65 lines
1.9 KiB
Markdown
65 lines
1.9 KiB
Markdown
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# Quantum Gate
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{ Currently studying this, there may be errors. ~drummyfish }
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Quantum (logic) gate is a [quantum computing](quantum.md) equivalent of a traditional [logic gate](logic_gate.md). A quantum gate takes as an input *N* [qubits](qubit.md) and transforms their states to new states (this is different from classical logical gates that may potentially have a different number of input and output values).
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Quantum gates are represented by [complex](complex_number.md) [matrices](matrix.md) that transform the qubit states (which can be seen as points in multidimensional space, see Bloch sphere). A gate operating on *N* qubits is represented by a *2^N*x*2^N* matrix. These matrices have to be **unitary**. Operations performed by quantum gates may be reversed, unlike those of classical logic gates.
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We normally represent a single qubit state with a **column** [vector](vector.md) *|a> = a0 * |0> + a1 * |1> => [a0, a1]* (look up bra-ket notation). Multiple qubit states are represented as a [tensor product](tensor_product.md) of the individual state, e.g. *|a,b> = [a0 * b0, a0 * b1, a1 * b0, a1 * b1]*. Applying a quantum gate *G* to such a qubit vector *q* is performed by simple matrix multiplication: *G * v*.
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## Basic gates
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Here are some of the most common quantum gates.
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### Identity
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Acts on 1 qubit, leaves the qubit state unchanged.
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```
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1 0
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0 1
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```
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## Pauli Gates
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Act on 1 qubit. There are three types of Pauli gates: X, Y and Z, each one rotates the qubit about the respective axis by [pi](pi.md) radians.
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The X gate is:
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```
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0 1
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1 0
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```
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The Y gate is:
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```
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0 -i
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i 0
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```
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The Z gate is:
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```
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1 0
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0 -1
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```
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## NOT
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The not gate is identical to the Pauli X gate. It acts on 1 qubit and switches the probabilities of measuring 0 vs 1.
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## CNOT
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Controlled NOT, acts on 2 qubits. Performs NOT on the second qubit if the first qubit is *|1>*.
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```
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1 0 0 0
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0 1 0 0
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0 0 0 1
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0 0 1 0
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```
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TODO
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