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Snapshot-QAOA: Extending QAOA to Quantum Hamiltonian Simulation

Reuben Tate, Quinn Langfitt, Elijah Pelofske, Ammar A. Kirmani, Andreas Bärtschi, John K. Golden, Stephan Eidenbenz

Published Mar 22, 2026• 0 citations
Publisher

Abstract

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. 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. Snapshot-QAOA retains the desirable variational-algorithm qualities of QAOA such as a small parameter count and relatively shallow circuit depth. Snapshot-QAOA is thus a better trainable alternative to the (Near Intermediate-Scale Quantum) NISQ-era Variational Quantum Eigensolver (VQE) algorithm, while retaining a significant circuit-depth advantage over the (Quantum Error Corrected) 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. 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. In this way, the cost of the quantum Hamiltonian is encoded by the phase separator to address the diagonal terms in the problem Hamiltonian and by the mixer to address the non-diagonal terms in the problem Hamiltonian. We focus on QAOA with the transverse field mixer, and thus simulations of quantum Hamiltonians that contain single-site Pauli-X terms. By measuring the expectation value of the system in both X and Z-basis, we can estimate the ground-state energy of transverse field quantum Hamiltonians. The accuracy of ground-state energy finding with snapshot-QAOA is evaluated using extensive numerical simulations on a 16 qubit J1-J2 frustrated transverse field Ising model.

DOI
10.21203/rs.3.rs-9171925/v1

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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. 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. Snapshot-QAOA retains the desirable variational-algorithm qualities of QAOA such as a small parameter count and relatively shallow circuit depth. Snapshot-QAOA is thus a better trainable alternative to the (Near Intermediate-Scale Quantum) NISQ-era Variational Quantum Eigensolver (VQE) algorithm, while retaining a significant circuit-depth advantage over the (Quantum Error Corrected) 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. 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. In this way, the cost of the quantum Hamiltonian is encoded by the phase separator to address the diagonal terms in the problem Hamiltonian and by the mixer to address the non-diagonal terms in the problem Hamiltonian. We focus on QAOA with the transverse field mixer, and thus simulations of quantum Hamiltonians that contain single-site Pauli-X terms. By measuring the expectation value of the system in both X and Z-basis, we can estimate the ground-state energy of transverse field quantum Hamiltonians. The accuracy of ground-state energy finding with snapshot-QAOA is evaluated using extensive numerical simulations on a 16 qubit J1-J2 frustrated transverse field Ising model.
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 (Near Intermediate-Scale Quantum) NISQ-era Variational Quantum Eigensolver (VQE) algorithm, while retaining a significant circuit-depth advantage over the (Quantum Error Corrected) 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 particular 16 qubit 2D J1-J2 frustrated square transverse field Ising model with periodic boundary conditions.
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.
Quantum optimization allows for up to exponential quantum speedups for specific, possibly industrially relevant problems. As the key algorithm in this field, we motivate and discuss the Quantum Approximate Optimization Algorithm (QAOA), which can be understood as a slightly generalized version of Quantum Annealing for gate-based quantum computers. We delve into the quantum circuit implementation of the QAOA, including Hamiltonian simulation techniques for higher-order Ising models, and discuss parameter training using the parameter shift rule. An example implementation with Pennylane source code demonstrates practical application for the Maximum Cut problem. Further, we show how constraints can be incorporated into the QAOA using Grover mixers, allowing to restrict the search space to strictly valid solutions for specific problems. Finally, we outline the Variational Quantum Eigensolver (VQE) as a generalization of the QAOA, highlighting its potential in the NISQ era and addressing challenges such as barren plateaus and ansatz design.
Determining Hamiltonian ground states and energies is a challenging task with many possible approaches on quantum computers. While variational quantum eigensolvers are popular approaches for near term hardware, adiabatic state preparation is an alternative that does not require noisy optimization of parameters. Beyond adiabatic schedules, QAOA is an important method for optimization problems. In this work we modify QAOA to apply to finding ground states of molecules and empirically evaluate the modified algorithm on several molecules. This modification applies physical insights used in classical approximations to construct suitable QAOA operators and initial state. We find robust qualitative behavior for QAOA as a function of the number of steps and size of the parameters, and demonstrate this behavior also occurs in standard QAOA applied to combinatorial search. To this end we introduce QAOA phase diagrams that capture its performance and properties in various limits. In particular we show a region in which non-adiabatic schedules perform better than the adiabatic limit while employing lower quantum circuit depth. We further provide evidence our results and insights also apply to QAOA applications beyond chemistry.
Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly solving exponentially hard problems, such as optimization and satisfiability. Here we report the first implementation of a shallow-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator to estimate the ground state energy of the transverse field Ising model with tunable long-range interactions. First, we exhaustively search the variational control parameters to approximate the ground state energy with up to 40 trapped-ion qubits. We then interface the quantum simulator with a classical algorithm to more efficiently find the optimal set of parameters that minimizes the resulting energy of the system. We finally sample from the full probability distribution of the QAOA output with single-shot and efficient measurements of every qubit.
Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.