High Energy Physics

A quantum state is a map from operators to real numbers that are their expectation values. Evaluating this map always entails using some algorithm, for example contracting a tensor network. I propose a novel way of quantifying the complexity of a quantum state in terms of "query complexity": the number of times an efficient algorithm for computing correlation functions in the given state calls a certain subroutine. I construct such an algorithm for a general "state at a cutoff" in 1+1-dimensional field theory. The algorithm scans cutoff-sized intervals for operators whose expectation values will be computed. It can be written as a Matrix Product State, with individual matrices performing translations in the space of (cutoff-sized) intervals and reading off consecutive operator inputs. If we take the queried subroutine to be a translation in the space of intervals, query complexity counts "how many" intervals the algorithm visits--a notion of distance in the space of intervals. A unique distance function is consistent with the requisite notion of translations; therefore the query complexity of a state at a cutoff is unambiguously defined. In holographic theories, the query complexity evaluates to the integral of the Ricci scalar on a spatial slice enclosed by the bulk cutoff, which in pure AdS3 agrees with the volume proposal but otherwise departs from it.

"Besides tensor networks, quantum computations (QC) as well use a Hamiltonian formulation to solve physical problems. Although QC are presently very limited, since only small number of qubits are available, they have the principal advantage that they straightforwardly scale to higher dimensions. A standard tool in the QC approach are Variational Quantum Simulations (VQS) which form a class of hybrid quantum-classical algorithms for solving optimization problems. For example, the objective may be to find the ground state of a Hamiltonian by minimizing the energy. As such, VQS use parametric quantum circuit designs to generate a family of quantum states (e.g., states obeying physical symmetries) and efficiently evaluate a cost function for the given set of variational parameters (e.g., energy of the current quantum state) on a quantum device. The optimization is then performed using a classical feedback loop based on the measurement outcomes of the quantum device. In the case of energy minimization, the optimal parameter set therefore encodes the ground state corresponding to the given Hamiltonian provided that the parametric quantum circuit is able to encode the ground state. Hence, the design of parametric quantum circuits is subject to two competing drivers. On one hand, the set of states, that can be generated by the parametric quantum circuit, has to be large enough to contain the ground state. On the other hand, the circuit should contain as few quantum gates as possible to minimize noise from the quantum device. In other words, when designing a parametric quantum circuit we want to ensure that there are no redundant parameters. In this talk, I will consider the parametric quantum circuit as a map from parameter space to the state space of the quantum device. Using this point of view, the set of generated states forms a manifold. If the quantum circuit is free from redundant parameters, then the number of parameters is precisely the dimension of the manifold of states. This leads us to the notion of dimensional expressivity analysis. I will discuss means of analyzing a given parametric design in order to remove redundant parameters as well as any unwanted symmetries (e.g., a gate whose only effect is a change in global phase). Time permitting, I may discuss the manifold of physical states as well since this will allow us to decide whether or not a parametric quantum circuit can express all physical states (thereby ensuring that the ground state can be expressed as well)."

In contrast to the 4D case, there are well understood theories of quantum gravity for the 3D case. Indeed, 3D general relativity constitutes a topological field theory (of BF or equivalently Chern-Simons type) and can be quantized as such. The resulting quantum theory of gravity offers many interesting lessons for the 4D case. In this talk I will discuss the quantum theory which results from quantizing 3D gravity as a topological field theory. This will also allow a derivation of a holographic boundary theory, together with a geometric interpretation of the boundary observables. The resulting structures can be interpreted in terms of tensor networks, which provide states of the boundary theory. I will explain how a choice of network structure and bond dimensions constitutes a complete gauge fixing of the diffeomorphism symmetry in the gravitational bulk system. The theory provides a consistent set of rules for changing the gauge fixing and with it the tensor network structure. This provides an example of how diffeomorphism symmetry can be realized in a tensor network based framework. I will close with some remarks on the 4D case and the challenges we face there.

The success of the Ryu-Takayanagi formula suggests a profound connection between the AdS/CFT correspondence and tensor networks. There are since many works on constructing examples, although it is very difficult to make them explicit and quantitative. We will discuss some new progress in the toy example of p-adic CFT where its tensor network dual was previously constructed explicitly [ arXiv:1703.05445 , arXiv:1812.06059, arXiv:1902.01411], and how some analogue of Einstein equation on the graph emerges as we consider RG flow of these CFTs. These progresses inspire us to find a way to construct explicit holographic tensor networks of more realistic CFTs based on their connection with topological models with one higher dimension, at least in CFT2. We will present some preliminary results and works in progress.

The search for applications of quantum computers has highlighted the field of quantum chemistry, where one can also apply tensor network methods. There are several challenges in getting useful results for molecules compared to simulating a model Hamiltonian in condensed matter physics. The first issue is in descretizing continuum space to get a finite Hamiltonian which is amenable to tensor network techniques. Another is the need for high accuracy, particularly in energies, to compare with experiments. I will give an overview of the approaches used in this field, and then focus on our work using grid and wavelet-based discretizations coupled with DMRG methods.

Multi-scale tensor networks offer a way to efficiently represent ground states of critical systems and may be adapted for state-preparation on a quantum computer. The tensor network for a single scale specifies a quantum channel whose fixed-point is a subregion of the approximate critical ground state. The fixed-point of a noisy channel is perturbed linearly in the noise parameter from the ideal state, making local observables stable against errors for these iterative algorithms. We consider the wavelet-designed circuit for the 1+1D critical Ising ground state as a concrete example to numerically test the noise robustness against our error models and compare the smallest instance case with an implementation on a present-day ion-trap quantum computer.

We examine holographic complexity in the doubly holographic model to study quantum extremal islands. We focus on the holographic complexity=volume (CV) proposal for boundary subregions in the island phase. Exploiting the Fefferman-Graham expansion of the metric and other geometric quantities near the brane, we derive the leading contributions to the complexity and interpret these in terms of the generalized volume of the island derived from the higher curvature action for the brane gravity. Motivated by these results, we propose a generalization of the CV proposal for higher curvature theories of gravity. Further, we provide two consistency checks of our proposal by studying Gauss-Bonnet gravity and f(R) gravity in the bulk.

"I will introduce a tensor-network based language for classifying topological phases via fixed-point models. The "models" will be tensor networks formalizing a discrete Euclidean path integral living in a topological space-time, and can be obtained from Hamiltonian models by Trotterizing the imaginary time evolution. Topological fixed-point models are invariant under topology-preserving space-time deformations. Space-time manifolds and homeomorphisms can be combinatorially represented by graph-like "networks", which together with "moves" form a "liquid". The networks can be interpreted as tensor networks, and the moves as equations which determine the fixed-point models. Different combinatorial representations of the same space-times yield new kinds of fixed-point models. Given the limited time, I will stick to very simple examples in 1+1 dimensions for this talk."