In computational complexity theory, BQP ( bounded error quantum polynomial time ) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1 / 3 for all instances.

In other words, there is an algorithm for a quantum computer ( a quantum algorithm ) that solves the decision problem with high probability and is guaranteed to run in polynomial time.

Usually, computation on a quantum computer ends with a measurement.

This class is defined for a quantum computer and its natural corresponding class for an ordinary computer ( or a Turing machine plus a source of randomness ) is BPP.

These newer concerns are among the many factors causing researchers to investigate new methods of computing such as the quantum computer, as well as to expand the usage of parallelism and other methods that extend the usefulness of the classical von Neumann model.

If a suitably sized quantum computer capable of running Grover's algorithm reliably becomes available, it would reduce a 128-bit key down to 64-bit security, roughly a DES equivalent.

In other words, it takes no more time to break RSA on a quantum computer ( up to a multiplicative constant ) than to use it legitimately on a classical computer.

Bennett, Bernstein, Brassard, and Vazirani proved in 1996 that a brute-force key search on a quantum computer cannot be faster than roughly 2 < sup > n / 2 </ sup > invocations of the underlying cryptographic algorithm, compared with roughly 2 < sup > n </ sup > in the classical case.

This implies that at least a 160-bit symmetric key is required to achieve 80-bit security rating against a quantum computer.

He believes that topological quantum computing is about to revolutionize computer science, and hopes that his teaching will help his students to understand its principles.

Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.

For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time.

This will have significant implications for cryptography if a large quantum computer is ever built.

In 2001, the first seven-qubit quantum computer became the first to run Shor's algorithm.

That is, problems likely exist that an NTM could efficiently solve but that a quantum computer cannot.

showed that a quantum computer can find a product representation, as described above, for the solution to Pell's equation in polynomial time.

A quantum computer is a computation device that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data.

A theoretical model is the quantum Turing machine, also known as the universal quantum computer.

A quantum computer with spins as quantum bits was also formulated for use as a quantum space-time in 1969.

The standard ampere is most accurately realized using a watt balance, but is in practice maintained via Ohm's Law from the units of electromotive force and resistance, the volt and the ohm, since the latter two can be tied to physical phenomena that are relatively easy to reproduce, the Josephson junction and the quantum Hall effect, respectively.

Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers.

If a particle and antiparticle are in the appropriate quantum states, then they can annihilate each other and produce other particles.

In quantum field theory, this process is allowed only as an intermediate quantum state for times short enough that the violation of energy conservation can be accommodated by the uncertainty principle.

The classical definition of angular momentum as can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator.

In quantum computing, a quantum bit or qubit is a quantum system that can exist in superposition of two bit values, " true " and " false ".

BQP can also be viewed as a bounded-error uniform family of quantum circuits.

It can be said that the quantum state is measured to be in the correct state with high probability.

The Boltzmann distribution applies only to particles at a high enough temperature and low enough density that quantum effects can be ignored, and the particles are obeying Maxwell – Boltzmann statistics.

In fact the distribution applies whenever quantum considerations can be ignored.

All of this cosmic evolution after the inflationary epoch can be rigorously described and modeled by the ΛCDM model of cosmology, which uses the independent frameworks of quantum mechanics and Einstein's General Relativity.

* Wave functions and other quantum states can be represented as vectors in a complex Hilbert space.

Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often does involve, bra-ket notation.

Since these are a basis, any quantum state of the particle can be expressed as a linear combination ( i. e., quantum superposition ) of these two states:

The quantum hall effect is another example of measurements with high magnetic fields where topological properties such as Chern-Simons angle can be measured experimentally.

Cold atoms in optical lattices are used as " quantum simulators ", that is, they act as controllable systems that can model behavior of more complicated systems, such as frustrated magnets.

Condensed matter systems can be tuned to provide the conditions of coherence and phase-sensitivity that are essential ingredients for quantum information storage.

Thus in the presence of large quantum computers an n-bit key can provide at least n / 2 bits of security.

There is a classical analogue to the quantum no-cloning theorem, which we might state as follows: given only the result of one flip of a ( possibly biased ) coin, we cannot simulate a second, independent toss of the same coin.

Given unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church – Turing thesis.

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.

** Richard Feynman in his talk at the First Conference on the Physics of Computation, held at MIT in May, observed that it appeared to be impossible in general to simulate an evolution of a quantum system on a classical computer in an efficient way.

Just as a Universal Turing machine can simulate any other Turing machine efficiently, so the universal quantum computer is able to simulate any other quantum computer with at most a polynomial slowdown.

The idea that quantum computers might be more powerful than classical computers originated in Richard Feynman's observation that classical computers seem to require exponential time to simulate many-particle quantum systems.

Since then, the idea that quantum computers can simulate quantum physical processes exponentially faster than classical computers has been greatly fleshed out and elaborated.

Quantum computers can also efficiently simulate topological quantum field theories.

In 1996, Seth Lloyd showed that a standard quantum computer can be programmed to simulate any local quantum system efficiently.

Since one need not simulate the world to a quantum level to make it appear completely real, a simulated reality which appears completely real to a user could be simulated on a computer much smaller than the simulated space.

Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems with the idea of solving the quantum many-body problem.