Snapshot-QAOA: Extending QAOA to Quantum Hamiltonian Simulation

Abstract

We present Snapshot-QAOA, a variation of the Quantum Approximate Optimization Algorithm (QAOA) that finds approximate minimum energy eigenstates of a large set of quantum Hamiltonians (i.e. Hamiltonians with non-diagonal terms). Traditionally, QAOA targets the task of approximately solving combinatorial optimization problems; Snapshot-QAOA enables a significant expansion of the use case space for QAOA to more general quantum Hamiltonians, where the goal is to approximate the ground-state. Such ground-state finding is a common challenge in quantum chemistry and material science applications. Snapshot-QAOA retains desirable variational-algorithm qualities of QAOA, in particular small parameter count and relatively shallow circuit depth. Snapshot-QAOA is thus a better trainable alternative to the NISQ-era Variational Quantum Eigensolver (VQE) algorithm, while retaining a significant circuit-depth advantage over the QEC-era Quantum Phase Estimation (QPE) algorithm. Our fundamental approach is inspired by the idea of Trotterization of a continuous-time linear adiabatic anneal schedule, which for sufficiently large QAOA depth gives very good performance. Snapshot-QAOA restricts the QAOA evolution to not phasing out the mixing Hamiltonian completely at the end of the evolution, instead evolving only a partial typical linear QAOA schedule, thus creating a type of snapshot of the typical QAOA evolution. As a test case, we simulate Snapshot-QAOA on a 16 qubit J1-J2 frustrated square transverse field Ising model with periodic boundary conditions.

Figures (15)

• Figure 1: Venn diagram depicting the various properties of Snapshot-QAOA that other alternative algorithms do not have. Regarding the $(*)$ in the diagram above, we do acknowledge that QPE and VQE are able to support a broader class of quantum Hamiltonians (e.g. any sum of Pauli strings) compared to Snapshot-QAOA. We also remark that, per layer, QAOA typically has much fewer variational parameters compared to VQE; the former typically has $O(1)$ parameters per layer whereas the latter typically has $\Omega(n)$ parameters per layer where $n$ is the number of qubits.
• Figure 2: A visualization connecting the $\vec{\gamma}$ and $\vec{\beta}$ parameters of Snapshot-QAOA to partial quantum annealing. The $\vec{\gamma}$ and $\vec{\beta}$ parameters of Snapshot-QAOA follow a linear ramp for varying $\hat{c}_1$ values ($\hat{c}_1=0.3,0.7,1.0$). The last subplot, with $\hat{c}_1=1.0$, corresponds to TQA QA_initialization_of_QAOA. Each subplot above uses $p=20$ layers with $T=1$. The parameters are scaled by $\Delta t = \tau / p = \hat{c_1} T / p = \hat{c_1}/20$ so that the values along the vertical axis range lie in the interval $[0,1]$. The horizontal axis represents the fraction of the total anneal schedule (from $t=0$ to $t=T$) that the parameter corresponds to. At each $k=1,2,\dots,p$, the scaled parameters $\beta_k/\Delta t$ and $\gamma_k/\Delta t$ add up to 1, so the vertical axis can be interpreted as the "relative influence" that each of the two alternating unitaries has at each layer of Snapshot-QAOA.
• Figure 3: The underlying graph $G$ for the sub-Hamiltonian $H_1$ of the $4\times 4$ frustrated TFIM $H^\star$ that defines toroidal periodic boundary conditions of the lattice. The black lines denote nearest-neighbor connections with weight $J_1=1$ and the red lines denote next-nearest-neighbor connections with weight $J_2$. Each solid line corresponds to exactly one edge in $G$. The periodic boundary conditions are represented via dashed lines; each dashed line has a corresponding dashed line that corresponds to the same underlying edge in the graph. The bolded red lines are one such example of a pair of dashed lines that correspond to the same underlying edge.
• Figure 4: Representative $[0, p]$ T parameter search space for Snapshot-QAOA. These are several examples of Hamiltonian energy (y-axis) vs. the total-anneal-time parameter $T$ (x-axis), for different values of $p$, $J_2$, and $B_x$. Each sub-plot is showing the Hamiltonian expectation value for $1000$ linearly spaced T values between $0$ and $p$.
• Figure 5: Snapshot-QAOA total-anneal-time $T$ parameter search space over the full time period in the interval from $[0, \rho]$ (defined by Eq. \ref{['eqn:period_formula']}). Hamiltonian energy (y-axis) vs. the total-anneal-time parameter $T$ (x-axis), for different values of $p$, $J_2$, and $B_x$. Each plot is showing the Hamiltonian expectation value for $1000$ linearly spaced T values between $0$ and the minimum T-period $\rho$. The x-axis index where $T=p$ is marked by a vertical green line. Notice that there is a clear mirror symmetry in each $T$ search space.

Theorems & Definitions (2)

• Remark 1
• Remark 2