U(1) fields from qubits: An approach via D-theory algebra

A new quantum link microstructure was proposed for the lattice quantum chromodynamics (QCD) Hamiltonian, replacing the Wilson gauge links with a bilinear of fermionic qubits, later generalized to D-theory. This formalism provides a general framework for building lattice field theory algorithms for quantum computing. We focus mostly on the simplest case of a quantum rotor for a single compact U(1) field. We also make some progress for non-Abelian setups, making it clear that the ideas developed in the U(1) case extend to other groups. These in turn are building blocks for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>1</a:mn><a:mo>+</a:mo><a:mn>0</a:mn></a:math>-dimensional (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mn>1</c:mn><c:mo>+</c:mo><c:mn>0</c:mn></c:math>-D) matrix models, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>1</e:mn><e:mo>+</e:mo><e:mn>1</e:mn></e:math>-D sigma models and non-Abelian gauge theories in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>2</g:mn><g:mo>+</g:mo><g:mn>1</g:mn></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>3</i:mn><i:mo>+</i:mo><i:mn>1</i:mn></i:math> dimensions. By introducing multiple flavors for the U(1) field, where the flavor symmetry is gauged, we can efficiently approach the infinite-dimensional Hilbert space of the quantum O(2) rotor with increasing flavors. The emphasis of the method is on preserving the symplectic algebra exchanging fermionic qubits by sigma matrices (or hard bosons) and developing a formal strategy capable of generalization to a <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>SU</k:mi><k:mo stretchy="false">(</k:mo><k:mn>3</k:mn><k:mo stretchy="false">)</k:mo></k:math> field for lattice QCD and other non-Abelian <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mn>1</o:mn><o:mo>+</o:mo><o:mn>1</o:mn></o:math>-D sigma models or <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mn>3</q:mn><q:mo>+</q:mo><q:mn>1</q:mn></q:math>-D gauge theories. For U(1), we discuss briefly the qubit algorithms for the study of the discrete <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mn>1</s:mn><s:mo>+</s:mo><s:mn>1</s:mn></s:math>-D sine-Gordon equation.