Quantum computers
Quantum logic gates can only be applied to qubits. The Hadamard gate places classical values into a superposition with a 50% chance of being measured in the 1 or 0 state.
1 √2 1 √2
1 √2
1 √2 1 √2
1 √2
1 √2
1 √2 1 √2
0 1
1 0
(
) = (
)
(
) = (
)
1 √2 )
1 √2 )
1 √2
−
−
−
(
(
As the Euclidean norm ignores signs, there is no difference between the measured probabilities above. Nonetheless, these are distinct quantum states that behave differently under other operations, providing more calculation possibilities.
All quantum gates are their own inverse. Applying the Hadamard gate to 1 or 0 twice returns the original value. This allows some quantum algorithms to be deterministic by obtaining a 100% certain classical representation, but not all algorithms can reach a state in this form. Instead, the probabilities of a qubit collapsing to 1 or 0 are observed by running the computation multiple times.
1 √2 1 √2
1 √2
1 √2
0 1
(
) = (
)
1 √2 )
1 √2
−
−
(
All gates applied to quantum states must be reversible – a known gate’s input must be able to be deduced from a given output. Constant-0 and Constant-1 are reversible if the input value is tracked, so quantum gates always use an additional wire, the ‘ Output ’ wire. The ‘ Input ’ wire never changes value – the operation applies to end of the Output wire leaving the gate. Constant-0 does nothing to bits in superposition; Constant-1 negates the Output. Identity uses the Input as the control bit for CNOT applied to the Output. Negation flips Identity’s Outp ut using a NOT.
Certain problems could be solved on quantum computers many orders of magnitude faster than what current computers achieve. They use fewer gates and bits as they can do more with what they have. As a result, billions of pounds of research are being poured into quantum computing annually, by companies such as IBM and Microsoft. 18
The Deutsch Oracle is the simplest problem where a Quantum computer outperforms a classical one. Given an unknown gate (the oracle) out of Constant-0, Constant-1, Identity and Negation, it must be found whether the function depends on the input value (it is variable) or doesn’t (it is constant). A classical computer takes two tests to find the solution: an output of 1 from an input of 0 indicates Constant-1 or Negation; an output of 0 from inputting 0 suggests Identity or Constant-0. It is similar for an input of 1; both input values need to be tested. By comparison, a quantum computer only needs one test. As the Euclidean norm doesn’t distinguish between signs, Constant-0 and Constant-1 give identical results when applied to qubits in superposition. The same goes for Identity and Negation.
18 Helwer (2018).
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