Examples of Quantum Walk Algorithms

Examples of quantum walk algorithms, specifically two, are the Element Distinctness Problem and Glued Trees Graph Traversal.

• Element Distinctness Problem: Ambainis’s method for the element distinctiveness problem makes a noteworthy use of quantum walks. This method does not need many searches to update this information for a nearby vertex; it keeps information from multiple searches at every vertex.
• Glued Trees Graph Traversal: By a random cycle exponentially quicker than any classical method, a quantum walk may solve a black-box issue involving a graph derived by combining two binary trees.

Quantum walk algorithm for element distinctness

The element distinctiveness problem—that is, whether a list contains any duplicate objects—can be solved using a quantum walk method.

Using a graph structure, the quantum walk method to element distinctness effectively explores the input space and detects collisions, hence accelerating over conventional methods. The fundamental concept is to maximize the search process by examining the information from several searches in a way that reduces the total number of searches required by use of quantum walks.

 How quantum walks apply to Element Distinctness Problem

• The Element Distinctness Problem: The goal is to determine whether there are two distinct inputs x and y such that f(x) = f(y).
• Classical Lower Bound: Classical algorithms require Ω(n) queries to solve the problem, since deciding whether there is such a pair is at least as hard as unstructured search.
• Quantum Lower Bound: There is a quantum lower bound for element distinctness of Ω(√n).
• Quantum Walk Approach: Ambainis’s algorithm uses a quantum walk on the Johnson graph J(n,m), or a related graph, to solve the element distinctness problem. The vertices of the graph correspond to subsets of {1, 2, . . . , n}.
  • A vertex R in the graph is marked if there exist i ∈ R and j ∈ [n] \ R such that xi = xj.
• Data Storage: The algorithm stores data about the queried elements at each vertex, but does not require many queries to update this information for an adjacent vertex.
• Query Complexity: The quantum walk algorithm achieves a query complexity of O(n^2/3), which is better than the classical algorithms that require O(n) queries.
• Quantum Speedup: Quantum walks can achieve a speedup over classical algorithms for element distinctness by utilizing superposition and interference.
• Collision Problem: The element distinctness algorithm can be used to solve the collision problem, where the goal is to find two inputs that map to the same value, by running the element distinctness algorithm on a set of inputs. A k-query element distinctness algorithm implies an O(√k)-query collision algorithm.
• Amplitude Amplification: Amplitude amplification can be used in more subtle ways to solve the element distinctness problem more efficiently than any classical algorithm.

Concepts of the Quantum Walk Algorithm in Element Distinctness Problem

• Johnson Graph: The Johnson graph J(n,m) has vertices corresponding to subsets of {1, 2, . . . , n} of size m, with edges between vertices that differ by exactly one element.
• Hamming Graph: A related graph is the Hamming graph H(n,m) whose vertices are m-tuples of values from {1, 2, . . . , n}. Two vertices are connected by an edge if they differ in exactly one coordinate. The Hamming graph allows for repeated elements, and the order of elements is significant, but neither significantly impacts performance.
• Marked Vertices: A vertex is considered marked if it represents a subset where a collision (duplicate) exists.
• Query Cost: The quantum algorithm balances the cost of setting up the data, updating the data, and checking if the data is marked.
• Optimized Query Complexity: The algorithm optimizes the number of queries by storing some data about the input at each vertex, and minimizing the number of queries needed to update the information. This balance is what allows the quantum algorithm to achieve a query complexity of O(n^2/3), instead of a less efficient query complexity of m + O(n²/m).
• Data Storage and Updates: The quantum walk stores information at each vertex, and takes into account the operations on this data when analyzing the walk.

Quantum walk algorithm for black box problem

Black box problems—where the input is obtained through a black box or oracle and the internal operations of the black box are unknown—can be solved using quantum walk techniques. Using either continuous or discrete time formulations, quantum walks give a strong foundation for addressing black box issues and have the possibility to greatly speed up over conventional methods. The individual challenge and the black box’s construction determine the particular method and speed-up attained.

Applying quantum walks for black box problems

• Black Box Representation: The black box oracle in the black box model computes a function to access the input to the issue. The aim is to compute some function of the input with as few black box searches as feasible. One further name for this is query complexity.
• Graph Traversal: Graph traversal is one of quantum walks’ uses; it may be considered as a black box issue. In this sense, the black box offers details on the structure of the graph including vertex and edge names.
• Glued Trees Graph: Traversal of a glued trees graph is a fundamental example.
  • Two binary trees coupled by a random cycle build this graph.
  • Moving from the root of one tree to the root of the other a traditional random walk on this graph is rather ineffective.
  • Still, a quantum walk can accomplish this traverse considerably quicker than any conventional method.
• Query Complexity: Often expressed in terms of query complexity—that is, the number of times the quantum walk algorithm must search the black box—the efficiency of a quantum walk method for a black box problem.
• Continuous-Time Quantum Walk: Using a Hamiltonian constructed from the adjacency matrix of a graph, one may simulate a continuous-time quantum walk on that network.
  • The Hamiltonian controls the dynamics of the walk.
  • Black box challenges, including the graph traversal problem for the glued trees graph have been solved using this kind of walk.
• Discrete-Time Quantum Walk: Discrete-time quantum walks find the direction of the walk by means of a unitary operator comprising a “coin” operator.
  • Black box issues including unstructured search, can also be addressed with these walks.
• Unstructured Search: Unstructured search issues can be seen in terms of a black box function f and solved using a quantum walk method.
  • The best unstructured search solution is Grover’s method.
  • Grover’s method may be carried out with a quantum walk.