Adaptive Variational Quantum Dynamics Simulations

We propose a general-purpose, self-adaptive approach to construct a variational wave-function ansatz for highly accurate quantum dynamics simulations based on McLachlan’s variational principle. The key idea is to dynamically expand the variational ansatz along the time-evolution path such that the “McLachlan distance”, which is a measure of the simulation accuracy, remains below a set threshold. We apply this adaptive variational quantum dynamics simulation (AVQDS) approach to the integrable Lieb-Schultz-Mattis spin chain and the nonintegrable mixed-field Ising model, where it captures both finite-rate and sudden post-quench dynamics with high fidelity. The AVQDS quantum circuits that prepare the time-evolved state are much shallower than those obtained from first-order Trotterization and contain up to 2 orders of magnitude fewer cnot gate operations. We envision that a wide range of dynamical simulations of quantum many-body systems on near-term quantum-computing devices will be made possible through the AVQDS framework.

Popular Summary

Accurate and efficient quantum dynamics simulations for interacting systems can guide experimental efforts to control magnetism, superconductivity, chemical reactions, and photo-energy conversion on ultrafast timescales. Quantum computing offers a fresh view on this challenging problem with hybrid quantum-classical variational methods being particularly promising, given the shallow-circuit requirements of current noisy quantum hardware. A major open question is how to design efficient variational ansätze that can faithfully represent the quantum state during time-evolution. In this work, we propose a flexible design strategy using a dynamic circuit ansatz that automatically adapts during time-evolution.

This adaptive variational approach is built on the McLachlan variational principle for real-time quantum dynamics. The key novel idea is to automatically generate and dynamically expand the variational ansatz by constraining the McLachlan distance to be along a given threshold along the complete time-evolution path. We apply this approach to study finite-rate and sudden quantum quench dynamics of integrable and nonintegrable spin models. We demonstrate quantitative agreement with exact dynamics with a circuit complexity that is two orders of magnitude smaller than for Trotter evolution, making it feasible to simulate intermediate size systems on current devices.

The proposed method opens up many opportunities to study quantum dynamics on current quantum processing units, including the simulation of chemical reaction pathways via molecular quench dynamics, and can be generalized to imaginary time for molecular ground-state preparation.

Figure 1

Schematic illustration of the AVQDS algorithm. The flow chart of AVQDS is plotted in (a). The details of the MMD module, which measures the McLachlan distance for a given variational wave function $\mathrm{\Psi }[\mathit{\theta }]$ and time-dependent Hamiltonian ${\hat{\mathcal{H}}}_t$, are shown in (b). The circuits to measure matrix $M$ and vector $V$ can be found in Ref. [27] (see also Fig. 2). Note that in the ansatz adaptive procedure, one only needs to measure the incremental elements in $M$ and $V$, which are added in a given step.

Figure 2

Quantum-circuit implementation of the AVQDS algorithm. The left column lists the unique terms to be evaluated in Eqs. (5) and (7) of VQDS, with the terms (a)–(c) highlighted in red also involved in the ansatz adaptive procedure in AVQDS. The middle column specifies the expressions when the wave-function ansatz takes the pseudo-Trotter form of Eq. (10) with ${\hat{U}}_{j,k}[\mathit{\theta }]=\prod _{\mu =j}^k{e}^{-i{\theta }_{\mu }{\hat{\mathcal{A}}}_{\mu }}$ and $\left|{\mathrm{\Psi }}_0\right.⟩=\left|\mathrm{\varnothing }\right.⟩\equiv {\otimes }_{j=0}^{N-1}\left|0\right.⟩$ for an $N$-qubit system. Two types of quantum circuits are adopted: a green block for the direct measurement circuit, and a blue block for a generalized Hadamard test circuit [27, 45]. The direct measurement circuit includes optional Hadamard gate $H$ or Hadamard-phase gate $H{S}^{†}$ when measuring $X$ or $Y$-Pauli strings present in ${\hat{\mathcal{A}}}_{\mu }$, ${\hat{\mathcal{A}}}_{\mu }^2$, $\hat{\mathcal{H}}$, and ${\hat{\mathcal{H}}}^2$. Accordingly, $\hat{U}[\mathit{\theta }]$ can be ${\hat{U}}_{0,\mu -1}$ or ${\hat{U}}_{0,N_{\mathit{\theta }}-1}$ as highlighted in the middle column. In the generalized Hadamard test, the expectation value $\mathrm{Re}\left[⟨\mathrm{\varnothing }|{\hat{U}}_A^{†}[\mathit{\theta }]{\hat{U}}_B^{†}[\mathit{\theta }]{\hat{\sigma }}_B{\hat{U}}_B[\mathit{\theta }]{\hat{\sigma }}_A{\hat{U}}_A[\mathit{\theta }]|\mathrm{\varnothing }⟩\right]$ is given by $2P_{\left|0\right.⟩}-1$, with $P_{\left|0\right.⟩}$ the probability that the ancillary qubit is in state $\left|0\right.⟩$. ${\hat{U}}_A[\mathit{\theta }]$ can be ${\hat{U}}_{0,\mu -1}$ or ${\hat{U}}_{0,\nu -1}[\mathit{\theta }]$ as dark red color encoded in the blue box of the middle column, with similar color encoding for the other unitaries. ${\hat{\sigma }}_B$ represents a Pauli string in ${\hat{\mathcal{A}}}_{\mu }$ or $\hat{\mathcal{H}}$, which is typically defined on one or two qubits for spin models. Therefore, the controlled-$\sigma$ gate represents a controlled-single qubit or controlled two-qubit gate. Alternatively, the generalized Hadamard test circuit can be replaced with direct measurement circuit [46, 47].

Figure 3

Phase diagram of LSM model as a function of ($h_z$, $\gamma$) and the quench protocol. Two ferromagnetic phases (${\mathrm{FM}}_x$, ${\mathrm{FM}}_y$) and a paramagnetic phase are present in the ground-state phase diagram of the LSM model in the thermodynamic limit ($N\to \mathrm{\infty }$). The phase boundaries and tricritical points are also shown. Below we present results of AVQDS simulations for the two vertical parameter paths indicated here, which quenches the system from the ${\mathrm{FM}}_x$ to the ${\mathrm{FM}}_y$ phase at finite speed, together with postquench dynamics.

Figure 4

Variations of instantaneous total energy, spin correlations, and number of two-qubit gates for quantum-quench simulations of the eight-site LSM chain model with open boundary conditions. The simulation is composed of a linear ramp with speed defined by $T=3$, and postramp dynamics for another period of $T$. The results for $h_z=-0.7$ sites are shown in the upper panels, and that for $h_z=1.6$ in the lower panels. The exact results are shown in symbols, and AVQDS data in solid curves. The evolution of the instantaneous energy from numerically exact calculations and AVQDS is shown in (a),(b), along with the adiabatic results plotted as gray dotted lines for reference. The spin correlation function in (c),(d) includes $xx$ and $yy$ components for a pair of the first and second site and a pair of the first and last site. Compared with linear circuit growth of Trotter dynamics simulations, AVQDS circuits are much shallower, with a sublinear circuit growth rate that quickly decreases, as plotted in (e),(f). The ansatz at $t=T$ is composed of 50 two-qubit Pauli rotation gates (100 cnots) for both cases, which slightly increases to 53 two-qubit gates at $t=2T$. A multiplying factor of 2 for cnot gates is included, since each two-qubit Pauli rotation gate can be compiled to two cnots along with single-qubit gates assuming full connectivity [38]. Inset in (f): variation of Hamiltonian parameter $\gamma$ as a function of $t$.

Figure 5

Saturated number of variational parameters ($N_{\mathit{\theta }}$) as a function of LSM model size $N$. The total number of variational parameters for AVQDS simulations after saturation is reached (blue squares) grows quadratically with the number of sites $N$. The corresponding number of variational parameters associated with two-qubit Pauli rotation gates (red dots) also shows quadratic scaling behavior $\propto N^2$, yet with a smaller prefactor.

Figure 6

Quench dynamics of eight-site mixed-field Ising model with periodic boundary conditions. The time-dependent Loschmidt echo, infidelity $1-f\equiv 1-{\left|⟨\mathrm{\Psi }[\mathit{\theta }(t)]|{\mathrm{\Psi }}_{\mathrm{exact}}(t)⟩\right|}^2$, and number of cnots $N_{\mathrm{cx}}$ are plotted in the upper panels for $h_z=0$ (integrable), and the lower panels for $h_z=0.5$ (nonintegrable). The Loschmidt echoes (return probabilities) in (a),(d) from AVQDS show excellent agreement with exact calculations. The infidelity plots in (b),(e) further quantify the AVQDS accuracy with a fidelity $\ge 99.5\mathrm{\%}$. As shown in (c),(f), Trotter circuits are about 2 orders of magnitude deeper than AVQDS circuits for similar simulation accuracies, shown in (b),(e). At $t=3$, $N_{\mathrm{cx}}$ reaches 134 and 210 for the integrable and nonintegrable models, respectively.

Figure 7

Number of cnots $N_{\mathrm{cx}}$ in the AVQDS circuits for the MFIM as a function of simulation time $t$ and system size $N$. (a) The AVQDS circuit depth for MFIM initially grows linearly with $t$, followed by a slowdown toward saturation. Times up to $t=12$ are considered for $N\le 8$, with shorter final times shown for $N=9,10$. Clearly for $N\le 8$, the circuit begins to saturate at a critical time $t_s<12$. (b) Vertical cuts of (a) at constant times for $1\le t\le 7$ and $t=12$. Inset: quadratic scaling of the saturated $N_{\mathrm{cx}}$ for AVQDS simulations of the integrable TFIM.