Quantum simulation of thermal field theories

The gist: This work presents studies on thermal field theories using quantum algorithms, specifically exploring fermion and scalar fields in 1+1 dimensions via quantum simulators to understand thermal fixed points.

Quantum Simulation Approach

The study utilizes quantum algorithms to prepare thermal states and evaluate physical observables in fermionic [2] and scalar [2] thermal field theories The general procedure involves representing fields using qubits on general digital quantum computers alongside the quantum algorithms employed to evaluate thermal properties of generic quantum field theories For fermionic fields in 1+1 dimensions, the continuum theory is approximated with a discrete theory similar to lattice QCD but with continuous time The general procedure to represent fields using qubits involves first approximating the continuum theory with a discrete theory that’s similar to lattice QCD but with continuous time, so that, in the case of free fermions, the Hamiltionian adopts the form. Secondly, the fields are mapped into qubits.

Fermion Field Simulation

For free fermion fields in 1+1 D, the free Hamiltonian is written in terms of creation/annihilation operators The mapping onto qubits employs the Jordan-Wigner transformation where a n† = σn X − iσn Y 2 ∏ σi Z n−1 i=0, a n = σn X + iσn Y 2 ∏ σi Z n−1 i=0. The simulation is carried out in coordinate space, and the results for the thermal distribution of free fields obtained by evolving the free fermion Hamiltonian with the QITE algorithm show strong agreement with analytical lines, which are Fermi-Dirac distributions.

Interacting Fermion Fields

When investigating interacting fermionic systems coupled through four-fermion interactions, identical Majorana fields effectively vanish due to the Pauli exclusion principle This necessitates introducing another Majorana field, psiB = 1/√2 (1 1 i −i) (b b†), assumed to be homogeneous in space. The model introduces two phase-space distributions, f p0 and f p1, related to the partition functions Z β0 and Z β1. In the simulation, N = 4 qubits are used for psi and one qubit for psiB using the QITE algorithm. The simulation results using a n† and an in position space agree with the analytical calculations using the partition functions.

Scalar Field Theory Simulation

For scalar field theory, the Lagrangian density for the phi4 theory in (d + 1) dimensions takes a specific form. The Hamiltonian is discretized into Hlat, and then reduced to H̅ = ∑ [1/2 Π̅n2 + 1/2 m̅2Φ̅n2 + 1/2 (Φ̅n+1 − Φ̅n)2 + λ4!Φ̅n4] in 1+1 dimensions. To solve the problem numerically, the number of bosons is truncated by a cutoff number Nb, and continuous field operators are digitized into discrete values using a finite Hilbert space Hn. The representation on the qubit involves using nQ qubits on each register to represent N lattice points. The numerical results obtained in coordinate space approach the analytical Bose-Einstein distribution with increasing nQ.

Conclusion and Outlook

The work develops the QFT for fermionic and scalar fields in 1+1 D using qubits and investigates various thermal properties at finite temperature through quantum simulation algorithms. The numerical results are compared with analytical calculations and exact diagonalisation methods, showing strong agreement. This highlights the potential of quantum computing for field theory computations Our work represents a significant initial step toward understanding thermal fixed points via quantum simulation. The qubit-based representation effectively yields thermal states towards which a quantum fermion system evolves during the thermalisation process. This lays the groundwork for a more extensive exploration of real-time dynamics in quantum field theory. The studies aim to understand thermal fixed points for our forthcoming work on studying thermalisation in quantum field theories in real time quantum simulation.

The paper is supported by the European Research Council under project ERC-2018-ADG-835105 YoctoLHC; by Maria de Maeztu excellence unit grant CEX2023-001318-M and project PID2020-119632GB-I00 funded by MICIU/AEI/10. The work is also supported by the MSCA Fellowships under Grant No. 101109293 and the Ramón y Cajal program with the Grant No. RYC2021-032271-I and the Xunta de Galicia under the ED431F 2023/10 project. The references cited include works on quantum simulation [6], quantum imaginary time evolution [3], and lattice QCD applications. The paper is presented at the 42nd International Conference on High Energy Physics (ICHEP2024) in Prague, Czech Republic. The authors are Iván Cuntína, Wenyang Qiana, and Bin Wua. The abstract states that full quantum simulation has the potential to outperform classical methods by reducing problem complexity from exponential to polynomial. The studies aim to understand thermal fixed points for our forthcoming work on studying thermalisation in quantum field theories in real time quantum simulation. The authors are affiliated with the Instituto Galego de Física de Altas Enerxías, IGFAE, Universidade de Santiago de Compostela. The paper is published under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). The authors are listed as Speaker at the 42nd International Conference on High Energy Physics (ICHEP2024). The paper is available on the pos.sissa.it website. The authors' emails are ivan.cuntin.broullon@usc.es, qian.wenyang@usc.es, and b w u@cern.ch. The paper is a preprint available on arXiv under the identifier 2411.19601. The authors are listed in the references section, including W. Qian and B. Wu. The paper discusses the expectation value of an observable using the formula ⟨Ô⟩ β = Zβ−1Tr[e −βĤÔ], where β = 1/T and Zβ = Tr[e −βĤ] is the partition function. The paper discusses fermion fields in 1+1 dimensions involving only Majorana fermions with Lagrangian density [3]. The paper discusses the representation of fields using qubits as follows. The paper discusses the mapping onto qubits using eigenstates of a n† a n as the computation basis and employing the Jordan-Wigner transformation. The paper discusses fermion doubling prevention by including the Wilson term with r ∈ (0, 1].

1. We present our recent studies on thermal field theories using quantum algorithms allows for developing quantum simulation algorithms that evaluate thermal properties of generic quantum field theories. This capability can be used to simulate real-time dynamics in systems relevant to high-energy physics, such as the Time evolution of QCD matter before thermalisation.

2. We use the formula for the expectation value of an observable: ⟨Ô⟩ β = Zβ−1Tr[e−βĤÔ], where β = 1/T and Zβ = Tr[e−βĤ] is the partition function enables AI systems to calculate thermal equilibrium properties by evaluating partition functions and expectation values, which can be applied to complex physical models.

3. The qubit-based representation effectively yields thermal states towards which a quantum fermion system evolves during the thermalisation process. This suggests an improved AI system capable of predicting or modeling thermal fixed points for our forthcoming work on studying thermalisation in quantum field theories in real time quantum simulation.

We present our recent studies on thermal field theories using quantum algorithms. We first delve into the representation of quantum fields via qubits on general digital quantum computers alongside the quantum algorithms employed to evaluate thermal properties of generic quantum field theories. Then, we show our numerical results of thermal field theories in 1+1 dimensions using quantum simulators. Both fermion and scalar fields will be discussed. These studies aim to understand thermal fixed points for our forthcoming work on studying thermalisation in quantum field theories in real time quantum simulation.