[Paper Review] Lattice Quantum Chromodynamics and Electrodynamics on a Universal Quantum Computer

Angus Kan, Yunseong Nam|arXiv (Cornell University)|Jul 27, 2021

Advanced Data Storage Technologies28 citations

TL;DR

The paper provides a gate-by-gate construction to simulate U(1), SU(2), and SU(3) lattice gauge theories on a quantum computer, with resource estimates and a second-order Trotter-based quantum algorithm.

ABSTRACT

It is widely anticipated that a large-scale quantum computer will offer an evermore accurate simulation of nature, opening the floodgates for exciting scientific breakthroughs and technological innovations. Here, we show a complete, instruction-by-instruction rubric to simulate U(1), SU(2), and SU(3) lattice gauge theories on a quantum computer. These theories describe quantum electrodynamics and chromodynamics, the key ingredients that form the fabric of our universe. We further provide a concrete estimate of the quantum computational resources required for an accurate simulation of lattice gauge theories using a second-order product formula. We show that lattice gauge theories in any spatial dimension can be simulated using $ ilde{O}(T^{3/2}N^{3/2}Λ/ε^{1/2})$ T gates, where $N$ is the number of lattice sites, $Λ$ is the bosonic gauge field truncation, and $T$ is the simulation time.

Motivation & Objective

• Motivate and enable real-time, first-principles simulations of QED and QCD on quantum hardware.
• Present a complete, instruction-level quantum circuit framework for U(1), SU(2), and SU(3) lattice gauge theories across arbitrary spatial dimensions.
• Provide rigorous upper bounds on quantum resources (gates, errors) for fault-tolerant simulation using second-order Trotterization.
• Show how binary encoding of gauge fields reduces qubit counts and improves gate efficiency.
• Compare with prior work and discuss practical implications for lattice gauge theory simulations on quantum computers.

Proposed method

• Use the Kogut-Susskind Hamiltonian for lattice gauge theories including both fermionic matter and bosonic gauge fields.
• Encode bosonic gauge-field quantum numbers in binary, truncating to Λ values per quantum number, enabling qubit-efficient representations.
• Apply a second-order Suzuki–Trotter formula to approximate e^{iHT}, with H decomposed into mass, electric, kinetic, and magnetic subterms.
• Design gate-by-gate circuits for mass, electric, kinetic, and magnetic evolutions across U(1), SU(2), and SU(3) cases, using fixed-point arithmetic and phase oracles where needed.
• Optimize circuit depth via parallelization of commuting terms and the weight-sum trick to layer same-angle Rz gates, reducing Rz counts.
• Provide explicit resource scalings: U(1) ~ O(T^{3/2} d Λ^{d/2} L^{d} ϵ^{-1/2} …) and SU(2)/SU(3) ~ O(T^{3/2} d Λ^{d/2} L^{d} …) with detailed constants in the supplementary material.

Experimental results

Research questions

• RQ1 Can a gate-level, fault-tolerant quantum circuit be constructed to simulate real-time dynamics of U(1), SU(2), and SU(3) lattice gauge theories?
• RQ2 What are the quantum computational resource requirements (gates, qubits) for simulating lattice gauge theories in arbitrary spatial dimensions using a second-order Trotter scheme?
• RQ3 How does binary encoding of bosonic gauge-field quantum numbers impact qubit counts and gate complexity compared to unary encoding?
• RQ4 How do the U(1), SU(2), and SU(3) simulations compare in terms of circuit depth, error budgets, and scalability with lattice size Λ, dimension d, and volume L^d?
• RQ5 What are the practical implications and potential applications of quantum lattice gauge theory simulations for QED/QCD and related physics?

Key findings

• A complete, gate-level construction is provided for simulating real-time dynamics of U(1), SU(2), and SU(3) lattice gauge theories on a universal quantum computer.
• Binary encoding of bosonic gauge-field numbers yields exponential reductions in qubit counts for SU(2) and SU(3) compared to unary encoding.
• Second-order Trotterization with careful term ordering and parallelization achieves favorable resource scaling; the paper provides explicit complexity bounds for U(1) and SU(N) cases.
• For SU(2) and SU(3), the complexity scales as O(T^{3/2} d Λ^{d/2} L^{d} ϵ^{-1/2}) with additional factors depending on encoding and arithmetic; U(1) has similar scaling with ε and Λ but different constants.
• The work includes detailed circuit constructions and supplementary material that quantify gate counts and qubit requirements for realistic parameters and error budgets.
• Applications discussed include computing transport coefficients, hadronic tensors, and potentially exploring QCD phase diagrams and beyond-Standard-Model phenomena on quantum hardware.