# Adam R. Brown's research while affiliated with Stanford University and other places

## Publications (33)

Preprint
This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group -- often called complexity geometries' following the definition of quantum complexity proposed by Nielsen -- and delineate the equivalence class of metrics that have the same comput...
Preprint
The Bishop-Gromov bound -- a cousin of the focusing lemmas that Hawking and Penrose used to prove their black hole singularity theorems -- is a differential geometry result that upperbounds the rate of growth of volume of geodesic balls in terms of the Ricci curvature. In this paper, I apply the Bishop-Gromov bound to Nielsen's complexity geometry...
Preprint
Post-Wilsonian physics views theories not as isolated points but elements of bigger universality classes, with effective theories emerging in the infrared. This paper makes initial attempts to apply this viewpoint to homogeneous geometries on group manifolds, and complexity geometry in particular. We observe that many homogeneous metrics on low-dim...
Preprint
In [1] we discussed how quantum gravity may be simulated using quantum devices and gave a specific proposal -- teleportation by size and the phenomenon of size-winding. Here we elaborate on what it means to do 'Quantum Gravity in the Lab' and how size-winding connects to bulk gravitational physics and traversable wormholes. Perfect size-winding is...
Article
In Playing Pool with π '' \cite{Galperin}, Galperin invented an extraordinary method to learn the digits of π by counting the collisions of billiard balls. Here I demonstrate an exact isomorphism between Galperin's bouncing billiards and Grover's algorithm for quantum search. This provides an illuminating way to visualize Grover's algorithm.
Article
Full-text available
A bstract According to Harlow and Hayden [ arXiv:1301.4504 ] the task of distilling information out of Hawking radiation appears to be computationally hard despite the fact that the quantum state of the black hole and its radiation is relatively un-complex. We trace this computational difficulty to a geometric obstruction in the Einstein-Rosen brid...
Preprint
In "Playing Pool with $\pi$'', Galperin invented an extraordinary method to learn the digits of $\pi$ by counting the collisions of billiard balls. Here I demonstrate an exact isomorphism between Galperin's bouncing billiards and Grover's algorithm for quantum search. This provides an illuminating way to visualize what Grover's algorithm is actuall...
Preprint
According to Harlow and Hayden [arXiv:1301.4504] the task of distilling information out of Hawking radiation appears to be computationally hard despite the fact that the quantum state of the black hole and its radiation is relatively un-complex. We trace this computational difficulty to a geometric obstruction in the Einstein-Rosen bridge connectin...
Preprint
Full-text available
With the long-term goal of studying quantum gravity in the lab, we propose holographic teleportation protocols that can be readily executed in table-top experiments. These protocols exhibit similar behavior to that seen in recent traversable wormhole constructions: information that is scrambled into one half of an entangled system will, following a...
Article
Full-text available
The computational complexity of a quantum state quantifies how hard it is to make. Complexity geometry, first proposed by Nielsen, is an approach to defining computational complexity using the tools of differential geometry. Here we demonstrate many of the attractive features of complexity geometry using the example of a single qubit, which turns o...
Preprint
The computational complexity of a quantum state quantifies how hard it is to make. Complexity geometry', first proposed by Nielsen, is an approach to defining computational complexity using the tools of differential geometry. Here we demonstrate many of the attractive features of complexity geometry using the example of a single qubit, which turns...
Article
Full-text available
The Jackiw-Teitelboim (JT) model arises from the dimensional reduction of charged black holes. Motivated by the holographic complexity conjecture, we calculate the late-time rate of change of action of a Wheeler-DeWitt patch in the JT theory. Surprisingly, the rate vanishes. This is puzzling because it contradicts both holographic expectations for...
Article
Full-text available
A model of cosmological inflation is proposed in which field space is a hyperbolic plane. The inflaton never slow-rolls, and instead orbits the bottom of the potential, buoyed by a centrifugal force. Though initial velocities redshift away during inflation, in negatively curved spaces angular momentum naturally starts exponentially large and remain...
Preprint
Full-text available
The Jackiw-Teitelboim (JT) model arises from the dimensional reduction of charged black holes. Motivated by the holographic complexity conjecture, we calculate the late-time rate of change of action of a Wheeler-DeWitt patch in the JT theory. Surprisingly, the rate vanishes. This is puzzling because it contradicts both holographic expectations for...
Article
Full-text available
The growth of the "size" of operators is an important diagnostic of quantum chaos. In arXiv:1802.01198 [hep-th] it was conjectured that the holographic dual of the "size" of an operator is the radial component of the momentum of the particle created by the operator. Thus the growth of operators in the background of a black hole is nothing but the a...
Article
Full-text available
The 'thin-wall approximation' gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the approximation is uncontrolled. In this paper I show that there are actually two different thin-wall approximations and that they bracket the true decay rate: I prove that one is an upper bound and the other a lower bound. In the...
Article
A model of cosmological inflation is proposed in which field space is a hyperbolic plane. The inflaton never slow-rolls, and instead orbits the bottom of the potential, buoyed by a centrifugal force. Though initial velocities redshift away during inflation, in negatively curved spaces angular momentum naturally starts exponentially large and remain...
Article
Full-text available
We give arguments for the existence of a thermodynamics of quantum complexity that includes a "Second Law of Complexity". To guide us, we derive a correspondence between the computational (circuit) complexity of a quantum system of $K$ qubits, and the positional entropy of a related classical system with $2^K$ degrees of freedom. We also argue that...
Article
As time passes, once simple quantum states tend to become more complex. For strongly coupled k-local Hamiltonians, this growth of computational complexity has been conjectured to follow a distinctive and universal pattern. In this paper we show that the same pattern is exhibited by a much simpler system: classical geodesics on a compact two-dimensi...
Article
We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in anti–de Sitter spacetime, as well as black holes perturbed with static shells and with shock w...
Article
Full-text available
Electric fields may decay by quantum tunneling: as calculated by Schwinger, an electron-positron pair may be summoned from the vacuum. In this paper I calculate the pair-production rate at nonzero temperatures. I find that at high temperatures the decay rate is dominated by a new instanton that involves both thermal fluctuation and quantum tunnelin...
Article
Our earlier paper "Complexity Equals Action" conjectured that the quantum computational complexity of a holographic state is given by the classical action of a region in the bulk (the `Wheeler-DeWitt' patch). We provide calculations for the results quoted in that paper, explain how it fits into a broader (tensor) network of ideas, and elaborate on...
Article
We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in AdS, as well as black holes perturbed with static shells and with shock waves. This conjecture...
Article
The non-perturbative instabilities of hot Kaluza-Klein spacetime are investigated. In addition to the known instability of hot space (the nucleation of 4D black holes) and the known instability of KK space (the nucleation of bubbles of nothing by quantum tunneling), we find two new instabilities: the nucleation of 5D black holes, and the nucleation...
Article
We investigate a simple extra-dimensional model and its four-dimensional vacua. This model has a two-form flux and a positive cosmological constant, and the extra dimensions are compactified as the product of $N$ two-spheres. The theory is an interesting laboratory because it is at once simple enough to be soluble but rich enough to exhibit many fe...
Article
We study compactifications of $D$-dimensional de Sitter space with a $q$-form flux down to $D-Nq$ dimensions. We show that for $(N-1)(q-1)\geq 2$ there are double-exponentially or even infinitely many compact de Sitter vacua, and that their effective cosmological constants accumulate at zero. This population explosion of $\Lambda \ll 1$ de Sitters...
Article
We study the spectrum and perturbative stability of Freund-Rubin compactifications on $M_p \times M_{Nq}$, where $M_{Nq}$ is itself a product of $N$ $q$-dimensional Einstein manifolds. The higher-dimensional action has a cosmological term $\Lambda$ and a $q$-form flux, which individually wraps each element of the product; the extended dimensions \$M...
Article
Nothing—the absence of spacetime—can be either an endpoint of tunneling, as in the bubble of nothing, or a starting point for tunneling, as in the quantum creation of a universe. We argue that these two tunnelings can be treated within a unified framework, and that, in both cases, nothing should be thought of as the limit of anti-de Sitter space in...
Article
Every de Sitter vacuum can transition to every other de Sitter vacuum despite any obstacle, despite intervening anti-de Sitter sinks, despite not being connected by an instanton. Eternal inflation populates the whole landscape.
Article
There is a standard story about decay in multidimensional flux landscapes: that from any state, the fastest decay is to take a small step, discharging one flux unit at a time; that fluxes with the same coupling constant are interchangeable; and that states with N units of a given flux have the same decay rate as those with -N. We show that this sta...
Article
Instantons are tunneling solutions that connect two vacua, and under a small change in the potential, instantons sometimes disappear. We classify these disappearances as smooth (decay rate goes to 0 at disappearance) or abrupt (decay rate not equal to 0 at disappearance). Abrupt disappearances mean that a small change in the parameters can produce...
Article
The rate and manner of vacuum decay are calculated in an explicit flux compactification, including all thick-wall and gravitational effects. For landscapes built of many units of a single flux, the fastest decay is usually to discharge just one unit. By contrast, for landscapes built of a single unit each of many different fluxes, the fastest decay...
Article
For landscapes of field theory vacua, we identify an effect that can greatly enhance the decay rates to wildly distant minima--so much so that such transitions may dominate over transitions to near neighbors. We exhibit these 'giant leaps' in both a toy two-field model and, in the thin-wall approximation, amongst the four-dimensional vacua of 6D Ei...

## Citations

... A particularly interesting question is how this information storage capability compares to quantum systems [9] which typically involve entanglement [10]. Recent work has shown that certain non-quantum entanglements [11] can be realized in classical light fields [12][13][14][15] and coupled billiard balls [16]. However, the problem of creating analogues of more general entanglements in classical continuum systems continues to present challenges [10,17,18]. ...
... This also connects to recent discussions of complexity and the 'python's lunch' [43,44]. The spatial slice depicted in (1.4) is an example of a python, and the region enclosed by the extremal surface γ is the 'lunch'. ...
... (For a pedagogical introduction see [8]; for other recent work see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]) This metric assigns a length to infinitesimal paths through the space of unitaries; the length of a noninfinitesimal path is the sum of the lengths of its segments; and the distance between two unitaries is the length of the shortest path that connects them. This expression is manifestly still right-invariant but is in general not left-invariant. ...
... JT gravity also describes the near extremal behaviour of the higher dimensional black holes upon dimensional reduction. Following the CA conjecture, complexity growth of the two dimensional JT gravity has been discussed widely [50,51]. It has been shown that the late time complexity growth rate for the two dimensional dilatonic model is non vanishing only when we consider them upon dimensional reduction of higher dimensional action with the Maxwell boundary term. ...
... In this chapter, we consider a cosmological model in which the gravitational action integral is that of Brans-Dicke theory with an additional scalar field minimally coupled to gravity [227,228]. This two-scalar field model belongs to the family of multi-scalar field models, which have been used as unified dark energy models [229,230,231] or as alternative models for the description of the acceleration phases of the universe [232,233,234,235]. Indeed, multifield inflationary models provide an alternative mechanism for the description of the early acceleration phase of the universe. ...
... Quantum error correction is intimately linked to the appearance of gravity, geometry and bulk locality from holographic quantum mechanical systems, as was emphasized in the pioneering work of [2][3][4]. On the other hand, much work has been done to understand the emergence of a bulk dual in terms of operator size growth and complexity [1,[5][6][7][8][9]. In this work, we attempt to relate these two concepts, using the Sachdev-Ye-Kitaev (SYK) model as an illustrative example [10][11][12]. ...
... While the thin-wall prescription provides a solid upper bound on the bounce action [43], it does not provide any useful estimation on the actual value on the bounce in our case. This fact calls for an alternative way to estimate the action, mostly for higher-dimensional landscapes. ...
... Therefore, it is important to understand which properties of multi-field models of inflation lead to predictions compatible with the observational data. At the same time, it would be very desirable to have a simple framework for multi-field inflation in highly curved field spaces and trajectories (see this list for recent developments [7][8][9][10][11][12][13][14][15][16][17][18][19][20].) ...
Citing article
... The ability to sample from the Haar distribution of unitaries on a Hilbert space H is a widely useful capability [1-3] and is often a crucial ingredient in modern quantum information processing tasks including randomized benchmarking [4][5][6] and demonstrations of quantum advantage [7] in noisy intermediate-scale quantum processing devices. More generally, Haar-random unitaries and related matrix distributions are foundational to many areas of modern quantum information science, where they play central roles in our understanding of quantum information scrambling [8,9], quantum chaos in many-body thermalizing systems [10][11][12][13], and models of black hole dynamics in holographic quantum gravity [14][15][16][17][18]. However, it is known that for N qubits, the number of elementary 2-qubit gates needed to generate samples from the Haar distribution is exponential in N [19]. ...
... (1. 6) Recently in [13], as possible candidates for a gravitational dual of complexity an infinite family of gravitational observables on codimension-one slices of the geometry is introduced. Holographic complexity has been recently studied in various asymptotically AdS backgrounds [14][15][16][17][18][19][20][21] and also in their deformations [22][23][24][25][26]. It was also computed in the presence of defects and boundaries [27][28][29][30][31][32]. ...