figure 2

Magnetization (top) and infidelities (bottom) measured on the VFK history state. We considered a system of $n_s=6$ spins initialized in ${|0\rangle }^{\otimes 6}$ and different auxiliary qubits $n_a$ for a simulation time $T_e=3.0$. $F_{\text{VFK}}$ at time $t_i$ indicates the fidelity measured between the Trotter-Suzuki state with $i$ steps and the VFK state projected on the $|i\rangle$ subspace.

Figure 3

Final energy obtained with the VFK algorithm ($E_{\text{VFK}}$) over the energy of the first excited state ($E_1$) obtained using a classical simulator. The plot shows the final energy obtained for a system of $n_s$ spins and $2^{n_a-1}$ time steps as a function of the depth of ansatz in Eq. (13). The gray, dashed lines indicate the energies we require to consider the VFK converged. We fixed the simulation time $T_e=3.0$. In this case, increasing the number of auxiliary qubits will reduce the time step and the Trotter-Suzuki approximation error, accordingly. The corresponding fidelities can be found in Appendix pp3.

Figure 4

Comparison between two-qubit gates required by the Trotter-Suzuki circuit ($N_{\text{CX}}^{\mathrm{TS}}$) and by the variational ansatz ($N_{\text{CX}}^{\text{VFK}}$), given the same system size and Trotter steps. $N_{\text{CX}}^{\text{VFK}}$ corresponds to the minimal circuit required to make VFK converge and depends on ansatz type and depth. $N_{\text{CX}}^{\text{TS}}$ is fixed, given the system size and number of steps. We considered the ansatz in Eq. (13) and a simulation time $T_e=3.0$.

Figure 5

Rate function of the Loschmidt echo in the transverse-field Ising model measured using the VFK variational state. The plot shows a dynamical phase transition for an open chain initialized in the ${|0\rangle }^{\otimes n_s}$ state and evolved under the Hamiltonian in Eq. (12). The nonanalyticity appears when the number of spins is increased. We considered a number of auxiliary qubits $n_a=6$ for $n_s=2,4$ and $n_a=7$ for $n_s=6$.

Figure 6

Comparison between the mean infidelity of noisy VFK and Trotter-Suzuki circuits through a time evolution to $T_e=3.0$. Markers indicate the mean value, while the bars show the standard deviation over a time evolution. We fixed the error probability for single-qubit gates $p_1=2\times {10}^{-4}$. We considered a system with $n_s=2$ and a different number of auxiliary qubits $n_a$, and Trotter steps are varied accordingly. The horizontal line indicates equal mean fidelity between VFK and Trotter-Suzuki circuits. Vertical lines indicate the mean $p_2$ error probability for reference devices.

Figure 7

Mean infidelity over an entire time evolution as a function of the number of spins and auxiliary qubits. The plot compares the mean infidelity between the variational state obtained with VFK (after projecting to $|t\rangle$ subspace) or p-VQD and the exact state. For the VQE we used the ansatz in Eq. (13), while for p-VQD we use the alternating rotations ansatz presented in Ref. [29]. We considered a total simulation time $T_e=3.0$ and a number of optimization steps comparable between the two methods. The analysis is performed using a state-vector simulator and the Adam optimizer for both methods.

Figure 8

Final infidelities of the states obtained with the VFK algorithm ($1-F_{\text{VFK}}$) using a classical simulator with respect to the exact history state. The plot reports the infidelities corresponding to the calculations presented in Fig. 3.