Hamiltonian Simulation Techniques

Amey Meher, Yuan Liu, Huiyang Zhou

Quantum computing has emerged as a revolutionary paradigm with the potential to solve computationally intractable problems. However, the practical realization of quantum al-gorithms faces significant challenges due to noise and errors inherent in quantum systems. Hamiltonian simulations serve as a cornerstone in quantum computing, providing insights into quantum dynamics and enabling the study of quantum algorithms. This paper studies error mitigation in analog-based Hamiltonian simulations as they have native support for generating Hamiltonians as opposed to digital-based simulations, which results in shorter pulse schedules to simulate the Hamiltonians. We further aim to enhance the accuracy and reliability of these simulations through error mitigation techniques. This research explores several error mitigation techniques tailored for analog-based Hamiltonian simulations. Among them, Zero-Noise Extrapolation (ZNE) mitigates errors by extrapolating simulation results to the ideal noise-free scenario. Building upon ZNE, Folding-Free ZNE (FFZNE) introduces novel use of the infinite error point to address the challenges posed by folding techniques. Additionally, Reliability-Based ZNE (RBZNE) leverages a mix of both techniques, utilizing the folding method and the infinite error point. Through empirical studies and rigorous analysis, this paper evaluates the efficacy and limitations of these error mitigation techniques in analog-based Hamiltonian simulations on both noise simulator and real quantum hardware. The experimental results shed light on the comparative performance of ZNE, FFZNE, and RBZNE, understanding their strengths and weak-nesses in mitigating errors across different quantum systems and noise conditions. By providing the understanding of these techniques, this research contributes to the advancement of error mitigation strategies in quantum computing, paving the way for more reliable and accurate quantum simulations.

Daniel Ayomide Olanrewaju

Chi Zhang

Christopher T Kang, Micheline B. Soley, Eleanor Crane +2 more

Abstract Circuit quantum electrodynamics enables the combined use of qubits and oscillator modes. Despite a variety of available gate sets, many hybrid qubit-boson (i.e. qubit-oscillator) operations are realizable only through optimal control theory, which is oftentimes intractable and uninterpretable. We introduce an analytic approach with rigorously proven error bounds for realizing specific classes of operations via two matrix product formulas commonly used in Hamiltonian simulation, the Lie–Trotter–Suzuki and Baker–Campbell–Hausdorff product formulas. We show how this technique can be used to realize a number of operations of interest, including polynomials of annihilation and creation operators, namely ( a ) p ( a † ) q for integer p , q . We show examples of this paradigm including obtaining universal control within a subspace of the entire Fock space of an oscillator, state preparation of a fixed photon number in the cavity, simulation of the Jaynes–Cummings Hamiltonian, and simulation of the Hong-Ou-Mandel effect. This work demonstrates how techniques from Hamiltonian simulation can be applied to better control hybrid qubit-boson devices.

Andrew M. Childs, Nathan Wiebe

We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation algorithms based on product formulas and, most notably, scales better with the simulation error than any known Hamiltonian simulation technique. Our main tool is a general method to nearly deterministically implement linear combinations of nearby unitary operations, which we show is optimal among a large class of methods.