• Corpus ID: 15947511

The Heisenberg Representation of Quantum Computers

  title={The Heisenberg Representation of Quantum Computers},
  author={Daniel Gottesman},
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… 

Figures and Tables from this paper

Efficient classical simulation of noisy quantum computation
Understanding the boundary between classical simulatability and the power of quantum computation is a fascinating topic. Direct simulation of noisy quantum computation requires solving an open
Stabilizer states, conditional Clifford operations, and their possible role in quantum computation
We work with the following (fairly standard) setting for quantum computation. The computer consists of n qudits, i.e. quantum systems with a d-dimensional Hilbert space. We will mostly focus on
High-performance QuIDD-based simulation of quantum circuits
An improved implementation of QuIDDs is presented which can simulate Grover's algorithm for quantum search with the asymptotic runtime complexity of an ideal quantum computer up to negligible overhead.
Classical simulation of noninteracting-fermion quantum circuits
It is shown that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant corresponds to a physical model of noninteracting fermions in one dimension.
A quantum computer only needs one universe
Abstract The nature of quantum computation is discussed. It is argued that, in terms of the amount of information manipulated in a given time, quantum and classical computation are equally efficient.
Quantum computers that can be simulated classically in polynomial time
It is shown that two-bit operations characterized by 4 \times 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynometric time.
Quantum abstract interpretation
An abstract interpretation of quantum programs is presented and it is used to automatically verify assertions in polynomial time to let an abstract state be a tuple of projections.
Gate-level simulation of quantum circuits
Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with an increase in the
Gate-level simulation of quantum circuits
This work implemented a general-purpose quantum computing simulator in C++ called QuIDDPro and tested it on Grover's algorithm and found that it asymptotically outperforms other known simulation techniques.
Gottesman Types for Quantum Programs
It is shown that Gottesman’s semantics for quantum programs can be treated as a type system, allowing us to efficiently characterize a common subset of quantum programs and extend beyond the Clifford set to partially characterize a broad range of programs.


Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
  • P. Shor
  • Computer Science, Mathematics
    SIAM 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
In order to use quantum error-correcting codes to improve the performance of a quantum computer, it is necessary to be able to perform operations fault-tolerantly on encoded states. I present a
Perfect Quantum Error Correcting Code.
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.
Expressing the operations of quantum computing in multiparticle geometric algebra
We show how the basic operations of quantum computing can be expressed and manipulated in a clear and concise fashion using a multiparticle version of geometric (aka Clifford) algebra. This algebra
Codes for the quantum erasure channel
The quantum erasure channel (QEC) is considered. Codes for the QEC have to correct for erasures, i.e., arbitrary errors at known positions. We show that four quantum bits are necessary and sufficient
Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.
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.
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.
  • Gottesman
  • Physics, Medicine
    Physical 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
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.
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.