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arXiv:quant-ph/0206066

Quantum searching a classical database (or how we learned to stop worrying and love the bomb)

Terry Rudolph, Dr., Lov K. Grover

Published Jun 11, 2002• 12 citations
arXivHTMLPDF

Abstract

We show how to perform a quantum search for a classical object, specifically for a classical object which performs no coherent evolution on the quantum computer being used for the search. We do so by using interaction free measurement as a subroutine in a quantum search algorithm. In addition to providing a simple example of how non-unitary processes which approximate unitary ones can be useful in a quantum algorithm, our procedure requires only one photon regardless of the size of the database, thereby establishing an upper bound on the amount of energy required to search an arbitrarily large database. Alternatively, our result can be interpreted as showing how to perform an interaction free measurement with a single photon on an arbitrarily large number of possible bomb positions simultaneously. We also provide a simple example demonstrating that in terms of the number of database queries, the procedure outlined here can outperform the best classical one.

DOI
10.48550/arxiv.quant-ph/0206066

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We show how to perform a quantum search for a classical object, specifically for a classical object which performs no coherent evolution on the quantum computer being used for the search. We do so by using interaction free measurement as a subroutine in a quantum search algorithm. In addition to providing a simple example of how non-unitary processes which approximate unitary ones can be useful in a quantum algorithm, our procedure requires only one photon regardless of the size of the database, thereby establishing an upper bound on the amount of energy required to search an arbitrarily large database. Alternatively, our result can be interpreted as showing how to perform an interaction free measurement with a single photon on an arbitrarily large number of possible bomb positions simultaneously. We also provide a simple example demonstrating that in terms of the number of database queries, the procedure outlined here can outperform the best classical one.
This paper shows that a quantum mechanical algorithm that can query information relating to multiple items of the database can search a database for a unique item satisfying a given condition, in a single query [a query is defined as any question to the database to which the database has to return a (yes/no answer]. A classical algorithm will be limited to the information theoretic bound of at least log2N{log}_{2}Nlog2​N queries, which it would achieve by using a binary search.
A proposed quantum-computing algorithm would perform a search for an item of information in a database stored in a Hilbert-space memory structure. The algorithm is intended to make it possible to search relatively quickly through a large database under conditions in which available computing resources would otherwise be considered inadequate to perform such a task. The algorithm would apply, more specifically, to a relational database in which information would be stored in a set of N complex orthonormal vectors, each of N dimensions (where N can be exponentially large). Each vector would constitute one row of a unitary matrix, from which one would derive the Hamiltonian operator (and hence the evolutionary operator) of a quantum system. In other words, all the stored information would be mapped onto a unitary operator acting on a quantum state that would represent the item of information to be retrieved. Then one could exploit quantum parallelism: one could pose all search queries simultaneously by performing a quantum measurement on the system. In so doing, one would effectively solve the search problem in one computational step. One could exploit the direct- and inner-product decomposability of the unitary matrix to make the dimensionality of the memory space exponentially large by use of only linear resources. However, inasmuch as the necessary preprocessing (the mapping of the stored information into a Hilbert space) could be exponentially expensive, the proposed algorithm would likely be most beneficial in applications in which the resources available for preprocessing were much greater than those available for searching.
A quantum mechanical computer can search a database consisting of N items in O(sqrt(N)) elementary quantum queries (a query is defined as any question to the database to which a binary (YES/NO) answer is required; an elementary query is a query pertaining to only one of the N possible states). A classical computer would take at least O(N) elementary queries. This paper shows that a quantum mechanical algorithm that can query information relating to multiple states, can search the database in a single query. A classical algorithm will be limited to the information theoretic bound of at least O(log N)queries (which it would achieve by using a binary search). A similar result was recently independently obtained by Terhal & Smolin by a different method.
Quantum computers are proven to be more efficient at solving a specific class of problems compared to traditional digital computers. Superposition of states and quantum entanglement are the two key ingredients that make quantum computing so powerful. However, not all quantum algorithms require quantum entanglement (e.g., search through an unsorted database). Is it possible to utilize classical wave superposition to speed up database searching as much as by using quantum computers? There were several attempts to mimic quantum computers using classical waves. It was concluded that the use of classical wave superposition comes with the cost of an exponential increase in resources. In this work, we consider the feasibility of building classical wave-based devices able to provide fundamental speedup over digital counterparts without the exponential overhead. We present experimental data on database searching through a magnetic database using spin wave superposition. The results demonstrate the same speedup as expected for quantum computers. Also, we present examples of numerical modeling demonstrating classical wave interference for period finding. This approach may not compete with quantum computers with efficiency but outperform classical digital computers. We argue that classical wave-based devices can perform some of the quantum algorithms with the same efficiency as quantum computers as long as quantum entanglement is not required.
Sorting is a fundamental computational process that facilitates subsequent searching of a database. It can be thought of as factorization of the search process. The location of a desired item in a sorted database can be found by classical queries that inspect one letter of the label at a time. For an unsorted database, no such classical quick search algorithm is available. If the database permits quantum queries, however, then mere digitization is sufficient for efficient search. Sorting becomes redundant with the quantum superposition of states. A quantum algorithm is written down that locates the desired item in an unsorted database a factor of two faster than the best classical algorithm can in a sorted database. This algorithm has close resemblance to the assembly process in DNA replication.
Abstract Quantum search algorithms have important applications in quantum information, which have been experimentally demonstrated in several quantum systems with database sizes up to 16. We first propose an efficient quantum search scheme based on an optimized Grover’s algorithm and a novel encoding method. With this scheme, a target element can be deterministically searched in a large database using only a single qubit. Then we implement this search scheme in linear optics with database sizes up to 40. Experimental results fit well with the theoretical results, with all fidelities greater than 99.7%. The limitation of the number of elements encoded in a single qubit is also investigated in the experiment. The experimental results indicate that a target state can be successfully searched from a database of size 36 using a single qubit in our laboratory. Since only one qubit and one-qubit gates are required, this search scheme is quite efficient and general, and can be adopted to implement the same task in most quantum systems.