We now start what will be the subject of this book: quantum field theory at non-zero temperature and/or non-zero chemical potential. Two competing formalisms have been used at zero temperature in order to study field theory: the operator formalism, which is the older one, and the path integral formalism, which represents the more modern approach. Actually, it may often be illuminating to look at a given problem from both points of view, although in some cases one of the formalisms may prove to be definitely superior to the other: for example, path integrals are much simpler when quantizing gauge theories. At finite temperature, both formalisms are useful, and it is instructive to be able to switch from one approach to the other. Since the simplest case of field theory is field theory with zero space dimension, or, in other words, quantum mechanics, we shall begin with a short description of quantum mechanics at finite temperature, or, equivalently, quantum statistical mechanics. We first deal with the path integral approach, and later on revert to the more conventional operator formalism. We shall be particularly interested in time-ordered products of position operators, which will generalize to time-ordered products of field operators in field theory.
To study quantum field theories on a quantum computer, we must begin with Hamiltonians defined on a finite-dimensional Hilbert space and then take appropriate limits. This approach can be seen as a new type of regularization for quantum field theories, which we refer to as qubit regularization. A related finite-dimensional regularization, known as the D-theory approach, was proposed long ago as a general framework for all quantum field theories. In this framework, the dimensionality of the local Hilbert space at each spatial point can increase as needed through an additional flavor index. To reproduce asymptotically free QFTs, most studies assume that qubit-regularized theories require extending the local Hilbert space to infinity. However, contrary to this common belief, recent discoveries in (1+1) dimensions have revealed two examples where asymptotic freedom appears to emerge within a strictly finite-dimensional local Hilbert space through a novel renormalization group (RG) flow. These findings motivate further investigation into whether asymptotically free gauge theories could also emerge within a strictly finite-dimensional local Hilbert space. To support these explorations, we propose an orthonormal basis called the monomer-dimer-tensor-network (MDTN) basis and use it to construct new types of qubit-regularized lattice gauge theories.
Gauge-invariant QMETTS with mutually unbiased physical bases for
Z2 lattice gauge theories at finite temperature and density
In quantum computations of gauge theories at finite temperature and finite density, enforcing Gauss's law for all states contributing to the thermal ensemble is a nontrivial challenge. In this work, we adopt the Quantum Minimally Entangled Typical Thermal States (QMETTS) algorithm for
Z2 gauge-constrained systems and propose a method for computing finite-temperature and finite-density expectation values without eliminating gauge degrees of freedom. To preserve gauge invariance while maintaining efficient sampling, we introduce measurement bases that are gauge invariant and mutually unbiased within the physical subspace. We show that such measurement bases can be constructed efficiently for
Z2 lattice gauge theories in general dimensions and arbitrary boundary conditions by exploiting the correspondence between
Z2 lattice gauge theories and the stabilizer formalism. Furthermore, since expectation-value estimation on quantum hardware is inherently affected by shot noise, we explicitly incorporate shot noise into the analysis. We find that the single-shot strategy is near optimal under a fixed total number of circuit executions in terms of the variance. This result indicates that it is generally more efficient to generate more QMETTS samples than to accurately estimate the expectation value for each individual pure state. We validate the proposed method numerically in a
(1+1)-dimensional
Z2 lattice gauge theory coupled to staggered fermions. Our results provide a gauge-invariant framework for finite-temperature and finite-density calculations on quantum devices.