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arXiv:2109.09759 · Symmetry

Circuit Complexity in Z2 EEFT

Kiran Adhikari, Sayantan Choudhury, Sourabh Kumar, Saptarshi Mandal, Nilesh Pandey, Abhishek Roy, Soumya Sarkar, Partha Sarker, Saadat Salman Shariff

Published Dec 22, 2022• 10 citations
arXivHTMLPDF

Abstract

Motivated by recent studies of circuit complexity in weakly interacting scalar field theory, we explore the computation of circuit complexity in Z2 Even Effective Field Theories (Z2 EEFTs). We consider a massive free field theory with higher-order Wilsonian operators such as ϕ4, ϕ6, and ϕ8. To facilitate our computation, we regularize the theory by putting it on a lattice. First, we consider a simple case of two oscillators and later generalize the results to N oscillators. This study was carried out for nearly Gaussian states. In our computation, the reference state is an approximately Gaussian unentangled state, and the corresponding target state, calculated from our theory, is an approximately Gaussian entangled state. We compute the complexity using the geometric approach developed by Nielsen, parameterizing the path-ordered unitary transformation and minimizing the geodesic in the space of unitaries. The contribution of higher-order operators to the circuit complexity in our theory is discussed. We also explore the dependency of complexity on other parameters in our theory for various cases.

DOI
10.3390/sym15010031

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Motivated by recent studies of circuit complexity in weakly interacting scalar field theory, we explore the computation of circuit complexity in Z2 Even Effective Field Theories (Z2 EEFTs). We consider a massive free field theory with higher-order Wilsonian operators such as ϕ4, ϕ6, and ϕ8. To facilitate our computation, we regularize the theory by putting it on a lattice. First, we consider a simple case of two oscillators and later generalize the results to N oscillators. This study was carried out for nearly Gaussian states. In our computation, the reference state is an approximately Gaussian unentangled state, and the corresponding target state, calculated from our theory, is an approximately Gaussian entangled state. We compute the complexity using the geometric approach developed by Nielsen, parameterizing the path-ordered unitary transformation and minimizing the geodesic in the space of unitaries. The contribution of higher-order operators to the circuit complexity in our theory is discussed. We also explore the dependency of complexity on other parameters in our theory for various cases.
Abstract We consider circuit complexity in certain interacting scalar quantum field theories, mainly focusing on theϕ4theory. We work out the circuit complexity for evolving from a nearly Gaussian unentangled reference state to the entangled ground state of the theory. Our approach uses Nielsen’s geometric method, which translates into working out the geodesic equation arising from a certain cost functional. We present a general method, making use of integral transforms, to do the required lattice sums analytically and give explicit expressions for thed= 2,3 cases. Our method enables a study of circuit complexity in the epsilon expansion for the Wilson-Fisher fixed point. We find that with increasing dimensionality the circuit depth increases in the presence of theϕ4interaction eventually causing the perturbative calculation to breakdown. We discuss how circuit complexity relates with the renormalization group.
Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.
In this paper, we study circuit complexity for a free vector field of a [Formula: see text] gauge theory in Coulomb gauge, and Gaussian states. We introduce a quantum circuit model with Gaussian states, including reference and target states. Using Nielsen’s geometric approach, the complexity then can be found as the shortest geodesic in the space of states. This geodesic is based on the notion of geodesic distance on the Lie group of Bogoliubov transformations equipped with a right-invariant metric. We use the framework of the covariance matrix to compute circuit complexity between Gaussian states. We apply this framework to the free vector field in general dimensions where we compute the circuit complexity of the ground state of the Hamiltonian.
We evaluate the complexity of the free scalar field by the operator approach in which the transformation matrix between the second quantization operators of the reference state and target state is regarded as the quantum gate. We first examine the system in which the reference state is two noninteracting oscillators with the same frequency ${\ensuremath{\omega}}_{0}$ while the target state is two interacting oscillators with frequency ${\stackrel{\texttildelow{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\texttildelow{}}{\ensuremath{\omega}}}_{2}$. We calculate the geodesic length on the associated group manifold of gate matrix and reproduce the known value of ground-state complexity. Next, we study the complexity in the excited states. Although the gate matrix is very large we can transform it to a diagonal matrix and obtain the associated complexity. We explicitly calculate the complexity in several excited states and prove that the square of the geodesic length in the general state ∣n,m⟩|\mathrm{n},\mathrm{m}⟩∣n,m⟩ is ${D}_{(\mathrm{n},\mathrm{m})}^{2}=(n+1){(\mathrm{ln}\sqrt{\frac{{\stackrel{\texttildelow{}}{\ensuremath{\omega}}}_{1}}{{\ensuremath{\omega}}_{0}}})}^{2}+(m+1){(\mathrm{ln}\sqrt{\frac{{\stackrel{\texttildelow{}}{\ensuremath{\omega}}}_{2}}{{\ensuremath{\omega}}_{0}}})}^{2}$. The results are extended to the NNN couple harmonic oscillators which correspond to the lattice version of the free scalar field.
A bstract We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited energy eigenstates of the free Dirac field. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.
We define and calculate versions of complexity for free fermionic quantum field theories in 1+11+11+1 and 3+13+13+1 dimensions, adopting Nielsen's geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for fermions: viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cutoff dependence are discussed.