The unitary evolution under H B is generated by applying a global rotation around the y axis of the Bloch sphere.įig. The power-law exponent α ∼ 1 and the interaction strengths vary in the range J 0 / 2 π = (0.3–0.57) kHz, depending on the system size and the experimental realization (see SI Appendix for details). This gives rise to effective long-range Ising couplings that fall off approximately as J i j ≈ J 0 / | i − j | α ( 24). The unitary evolution under H A is realized by generating spin–spin interactions through spin-dependent optical dipole forces implemented by an applied laser field. Both systems are based on a linear radiofrequency Paul trap, where we store chains of up to N = 40 ions and initialize the qubits in the ground state of H B, namely, the product state ↑ ↑ ⋯ ↑ y ≡ + ⊗ N = ψ 0, where ↑ y ≡ ( ↑ z + i ↓ z ) / 2, and B is assumed to be negative. In this work, depending on the number of qubits and measurements required, we employ two different quantum-simulation apparatus to run the QAOA, which will herein be referred to as system 1 ( 22) and system 2 ( 23) ( SI Appendix). In order to implement the quantum-optimization algorithm, each spin in the chain is encoded in the 2 S 1 / 2 F = 0, m F = 0 ≡ ↓ z and F = 1, m F = 0 ≡ ↑ z hyperfine “clock” states of a 171Yb + ion ( SI Appendix). The state obtained after p layers of the QAOA is: 1) that are applied to a known initial state ψ 0. 1) under the noncommuting Hamiltonians H A and H B (defined under Eq.
The realization of the QAOA entails a series of unitary quantum evolutions ( Fig. One of the goals of this work is to find an approximation of the ground-state energy both at the critical point ( B / J 0 ) c, where J 0 is the average nearest-neighbor coupling, and in the case of B = 0, optimizing the QAOA output for the classical Hamiltonian H A. 1) exhibits a quantum-phase transition for antiferromagnetic interactions with power-law decay. It is well known ( 21) that the Hamiltonian ( Eq. Here, we set the reduced Planck’s constant ℏ = 1 σ i γ ( γ = x, y, z) is the Pauli matrix acting on the i th spin along the γ direction of the Bloch sphere J i j > 0 is the Ising coupling between spins i and j, which, in our case, falls off as a power law in the distance between the spins and B denotes the transverse magnetic field. 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.
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 benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. 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 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. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator.
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.