Exploring Entanglement and Optimization within the Hamiltonian Variational Ansatz

Quantum variational algorithms are one of the most promising applications of near-term quantum computers; however, recent studies have demonstrated that unless the variational quantum circuits are configured in a problem-specific manner, optimization of such circuits will most likely fail. In this paper, we focus on a special family of quantum circuits called the Hamiltonian variational ansatz (HVA), which its takes inspiration from the quantum approximate optimization algorithm and adiabatic quantum computation. Through the study of its entanglement spectrum and energy-gradient statistics, we find that the HVA exhibits favorable structural properties such as mild or entirely absent barren plateaus and a restricted state space that eases their optimization in comparison to the well-studied “hardware-efficient ansatz.” We also numerically observe that the optimization landscape of the HVA becomes almost trap free, i.e., there are no suboptimal minima, when the ansatz is overparametrized. We observe a size-dependent “computational phase transition” as the number of layers in the HVA circuit is increased where the optimization crosses over from a hard to an easy region in terms of the quality of the approximations and the speed of convergence to a good solution. In contrast to the analogous transitions observed in the learning of random unitaries, which occur at a number of layers that grows exponentially with the number of qubits, our variational-quantum-eigensolver experiments suggest that the threshold to achieve the overparametrization phenomenon scales at most polynomially in the number of qubits for the transverse-field Ising and XXZ models. Lastly, as a demonstration of its entangling power and effectiveness, we show that the HVA can find accurate approximations to the ground states of a modified Haldane-Shastry Hamiltonian on a ring, which has long-range interactions and has a power-law entanglement scaling.

Popular Summary

Quantum computers in the near term will consist of few qubits and can maintain quantum coherence for only a short period of time. Various quantum-classical hybrid algorithms have been proposed for these near-term quantum computers to perform nontrivial tasks that are intractable with existing classical computers. In a quantum-classical hybrid algorithm, a classical optimizer instructs a quantum computer to execute various quantum circuits, with the goal of finding good solutions to problems in quantum many-body physics and combinatorial optimization. A fundamental question is to understand when and why these hybrid algorithms succeed.

In the current work, we make important progress toward understanding one of the most promising quantum-classical hybrid algorithms. The idea of the algorithm is to use “time” as a variational parameter in different layers of the quantum circuit and optimize such “time” in different layers using the classical optimizer. This algorithm is called the Hamiltonian variational ansatz (HVA) because the Hamiltonian, the quantum-mechanical energy operator, is responsible for the “time” evolution. With the goal of understanding why the HVA is effective in practice, we probe the structure and dynamics of the HVA and uncover striking features about how entanglement evolves in the quantum circuits, as well as unveiling an interesting connection to deep-neural-network algorithms.

Our work represents an important step toward the realization of a promising quantum-classical algorithm for near-term quantum computers. In doing so, we find an important connection to neural-network research. This paves a novel pathway toward further explorations on unexpected connections between these two research areas.

Figure 1

The HVA quantum circuit for the TFIM with $p=1$. The first layer of Hadamard gates, represented by $H$, are used to construct the initial $|+⟩$ state. The ZZ gates are 2-local qubit rotation gates of the form $\mathrm{ZZ}=\exp \{i{\beta }_l/2{\sigma }_i^z{\sigma }_j^z\}$. The RX gates are single-qubit rotation gates $\mathrm{RX}=\exp \{i{\gamma }_l/2{\sigma }_i^x\}$.

Figure 2

The HVA quantum circuit for the XXZ model with $p=1$. Here, the X gates are given by $\mathrm{X}={\sigma }_i^x$. Together with a single Hadamard gate and a controlled-not on even links, we prepare the $|{\mathrm{\Psi }}^{-}⟩$ Bell state. The 2-local qubit rotations are all of the form $\mathrm{AA}=\exp \{-ix/2{\sigma }_i^a{\sigma }_j^a\}$, with $x=\theta ,\varphi ,\beta ,\gamma$ depending on whether the links are even or odd and ZZ or XX, YY [see Eq.  (7)].

Figure 3

The division of the full system into two subsystems $A$ (blue) and $B$ (red) on a 1D chain.

Figure 4

The average entanglement spectrum of HVA quantum states from layer $p=1$ (bottom line in purple) to $p=N$ (top line in yellow) over $5000$ random parameter initializations: (a) TFIM; (b) XXZ model. ${\xi }_k$ denotes the $k\mathrm{th}$ eigenvalue of $H_{\mathrm{ent}}$. The eigenvalues are arranged in descending order and cut off at ${\xi }_k=-30$. The black lines in the insets show how close the average entanglement entropy is to the Page entropy (purple dashed line) as a function of the increasing circuit depth. The lower blue dashed line in the inset indicates the entanglement entropy of the ground state. We see that the average HVA state is more entangled than the ground states of interest.

Figure 5

The change of the entanglement spectrum of the final layer during the optimization. Both figures are for a 16-qubit XXZ model with a depth $p=N/2$ circuit and the times are percentages of the total optimization time: (a) corresponds to a converged state of fidelity, whereas (b) corresponds to an approximately 70% fidelity state. (a) The identity-state initialization remains far away from the MP distribution at all times during the optimization and convergence to state with $>99.9\mathrm{\%}$ with the ground state. Since this initialization strategy starts with the identity circuit, we find the $t=0\mathrm{\%}$ state to be a product state, as indicated by the single eigenvalue. (b) The random initialization starts close to the MP distribution and converges to a local minimum with approximately 70% fidelity.

Figure 6

Overparametrization in the HVA: (a) TFIM; (b) XXZ model. Each line corresponds to the VQE optimization at depth $p$ that takes the most iterations to converge out of 100 random initializations. Both figures are for $N=12$ qubits. The rapid oscillations in (b) are artifacts of the adam optimizer and are less severe as the circuit depth increases. Due to our stopping criterion, we know that if the number of iterations is smaller than $3000$, then ${ϵ}_{\mathrm{res}}\le 1e-4$ and so the model does converge to a good ground-state approximation.

Figure 7

The mean iteration time to convergence as a function of depth: (a) TFIM; (b) XXZ model. The error bars indicate the standard deviation over $100$ different initializations. For both models, there is a clear cutoff where the number of iterations saturates. Note that if the number of iterations is smaller than $3000$, then we know that ${ϵ}_{\mathrm{res}}\le 1e-4$, indicating that the optimization has converged to a good ground-state approximation. We see that the error bars decrease systematically with depth. For both models, there is a critical $p$ after which all random initializations converge to a good ground-state approximation. Moreover, for depth $p=34$ and $p=52$ for the TFIM and the XXZ model, respectively, the number of iterations to find the ground state is of the order of $100$ iterations for every starting point.

Figure 8

The variance of the gradients of a single $Z_1{Z}_2$ term with respect to ${\theta }_0$ as a function of the number of qubits at initialization. The number of samples used per $N$ for each $p$ is 20. (a) For the TFIM, the gradient-variance decay is almost constant for all $N$. (b) The XXZ-model gradient variance is still exponential, although the effect is not as pronounced as for the RQCs of Ref. [5], where the $N=16$ variance is 2 orders of magnitude smaller.

Figure 9

The infidelities found after optimization for the MHS Hamiltonian. The circuit is initialized with an identity start. For the four-qubit case, we get close to machine precision and hence the fidelities are unstable.

Figure 10

The infidelities as a function of the order values $g$ and $\mathrm{\Delta }$ for (a) the TFIM and (b) the XXZ model, respectively. For these results, we use the identity initialization combined with a depth $p=N/2$ circuit. The dashed black line indicates the cutoff for $99.9\mathrm{\%}$ fidelity. (a) For the TFIM, we can obtain machine-precision results, except in the region $g>1.24$ for $N=16$. In this region, the optimization has not fully converged but is stopped after $15000$ iterations. We note that for increasing $N$, the time until convergence is polynomial in $N$ (not shown here), similar to what has been observed in Ref. [23]. Additionally, we observe a worsening of this scaling with increased $g$. (b) For the XXZ model, we are unable to consistently reach machine-precision fidelities. In addition, the fidelities that we find become worse as $N$ increases. However, except for a couple of outliers, we are able to obtain $>99.9\mathrm{\%}$ fidelities for $N\le 16$ or all order values.

Figure 11

The ratio of random initializations that converge to the ground state: (a) TFIM; (b) XXZ model. We consider a run converged if ${ϵ}_{\mathrm{res}}\le 1e-4$.

Figure 12

The dynamics of the entanglement entropy at each layer during optimization: (a),(c) TFIM, $p=4$; (b),(d) XXZ model, $p=8$. Each separate line indicates the entanglement entropy of the state in layer $l$. The gray dashed line denotes the maximum possible entanglement and the purple line gives the Page entropy. For all figures, the final state is a $>99.9\mathrm{\%}$ fidelity state. (a) The identity initialization for an eight-qubit TFIM with $g=1.0$ and $p=4$. (b) The same TFIM with a random-state initialization and overparametrization $p=8$. (c) Random-state initialization for an eight-qubit XXZ model with $\mathrm{\Delta }=1.0$ and $p=4$. (d) Typical XXZ-model dynamics for a random-state initialization and overparametrization $p=8$.

Figure 13

The scaling of the entanglement entropy of the converged state after $p/2$ and $p$ layers: (a) TFIM, $g=1.0$; (b) TFIM, $g=0.5$; (c) XXZ model, $\mathrm{\Delta }=1.0$. (a) For the TFIM at the critical point, the ground-state entanglement entropy has a logarithmic correction with increasing $N$. The entanglement halfway through the circuit is larger than in the final layer. (b) For a noncritical point, the ground-state entanglement entropy is constant but the entanglement entropy halfway through the circuit scales linearly with system size. (c) For the XXZ model, in addition to the logarithmic scaling of the entanglement entropy, the final layer entanglement is consistently higher than in the $p/2$ depth layer.