The most common quantum gates operate on spaces of one or two qubits. This means that as matrices, quantum gates can be described by 2 × 2 or 4 × 4 matrices with orthonormal rows.

Remark. "Quantum logic" can refer either to the performance of quantum logic gates or to a foundational formalism for quantum mechanics called quantum logic based on a modification of some of the rules of propositional logic.

Examples

Hadamard gate. This gate operates on a single qubit. It is represented by the Hadamard matrix:

Since the rows of the matrix are orthogonal, H is indeed a unitary matrix.

Phase shifter gates. Gates in this class operate on a single qubit. They are represented by 2 × 2 matrices of the form

where θ is the phase shift.

Controlled gates. Suppose U is a gate that operates on single qubits with matrix representation

The controlled-U gate is a gate that operates on two qubits in such a way that the first qubit serves as a control.

$|\; 0\; 0\; rangle\; mapsto\; |\; 0\; 0\; rangle$

$|\; 0\; 1\; rangle\; mapsto\; |\; 0\; 1\; rangle$

$|\; 1\; 0\; rangle\; mapsto\; |\; 1\; rangle\; U\; |0\; rangle\; =\; |\; 1\; rangle\; left(x\_\{00\}\; |0\; rangle\; +\; x\_\{10\}\; |1\; rangleright)$

$|\; 1\; 1\; rangle\; mapsto\; |\; 1\; rangle\; U\; |1\; rangle\; =\; |\; 1\; rangle\; left(x\_\{01\}\; |0\; rangle\; +\; x\_\{11\}\; |1\; rangleright)$

Thus the matrix of the controlled U gate is as follows:

Uncontrolled gate. We note the difference between the controlled-U gate and an uncontrolled 2 qubit gate

$I\; otimes\; U$ defined as follows:

$|\; 0\; 0\; rangle\; mapsto\; |\; 0\; rangle\; U\; |0\; rangle$

$|\; 0\; 1\; rangle\; mapsto\; |\; 0\; rangle\; U\; |1\; rangle$

$|\; 1\; 0\; rangle\; mapsto\; |\; 1\; rangle\; U\; |0\; rangle$

$|\; 1\; 1\; rangle\; mapsto\; |\; 1\; rangle\; U\; |1\; rangle$

represented by the unitary matrix

Since this gate is reducible to more elementary gates it is usually not included in the basic repertoire of quantum gates. It is mentioned here only to contrast it with the previous controlled gate.

Universal quantum gates

A set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced, that is, any other unitary operation can be expressed as a finite sequence of gates from the set. Equivalently, a set of universal quantum gates is a set of generators for the group of unitary matrices. One simple set of two-qubit universal quantum gates is the Hadamard gate ($H$), a phase rotation gate $R(cos^\{-1\}begin\{matrix\}\; frac\{3\}\{5\}\; end\{matrix\}))$, and the controlled NOT gate, a special case of controlled-U such that

A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate, $D(theta)$

The universal classical logic gate, the Toffoli gate, is reducible to the Deutsch gate, $D(begin\{matrix\}\; frac\{pi\}\{2\}\; end\{matrix\})$, thus showing that all classical logic operations can be performed on a universal quantum computer.

See also

• Pauli matrices
• Quantum finite automata

References

• M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000

Greg Roach wrote and designed the title which follows the story of Drew Griffin, a military recruit in a base on a far off planet, preparing for a war with an alien species. A sequel: Quantum Gate 2: The Vortex was also developed by Hyperbole Studios. Both titles suffered from the expectations of gamers, who, by and large failed to understand the goals and intentions of the creators - to create a new kind of participatory and immersive story - rather than a traditional computer game filled with puizzles and action sequences.

Quantum Gate was celebrated for its superior graphics, powerful story and computer-generated imagery, but criticised for having limited gameplay. Computer Gaming World referred to it as "A future shock film on par with Alien."

Originally the two releases (Quantum Gate/The Vortex) were written and designed as a single story/experience, but the publisher, Media Vision, exercised an acceleration clause in the development contract and demanded the title months ahead of the original release date - causing the developers to have to split the title into two. As a result, The Vortex was redesigned to allow its story to stand alone.

References