A quantum computer is any device for computation that makes direct use of distinctively quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. In a classical (or conventional) computer, the amount of data is measured by bits; in a quantum computer, it is measured by qubits. The basic principle of quantum computation is that the quantum properties of particles can be used to represent and structure data and that quantum mechanisms can be devised and built to perform operations with these data.
It is widely believed that if large-scale quantum computers can be built, they will be able to solve certain problems faster than any classical computer. Quantum computers are different from classical computers such as DNA computers and computers based on transistors, even though these may ultimately use some kind of quantum mechanical effect (for example covalent bonds). Some computing architectures such as optical computers may use classical superposition of electromagnetic waves, but without some specifically quantum mechanical resource such as entanglement, they do not share the potential for computational speed-up of quantum computers.
(The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers.)
The basis of quantum computing:
In quantum mechanics, the state of a physical system (such as an electron or a photon) is described by a vector in a mathematical object called a Hilbert space. The realization of the Hilbert space depends on the particular system. For instance, in the case of a single particle system in three dimensions, the state can be described by a complex-valued function defined on R3 (three-dimensional space) called a wave function. As described in the article on quantum mechanics, this function has a probabilistic interpretation; of particular significance is that quantum states can be in a superposition of the basis states. The time evolution of the system state vector is assumed to be unitary, meaning that it is reversible.
A classical computer has a memory made up of bits, where each bit holds either a one or a zero. The device computes by manipulating those bits, i.e. by transporting these bits from memory to (possibly a suite of) logic gates and back. A quantum computer maintains a set of qubits. A qubit can hold a one, or a zero, or a superposition of these. A quantum computer operates by manipulating those qubits, i.e. by transporting these bits from memory to (possibly a suite of) quantum logic gates and back.
Qubits for a quantum computer can be implemented using particles with two spin states: "up" and "down" (typically written 
and 
) in fact, any system possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit, since any such system can be mapped onto an effective spin-1/2..
and ) in fact, any system possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit, since any such system can be mapped onto an effective spin-1/2..Consider first a classical computer that operates on a 3 bit register. At any given time, the bits in the register are in a definite state, such as 101. In a quantum computer, however, the qubits can be in a superposition of all the classically allowed states. In fact, the register is described by a wave function:
where the coefficients α, β, γ,... are complex numbers whose amplitudes squared are the probabilities to measure the qubits in each state. Consequently, | γ | 2- is the probability to measure the register in the state 010. That these numbers are complex is important because the phases of the numbers can constructively and destructively interfere with one another, an important feature for quantum algorithms.
For an n qubit quantum register, recording the state of the register requires 2n complex numbers (the 3-qubit register requires 23 = 8 numbers). Consequently, the number of classical states encoded in a quantum register grows exponentially with the number of qubits. For n=300, this is roughly 1090, more states than there are atoms in the known universe. Note that the coefficients are not all independent, since the probabilities must sum to 1. The representation is also (for most practical cases) non-unique, since there is no way to physically distinguish between a particular quantum register and a similar one where all of the amplitudes have been multiplied by the same phase such as −1, i, or in general any number on the complex unit circle. One can show the dimension of the set of states of an n qubit register is 2n+1 − 2.
A qubit registers can be thought of as an 8-dimensional complex vector. An algorithm for a quantum computer must initialize this vector in some specified form (dependent on the design of the quantum computer). In each step of the algorithm, that vector is modified by multiplying it by a unitary matrix. The matrix is determined by the physics of the device. The unitary character of the matrix ensures the matrix is invertible
Upon termination of the algorithm, the 8-dimensional complex vector stored in the register must be somehow read off from the qubit register by a quantum measurement. However, by the laws of quantum mechanics, that measurement will yield a random 3 bit string (and it will destroy the stored state as well). This random string can be used in computing the value of a function because (by design) the probability distribution of the measured output bitstring is skewed in favor of the correct value of the function. By repeated runs of the quantum computer and measurement of the output, the correct value can be determined, to a high probability, by majority polling of the outputs. See quantum circuit for a more precise formulation. In brief, quantum computations are probabilistic.
A quantum algorithm is implemented by an appropriate sequence of unitary operations. Note that for a given algorithm, the operations will always be done in exactly the same order. There is no "IF THEN" statement to vary the order, since there is no way to read the state of a qubit before the final measurement. There are, however, conditional gate operations such as the controlled NOT gate, or CNOT.
This section surveys what is currently known mathematically about the power of quantum computers. It describes the known results from computational complexity theory and the theory of computation dealing with quantum computers.
The class of problems that can be efficiently solved by quantum computers is called BQP, for "bounded error, quantum, polynomial time". Quantum computers only run randomized algorithms, so BQP on quantum computers is the counterpart of BPP on classical computers. It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one quarter (Nielsen & Chuang 2000). A quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in BQP.
BQP is suspected to be disjoint from NP-complete and a strict superset of P, but that is not known. Both integer factorization and discrete log are in BQP. Both of these problems are NP problems suspected to be outside BPP, and hence outside P. Both are suspected to not be NP-complete. There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally suspected to be false.
An operator for a quantum computer can be thought of as changing a vector by multiplying it with a particular matrix. Multiplication by a matrix is a linear operation. It has been shown that if a quantum computer could be designed with nonlinear operators, then it could solve NP-complete problems in polynomial time. It could even do so for #P-complete problems. It is not yet known whether such a machine is possible.
Although quantum computers are sometimes faster than classical computers, ones of the types described above can't solve any problems that classical computers can't solve, given enough time and memory (albeit possibly an amount that could never practically be brought to bear). A Turing machine can simulate these quantum computers, so such a quantum computer could never solve an undecidable problem like the halting problem. The existence of "standard" quantum computers does not disprove the Church-Turing thesis (Nielsen and Chuang 2000).
Very recently, some researchers have begun to investigate the possibility of using quantum mechanics for hypercomputation - that is, solving undecidable problems. Such claims have been met with very considerable skepticism as to whether it is even theoretically possible; see the hypercomputation article for more details.
- David P. DiVincenzo (2000). "The Physical Implementation of Quantum Computation". Experimental Proposals for Quantum Computation. arXiv:quant-ph/0002077.
- D.P. DiVincenzo (1995). "Quantum Computation". Science 270 (5234): 255–261. Table 1 lists switching and dephasing times for various systems.
- Richard Feynman (1982). "Simulating physics with computers". International Journal of Theoretical Physics 21: 467.
- Gregg Jaeger (2006). Quantum Information: An Overview. Berlin: Springer. ISBN 0-387-35725-4."
- Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9.
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