The simulation of adiabatic evolution has deep connections with Adiabatic Quantum Computation, the Quantum Approximate Optimization Algorithm and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time $T$ . The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed as the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: 1. the fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. 2. because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. 3. as the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.The simulation of adiabatic evolution has deep connections with Adiabatic Quantum Computation, the Quantum Approximate Optimization Algorithm and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time $T$. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed as the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: 1. the fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. 2. because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. 3. as the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.

Jul 13, 20211 citation

Realizations of standard quantum computational circuits by adiabatic evolution

V. E. Zobov, Alexander S. Ermilov

Theoretical and Mathematical PhysicsMar 1, 20072 citations

Success of digital adiabatic simulation with large Trotter step

Changhao Yi

The simulation of adiabatic evolution has deep connections with adiabatic quantum computation, the quantum approximate optimization algorithm, and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time $T$ . The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: (1) The fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. (2) Because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. (3) As the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.The simulation of adiabatic evolution has deep connections with adiabatic quantum computation, the quantum approximate optimization algorithm, and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time $T$. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: (1) The fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. (2) Because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. (3) As the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.

arXiv:2107.06404Physical Review ANov 8, 202115 citations

Simulating quantum circuits by adiabatic computation: Improved spectral gap bounds

Shane Dooley, Graham Kells, Hosho Katsura +1 more

Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$ . In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L\ensuremath{\gg}1$ is ${min}_{s}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\gtrsim}{\ensuremath{\pi}}^{2}/[8{(L+1)}^{2}]$, an improvement over previous bounds. Our first method is a direct approach based on an eigenstate ansatz, while the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L\ensuremath{\gg}1$ is ${min}_{s}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\gtrsim}{\ensuremath{\pi}}^{2}/[8{(L+1)}^{2}]$, an improvement over previous bounds. Our first method is a direct approach based on an eigenstate ansatz, while the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.

arXiv:1906.05233Physical Review AApr 1, 20205 citations

T

. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed as the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: 1. the fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. 2. because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. 3. as the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.

T

. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: (1) The fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. (2) Because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. (3) As the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.

L

. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L\ensuremath{\gg}1$ is ${min}_{s}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\gtrsim}{\ensuremath{\pi}}^{2}/[8{(L+1)}^{2}]$, an improvement over previous bounds. Our first method is a direct approach based on an eigenstate ansatz, while the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.