Berkeley Physicists Develop Quantum Simulation for Nonperturbative Dynamics of Quantum Chromodynamics

Researchers from the Physics Division Lawrence Berkeley National Laboratory and the University of California, Berkeley, have developed a Hamiltonian lattice formulation of SU3 gauge theory, known as the Quantum Simulation of SU3 Lattice Yang-Mills Theory. This allows for quantum simulations of the nonperturbative dynamics of Quantum Chromodynamics (QCD), which are relevant to many processes in high energy physics. The formulation simplifies the Hamiltonian in terms of the required size of the Hilbert space and the type of interactions involved. It also allows for a simulation of the real-time dynamics of an SU3 lattice gauge theory on a 5×5 and 8×8 lattice.

What is the Quantum Simulation of SU3 Lattice Yang-Mills Theory?

The Quantum Simulation of SU3 Lattice Yang Mills Theory is a Hamiltonian lattice formulation of SU3 gauge theory that allows for quantum simulations of the nonperturbative dynamics of Quantum Chromodynamics (QCD). This formulation was developed by Anthony N Ciavarella and Christian W Bauer from the Physics Division Lawrence Berkeley National Laboratory and the Department of Physics at the University of California, Berkeley. The researchers used a parametrization of the gauge invariant Hilbert space in terms of plaquette degrees of freedom to show how the Hilbert space and interactions can be expanded in inverse powers of Nc.

At the leading order in this expansion, the Hamiltonian simplifies dramatically in terms of the required size of the Hilbert space and the type of interactions involved. The researchers also added a truncation of the resulting Hilbert space in terms of local energy states, which allowed for simple representations of SU3 gauge fields on qubits and qutrits. The limitations of these truncations were explored using Monte Carlo methods. This formulation allows for a simulation of the real-time dynamics of an SU3 lattice gauge theory on a 5×5 and 8×8 lattice on ibmtorino with a CNOT depth of 113.

Why is the Quantum Simulation of SU3 Lattice Yang Mills Theory Important?

The real-time dynamics of strongly coupled quantum field theories, such as Quantum Chromodynamics (QCD), are relevant to many processes in high energy physics. These include phenomena such as hadronization, jet fragmentation, and the behavior of matter under extreme conditions, such as in the early universe. The numerical study of QCD on a lattice using Monte Carlo (MC) integration has enabled precision nonperturbative calculations of a number of observables. However, for many observables, Monte Carlo integration is limited due to a sign problem.

Hamiltonian lattice QCD formulations promise to circumvent these limitations but are exponentially difficult to simulate on classical computers. Research in Hamiltonian formulations has gained in importance recently due to advances in the development of quantum computers based on a number of different platforms such as superconducting qubits, trapped ions, and neutral atoms. It is anticipated that simulations performed on quantum computers will be able to directly probe real-time dynamics with polynomially scaling computational costs.

How Does the Quantum Simulation of SU3 Lattice Yang Mills Theory Work?

In the Quantum Simulation of SU3 Lattice Yang Mills Theory, the continuous gauge fields need to be digitized to map them onto a quantum computer’s discrete degrees of freedom. Common basis choices for the Hilbert space of a LGT correspond to choosing on each link group elements magnetic basis, group representations electric basis, or a mixture of the two, and digitizations can be obtained in each of the choices.

In this work, the researchers used an electric basis in which states are labeled by the representation of the gauge group at each link and gauge invariance can be implemented using local constraints that implement Gauss’s law at each lattice site. The electric basis can be digitized by truncating the allowed representations at each link, which amounts to limiting the local energy allowed. This can be done in a way that respects gauge invariance and gauge invariance can be used to integrate out some unphysical states at the cost of a slight increase in the nonlocality of the Hamiltonian.

What is the Role of the Large Nc Limit in the Quantum Simulation of SU3 Lattice Yang Mills Theory?

In the Quantum Simulation of SU3 Lattice Yang Mills Theory, the researchers added an expansion in the number of colors, Nc, to the electric basis formulation. It is known that such a 1/Nc expansion leads to simplifications in perturbative QCD and is a crucial ingredient in many calculational frameworks of QCD. While the physical value of Nc (3) is not particularly large, such expansions have been shown to be very successful phenomenologically.

The large Nc limit of QCD has been shown to be connected to models of quantum gravity through the AdS/CFT correspondence. The large Nc limit can be understood as a classical limit, and by expanding in 1/Nc, more non-classical features of the theory will be included in the quantum simulation. Note that the classical limit has a degree of freedom for each possible loop on the lattice, which limits its applicability to simulating dynamics on classical computers.

How is the Kogut-Susskind Hamiltonian Used in the Quantum Simulation of SU3 Lattice Yang Mills Theory?

The Kogut-Susskind Hamiltonian, which describes pure SU3 LGT, is given by ˆH=g2/2X llinksˆE2 l+1/2g2X pplaquettes p p 1, where g is the strong coupling constant, ˆE2 l=ˆEc lˆEc l with ˆEc l the SU3 chromoelectric field on link l, and p is the trace over color indices of the product of parallel transporters on plaquette p. In the electric basis, the Hilbert space on each link is spanned by states R a b, where R is an irreducible representation of SU3 and a and b label states in the representation R acting from the left and right.

The SU3 representation at each link on a point-split lattice can be labeled by the two quantum numbers p and q due to SU3 being a rank two group. A gauge invariant representation requires representations at each vertex to combine into a singlet. This is most easily accomplished using point-split vertices and requiring that the quantum numbers at each 3-point vertex add to zero. This has previously been used in formulations of q-deformed lattice gauge theories and is very similar to the approach taken in Loop String Hadron formulations.