# The Heisenberg Representation of Quantum Computers

@inproceedings{Gottesman1998TheHR, title={The Heisenberg Representation of Quantum Computers}, author={Daniel Gottesman}, year={1998} }

Since Shor`s discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers--the difficulty of describing them on classical computers--also makes it difficult to describe and understand precisely what can be done with them. A formalism describing the evolution of operators rather than states has proven extremely fruitful in understanding an important…

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## References

SHOWING 1-10 OF 11 REFERENCES

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 which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems.

Theory of fault-tolerant quantum computation

- Computer Science, Physics
- 1998

It is demonstrated that fault-tolerant universal computation is possible for any stabilizer code, including the five-quantum-bit code.

Perfect Quantum Error Correcting Code.

- Computer Science, PhysicsPhysical review letters
- 1996

A quantum error correction code which protects a qubit of information against general one qubit errors and encode the original state by distributing quantum information over five qubits, the minimal number required for this task.

Codes for the quantum erasure channel

- Computer Science, Physics
- 1997

A family of quantum codes for the QEC, the quantum Bose-Chaudhuri-Hocquenghem codes, that can be efficiently decoded is introduced.

Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.

- PhysicsPhysical review letters
- 1993

An unknown quantum state \ensuremath{\Vert}\ensuremath{\varphi}〉 can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen…

Mixed-state entanglement and quantum error correction.

- Computer SciencePhysical review. A, Atomic, molecular, and optical physics
- 1996

It is proved that an EPP involving one-way classical communication and acting on mixed state M (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa, and it is proved Q is not increased by adding one- way classical communication.

Class of quantum error-correcting codes saturating the quantum Hamming bound.

- Physics, Computer SciencePhysical review. A, Atomic, molecular, and optical physics
- 1996

Methods for analyzing quantum error-correcting codes are developed, and these methods are used to construct an infinite class of codes saturating the quantum Hamming bound.

Quantum Error Correction and Orthogonal Geometry

- Physics
- 1997

A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Codes are given which map 3…

Error prevention scheme with four particles.

- PhysicsPhysical review. A, Atomic, molecular, and optical physics
- 1996

It is shown that a simplified version of the error correction code recently suggested by Shor exhibits manifestation of the quantum Zeno effect and protection of an unknown quantum state is achieved.