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In this talk I will present recent results about the computational complexity of determining homology groups of simplicial complexes, a fundamental task in computational topology. In arXiv:2209.11793 we showed that this decision problem is QMA1-hard. Moreover, we showed that a version of the problem satisfying a suitable promise is contained in QMA. This suggests that the seemingly classical problem may in fact be quantum mechanical. In fact, we were able to significantly strengthen this by showing that the problem remains QMA1-hard in the case of clique complexes, a family of simplicial complexes specified by a graph which is relevant to the problem of topological data analysis. The proof combines a number of techniques from Hamiltonian complexity and homological algebra, and is inspired by a link with supersymmetric quantum mechanics. In this talk I will focus on how the link with supersymmetry inspired the result, and explain the intuition behind the proof.

Recently, Akers et al. proposed a non-isometric holographic map from the interior of a black hole to its exterior. Within this model, we study properties of the black hole S-matrix, which are in principle accessible to observers who stay outside the black hole. Specifically, we investigate a scenario in which an infalling agent interacts with radiation both outside and inside the black hole. Because the holographic map involves postselection, the unitarity of the S-matrix is not guaranteed in this scenario, but we find that unitarity is satisfied to very high precision if suitable conditions are met. If the internal black hole dynamics is described by a pseudorandom unitary transformation, and if the operations performed by the infaller have computational complexity scaling polynomially with the black hole entropy, then the S-matrix is unitary up to corrections that are superpolynomially small in the black hole entropy. Furthermore, while in principle quantum computation assisted by postselection can be very powerful, we find under similar assumptions that the S-matrix of an evaporating black hole has polynomial computational complexity.

The JLMS formula is a cornerstone in our understanding of bulk reconstruction in holographic theories of quantum gravity, best interpreted as a quantum error-correcting code. Moreover, recent work has highlighted the importance of understanding holography as an approximate and perhaps non-isometric code. In this work, we construct an enlarged code subspace for the bulk theory that contains multiple non-perturbatively different background geometries. In such a large holographic code, we carefully derive an approximate version of the JLMS formula from an approximate FLM formula for a class of nice states. We do not assume that the code is isometric, but interestingly find that approximate FLM forces the code to be approximately isometric. Furthermore, we show that the bulk modular Hamiltonian of the entanglement wedge makes important contributions to the JLMS formula and cannot in general be neglected even when the bulk state is semiclassical. Nevertheless, when acting on states with the same background geometry, we find that the modular flow is well approximated by the area flow which takes the geometric form of a boundary-condition-preserving kink transform. We also generalize the results to higher derivative gravity, where area is replaced by the geometric entropy. We conjecture that a Lorentzian definition of the geometric entropy is equivalent to its original, Euclidean definition, and we verify this conjecture in a dilaton theory with higher derivative couplings. Thus we find that the flow generated by the geometric entropy takes the universal form of a boundary-condition-preserving kink transform.

We construct approximately local observables, or "overlapping qubits", using non-isometric maps and show that processes in local effective theories can be spoofed with a quantum system with fewer degrees of freedom, similar to our expectation in holography. Furthermore, the spoofed system naturally deviates from an actual local theory in ways that can be identified with features in quantum gravity. For a concrete example, we construct two MERA toy models of de Sitter space-time and explain how the exponential expansion in global de Sitter can be spoofed with many fewer quantum degrees of freedom and that local physics may be approximately preserved for an exceedingly long time before breaking down. Conceptually, we comment on how approximate overlapping qubits connect Hilbert space dimension verification, degree-of-freedom counting in black holes and holography, approximate locality in quantum gravity, non-isometric codes, and circuit complexity.

In this paper, we explore the possibility of building a quantum memory that is robust to thermal noise using large N matrix quantum mechanics models. First, we investigate the gauged SU(N) matrix harmonic oscillator and different ways to encode quantum information in it. By calculating the mutual information between the system and a reference which purifies the encoded information, we identify a transition temperature, Tc, below which the encoded quantum information is protected from thermal noise for a memory time scaling as N^2. Conversely, for temperatures higher than T_c, the information is quickly destroyed by thermal noise. Second, we relax the requirement of gauge invariance and study a matrix harmonic oscillator model with only global symmetry. Finally, we further relax even the symmetry requirement and propose a model that consists of a large number N^2 of qubits, with interactions derived from an approximate SU(N) symmetry. In both ungauged models, we find that the effects of gauging can be mimicked using an energy penalty to give a similar result for the memory time. The final qubit model also has the potential to be realized in the laboratory.

The Sachdev-Ye-Kitaev (SYK) model is a simple toy model of holography that has seen widespread study in the area of quantum gravity. It is particularly notable for its feasibility of simulation on near-term quantum devices. Recently, Swingle et al. introduced a sparsified version of the SYK model and analyzed its holographic properties, which are remarkably robust to deletion of Majorana interaction terms. Here we analyze its spectral and quantum chaotic properties as they pertain to holography as well as how they scale with sparsity and system size through large scale numerics. We identify at least two transition points at which features of chaos and holography are lost as the model is sparsified, and above which all important features are preserved, which may serve as guidelines for future experiments to simulate quantum gravity. Additionally, we apply these analyses to the SYK model that was recently experimentally simulated on the Google Sycamore quantum processor, which itself was a highly sparsified SYK model obtained through a machine learning algorithm incorporating mutual information signatures of a traversable wormhole.

66 - It has been proposed that the exponential decay and subsequent power law saturation of out-of-time-order correlation functions can be universally described by collective 'scramblon' modes. We develop this idea from a path integral perspective in several examples, thereby establishing a general formalism. After reformulating previous work on the Schwarzian theory and identity conformal blocks in two-dimensional CFTs relevant for systems in the infinite coupling limit with maximal quantum Lyapunov exponent, we focus on theories with sub-maximal chaos: we study the large-q limit of the SYK quantum dot and chain, both of which are amenable to analytical treatment at finite coupling. In both cases we identify the relevant scramblon modes, derive their effective action, and find bilocal vertex functions, thus constructing an effective description of chaos. The final results can be matched in detail to stringy corrections to the gravitational eikonal S-matrix in holographic CFTs, including a stringy Regge trajectory, bulk to boundary propagators, and multi-string effects that are unexplored holographically.

The Spectral Form Factor is an important diagnostic of level repulsion Random Matrix Theory (RMT) and quantum chaos. The short-time behavior of the SFF as it approaches the RMT result acts as a diagnostic of the ergodicity of the system as it approaches the thermal state. In this work we observe that for systems without time-reversal symmetry, there is a second break from the RMT result at late times: specifically at the Heisenberg Time $T_H=2\pi \rho$. That is to say that after agreeing with the RMT result to exponential precision for an amount of time exponential in the system size, the spectral form factor of a large system will very briefly deviate in a way exactly determined by its early time thermalization properties. The conceptual reason for this is the Riemann-Siegel Lookalike formula, a resummed expression for the spectral determinant relating late time behavior to early time spectral statistics. We use the lookalike formula to derive a precise expression for the late time SFF for semiclassical systems, and then confirm our results numerically. We find that at late times, the various modes act on the SFF via repeated, which may give hints as to the analogous behavior for systems with time-reversal symmetry.

We consider the quantum gravity partition function that counts the dimension of the Hilbert space of a spatial region with topology of a ball and fixed proper volume, and evaluate it in the leading order saddle point approximation. The result is the exponential of the Bekenstein-Hawking entropy associated with the area of the saddle ball boundary, and is reliable within effective field theory provided the mild curvature singularity at the ball boundary is regulated by higher curvature terms. This generalizes the classic Gibbons-Hawking computation of the de Sitter entropy for the case of positive cosmological constant and unconstrained volume, and hence exhibits the holographic nature of nonperturbative quantum gravity in generic finite volumes of space.

We present a construction in which the origin of black hole entropy gets clarified. We start by building an infinite family of geometric microstates for black holes in general relativity. This construction naively overcounts the Bekenstein-Hawking entropy. We then describe how wormholes in the Euclidean path integral for gravity cause these states to have exponentially small, but universal, overlaps. These overlaps recontextualize the Gibbons-Hawking thermal partition function. We finally show how these results imply that the microstates span a Hilbert space of log dimension equal to the Bekenstein-Hawking entropy, and how they clarify the nature of the volumes of Eisntein-Rosen bridges.