Quantum computing

File:NMR implementation alanine.png

Molecule of alanine used in NMR implementation of error correction. Qubits are implemented by spin states of carbon atoms (numbered)

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 this data.Template:Fn

Though quantum computing is still in its infancy, experiments have been carried out in which quantum computational operations were executed on a very small number of qubits. Research in both theoretical and practical areas continues at a frantic pace, and many national government and military funding agencies support quantum computing research, to develop quantum computers for both civilian and national security purposes, such as cryptanalysis. Template:Fn (See Timeline of quantum computing for details on current and past progress.)

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 basis of quantum computing

Unsolved problem in physics:

Is it possible to construct a practical computer that performs calculations on qubits (quantum bits)?

(more unsolved problems in physics)

In quantum mechanics, the state of a physical system (such as an electron or a photon) is described by an element of 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, 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 have a property called superposition. A similar realization of the Hilbert space exists for systems of interacting particles. The time evolution of the system state is given by a family {U_t} (with t denoting time) of unitary transformations of H. Thus if φ is the state at time 0, then U_t φ is the state at time t. Note that this is true only if the system is isolated and the phenomenon of decoherence does not occur.

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"; 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.

For discussion of foundational aspects of quantum computing, see the article on quantum circuits.

Bits vs qubits

Consider first a classical computer that operates on a 3 bit register. At a given time, the state of the register is determined by a single string of 3 bits, such as "101". This is usually expressed by saying that the register contains a single string of 3 bits. A quantum computer, on the other hand, can be in a state which is a mixture of all the classically allowed states. The particular state is determined by 8 complex numbers. In quantum mechanics notation we would write:

|\psi \rangle =a|000\rangle +b|001\rangle +c|010\rangle +d|011\rangle +e|100\rangle +f|101\rangle +g|110\rangle +h|111\rangle

where a, b, c, d, e, f, g, and h are complex. Let us consider a particular example:

State	Amplitude	Probability
*	(α+i β)	(\|α\|²+\|β\|²)
000	a = 0.37 + i 0.04	0.14
001	b = 0.35 + i 0.43	0.31
010	c = 0.09 + i 0.31	0.10
011	d = 0.30 + i 0.30	0.18
100	e = 0.11 + i 0.18	0.04
101	f = 0.40 + i 0.01	0.16
110	g = 0.09 + i 0.12	0.02
111	h = 0.15 + i 0.16	0.05

For an n qubit quantum register, this table would have had 2ⁿ rows; for n=300, this is roughly 10⁹⁰, more rows than there are atoms in the known universe. Note that these values are not all independent, since the probability constraint must be met. The representation is also non-unique, since there is no way to physically distinguish between this 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. See Bloch sphere.

The first column shows all classically allowed states for three bits. Whereas a classical computer can hold only one such pattern at a time, a quantum computer can be in a superposition state of all 8 patterns. The second column shows the "amplitude" for each of the 8 states. These 8 complex numbers are a snapshot of the register at a given time. In this sense, a 3-qubit quantum computer has far more memory than a 3-bit classical computer because it can simultaneously represent all possible states of the classical computer.

When the qubit is measured, it is projected onto one of the classically allowed states. The absolute value squared of the amplitude of each classical state gives the probability that the qubit will be measured in that state. Looking at the table, the third column gives the probability for measuring each possible register configuration. In this example, there is a 14% chance that the returned string will be "000", a 31% chance it will be "001", and so on. Each complex number (α+βi) is called an (complex valued) amplitude, and each probability (|α|²+|β|²) is the absolute square of the amplitude, because it equals |α+ βi|². The probabilities must sum to 1.

Initialization, execution and termination

In our example, the contents of the 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 (so each step is reversible).

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.

Let us consider the case of the NMR implementation of a quantum circuit. NMR differs from other implementations on quantum computers in that it uses an ensemble of systems, in this case molecules. The ensemble is initialized to be the thermal equilibrium state (see quantum statistical mechanics). In mathematical parlance, this state is given by the density matrix:

\rho ={\frac {e^{-\beta H}}{\operatorname {Tr} (e^{-\beta H})}},

where H is the hamiltonian matrix of an individual molecule and

\beta =({\mbox{Boltzmann constant }}\times {\mbox{ temperature }})^{-1}.

The unitary operations are performed by shooting a short pulse of radiation at the container of molecules. Different types of pulses result in different unitary matrices. The quantum algorithm is implemented by an appropriate sequence of pulses. 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 molecule before the final measurement.

For more details on the sequences of operations used for various algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa algorithm, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction. Also refer to the growing field of quantum programming.

The power of quantum computers

Integer factorization is believed to be computationally infeasible with an ordinary computer for large numbers that are the product of two prime numbers of roughly equal size (e.g., products of two 300-digit primes). By comparison, a quantum computer could solve this problem much more quickly. If a number has n bits (is n digits long when written in the binary numeral system), then a quantum computer with just over 2n qubits can use Shor's algorithm to find its factors. It can also solve a related problem called the discrete log problem. This ability would allow a quantum computer to "break" many of the cryptographic systems in use today, in the sense that there would be a relatively fast (polynomial time in n) algorithm for solving the problem. In particular, most of the popular public key ciphers could be much more quickly broken, including forms of RSA, ElGamal and Diffie-Hellman. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security. The only way to increase the security of an algorithm like RSA would be to increase the key size and hope that an adversary does not have the resources to build and use a powerful enough quantum computer. It seems plausible that it will always be possible to build classical computers that have more bits than the number of qubits in the largest quantum computer. If that's true, then algorithms like RSA could be made secure by ensuring that keylengths exceed the storage capacities of quantum computers.

There is one digital signature scheme that is secure against quantum computers: Lamport signatures.

Perhaps not as surprisingly, quantum computers could also be useful for running simulations of quantum mechanics. The speedup could be just as large as for factoring. This could be a great boon to physics, chemistry, materials science, nanotechnology, biology and medicine, all of which are limited today by the slow speed of quantum mechanical simulations.

This dramatic advantage of quantum computers is currently known to exist for only those three problems: factoring, discrete log, and quantum physics simulations. However, there is no proof that the advantage is real: an equally fast classical algorithm may still be discovered (though some consider this unlikely). There is one other problem where quantum computers have a smaller, though significant (quadratic) advantage. It is quantum database search, and can be solved by Grover's algorithm. In this case the advantage is provable. This establishes beyond doubt that (ideal) quantum computers are superior to classical computers.

Consider a problem that has these four properties:

The only way to solve it is to guess answers repeatedly and check them,
There are n possible answers to check,
Every possible answer takes the same amount of time to check, and
There are no clues about which answers might be better: generating possibilities randomly is just as good as checking them in some special order.

An example of this is a password cracker that attempts to guess the password for an encrypted file (assuming that the password has a maximum possible length).

For problems with all four properties, it will take an average of (n + 1)/2 guesses to find the answer using a classical computer. The time for a quantum computer to solve this will be proportional to the square root of n. That can be a very large speedup, reducing some problems from years to seconds. It can be used to attack symmetric ciphers such as Triple DES and AES by attempting to guess the secret key. But it is also easy to defend against, by doubling the size of this key. There are also more complicated methods for secure communication, such as using quantum cryptography.

There are currently no other practical problems known where quantum computers give a large speedup over classical computers. Research is continuing, and more problems may yet be found.

Problems and practicality issues

There are a number of practical difficulties in building a quantum computer, and thus far quantum computers have only solved trivial problems. David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer:

scalable physically to increase the number of qubits
qubits can be initialized to arbitrary values
quantum gates faster than decoherence time
Turing-complete gate set
qubits can be read easily

Quantum decoherence

One major problem is keeping the components of the computer in a coherent state, as the slightest interaction with the external world would cause the system to decohere. This effect causes the unitary character (and more specifically, the invertibility) of quantum computational steps to be violated. Decoherence times for candidate systems, in particular the transverse relaxation time T₂ (terminology used in NMR and MRI technology), typically range between nanoseconds and seconds at low temperature. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much quicker than the decoherence time. If the error rate is small enough, it is possible to use quantum error correction, which corrects errors due to decoherence, thereby allowing the total calculation time to be longer than the decoherence time. An often cited (but rather arbitrary) figure for required error rate in each gate is 10⁻⁴. This implies that each gate must be able to perform its task 10,000 times faster than the decoherence time of the system.

Meeting this scalability condition is possible for a wide range of systems. However the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L⁴ and L⁶, where L is the number of bits in the number to be factored. For a 1000 bit number, this implies a need for 10¹² to 10¹⁸ qubits. Fabrication and control of this large number of qubits is non-trivial for any of the proposed designs.

One approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on knot theory to form stable logic gates.

Candidates

There are a number of quantum computing candidates, among those:

Superconductor-based quantum computers (including SQUID-based quantum computers)
Trapped ion quantum computers
Electrons on helium quantum computers
"Nuclear magnetic resonance on molecules in solution"-based
"Quantum dot on surface"-based
"Cavity quantum electrodynamics" (CQED)-based
"Molecular magnet"-based
Fullerene-based ESR quantum computer
Solid state NMR Kane quantum computers
Optic-based quantum computers (Quantum optics)
Topological quantum computer

In 2005, researchers at the University of Michigan built a semiconductor chip which functioned as an ion trap. Such devices, produced by standard lithography techniques, may point the way to scalable quantum computing tools. [1]

Quantum computing in computational complexity theory

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 half. 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.

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.

Further information

Quantum computing is part of quantum information processing.

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Quantum Computing with Molecules article in Scientific American by Neil Gershenfeld and Isaac L. Chuang - a generally accessible overview of quantum computing.