Universal imaginary-time critical dynamics on a quantum computer

Quantum computers promise a highly efficient approach to investigate quantum phase transitions, which describe abrupt changes between different ground states of many-body systems. At quantum critical points, the divergent correlation length and entanglement entropy render the ground state preparation difficult. In this work, we explore the imaginary-time evolution for probing the universal critical behavior as the universal information of the ground state can be extracted in the early time relaxation process. We propose a systematic and scalable scheme to probe the universal behaviors via imaginary-time critical dynamics on quantum computers and demonstrate the validness of our approach by both numerical simulation and quantum hardware experiments. With the full form of the universal scaling function in terms of imaginary time, system size, and circuit depth, we successfully probe the universality by scaling analysis of the critical dynamics at an early time and with shallower quantum circuit depth. Equipped with quantum error mitigation, we also confirm the expected scaling behavior from experimental results on a superconducting quantum processor which stands as the first experimental demonstration of universal imaginary-time quantum critical dynamics.

Figure 1

The variational circuit ansatz for simulating imaginary time evolved state under 1D TFIM Hamiltonian.

Figure 2

(a) Unscaled and (b) finite-size scaled data for variational imaginary-time dynamics using variational circuits of different size and depth $D=N/2+1$ starting from $M_0=1$ initial state. The scaling regime begins when $\tau \gtrsim 0.1N$. The critical exponent is estimated as $\beta /\nu \approx 0.124\pm 0.03$ for the best collapse of rescaled data.

Figure 3

Imaginary-time dynamics data with several $\tau /N$ and different $D$ (a) unscaled and (b) rescaled according to the short-time finite-depth scaling. Points in the same color should fall into the same curve with the rescaled axis according to the finite depth exponent. (c) The data collapse quality for different guesses of finite-depth exponent $\alpha$ and critical exponent $\beta /\nu$. The best fit is estimated at $\alpha =1.19\pm 0.03$ and $\beta /\nu =0.126\pm 0.04$. The grey contour indicates the region where the data collapse quality is no worse than $5\%$ compared to the optimal estimation.

Figure 4

Experimental demonstration of the universal short-time quantum critical dynamics in imaginary time: the experimental data are collected from a programmable superconducting quantum processor. In (a), (b), and (c), we show experimental results with only readout error mitigation (raw) and with extended Clifford data regression mitigation (mit) for $N=6,7,8$ qubits and $D=3,3,4$ variational circuit blocks, respectively. The exact results (exact) and numerical simulation results (sim) for the variational imaginary time evolution algorithm are also shown as lines for guidance. In (d), we apply data collapse on error mitigated data points to extract universal critical scaling from short-time dynamics, and the results coincide well with the theory prediction.

Figure 5

The absolute error in terms of $M^2$ for $N=10$ system variational quantum dynamics simulation of different circuit ansatz depth quenched from $|{\uparrow }^n\rangle$. The error is relatively small and saturates when the circuit depth exceeds half of the system size $N$.

Figure 6

Imaginary-time correlator $A$ (a) unscaled and (b) finite-size scaled. The imaginary-time dynamics related critical exponent $\theta \approx 0.373$.

Figure 7

(a) $M^2$ of converged quantum state after VQE optimization on quantum computers of different sizes $N$ and depths $D$. (b) Data collapse for the converged state on quantum computers of different sizes and depths. We estimate the critical exponent for the depth scaling as $\alpha =1.28\pm 0.06$.

Figure 8

The alternative variational circuit ansatz for VQE to demonstrate the universal finite-depth scaling. Each block is composed of only one layer of Rx gates and one layer of Rzz gates in the brickwall layout.

Figure 9

(a) $M^2$ of converged quantum state after VQE optimization on quantum computers of different sizes $N$ and depth $D$ with alternative ansatz. (b) Data collapse for the converged state on quantum computers of different sizes and depths. The critical exponent for the depth scaling is $\alpha =1.28\pm 0.07$, consistent with the circuit ansatz in the main text.

Figure 10

Data collapse quality grid search with variational imaginary time dynamics data of size $N=8,10,12$, circuit depth $D=2,3,4,5$, and time $\tau /N=0.1,0.2,0.3,0.4$. The optimal exponents estimate is $\alpha =1.06\pm 0.03$ and $\beta /\nu =0.14\pm 0.05$. The grey contour indicates the boundary where the cost function is $10\%$ worse than the optimal estimate.

Figure 11

The qubit layout and coupling map for the 20-qubit device. The qubits with blue circles are used in the experiment of this work.

Figure 12

(a) Experimental results from quantum computers via error mitigation. (b) The data collapse. All data points are expected to fall into the same curve in the ideal case.