# Quantum walks on hierarchical graphs

TLDRAaron Harrow from MIT discusses quantum algorithms derived from quantum walks on hierarchical graphs, which offer potential speedups over classical algorithms. He explains the concept of quantum walks as a quantum analog to classical random walks and how they can exploit the symmetry in graphs like hypercubes for faster traversal. Harrow's team's research suggests that under certain conditions, quantum walks can achieve super-polynomial or even exponential speedups, drawing parallels to the localization transition in many-body physics.

### Takeaways

- 😀 Aaron Harrow from MIT discusses new quantum algorithms derived from quantum walks, which could offer significant speedups for certain computational tasks.
- 🎓 Quantum walks are quantum analogs of classical random walks, which are fundamental in classical algorithms.
- 🔍 The graph Laplacian, a mathematical tool used to describe random walks, is introduced and explained in the context of regular graphs.
- 🌐 The script explains how quantum walks can exploit the symmetry of certain graphs, like the hypercube, to achieve faster traversal times compared to classical walks.
- 🌟 A key concept is the difference between quantum and classical walks: quantum walks reflect and apply different phases, rather than decaying like classical walks.
- 🌱 The potential for quantum walks to provide more than quadratic speedups is explored, which is crucial for justifying the construction of quantum computers.
- 🌳 The talk references a paper by Childs et al. that demonstrated exponential speedups using a quantum lock on a welded tree graph, highlighting the potential of quantum walks.
- 🔬 Harrow's team's research focuses on hierarchical graphs, a family of graphs where quantum walks can be particularly effective, offering either super-polynomial or exponential speedups.
- 🔄 The concept of 'localization transition' from condensed matter physics is connected to the performance of quantum walks on hierarchical graphs.
- 🔍 The script provides a mathematical framework for understanding when quantum algorithms are likely to outperform classical algorithms on hierarchical graphs.
- 🚀 Ongoing work includes applying these concepts to optimization landscapes and exploring the existence of hierarchical graphs in simulated annealing and adiabatic evolution.

### Q & A

### What is the main topic of Aaron Harrow's presentation?

-The main topic of Aaron Harrow's presentation is about achieving new quantum speedups using quantum walks on hierarchical graphs.

### Who are the collaborators mentioned in Aaron Harrow's presentation?

-Aaron Harrow mentions that the work is a collaboration with his grad student Shaun Karbala Subramanian and former postdoc Tongan Lee, who is now a faculty at Peking University.

### What is the classical random walk and how does it relate to quantum walks?

-A classical random walk is a tool used in classical algorithms, where a particle moves from one vertex to a neighboring vertex on a graph. Quantum walks are the quantum analog of classical random walks, where instead of probabilities, amplitudes are used, and the process involves quantum superposition and interference.

### What is the graph Laplacian and how does it relate to random walks?

-The graph Laplacian is a matrix that generalizes the Laplacian from calculus to graphs. It is defined as the degree matrix minus the adjacency matrix. For regular graphs, it helps in formulating the differential equation that describes the continuous time random walk on the graph.

### How does the behavior of quantum walks differ from classical random walks?

-Classical random walks converge to a uniform distribution over time, with high-frequency information decaying quickly. In contrast, quantum walks do not decay; instead, they oscillate, with each eigenfunction picking up phase at a different rate.

### What is the significance of the hypercube graph in the context of quantum walks?

-The hypercube graph is significant because it has a stratified structure by Hamming weights, which allows quantum walks to traverse it much more quickly than classical walks, effectively seeing a 1D chain rather than the entire exponential-sized Hilbert space.

### What is the welded tree graph and how does it provide an exponential speedup for quantum walks?

-The welded tree graph consists of two binary trees of depth n connected by a random weld in the middle. Classical walks get stuck in the middle, but quantum walks can traverse it efficiently because they exploit the hierarchical structure of the graph, which is not discernible to classical algorithms until most of the graph has been explored.

### What is the concept of localization transition in many-body physics and how does it relate to quantum walks?

-The localization transition in many-body physics refers to the behavior of eigenstates in disordered materials, determining whether they are delocalized over the whole system or localized to a specific spot. Aaron Harrow's work connects this concept to quantum walks, suggesting that under certain conditions, quantum walks can either be super-polynomial or exponential in speed, depending on the variance of the log of the hopping amplitudes.

### How does the structure of a 1D super graph influence the performance of quantum walks?

-In a 1D super graph, the size of the super vertices and the wiring between them influence the performance of quantum walks. If the super vertices are of varying sizes and the wiring satisfies regularity assumptions, quantum walks can traverse the graph with super-polynomial speedups, leveraging the zero mode of the Hamiltonian associated with the graph.

### What are some potential applications of the research on quantum walks on hierarchical graphs?

-The research on quantum walks on hierarchical graphs could potentially be applied to optimization landscapes, more general graphs, and simulated annealing. It may also provide alternate constructions for exponential speedups in ground state stoquastic adiabatic evolution compared to classical algorithms.

### Outlines

### 🔬 Introduction to Quantum Walks and Hierarchical Graphs

Aaron Harrow introduces his work at MIT, focusing on quantum algorithms and quantum walks. He collaborates with Shaun Karbalasubramanian and Tongan Lee to explore quantum speedups using quantum walks on hierarchical graphs. The video begins with a discussion on the classical random walk as a fundamental tool in algorithms, using the hypercube graph as an example. The continuous-time random walk is explained through the adjacency matrix and graph Laplacian. Harrow then contrasts this with the quantum analog, highlighting the differences in eigenvalue behavior and the implications for algorithmic applications. The quantum walk's potential for speedups, especially in graphs with symmetry, is introduced, setting the stage for the rest of the talk.

### 🌐 Exploring Quantum Speedups Beyond Quadratic

The speaker delves into the concept of quantum speedups, questioning whether quadratic speedups are sufficient to justify building a quantum computer. He discusses the potential for quantum walks to achieve more than quadratic speedups, especially in graphs with specific symmetries like the hypercube. Harrow explains how quantum walks can traverse the hypercube efficiently due to its hierarchical structure, contrasting this with classical walks that take much longer. He references a paper by Childs, Cleve, DiVincenzo, Farhi, Gutmann, and Spielman that demonstrated an exponential speedup with a quantum lock on a welded tree graph. The talk aims to explore if such speedups can be generalized and applied to more useful problems.

### 🌳 Generalizing Quantum Walks to Super Graphs

Harrow extends the discussion to general walks on super graphs, providing examples of 1D, 2D, and higher-dimensional graphs. He explains that classical algorithms often struggle to efficiently traverse these graphs due to the need to explore most of the graph, while quantum algorithms can sometimes succeed. The concept of localization transition from many-body physics is introduced to explain when quantum algorithms are likely to outperform classical ones. The speaker presents a specific example of a 1D super graph and discusses how the structure of super vertices and their connections can be used to traverse the graph efficiently with a quantum walk. The talk highlights the potential for super polynomial or exponential speedups under certain conditions.

### 🔍 Traversing Graphs with Zero Modes and Future Work

The speaker discusses the use of zero modes to traverse graphs, explaining how the zero mode's eigenstate can be used to move through the graph with a super polynomial speedup. He provides a mathematical framework for understanding the traversal process, including the fluctuation of amplitudes and the significance of the gap near zero. Harrow also touches on the limitations of this approach, particularly when there are many edges within super vertices or when there is diagonal disorder. He concludes by mentioning ongoing work to apply these concepts to optimization landscapes and simulated annealing, as well as recent developments in exponential speedups for ground state stoquastic adiabatic evolution relative to classical algorithms.

### Mindmap

### Keywords

### 💡Quantum walks

### 💡Hierarchical graphs

### 💡Graph Laplacian

### 💡Spectral gap

### 💡Hypercube

### 💡Welded tree graph

### 💡Localization transition

### 💡Off-diagonal disorder

### 💡Hamiltonian

### 💡Simulated annealing

### Highlights

Aaron Harrow discusses new quantum speedups from quantum walks on hierarchical graphs.

The research is a collaboration with grad student Shaun Karbala Subramanian and former postdoc Tongan Lee.

Quantum walks are a quantum analog of classical random walks, which are crucial in classical algorithms.

Classical random walks on a graph can be described by a differential equation using the adjacency matrix.

The graph Laplacian is introduced as a combinatorial generalization of the usual Laplacian from calculus.

Continuous time random walks converge exponentially to the uniform distribution on a graph.

The Laplacian measures how much a vector varies across the edges of a graph.

Quantum walks generalize classical walks to continuous space using the Laplacian from calculus.

Quantum walks apply a phase to the stationary state and other eigenstates, unlike classical walks that decay.

Quantum walks can traverse hierarchical graphs much more quickly than classical walks.

The hypercube graph is stratified by Hamming weights, which allows quantum walks to explore a subspace of dimension n+1.

Quantum walks on the hypercube graph traverse it effectively as a 1D chain, achieving polynomial time complexity.

Classical algorithms struggle with hierarchical graphs due to the absence of short cycles that reveal the graph's structure.

The welded tree graph is an example where quantum walks achieve an exponential speedup over classical algorithms.

The research explores general walks on super graphs, which can be 1D, 2D, or higher dimensional.

The performance of quantum walks on super graphs is connected to the topic of localization transition in many-body physics.

The zero mode of the Hamiltonian is used to traverse the graph with a super polynomial speedup.

Quantum runtime depends on the variance of the log of hopping term ratios, while classical runtime depends on their expectation.

For n-by-n-by-n super graphs in higher dimensions, the quantum runtime is polynomial in n, and the classical runtime remains exponential.

Ongoing work includes applying these findings to optimization landscapes and more general graphs.

Simulated annealing is a potential concrete example where hierarchical graphs can be accessed algorithmically.

The method may provide an alternate construction for recent exponential speedups of ground state stoquastic adiabatic evolution.