Surely it is hubris on my part to presume to co-opt the title of the venerable online column “This Week’s Finds in Mathematical Physics” written for nearly seventeen years by the great doyen of mathematical communication – John Baez. However, not finding any other takers for this task I figured … why not? So here is the first in a hopefully long line of posts on “*This Week-ish in Theoretical Physics*“.

# Observing Quantum Mechanical Collapse

In [1], Minev and collaborators have accomplished something quite remarkable: observing – yes, that’s right, observing – the collapse of a quantum mechanical wavefunction into an eigenstate and demonstrating that, contrary to the Copenhagen interpretation – wavefunction collapse is a gradual, continuous (at least on the resolution scales of the experiment), reversible process. It is hard to overstate the significance of this work. It clearly demonstrates that contrary to Bohr and in agreement with Einstein, quantum mechanical collapse is not a magical event which can never be described in physical terms. Instead, they show that wavefunction collapse is a physical *process*, like any other, and therefore, like any other process, it can be observed, monitored, controlled and even reversed.

There is a substantial amount of literature on this model of wavefunction collapse which goes by the name of “quantum trajectories” (see e.g. [2] for an accessible introduction) and is also studied under the title of “quantum state diffusion” [3, 4]

Now the million, nay – billion, dollar question is what is the *governing dynamics* of wavefunction collapse. Phenomenologically this process can be modeled using the framework of open quantum systems and the Lindblad equation [5]. However, this description is only at the level of an “effective theory”. Ultimately some deeper physical principles should determine the manner in which collapse happens. It is possible that the answer may lie in the quantum geometric picture of spacetime which arises from LQG (loop quantum gravity).

In any case, this work is sure to set off a frenzy of work in models of stochastic collapse related to quantum gravity in some way or another.

# Quantum Computation as Gravity

That is the title of a recent, very elegant paper [6] by Pawel Caputa and Javier Magan. It didn’t come out this week-ish, but I learnt of it this week-ish, therefore it qualifies! Let me quote the abstract:

We formulate Nielsen’s geometric approach to complexity in the context of two-dimensional conformal field theories, where series of conformal transformations are interpreted as ”unitary circuits”. We show that the complexity functional can be written as the Polyakov action of two-dimensional gravity or, equivalently, as the geometric action on the coadjoint orbits of the Virasoro group. This way, we argue that gravity sets the rules for optimal quantum computation in conformal field theories.

This work is about one of the hottest quantum gravity related lines of research which involves notions of complexity, quantum computation and, of-course, gravity.

## Background: Optimal Quantum Circuits

It all goes back to a paper [7] by Michael Nielsen – of “Nielsen and Chuang” [8] fame – from 2005. In this work Nielsen addressed the question of quantifying the minimum complexity – measured in terms of the number of primitive gates – a quantum computational circuit must have in order to implement a generic unitary operation on an $n$ qubit state.

Assuming the qubits in question are spin 1/2 systems, the space of unitary operations on a single qubit is given by the group $SU(2)$ and therefore the space of unitary operations on $n$ such qubits lives in the manifold $\mc{M} = SU(2^n)$ ^{1}. The question of finding the optimal way to generate an arbitrary element of $\mc{U} \in \mc{M}$ becomes equivalent to finding the shortest path from the identity element $\mbb{1}_n$ (the “origin” of the manifold) to the point $\mc{U}$.

To determine distances between different points of $\mc{M}$ one needs a metric on this manifold. Nielsen showed that one can define a suitable Finsler metric ^{2} which satisfies the desired properties (continuity, positivity, triangle inequality, etc.) on $\mc{M}$, such that a geodesic connecting any two points $\mc{U}, \mc{U’}$ of $\mc{M}$ can be viewed as the shortest possible “path” one can follow to generate the unitary $\mc{U’}$ starting from the unitary $\mc{U}$. That this is a remarkable result needs hardly be emphasized.

## Complexity = Action

For one, it immediately suggests a direct link between the geometry encoded in spin-networks to quantum computation and quantum control theory (more on this in an upcoming post). Secondly, it suggests that it should be possible to write down an action principle whose extremization yields an equation of motion whose trajectory yields the optimal path for generating any unitary operator starting from any other unitary operator. Third, these ideas have recently been exploited in work by Susskind and collaborators in order to formulate the so-called “Complexity = Action” conjecture [9, 10, 11, 12, 13, 14, 15] according to which the complexity of the holographic dual of a bulk geometry is equal to the action defined over a region of the bulk known as the Wheeler-deWitt patch. The connection between the second and third points is quite clear.

## CFT Complexity = 2D Gravity

This brief discussion brings us to the Caputa-Magan paper [6]. What the duo have shown can be summarized as follows:

- Conformal transformations in 2D CFTs can be viewed as being composed of a series of gates which belong to the Virasoro group.
- The corresponding action for the Nielsen complexity of a conformal transformation can be expressed entirely in terms of the central charge $c$ of the CFT.
- For large $c$, the Nielsen complexity becomes identical to the Polyakov action for two-dimensional gravity.

This represents a concrete realization of the hope expressed in the last section that there should exist an action whose extremization yields the Nielsen complexity. This paper is a very concrete link for the correspondence, as advocated by Susskind: $ GR = QM $.

That’s it for this first episode of “This Week-ish in Theoretical Physics”. Ciao!

- If the Hilbert space of a single system is $d$ dimensional, then the Hilbert space of $n$ such systems is $d^n$ dimensional. Thus, a vector in the $n$ spin 1/2 qubit state space will be of length $2^n$ and operators acting on these states will be represented by $ 2^n \times 2^n$ dimensional matrices. A unitary operator acting on such $n$ qubit states would therefore be an element of $SU(2^n)$. ↩
- A Finsler geometry is a generalization of Riemannian geometry where the restriction that the metric should be a quadratic form, on the tangent space of the manifold, is dropped. ↩

[Bibtex]

```
@article{Minev2019To-catch,
Abstract = {Quantum physics was invented to account for two fundamental features of measurement results -- their discreetness and randomness. Emblematic of these features is Bohr's idea of quantum jumps between two discrete energy levels of an atom. Experimentally, quantum jumps were first observed in an atomic ion driven by a weak deterministic force while under strong continuous energy measurement. The times at which the discontinuous jump transitions occur are reputed to be fundamentally unpredictable. Can there be, despite the indeterminism of quantum physics, a possibility to know if a quantum jump is about to occur or not? Here, we answer this question affirmatively by experimentally demonstrating that the jump from the ground to an excited state of a superconducting artificial three-level atom can be tracked as it follows a predictable "flight," by monitoring the population of an auxiliary energy level coupled to the ground state. The experimental results demonstrate that the jump evolution when completed is continuous, coherent, and deterministic. Furthermore, exploiting these features and using real-time monitoring and feedback, we catch and reverse a quantum jump mid-flight, thus deterministically preventing its completion. Our results, which agree with theoretical predictions essentially without adjustable parameters, support the modern quantum trajectory theory and provide new ground for the exploration of real-time intervention techniques in the control of quantum systems, such as early detection of error syndromes.},
Archiveprefix = {arXiv},
Arxivid = {1803.00545},
Author = {Minev, Z. K. and Mundhada, S. O. and Shankar, S. and Reinhold, P. and Guti{\'{e}}rrez-J{\'{a}}uregui, R. and Schoelkopf, R. J. and Mirrahimi, M. and Carmichael, H. J. and Devoret, M. H.},
Date-Added = {2019-06-04 11:16:40 +0530},
Date-Modified = {2019-06-04 11:16:41 +0530},
Doi = {10.1038/s41586-019-1287-z},
Eprint = {1803.00545},
File = {:Users/deepak/ownCloud/root/research/mendeley/Minev et al.{\_}To catch and reverse a quantum jump mid-flight{\_}2019.pdf:pdf},
Issn = {0028-0836},
Journal = {Nature},
Month = {jun},
Pages = {1},
Title = {{To catch and reverse a quantum jump mid-flight}},
Url = {http://www.nature.com/articles/s41586-019-1287-z https://www.nature.com/articles/s41586-019-1287-z http://arxiv.org/abs/1803.00545},
Year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1038/s41586-019-1287-z}}
```

[Bibtex]

```
@article{Brun2002A-simple,
Abstract = {Quantum trajectory theory, developed largely in the quantum optics community to describe open quantum systems subjected to continuous monitoring, has applications in many areas of quantum physics. In this paper I present a simple model, using two-level quantum systems (q-bits), to illustrate the essential physics of quantum trajectories and how different monitoring schemes correspond to different ``unravelings'' of a mixed state master equation. I also comment briefly on the relationship of the theory to the Consistent Histories formalism and to spontaneous collapse models.},
Archiveprefix = {arXiv},
Arxivid = {quant-ph/0108132},
Author = {Brun, Todd A.},
Date-Added = {2019-06-04 11:31:30 +0530},
Date-Modified = {2019-06-04 11:31:31 +0530},
Doi = {10.1119/1.1475328},
Eprint = {0108132},
File = {:Users/deepak/ownCloud/root/research/mendeley/Brun{\_}A simple model of quantum trajectories{\_}2002.pdf:pdf},
Issn = {0002-9505},
Journal = {American Journal of Physics},
Mendeley-Groups = {Random Walks Spin Networks,Quantum Foundations},
Month = {jul},
Number = {7},
Pages = {719--737},
Primaryclass = {quant-ph},
Title = {{A simple model of quantum trajectories}},
Url = {http://arxiv.org/abs/quant-ph/0108132 http://dx.doi.org/10.1119/1.1475328 http://arxiv.org/abs/quant-ph/0108132{\%}0Ahttp://dx.doi.org/10.1119/1.1475328 http://aapt.scitation.org/doi/10.1119/1.1475328},
Volume = {70},
Year = {2002},
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Bdsk-Url-1 = {https://doi.org/10.1119/1.1475328}}
```

[Bibtex]

```
@article{Gisin1997Quantum,
Abstract = {Deeper insight leads to better practice. We show how the
study of the foundations of quantum mechanics has led to
new pictures of open systems and to a method of computation
which is practical and can be used where others cannot. We
illustrate the power of the new method by a series of
pictures that show the emergence of classical features in a
quantum world. We compare the development of quantum
mechanics and of the theory of (biological) evolution.},
Archiveprefix = {arXiv},
Arxivid = {quant-ph/9701024},
Author = {Gisin, Nicolas and Percival, Ian C},
Date-Added = {2018-09-22 23:49:05 +0530},
Date-Modified = {2018-09-22 23:49:08 +0530},
Eprint = {9701024},
File = {:Users/deepak/mendeley/Gisin, Percival{\_}Quantum State Diffusion from Foundations to Applications{\_}1997.pdf:pdf;:Users/deepak/mendeley/Gisin, Percival{\_}Quantum State Diffusion from Foundations to Applications{\_}1997(2).pdf:pdf},
Mendeley-Groups = {Random Walks Spin Networks,Emergent QM},
Month = {jan},
Primaryclass = {quant-ph},
Title = {{Quantum State Diffusion: from Foundations to Applications}},
Url = {http://arxiv.org/abs/quant-ph/9701024},
Year = {1997},
Bdsk-File-1 = {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},
Bdsk-Url-1 = {http://arxiv.org/abs/quant-ph/9701024}}
```

[Bibtex]

```
@article{Pearle2012Simple,
Abstract = {The Lindblad equation is an evolution equation for the
density matrix in quantum theory. It is the general linear,
Markovian, form which ensures that the density matrix is
hermitian, trace 1, positive and completely positive. Some
elementary examples of the Lindblad equation are given. The
derivation of the Lindblad equation presented here is
"simple" in that all it uses is the expression of a
hermitian matrix in terms of its orthonormal eigenvectors
and real eigenvalues. Thus, it is appropriate for students
who have learned the algebra of quantum theory. Where
helpful, arguments are first given in a two-dimensional
hilbert space.},
Archiveprefix = {arXiv},
Arxivid = {1204.2016},
Author = {Pearle, Philip},
Date-Added = {2018-04-13 07:19:39 +0000},
Date-Modified = {2018-04-13 12:49:41 +0530},
Doi = {10.1088/0143-0807/33/4/805},
Eprint = {1204.2016},
File = {:Users/deepak/mendeley/Pearle{\_}Simple derivation of the Lindblad equation{\_}2012.pdf:pdf},
Isbn = {0143-0807$\backslash$n1361-6404},
Issn = {01430807},
Journal = {European Journal of Physics},
Keywords = {lindblad equation,lindbladian,markov process,open quantum systems,pedagogical},
Mendeley-Groups = {Emergent QM},
Mendeley-Tags = {lindbladian,open quantum systems,pedagogical,lindblad equation,markov process},
Month = {apr},
Number = {4},
Pages = {805--822},
Title = {{Simple derivation of the Lindblad equation}},
Url = {http://arxiv.org/abs/1204.2016 http://dx.doi.org/10.1088/0143-0807/33/4/805},
Volume = {33},
Year = {2012},
Bdsk-File-1 = {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},
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Bdsk-Url-2 = {https://doi.org/10.1088/0143-0807/33/4/805}}
```

[Bibtex]

```
@article{Caputa2018Quantum,
Abstract = {We formulate Nielsen's geometric approach to complexity in the context of two dimensional conformal field theories, where series of conformal transformations are interpreted as unitary circuits. We show that the complexity functional can be written as the Polyakov action of two dimensional gravity or, equivalently, as the geometric action on the coadjoint orbits of the Virasoro group. This way, we argue that gravity sets the rules for optimal quantum computation in conformal field theories.},
Archiveprefix = {arXiv},
Arxivid = {1807.04422},
Author = {Caputa, Pawel and Magan, Javier M.},
Date-Added = {2019-05-22 10:16:45 +0530},
Date-Modified = {2019-05-22 10:16:46 +0530},
Eprint = {1807.04422},
File = {:Users/deepak/ownCloud/root/research/mendeley/Caputa, Magan{\_}Quantum Computation as Gravity{\_}2018.pdf:pdf},
Mendeley-Groups = {Computational Universe},
Month = {jul},
Title = {{Quantum Computation as Gravity}},
Url = {http://arxiv.org/abs/1807.04422},
Year = {2018},
Bdsk-File-1 = {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},
Bdsk-Url-1 = {http://arxiv.org/abs/1807.04422}}
```

[Bibtex]

```
@article{Nielsen2005A-geometric,
Abstract = {What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on SU(2{\^{}}n). The geodesic curves of such a metric have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size, and give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call Pauli geodesics, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.},
Archiveprefix = {arXiv},
Arxivid = {quant-ph/0502070},
Author = {Nielsen, Michael A.},
Date-Added = {2019-05-20 19:26:21 +0530},
Date-Modified = {2019-05-20 19:26:22 +0530},
Eprint = {0502070},
File = {:Users/deepak/ownCloud/root/research/mendeley/Nielsen{\_}A geometric approach to quantum circuit lower bounds{\_}2005.pdf:pdf},
Mendeley-Groups = {Quantum Computation},
Month = {feb},
Primaryclass = {quant-ph},
Title = {{A geometric approach to quantum circuit lower bounds}},
Url = {http://arxiv.org/abs/quant-ph/0502070},
Year = {2005},
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```

[Bibtex]

```
@book{Nielsen2000Quantum,
Abstract = {{In this first comprehensive introduction to the main
ideas and techniques of quantum computation and
information, Michael Nielsen and Isaac Chuang ask the
question: What are the ultimate physical limits to
computation and communication? They detail such remarkable
effects as fast quantum algorithms, quantum teleportation,
quantum cryptography and quantum error correction. A wealth
of accompanying figures and exercises illustrate and
develop the material in more depth. They describe what a
quantum computer is, how it can be used to solve problems
faster than familiar "classical" computers, and the
real-world implementation of quantum computers. Their book
concludes with an explanation of how quantum states can be
used to perform remarkable feats of communication, and of
how it is possible to protect quantum states against the
effects of noise. }},
Author = {Nielsen, Michael A. and Chuang, Isaac L.},
Citeulike-Article-Id = {541803},
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Citeulike-Linkout-9 = {http://www.librarything.com/isbn/521635039},
Date-Added = {2010-09-26 17:38:59 -0400},
Date-Modified = {2010-09-27 03:08:59 +0530},
Day = {23},
Edition = {1},
Howpublished = {Paperback},
Isbn = {0521635039},
Keywords = {book, quantum\_computation, quantum\_information, textbook},
Month = {October},
Priority = {2},
Publisher = {Cambridge University Press},
Title = {Quantum Computation and Quantum Information},
Url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20\&path=ASIN/0521635039},
Year = {2000},
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```

[Bibtex]

```
@article{Brown2016aComplexity,
Abstract = {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 evolved from a previous conjecture that
complexity is dual to spatial volume, but appears to be a
major improvement over the original. In light of our
results, we discuss the hypothesis that black holes are the
fastest computers in nature.},
Archiveprefix = {arXiv},
Author = {Brown, Adam R. and Roberts, Daniel A. and Susskind, Leonard and Swingle, Brian and Zhao, Ying},
Citeulike-Article-Id = {14037419},
Citeulike-Linkout-0 = {http://arxiv.org/abs/1509.07876},
Citeulike-Linkout-1 = {http://arxiv.org/pdf/1509.07876},
Citeulike-Linkout-2 = {http://dx.doi.org/10.1103/physrevlett.116.191301},
Date-Added = {2016-07-05 01:08:11 +0000},
Date-Modified = {2016-07-05 09:13:12 +0530},
Day = {10},
Doi = {10.1103/physrevlett.116.191301},
Eprint = {1509.07876},
Issn = {0031-9007},
Journal = {Physical Review Letters},
Keywords = {adscft, black\_hole\_interior, brown\_adam, computational\_complexity, computational\_universe, entanglement\_entropy, quantum\_gravity, roberts\_daniel, susskind\_leonard, swingle\_brian, tensor\_networks, wheeler-dewitt\_patch, zhao\_ying},
Month = may,
Number = {19},
Pages = {191301+},
Posted-At = {2016-07-05 02:07:43},
Priority = {2},
Publisher = {American Physical Society},
Title = {Complexity Equals Action},
Url = {http://dx.doi.org/10.1103/physrevlett.116.191301},
Volume = {116},
Year = {2016},
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Bdsk-Url-1 = {http://dx.doi.org/10.1103/physrevlett.116.191301}}
```

[Bibtex]

```
@article{Brown2016bComplexity,
Abstract = {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 the hypothesis that black holes are the
fastest computers in nature.},
Archiveprefix = {arXiv},
Author = {Brown, Adam R. and Roberts, Daniel A. and Susskind, Leonard and Swingle, Brian and Zhao, Ying},
Citeulike-Article-Id = {13886236},
Citeulike-Linkout-0 = {http://arxiv.org/abs/1512.04993},
Citeulike-Linkout-1 = {http://arxiv.org/pdf/1512.04993},
Citeulike-Linkout-2 = {http://dx.doi.org/10.1103/physrevd.93.086006},
Date-Added = {2016-07-05 03:42:53 +0000},
Date-Modified = {2016-07-05 09:13:25 +0530},
Day = {10},
Doi = {10.1103/physrevd.93.086006},
Eprint = {1512.04993},
Issn = {2470-0010},
Journal = {Physical Review D},
Keywords = {black\_holes, brown\_adam, computational\_complexity, computational\_universe, entanglement\_entropy, quantum\_gravity, roberts\_daniel, scrambling, shock\_waves, susskind\_leonard, swingle\_brian, tensor\_networks, wheeler-dewitt\_patch, zhao\_ying},
Month = may,
Number = {8},
Posted-At = {2016-07-05 04:42:36},
Priority = {2},
Title = {Complexity, action, and black holes},
Url = {http://dx.doi.org/10.1103/physrevd.93.086006},
Volume = {93},
Year = {2016},
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```

[Bibtex]

```
@article{Brown2017Quantum,
Abstract = {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-dimensional geometry of uniform negative curvature. This striking parallel persists whether the system is allowed to evolve naturally or is perturbed from the outside.},
Archiveprefix = {arXiv},
Arxivid = {1608.02612},
Author = {Brown, Adam R. and Susskind, Leonard and Zhao, Ying},
Date-Added = {2019-05-20 19:20:51 +0530},
Date-Modified = {2019-05-20 19:21:49 +0530},
Doi = {10.1103/PhysRevD.95.045010},
Eprint = {1608.02612},
File = {:Users/deepak/ownCloud/root/research/mendeley/Brown, Susskind, Zhao{\_}Quantum complexity and negative curvature{\_}2017.pdf:pdf},
Issn = {24700029},
Journal = {Physical Review D},
Mendeley-Groups = {AdS/CFT},
Month = {aug},
Number = {4},
Title = {{Quantum complexity and negative curvature}},
Url = {http://arxiv.org/abs/1608.02612},
Volume = {95},
Year = {2017},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevD.95.045010},
Bdsk-Url-2 = {http://arxiv.org/abs/1608.02612}}
```

[Bibtex]

```
@article{Brown2017The-Second,
Abstract = {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 the kinetic entropy of the
classical system is equivalent to the Kolmogorov complexity
of the quantum Hamiltonian. We observe that the expected
pattern of growth of the complexity of the quantum system
parallels the growth of entropy of the classical system. We
argue that the property of having less-than-maximal
complexity (uncomplexity) is a resource that can be
expended to perform directed quantum computation. Although
this paper is not primarily about black holes, we find a
surprising interpretation of the uncomplexity-resource as
the accessible volume of spacetime behind a black hole
horizon.},
Archiveprefix = {arXiv},
Arxivid = {1701.01107},
Author = {Brown, Adam R. and Susskind, Leonard},
Date-Added = {2018-06-23 17:05:03 +0000},
Date-Modified = {2018-06-23 22:35:05 +0530},
Doi = {10.1103/PhysRevD.97.086015},
Eprint = {1701.01107},
File = {:Users/deepak/mendeley/Brown, Susskind{\_}The Second Law of Quantum Complexity{\_}2017.pdf:pdf},
Issn = {2470-0010},
Journal = {Physical Review D},
Month = {jan},
Number = {8},
Pages = {086015},
Title = {{The Second Law of Quantum Complexity}},
Url = {https://link.aps.org/doi/10.1103/PhysRevD.97.086015 http://arxiv.org/abs/1701.01107 http://dx.doi.org/10.1103/PhysRevD.97.086015},
Volume = {97},
Year = {2017},
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevD.97.086015}}
```

[Bibtex]

```
@article{Susskind2018PiTP,
Abstract = {This is the first of three PiTP lectures on complexity and
its role in black hole physics.},
Archiveprefix = {arXiv},
Arxivid = {1808.09941},
Author = {Susskind, Leonard},
Date-Added = {2018-10-19 23:14:46 +0530},
Date-Modified = {2019-03-12 15:45:46 +0530},
Eprint = {1808.09941},
File = {:Users/deepak/mendeley/Susskind{\_}PiTP Lectures on Complexity and Black Holes{\_}2018.pdf:pdf},
Title = {{PiTP Lectures on Complexity and Black Holes}},
Url = {https://arxiv.org/abs/1808.09941},
Year = {2018},
Bdsk-Url-1 = {https://arxiv.org/abs/1808.09941}}
```

[Bibtex]

```
@article{Susskind2018Three,
Abstract = {Given at PiTP 2018 summer program entitled "From Qubits to Spacetime." The first lecture describes the meaning of quantum complexity, the analogy between entropy and complexity, and the second law of complexity. Lecture two reviews the connection between the second law of complexity and the interior of black holes. I discuss how firewalls are related to periods of non-increasing complexity which typically only occur after an exponentially long time. The final lecture is about the thermodynamics of complexity, and "uncomplexity" as a resource for doing computational work. I explain the remarkable power of "one clean qubit," in both computational terms and in space-time terms. The lectures can also be found online at $\backslash$url{\{}https://static.ias.edu/pitp/2018/node/1796.html{\}} .},
Archiveprefix = {arXiv},
Arxivid = {1810.11563},
Author = {Susskind, Leonard},
Date-Added = {2019-05-08 00:55:32 +0530},
Date-Modified = {2019-05-08 00:55:33 +0530},
Eprint = {1810.11563},
File = {:Users/deepak/ownCloud/root/research/mendeley/Susskind{\_}Three Lectures on Complexity and Black Holes{\_}2018.pdf:pdf},
Mendeley-Groups = {Computational Universe},
Title = {{Three Lectures on Complexity and Black Holes}},
Url = {http://arxiv.org/abs/1810.11563 https://arxiv.org/abs/1810.11563},
Year = {2018},
Bdsk-Url-1 = {http://arxiv.org/abs/1810.11563%20https://arxiv.org/abs/1810.11563}}
```

[Bibtex]

```
@article{Susskind2018Why-do-Things,
Abstract = {This is the first of several short notes in which I will describe phenomena that illustrate GR=QM. In it I explain that the gravitational attraction that a black hole exerts on a nearby test object is a consequence of a fundamental law of quantum mechanics---the tendency for complexity to grow. It will also be shown that the Einstein bound on velocities is closely related to the quantum-chaos bound of Maldacena, Shenker, and Stanford.},
Archiveprefix = {arXiv},
Arxivid = {1802.01198},
Author = {Susskind, Leonard},
Date-Added = {2019-05-08 00:55:32 +0530},
Date-Modified = {2019-05-08 00:55:33 +0530},
Eprint = {1802.01198},
File = {:Users/deepak/ownCloud/root/research/mendeley/Susskind{\_}Why do Things Fall{\_}2018.pdf:pdf},
Mendeley-Groups = {Computational Universe},
Title = {{Why do Things Fall?}},
Url = {http://arxiv.org/abs/1802.01198 https://arxiv.org/abs/1802.01198},
Year = {2018},
Bdsk-Url-1 = {http://arxiv.org/abs/1802.01198%20https://arxiv.org/abs/1802.01198}}
```