# Strengths and Weaknesses of Quantum Computing

@article{Bennett1997StrengthsAW, title={Strengths and Weaknesses of Quantum Computing}, author={Charles H. Bennett and Ethan S. Bernstein and Gilles Brassard and Umesh V. Vazirani}, journal={SIAM J. Comput.}, year={1997}, volume={26}, pages={1510-1523} }

Recently a great deal of attention has been focused on quantum computation following a sequence of results [Bernstein and Vazirani, in Proc. 25th Annual ACM Symposium Theory Comput., 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339], [Simon, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 116--123, SIAM J. Comput., 26 (1997), pp. 1340--1349], [Shor, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 124--134] suggesting that quantumâ€¦Â

## 1,230 Citations

### Quantum Computability

- Computer ScienceSIAM J. Comput.
- 1997

It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in P and PSPACE.

### Fourier Transforms and Quantum Computation

- Computer ScienceTheoretical Aspects of Computer Science
- 2000

This survey paper focusses on presenting the essential ideas in a simple way, rather than getting the best results.

### Evolutionary Algorithms for Quantum Computers

- Computer ScienceAlgorithmica
- 2013

It turns out that the advantage of the respective quantum search heuristic over its classical counterpart varies with the problem structure and ranges from no speedup at all for the problem Discrepancy to exponential speedup for the problems TinyTrap.

### On quantum computation theory

- Computer Science
- 2002

The content of the Chapters 3 through 7 of this Ph.D. thesis corresponds with the following articles written by the author. Chapter 3: Quantum Oracle Interrogation â€¢ â€œQuantum Oracle Interrogation:â€¦

### Quantum Computation

- Physics
- 1998

In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, ifâ€¦

### Quantum algorithms and learning theory

- Computer Science
- 2018

A quantum algorithm to solve a search space of N elements using essentially sqrt{N} queries and other operations, improving over the gate count of Grover's algorithm is described.

### The complexity of the consistency and N-representability problems for quantum states - eScholarship

- Physics
- 2007

Quantum mechanics has important consequences for machines that store and manipulate information. In particular, quantum computers might be more powerful than classical computers; examples of thisâ€¦

### Quantum computing : pro and con BY JOHN PRESKILL

- Computer Science
- 1997

The potential of quantum computation is assessed, some of the known quantum algorithms and the prospects for finding new ones are reviewed, and the specifications that should be met by future hardware are commented on.

### INTRODUCTION TO QUANTUM ALGORITHMS

- Computer Science, Physics
- 2000

These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms. So far, we have only discovered a few techniques whichâ€¦

## References

SHOWING 1-10 OF 29 REFERENCES

### Quantum complexity theory

- Computer ScienceSTOC
- 1993

This paper gives the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis, and proves that bits of precision suffice to support a step computation.

### A fast quantum mechanical algorithm for database search

- Computer ScienceSTOC '96
- 1996

In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .

### On the power of quantum computation

- Computer ScienceProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

This work presents here a problem of distinguishing between two fairly natural classes of function, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class.

### The quantum challenge to structural complexity theory

- Physics, Computer Science[1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference
- 1992

There are cryptographic tasks that are demonstrably impossible to implement with unlimited computing power probabilistic interactive turning machines, yet they can be implemented even in practice by quantum mechanical apparatus.

### Quantum theory, the Churchâ€“Turing principle and the universal quantum computer

- Physics, PhilosophyProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1985

It is argued that underlying the Churchâ€“Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: â€˜every finitely realizibleâ€¦

### Algorithms for quantum computation: discrete logarithms and factoring

- Computer ScienceProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.

### Tight bounds on quantum searching

- Computer Science
- 1996

A lower bound on the efficiency of any possible quantum database searching algorithm is provided and it is shown that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table.

### Quantum Circuit Complexity

- Computer ScienceFOCS
- 1993

It is shown that any function computable in polynomial time by a quantum Turing machine has aPolynomial-size quantum circuit, and this result enables us to construct a universal quantum computer which can simulate a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them.

### Oracle Quantum Computing

- Computer ScienceWorkshop on Physics and Computation
- 1992

Oracles are constructed relative to which there is a decision problem that can be solved with certainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynometric time if errors are not tolerated.

### Learning DNF over the uniform distribution using a quantum example oracle

- Computer ScienceCOLT '95
- 1995

It is shown that disjunctive normal form (DNF) is efficiently learnable with respect to the uniform distribution by a quantum algorithm using a quantum example oracle, and it is proved that a quantum examples oracle with respectto uniform is less powerful than a membership oracle.