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arXiv:2004.00344 · Physical review. D/Physical review. D.

Circuit complexity for generalized coherent states in thermal field dynamics

Minyong Guo, Zhong-Ying Fan, Jie Jiang, Xiangjing Liu, Bin Chen

Published Jun 12, 2020• 32 citations
arXivHTMLPDF

Abstract

In this work, we study the circuit complexity for generalized coherent states in thermal systems by adopting the covariance matrix approach. We focus on the coherent thermal (CT) state, which is non-Gaussian and has a nonvanishing one-point function. We find that even though the CT state cannot be fully determined by the symmetric two-point function, the circuit complexity can still be computed in the framework of the covariance matrix formalism by properly enlarging the covariance matrix. Now the group generated by the unitary is the semiproduct of translation and the symplectic group. If the reference state is Gaussian, the optimal geodesic is still be generated by a horizontal generator such that the circuit complexity can be read from the generalized covariance matrix associated to the target state by taking the cost function to be ${F}_{2}$. For a single harmonic oscillator, we discuss carefully the complexity and its formation in the cases that the reference states are Gaussian and the target space is excited by a single mode or double modes. We show that the study can be extended to the free scalar field theory.

DOI
10.1103/physrevd.101.126007

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In this work, we study the circuit complexity for generalized coherent states in thermal systems by adopting the covariance matrix approach. We focus on the coherent thermal (CT) state, which is non-Gaussian and has a nonvanishing one-point function. We find that even though the CT state cannot be fully determined by the symmetric two-point function, the circuit complexity can still be computed in the framework of the covariance matrix formalism by properly enlarging the covariance matrix. Now the group generated by the unitary is the semiproduct of translation and the symplectic group. If the reference state is Gaussian, the optimal geodesic is still be generated by a horizontal generator such that the circuit complexity can be read from the generalized covariance matrix associated to the target state by taking the cost function to be F2{F}_{2}F2​. For a single harmonic oscillator, we discuss carefully the complexity and its formation in the cases that the reference states are Gaussian and the target space is excited by a single mode or double modes. We show that the study can be extended to the free scalar field theory.
Abstract In this paper, we first construct thermofield double states for bosonic string theory in the light-cone gauge. We then obtain a coherent-thermal string state and a thermal-coherent string state. We use the covariance matrix approach to calculate the circuit complexity of coherent-thermal string states. In this approach, we generate the optimal geodesics by a horizontal string generator, and then obtain the circuit complexity using the length of the minimal geodesics in the group manifold.
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.
Abstract We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [1]. The complexity of the coherent states have the same UV divergences as the vacuum state complexity and so we consider the finite increase of the complexity of these states over the vacuum state. One observation is that generally, the optimal circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen’s approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal circuits, derived from these two approaches, are different.
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.
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.