Quantum circuits with mixed states

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

Polynomial simulations of decohered quantum computers

  • D. AharonovM. Ben-Or
  • Physics, Computer Science
    Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
This work presents a simulation of decohered sequential quantum computers, on a classical probabilistic Turing machine, and proves that the expected slowdown of this simulation is polynomial in time and space of the quantum computation, for any non zero decoherence rate.

Eliminating intermediate measurements in space-bounded Quantum computation

This work exhibits a procedure to eliminate all intermediate measurements that is simultaneously space efficient and time efficient, and shows that the definition of a space-bounded quantum complexity class is robust to allowing or forbidding intermediate measurements.

Stone-Weierstrass Style Theorem for Quantum Operations

In the usual representation of quantum computational processes, a quantum circuit is identified with an appropriate composition of quantum gates, i.e. unitary operators acting on pure states (qubits

Computational quantum-classical boundary of noisy commuting quantum circuits

A computational quantum-classical boundary from the viewpoint of classical simulatability of a quantum system under decoherence is established and paves a way to an experimentally feasible verification of quantum mechanics in a high complexity limit beyond classically simulatable region.

Techniques for Quantum Computing

This thesis considers algorithmic cooling—a technique that is potentially important for making quantum computation using nuclear magnetic resonance (NMR) feasible and the main challenges and obstacles to implementing quantum computing fault tolerantly using globally controlled arrays.

Theory of measurement-based quantum computing

In the study of quantum computation, data is represented in terms of linear operators which form a generalized model of probability, and computations are most commonly described as products of

Lower bounds for quantum computation and communication

This dissertation establishes limitations on the ways in which the exponentially many degrees of freedom hidden in quantum states may be accessed, and derives nearly optimal lower bounds for several problems in this model, including that of approximating the median.

Noisy Quantum Computation Modeled by Quantum Walk

This work extends the quantum walk model to open noisy systems in order to provide such a tool for the study of NISQ computers, and model the application of general unitary gates and nonunitary channels with an explicit implementation protocol for channels that are commonly used in noise models.

Routed quantum circuits

It is proved that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries.

Computation in generalised probabilisitic theories

This work defines a circuit-based model of computation in a class of operationally-defined theories more general than quantum theory, and shows there exists a classical oracle relative to which efficient computation in any theory satisfying the causality assumption does not include .



Quantum complexity theory

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

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

  • Daniel R. Simon
  • Computer Science
    Proceedings 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

  • A. Yao
  • Computer Science
  • 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

  • D. Deutsch
  • Physics, Philosophy
    Proceedings 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

  • P. Shor
  • Computer Science
    Proceedings 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

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

  • P. Shor
  • Computer Science
    SIAM 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

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

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