Renormalization Group Circuits for Weakly Interacting Continuum Field Theories

Abstract

We develop techniques to systematically construct local unitaries which map scale-invariant, product state wavefunctionals to the ground states of weakly interacting, continuum quantum field theories. More broadly, we devise a "quantum circuit perturbation theory" to construct local unitaries which map between any pair of wavefunctionals which are each Gaussian with arbitrary perturbative corrections. Further, we generalize cMERA to interacting continuum field theories, which requires reworking the existing formalism which is tailored to non-interacting examples. Our methods enable the systematic perturbative calculation of cMERA circuits for weakly interacting theories, and as a demonstration we compute the 1-loop cMERA circuit for scalar ϕ 4 theory and analyze its properties. In this case, we show that Wilsonian renormalization of the spatial momentum modes is equivalent to a local position space cMERA circuit. This example provides new insights into the connection between position space and momentum space renormalization group methods in quantum field theory. The form of cMERA circuits derived from perturbation theory suggests useful ansatzes for numerical variational calculations. RG flows. [9] These "entanglement renormalization" tensor networks are comprised of sequences of spatially local quantum gates.1. Why unitaries with quadratic generators are special; 2. How to systematically treat unitaries with higher-order generators perturbatively in the higher order generators (i.e., "quantum circuit perturbation theory"); and 3. Special circumstances required for us to treat unitaries with higher-order generators non-perturbatively in the higher order generators.By developing quantum circuit perturbation theory, we can systematically construct unitaries which map scale-invariant product states to the ground states of weakly interacting quantum field theory, to any fixed order in perturbation theory. We can also construct unitaries which map between any pair of states that are each Gaussian with arbitrary perturbative corrections. In very special cases, unitaries between non-Gaussian states (or between a Gaussian state and a non-Gaussian state) can be constructed non-perturbatively, although we will not discuss such cases here. Next, we generalize cMERA to interacting fields, which involves generalizing our methods above to unitaries created by path-ordered exponentials of Hermitian generators. We discover new features of cMERA which are absent for the free field examples which have previously been studied. Our techniques enable systematic perturbative calculations of cMERA circuits for interacting field theories, and we calculate the 1-loop cMERA circuit for ϕ 4 theory as an example. Furthermore, the disentangler is constructed to disentangle spatial momentum modes in a manner exactly corresponding to perturbative Wilsonian RG (on spatial momentum modes), and yet in position space disentangles spatially local subsystems. This relationship between disentangling degrees of freedom in momentum space and position s...

Complexity and entanglement for thermofield double states

“…[23,51,55,56,58]. Such results can also have an interesting application in the field of tensor networks, namely if MERA [111] or cMERA [84,125,126], as tensor networks are in some sense an optimal way of preparing critical ground states from product states.…”

“…Recall that the geodesic ending at τ 1 = 0 is still the straight-line geodesic with ∆θ = 0, which remains in the τ = 0 plane for the entire trajectory. 25 Since for other values of φ we will have ρ 1 (φ) ≥ ρ 1 (0) and ∆θ 2 ≥ 0, we can use the inequality (125) to argue that the shortest geodesic is in fact still the simple straight-line geodesic (121), where the previous analysis indicates that θ 1 = ±π/2. Alternatively, if x and y have opposite signs, then the short axis is aligned with φ = π/2, and we should expect that the optimal geodesic will no longer be the simple one which remains in the τ = 0 plane.…”

“…It would be very interesting to understand if recent attempts of refs. [125,126] to include interactions in the continuous version of MERA (cMERA) [84] can help in addressing this problem. See also the recent work [65].…”

Renormalized Circuit Complexity

“…• Interacting scalar field theory: We consider first λφ 4 theory in D + 1 dimensions. A first principle analysis will require working out the algebra of operators systematically and computing the geodesic-this has not been done and appears difficult with current technology [73,74]. However, we can give a heuristic argument as follows.…”

Circuit complexity and 2D bosonisation

“…In most of the works on the complexity, the choice of gates was restricted to gates quadratic in the fields, in order not to depart from the space of Gaussian states. No fundamental reason underlies this choice except that it allows to do explicit computations, and it is not obvious how to make an alternative choice: in general it is difficult to find an algebraically closed set of gates that is larger than the set of quadratic operators but not as large as the full space [24,31]. But it turns out that such a set is provided by the magic of bosonisation.…”