# Quantum circuits with mixed states

@inproceedings{Aharonov1998QuantumCW, title={Quantum circuits with mixed states}, author={Dorit Aharonov and Alexei Y. Kitaev and Noam Nisan}, booktitle={Symposium on the Theory of Computing}, year={1998} }

We define the model of quantum circuits with density matrices, where non-unitary gates are allowed. Measurements in the middle of the computation, noise and decoherence are implemented in a natural way in this model, which is shown to be equivalent in computational power to standard quantum circuits.
The main result in this paper is a solution for the subroutine problem: The general function that a quantum circuit outputs is a probabilistic function, but using pure state language, such a…

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## References

SHOWING 1-10 OF 17 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.

### Fault-tolerant quantum computation with constant error

- Computer ScienceSTOC '97
- 1997

This paper shows how to perform fault tolerant quantum computation when the error probability, q, is smaller than some constant threshold, q.. the cost is polylogarithmic in time and space, and no measurements are used during the quantum computation.

### 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.

### 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.

### Quantum theory, the Church–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 Church–Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: ‘every finitely realizible…

### 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.

### Strengths and Weaknesses of Quantum Computing

- Computer ScienceSIAM J. Comput.
- 1997

It is proved that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$.

### Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

- Computer ScienceSIAM Rev.
- 1999

Efficient randomized algorithms are given for factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems.

### Quantum Error Correction with Imperfect Gates

- Physics
- 1997

Quantum error correction can be performed fault-tolerantly This allows to store a quantum state intact (with arbitrary small error probability) for arbitrary long time at a constant decoherence rate.

### Simulating physics with computers

- Education, Physics
- 1982

On the program it says this is a keynote speech--and I don't know what a keynote speech is. I do not intend in any way to suggest what should be in this meeting as a keynote of the subjects or…