Quantum Foundations

Quantum simulation of lattice gauge theory is expected to become a major application of near-term quantum devices. In this presentation, I will talk about a quantum simulation scheme for lattice gauge theories motivated by Measurement-Based Quantum Computation [1], which we call Measurement-Based Quantum Simulation (MBQS). In MBQS, we consider preparing a resource state whose entanglement structure reflects the spacetime structure of the simulated gauge theory. We then consider sequentially measuring qubits in the resource state in a certain adaptive manner, which drives the time evolution in the Hamiltonian lattice gauge theory. It turns out that the resource states we use for MBQS of Wegner’s models possess topological order protected by higher-form symmetries. These higher-form symmetries are also practically useful for error correction to suppress contributions that violate gauge symmetries. We also discuss the relation between the resource state and the partition function of Wegner’s model. This presentation is based on my work with Takuya Okuda [2]. [1] R. Raussendorf and H. J. Briegel, A One-Way Quantum Computer, Phys. Rev. Lett. 86, 5188 (2001) [2] H. Sukeno and T. Okuda, Measurement-based quantum simulation of Abelian lattice gauge theories, arXiv:2210.10908

Within the setting of the AdS/CFT correspondence, we ask about the power of computers in the presence of gravity. We show that there are computations on $n$ qubits which cannot be implemented inside of black holes with entropy less than $O(2^n)$. To establish our claim, we argue computations happening inside the black hole must be implementable in a programmable quantum processor, so long as the inputs and description of the unitary to be run are not too large. We then prove a bound on quantum processors which shows many unitaries cannot be implemented inside the black hole, and further show some of these have short descriptions and act on small systems. These unitaries with short descriptions must be computationally forbidden from happening inside the black hole.

We investigate the link between position-based quantum cryptography (PBQC) and holography established in [May19] using holographic quantum error correcting codes as toy models. If the "temporal" scaling of the AdS metric is inserted by hand into the toy model via the bulk Hamiltonian interaction strength we recover a toy model with consistent causality structure. This leads to an interesting implication between two topics in quantum information: if position-based cryptography is secure against attacks with small entanglement then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another.

In holographic CFTs satisfying eigenstate thermalization, there is a regime where the operator product expansion can be approximated by a random tensor network. The geometry of the tensor network corresponds to a spatial slice in the holographic dual, with the tensors discretizing the radial direction. In spherically symmetric states in any dimension and more general states in 2d CFT, this leads to a holographic error-correcting code, defined in terms of OPE data, that can be systematically corrected beyond the random tensor approximation. The code is shown to be isometric for light operators outside the horizon, and non-isometric inside, as expected from general arguments about bulk reconstruction. The transition at the horizon occurs due to a subtle breakdown of the Virasoro identity block approximation in states with a complex interior.

We study entanglement entropy in (2+1)-dimensional gravity as a window into larger open questions regarding entanglement entropy in gravity. (2+1)-dimensional gravity can be rewritten as a topological field theory, which makes it a more tractable model to study. In these topological theories, there remain key questions which we seek to answer in this work, such as the questions 1) What is the entropy of the physical algebra of observables in a subregion, 2) How do we define a factorization map such that the entropy of the resulting factors agrees with this algebraic entropy, and 3) Can we use these insights to build a tensor network that exhibits non-commuting areas? We investigate non-Abelian toric codes / Levin-Wen models as a toy model for black hole entropy in Chern Simons theory. These differ from the usual model in that the stabilizers are implemented as constraints. By enforcing constraints for both Gauss' Law and the flatness of the gauge field, we obtain a choice of algebra that contains only topological operators. The desirable properties of this model are twofold: first, we produce the finiteness of black hole entropy described in previous literature while providing a natural algebraic motivation for this result. Secondly, we obtain non-commuting area operators on a toy model with the topology of a torus.

Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.

Motivated by the ground state structure of quantum models with all-to-all interactions such as mean-field quantum spin glass models and the Sachdev-Ye-Kitaev (SYK) model, we propose a tensor network architecture which can accomodate volume law entanglement and a large ground state degeneracy. We call this architecture the non-local renormalization ansatz (NoRA) because it can be viewed as a generalization of MERA, DMERA, and branching MERA networks with the constraints of spatial locality removed. We argue that the architecture is potentially expressive enough to capture the entanglement and complexity of the ground space of the SYK model, thus making it a suitable variational ansatz, but we leave a detailed study of SYK to future work. We further explore the architecture in the special case in which the tensors are random Clifford gates. Here the architecture can be viewed as the encoding map of a random stabilizer code. We introduce a family of codes inspired by the SYK model which can be chosen to have constant rate and linear distance at the cost of some high weight stabilizers. We also comment on potential similarities between this code family and the approximate code formed from the SYK ground space.

We study Generalized Free Fields (GFF) from the point of view of information measures. We begin by reviewing conformal GFF, their holographic representation, and the multiple possible assignations of algebras to a single spacetime region that arise in these theories. We will focus on manifestations of these features present in the Mutual Information (MI) of holographic GFF. First, we show that the MI can be expected to be finite even if the AdS dual space is of infinite volume. Then, we present the long-distance limit of the MI for regions with arbitrary boundaries in the light cone for the causal and entanglement wedge algebras. The pinching limit of these surfaces shows the GFF behaves as an interacting model from the MI point of view. The entanglement wedge algebra choice allows these models to ``fake'' causality, giving results consistent with their role in the description of large N models. Finally, we explore the short distance limit of the MI. Interestingly, we find that the GFF has a leading volume term rather than an area term and a logarithmic term in any dimension rather than only for even dimensions as in ordinary CFTs. We also find the dependence of some subleading terms on the conformal dimension of the GFF.

In this talk, we will first present an analysis of infinitesimal null deformations for the entanglement entropy, which leads to a major simplification of the proof of the C, F and A-theorems in quantum field theory. Next, we will discuss the quantum null energy condition (QNEC) on the light-cone. Finally, we combine these tools in order to establish the irreversibility of renormalization group flows on planar d-dimensional defects, embedded in D-dimensional conformal field theories. This proof completes and unifies all known defect irreversibility theorems for defect dimensions below d=5. The F-theorem on defects (d=3) is a new result using information-theoretic methods. The geometric construction connects the proof of irreversibility with and without defects through the QNEC inequality in the bulk, and makes contact with the proof of strong subadditivity of holographic entropy taking into account quantum corrections.

We probe the multipartite entanglement structure of the vacuum state of a CFT in 1+1 dimensions, using recovery operations that attempt to reconstruct the density matrix in some region from its reduced density matrices on smaller subregions. We use an explicit recovery channel known as the twirled Petz map, and study distance measures such as the fidelity, relative entropy, and trace distance between the original state and the recovered state. One setup we study in detail involves three contiguous intervals A, B and C on a spatial slice, where we can view these quantities as measuring correlations between A and C that are not mediated by the region B that lies between them. We show that each of the distance measures is both UV finite and independent of the operator content of the CFT, and hence depends only on the central charge and the cross-ratio of the intervals. We evaluate these universal quantities numerically using lattice simulations in critical spin chain models, and derive their analytic forms in the limit where A and C are close using the OPE expansion. We also compare the mutual information between various subsystems in the original and recovered states, which leads to a more qualitative understanding of the differences between them. Further, we introduce generalizations of the recovery operation to more than three adjacent intervals, for which the fidelity is again universal with respect to the operator content.