Variational Quantum Gibbs State Preparation with a Truncated Taylor Series

The preparation of a quantum Gibbs state is an essential part of quantum computation and has wide-ranging applications in various areas, including quantum simulation, quantum optimization, and quantum machine learning. In this paper, we propose variational hybrid quantum-classical algorithms for quantum Gibbs state preparation. We first utilize a truncated Taylor series to evaluate the free energy and choose the truncated free energy as the loss function. Our protocol then trains the parameterized quantum circuits to learn the desired quantum Gibbs state. Notably, this algorithm can be implemented on near-term quantum computers equipped with parameterized quantum circuits. By performing numerical experiments, we show that shallow parameterized circuits with only one additional qubit can be trained to prepare the Ising chain and spin chain Gibbs states with a fidelity higher than 95%. In particular, for the Ising chain model, we find that a simplified circuit ansatz with only one parameter and one additional qubit can be trained to realize a 99% fidelity in Gibbs state preparation at inverse temperatures larger than $2$.

Figure 1

Schematic representation of the variational quantum Gibbs state preparation with truncation order $2$. First, we prepare the Hamiltonian $H$ and inverse temperature $\beta$ and then send them into the hybrid optimization. Second, we choose an ansatz and employ it to evaluate the loss functions $L_1,L_2,L_3$ on quantum devices. Then we calculate the difference $\mathrm{\Delta }{\mathcal{F}}_2(\mathit{\theta })$ using $L_1,L_2,L_3$. Next, if the condition $\mathrm{\Delta }{\mathcal{F}}_2\le ϵ$ is not satisfied then we perform classical optimization to update parameters $\mathit{\theta }$ of the ansatz and return to the loss evaluation. Otherwise, we output the current parameters ${\mathit{\theta }}^{\ast }$, which could be used to prepare Gibbs state ${\rho }_G$ via $U(\mathit{\theta })$. Here in the quantum device, registers $A_2,B_2,A_3,B_3$ are used to evaluate $\mathrm{tr}({\rho }_{B_2}{\rho }_{B_3})$ and registers $A_4,B_4,\dots ,A_6,B_6$ are used to evaluate $\mathrm{tr}({\rho }_{B_4}{\rho }_{B_5}{\rho }_{B_6})$.

Figure 2

Quantum circuit for implementing the destructive swap test. In the circuit, two states $\rho$ and $\sigma$ are prepared at different registers. Then cnot and Hadamard gates are performed as shown. The state overlap can be estimated via postprocessing.

Figure 3

Quantum circuit for computing $\mathrm{tr}({\rho }^3)$. In the circuit, $U(\mathit{\theta })$ denotes the state preparation circuit and $H$ denotes the Hadamard gate. Four registers are used to prepare states by $U(\mathit{\theta })$, and one ancillary qubit is used to perform the controlled swap operator. The qubit reset occurs on the bottom two registers, where the break in the wire represents the reset operation. Notably, the state on the bottom two registers is first implemented with a circuit $U(\mathit{\theta })$ and controlled swap operator and then reset to state $|0⟩$. Again, $U(\mathit{\theta })$ and the controlled swap operator are performed on the bottom registers. Finally, $\mathrm{tr}({\rho }^3)$ can be obtained via postprocessing the measurement results.

Figure 4

Two ansatzes for the Ising chain model. These ansatzes are composed of two registers $A$ and $B$, where one ancillary qubit is set in $A$ and five qubits are set in $B$. Notably, the qubits in $B$ are performed with rotations $R_y(\theta )$ and cnot gates in (a), and only cnot gates in (b).

Figure 5

Fidelity curves for the Ising chain Gibbs state preparation with different $\beta$. In (a), we use the ansatz with six parameters [cf. Fig. 4]; in (b), we use the ansatz with only one parameter [cf. Fig. 4]. We can see that they have almost the same performance, which indicates that one parameter is enough for this task.

Figure 6

Semilog plot of the fidelity versus the Ising Hamiltonian length ($L$) with different $\beta$ for the Ising chain model. Here, ${\log }_2$ is the logarithm to the base 2. We can see that the fidelity increases exponentially with increasing $\beta$.

Figure 7

The ansatz for the $XY$ spin-$\frac{1}{2}$ chain model. This ansatz contains one ancilla qubit in register $A$ and five qubits in register $B$. Rotation gates $R_y(\theta )$ are first applied on all qubits. Then, a basic circuit module (highlighted in the dashed box) composed of cnot gates and rotation gates $R_y(\theta )$ is repeatedly applied. Here, $d$ means repeating $d$ times.

Figure 8

Fidelity curves for the $XY$ spin-$\frac{1}{2}$ chain Gibbs state preparation with different $\beta$. The results of the fidelity obtained with different $\beta$ are represented by colored lines. In (a)–(d), numerical experiments are performed using different ansatzes. In each ansatz, the basic circuit module (cf. Fig. 7) is repeated different times, i.e., $d$. Note that each ansatz has $(n_A+n_B)(d+1)=6(d+1)$ parameters. Here better performance is obtained with larger $d$.

Figure 9

Boxplot of the fidelity versus the truncation order $K$ for different $\beta$ for the $XY$ spin-$\frac{1}{2}$ chain model. Here the ansatz is similar to Fig. 7 while $n_A=n_B=3$. Each box consists of 30 runs with different parameter initializations.

Figure 10

Variational quantum Gibbs state preparation with truncation order $2$

Figure 11

Variational quantum Gibbs state preparation with truncation order $K$