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arXiv:2308.03196 · EPJ Quantum Technology

Quantum gate sets for lattice QCD in the strong-coupling limit: $N_{f}=1$

Michael Fromm, Owe Philipsen, Wolfgang Unger, Christopher Winterowd

Published Mar 25, 2024• 7 citations
arXivHTMLPDF

Abstract

Abstract We derive the primitive quantum gate sets to simulate lattice quantum chromodynamics (LQCD) in the strong-coupling limit with one flavor of massless staggered quarks. This theory is of interest for studies at non-zero density as the sign problem can be overcome using Monte Carlo methods. In this work, we use it as a testing ground for quantum simulations. The key point is that no truncation of the bosonic Hilbert space is necessary as the theory is formulated in terms of color-singlet degrees of freedom (“baryons” and “mesons”). The baryons become static in the limit of continuous time and decouple, whereas the dynamics of the mesonic theory involves two qubits per lattice site. Lending dynamics also to the “baryons” simply requires to use the derived gate set in its controlled version.

DOI
10.1140/epjqt/s40507-024-00236-y

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Quantum gate sets for lattice QCD in the strong-coupling limit: Nf=1N_{f}=1Nf​=1
Abstract We derive the primitive quantum gate sets to simulate lattice quantum chromodynamics (LQCD) in the strong-coupling limit with one flavor of massless staggered quarks. This theory is of interest for studies at non-zero density as the sign problem can be overcome using Monte Carlo methods. In this work, we use it as a testing ground for quantum simulations. The key point is that no truncation of the bosonic Hilbert space is necessary as the theory is formulated in terms of color-singlet degrees of freedom (“baryons” and “mesons”). The baryons become static in the limit of continuous time and decouple, whereas the dynamics of the mesonic theory involves two qubits per lattice site. Lending dynamics also to the “baryons” simply requires to use the derived gate set in its controlled version.
Abstract We consider the strong-coupling limit of lattice QCD with massless staggered quarks and study the resource requirements for quantum simulating the theory in its Hamiltonian formulation. The bosonic Hilbert space of the color-singlet degrees of freedom grows quickly with the number of quark flavors NfN_{f}Nf​ N f , making it a suitable testing ground for resource considerations across different platforms. In particular, in addition to the standard model of computation with qubits, we consider mapping the theory to qudits (d>2)(d>2)(d>2) ( d > 2 ) and qumodes, as used on atomic systems and photonic devices, respectively. We subsequently derive the resource requirements to quantum simulate the theory, for varying number of quark flavors Nf=1N_{f}=1Nf​=1 N f = 1 and Nf=2N_{f} = 2Nf​=2 N f = 2 , using a first-order product formula.
We develop a framework for quantum simulations of lattice SU(NcN_cNc​) gauge theory with quarks. Staggered and Wilson lattice fermions are considered in two and three spatial dimensions and a theta angle is included in three dimensions. The physical, gauge-invariant Hilbert space is formulated in a representation basis, where gauge and fermionic degrees of freedom are encoded by irreducible representations of SU(NcN_cNc​) that tensor at each lattice site to contain a singlet. We discuss algorithms for simulating time evolution on quantum computers and carry out noiseless simulations of lattice quantum chromodynamics (Nc=3N_c=3Nc​=3) on small lattices with up to 32 qubits, showcasing theta angle effects, hadronic states, string dynamics, and a baryon chemical potential for the first time in three spatial dimensions.
Quantum simulations of lattice gauge theories are anticipated to directly probe the real time dynamics of QCD, but scale unfavorably with the required truncation of the gauge fields. Improved Hamiltonians are derived to correct for the effects of gauge field truncations on the SU(3) Kogut-Susskind Hamiltonian. It is shown in 1+1D1+1D1+1D that this enables low chromoelectric field truncations to quantitatively reproduce features of the untruncated theory over a range of couplings and quark masses. In 3+1D3+1D3+1D, an improved Hamiltonian is derived for lattice QCD with staggered massless fermions. It is shown in the strong coupling limit that the spectrum qualitatively reproduces aspects of two flavor QCD and simulations of a small system are performed on IBM's Perth quantum processor.
A Hamiltonian lattice formulation of lattice gauge theories opens the possibility for quantum simulations of the non-perturbative dynamics of QCD. By parametrizing the gauge invariant Hilbert space in terms of plaquette degrees of freedom, we show how the Hilbert space and interactions can be expanded in inverse powers of N. At leading order in this expansion, the Hamiltonian simplifies dramatically, both in the required size of the Hilbert space as well as the type of interactions involved. Adding a truncation of the resulting Hilbert space in terms of local electric energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits to leading order in large N.
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Tools necessary for quantum simulations of 1+11+11+1 dimensional quantum chromodynamics are developed. When formulated in axial gauge and with two flavors of quarks, this system requires 12 qubits per spatial site with the gauge fields included via nonlocal interactions. Classical computations and DDD-wave's quantum annealer advantage are used to determine the hadronic spectrum, enabling a decomposition of the masses and a study of quark entanglement. Color ``edge'' states confined within a screening length of the end of the lattice are found. IBM's seven-qubit quantum computers, ibmq_jakarta and ibm_perth, are used to compute dynamics from the trivial vacuum in one-flavor QCD with one spatial site. More generally, the Hamiltonian and quantum circuits for time evolution of 1+11+11+1 dimensional SU(Nc)SU({N}_{c})SU(Nc​) gauge theory with Nf{N}_{f}Nf​ flavors of quarks are developed, and the resource requirements for large-scale quantum simulations are estimated.