Lectures on Quantum Field Theory on a Quantum Computer

TL;DR

The notes present a pedagogical framework for simulating quantum field theory with quantum computers, starting from the anharmonic oscillator and moving through lattice φ^4 theory, Ising field theory, and bosonized Schwinger/Thirring models in 1+1D. They couple physics background with quantum-computing essentials, introducing gates, Hamiltonian simulation, and Pauli-string exponentials, while benchmarking with tensor networks (MPS) and adiabatic state preparation. A central theme is building and validating quantum pipelines for ground-state preparation, scattering, and resonance phenomena, including both elastic and inelastic processes, with explicit discussions of Wigner time delays, finite-volume spectra, and phase shifts. The material emphasizes practical computational strategies (ASP, Trotterization, LCU, qubitization) and connects lattice models to continuum QFT via bosonization and dimensional reductions, underscoring the potential for quantum hardware to illuminate real-time dynamics that challenge classical methods.

Abstract

The lecture notes cover the basics of quantum computing methods for quantum field theory applications. No detailed knowledge of either quantum computing or quantum field theory is assumed and we have attempted to keep the material at a pedagogical level. We review the anharmonic oscillator, using which we develop a hands-on treatment of certain interesting QFTs in $1+1D$: $φ^4$ theory, Ising field theory, and the Schwinger model. We review quantum computing essentials as well as tensor network techniques. The latter form an essential part for quantum computing benchmarking. Some error modelling on QISKIT is also done in the hope of anticipating runs on NISQ devices. These lecture notes are the expanded version of a one semester course taught by AS during August-November 2025 at the Indian Institute of Science and TA-ed by UB. The programs written for this course are available in a GitHub repository.

Figures (27)

• Figure 1.4.1: Google Gemini 3 pro's summary of these lectures. Courtesy: Chayanka Kakati.
• Figure 2.2.1: The anharmonic potential.
• Figure 2.2.2: Level spacings $\Delta E_n$ for $H=\tfrac{p^2}{2}+\tfrac{x^2}{2}+\lambda x^4$.
• Figure 2.3.1: Level spacings $\Delta E_n$ for $H=\frac{p^2}{2}-\frac{x^2}{2}+\lambda x^4$.
• Figure 2.5.1: The symmetric quartic double well. The instanton connects $-a$ to $+a$ in Euclidean time.