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arXiv:2401.01962 · International Symposium on Lattice Field Theory (LATTICE)

Simulating Field Theories with Quantum Computers

Muhammad Asaduzzaman, Simon M. Catterall, Yannick Meurice, Goksu Can Toga

Published May 4, 2024• 0 citations
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Abstract

In this study, we investigate Trotter evolution in the Gross-Neveu and hyperbolic Ising models in two spacetime dimensions, using quantum computers. We identify different sources of errors prevalent in various quantum processing units and discuss challenges to scale up the size of the computation. We present benchmark results obtained on a variety of platforms and employ a range of error mitigation techniques to address coherent and incoherent noise. By comparing these mitigated outcomes with exact diagonalization results and density matrix renormalization group calculations, we assess the effectiveness of our approaches. Moreover, we demonstrate the implementation of an out-of-time-ordered correlator (OTOC) protocol using IBM's quantum computers.

DOI
10.22323/1.453.0228

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Benchmarking quantum computers for real-time evolution of a (1+1)(1+1)(1+1) field theory with error mitigation
Quantum computers open the possibility of performing real-time calculations for quantum field theory scattering processes. We propose to use an index averaging the absolute value of the difference between the accurately calculated Trotter evolution of site occupations and their actual measurements on NISQ machines. The average is over all the qubits for a certain number of Trotter steps. We use this metric to quantify the progress made in successive state-of-the-art machines and error-mitigation techniques. We illustrate the concept with the transverse Ising model in one spatial dimension with four sites using three of IBM's quantum computers (Almaden, Boeblingen, and Melbourne). We discuss the size of the Trotter steps needed to achieve physics goals. Using the proposed metric, we show that readout mitigation methods and Richardson extrapolations of mitigated measurements are very effective for specific numbers of Trotter steps of a chosen size. This specific choice can be applied to other machines and noise mitigation methods. On the other hand, a reliable algorithmic mitigation would require a significantly larger number of smaller Trotter steps.
We present a scalable quantum simulation framework for real-time dynamics of the multi-flavor Gross-Neveu model in 1+1 dimensions. Using superconducting quantum processors at utility scale, we develop a hardware-efficient Trotterization whose per-step circuit depth scales with fermion flavor number rather than total system size, enabling simulations beyond 100 qubits. A central contribution of this work is the Localized Diagonal Operator Approximation (LDOA), which systematically reduces the overhead associated with quartic interactions. We formulate diagonal unitary synthesis as a structured least-squares problem in phase space and obtain analytic solutions via the Moore-Penrose pseudoinverse. This formulation provides a principled and quantitatively controlled approximation: in the small Trotter-step regime, the unitary error is directly linked to the phase reconstruction error and vanishes asymptotically as the Trotter step size decreases. This establishes a clear mathematical foundation for the LDOA while significantly reducing two-qubit gate counts and circuit depth, and is broadly applicable to diagonal quantum operators with long-range structure, making it particularly well suited for quantum hardware with limited qubit connectivity. Using these techniques, we run large-scale simulations on IBM superconducting processors and study real-time observables, including density-density correlators. We benchmark against exact diagonalization and tensor network-based methods, finding strong agreement across system sizes. These results show that combining hardware-aware circuit design with rigorous approximations enables practical near-term simulation of interacting fermionic field theories and provides a scalable pathway toward more complex quantum field theory simulations.
We discuss the use of quantum simulation to study an NNN-flavor theory of interacting relativistic fermions in (1+11+11+1) dimensions on noisy intermediate-scale quantum (NISQ)-era machines. The case of two flavors is particularly interesting as it can be mapped to the Hubbard model. We derive the appropriate qubit Hamiltonians and associated quantum circuits. We compare classical simulation and density matrix renormalization group/time-evolving blocked decimation calculations with the results of quantum simulation on various platforms for N=2N=2N=2 and 4-flavors. We demonstrate that the four steps needed for calculations of real-time scattering can be implemented using current NISQ devices.
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We discuss real-time evolution for the quantum Ising model in one spatial dimension with Ns{N}_{s}Ns​ sites. In the limit where the nearest-neighbor interactions JJJ in the spatial directions are small, there is a simple physical picture where qubit states can be interpreted as approximate particle occupations. Using exact diagonalization, for initial states with one or two particles, we show that for small JJJ, discrete Bessel functions provide very accurate expressions for the evolution of the occupancies corresponding to initial states with one and two particles. Boundary conditions play an important role when the evolution time is long enough. We discuss a Trotter procedure to implement the evolution on existing quantum computers and discuss the error associated with the Trotter step size. We discuss the effects of gate and measurement errors on the evolution of one- and two-particle states using four and eight-qubits circuits approximately corresponding to existing or near-term quantum computers.
Quantum many-body scars are an intriguing dynamical regime in which quantum systems exhibit coherent dynamics and long-range correlations when prepared in certain initial states. We use this combination of coherence and many-body correlations to benchmark the performance of present-day quantum computing devices by using them to simulate the dynamics of an antiferromagnetic initial state in mixed-field Ising chains of up to 19 sites. In addition to calculating the dynamics of local observables, we also calculate the Loschmidt echo and a nontrivial unequal-time connected correlation function that witnesses long-range many-body correlations in the scarred dynamics. We find coherent dynamics to persist over up to 40 Trotter steps even in the presence of various sources of error. To obtain these results, we leverage a variety of error-mitigation techniques including noise tailoring, zero-noise extrapolation, dynamical decoupling, and physically motivated postselection of measurement results. Crucially, we also find that using pulse-level control to implement the Ising interaction yields a substantial improvement over the standard controlled-not-based compilation of this interaction. Our results demonstrate the power of error-mitigation techniques and pulse-level control to probe many-body coherence and correlation effects on present-day quantum hardware.