Quantum Gaussian filter for exploring ground-state properties

Filter methods realize a projection from a superposed quantum state onto a target state, which can be efficient if two states have sufficient overlap. Here we propose a quantum Gaussian filter (QGF) with the filter operator being a Gaussian function of the system Hamiltonian. A hybrid quantum-classical algorithm feasible on near-term quantum computers is developed, which implements the quantum Gaussian filter as a linear combination of Hamiltonian evolution at various times. Remarkably, the linear combination coefficients are determined classically and can be optimized in the postprocessing procedure. Compared to the existing filter algorithms whose coefficients are given in advance, our method is more flexible in practice under given quantum resources with the help of postprocessing on classical computers. We demonstrate the quantum Gaussian filter algorithm for the quantum Ising model with numeral simulations under noises. We also propose an alternative full quantum approach that implements a QGF with an ancillary continuous-variable mode.

Figure 1

The QGF algorithm starts from preparing an initial quantum state and approximates the Gaussian filter operator as a linear combination of unitary operators. The next step is to evolve the initial state and measure the overlap matrices on a quantum computer. The estimated energy is a classical summation of the overlap information with corresponding weight. Finally, the minimum result is obtained by scanning the parameter space on a classical computer.

Figure 2

Demonstration of the ground-state energy estimation by the QGF algorithm for a transverse-field Ising model with $N=8$ qubits. The discrete parameters are set as ${\mathrm{\Delta }}_y=0.16$ and $M_y=50$. (a) Energy between the exact ground-state energy and the estimated energy for different negative shift energy $\mu$. Red, green, and blue lines correspond to the cases $1/{\sigma }^2=1$, 1.5, and 2, respectively. (b) Energy difference as a function of $1/{\sigma }^2$. Red, green, and blue lines correspond to the cases $\mu =-2, -3$, and $-4$, respectively.

Figure 3

Energy difference between estimated ground-state energy and the exact ground-state energy as a function of maximum evolution time. The red, green, and blue lines correspond to the cases of $N=6$, 8, and 10, respectively.

Figure 4

Numerical results for solving the approximate ground-state energy of the transverse-field Ising model with $N=4$ at different maximum evolution times in a noisy environment. For both figures, red and green lines correspond to the noiseless and noisy cases, respectively; the blue line is the result under noise, but a noise mitigation strategy is used. (a) Solution of the approximate ground-state energy using a QGF for bit-flip noise. (b) Solution of the approximate ground-state energy using a QGF for phase-flip noise.

Figure 5

Fidelity between the initial state and the ground state as a function of the number of qubits $N$. The red line represents the average result of 50 randomly generated initial states. The blue line corresponds to the initial state well prepared with prior knowledge of the system Hamiltonian. (a) Preparation of the initial state for solving the transverse-field Ising model with $J/g=2$. (b) Preparation of the initial state for solving the transverse-field Ising model with $g/J=2$.

Figure 6

Solution of an $N=4$ qubit transverse-field Ising model by iterative continuous-variable-assisted strategy. The red line is the energy difference between the estimated energy and the exact ground-state energy, corresponding to the left axis. The green line is the required measurement times corresponding to the right axis.

Figure 7

Numerical demonstration of the approximate Gaussian functions for different settings. The slice thickness is set to a sufficient low scale $\mathrm{\Delta }y=0.16$. The vertical dashed lines correspond to $\lambda ={\varphi }_m{\sigma }^2/2$. (a) Approximate Gaussian function for a fixed truncation ${\varphi }_m=8$ and three different inverse variances $1/{\sigma }^2=1$, 2, and 4. (b) Approximate Gaussian function for a fixed inverse variance $1/{\sigma }^2=2$ and three different truncations ${\varphi }_m=4$, 8, and 12.

Figure 8

Numerical comparison between the cosine filter and Gaussian filter. (a) Gaussian function of the eigenvalue approximated by the Gaussian filter and cosine filter, and the exact Gaussian function. We set the variance of the Gaussian ${\sigma }^2=\frac{1}{2}$ and the maximum evolution ${\varphi }_m=8$. The blue solid line, red solid line, and light blue dashed line correspond to the Gaussian filter, cosine filter, and exact Gaussian function, respectively. Both of the filters can approximate well the Gaussian function $e^{-2{\lambda }^2}$ for $\lambda \in [0,2]$ with a maximum evolution time ${\varphi }_m=2h_m/{\sigma }^2=8$. (b) Numerical result for solving the ground state of a four-qubit transverse-field Ising model for bit noise with $p_b=0.0001$ after each gate. The red and blue lines are the results solved by the cosine filter and Gaussian filter in the noisy environment, respectively. The green and light blue lines are the results solved by the cosine filter and Gaussian filter with an error mitigation strategy, respectively.