title: Qutuam Computation
#+STARTUP: overview
Basic states and operations
ketbra
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Ket:
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Bra:
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Production of Ket and Bra:
states
If a measurement will collapse either on |0⟩ or |1⟩, which means |0⟩ or |1⟩ is its eigentstate, we call it as Z-measurement, and denote as .
If a measurement will collapse either on |+⟩ or |-⟩, which means |+⟩ or |-⟩ is its eigentstate, we call it as X-measurement, and denote as .
If a measurement will collapse either on |+i⟩ or |-i⟩, which means |+i⟩ or |-i⟩ is its eigentstate, we call it as Y-measurement, and denote as .
The coefficient has to be because of normalisation.
Measurements
for states basis of{ } and { }, the measurement of state on { } is .
Single Qutantum circuits
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bit flip: :
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phase flip: :
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bit & phase filp: :
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Hadamard gate: , ,
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Phase gate: ,
Multipartite quantum states
we use tensor products to describe multiple states only such can to described by of other states are called uncorrelated, otherweise it's correlated, and when some fully correlated are called entangled.
XOR gate
#+header: :headers '("\\usepackage{qcircuit}")
\Qcircuit @C=1em @R=1em {
\lstick{\ket{x}} & \ctrl{1} & \rstick{\ket{x}} \qw \\
\lstick{\ket{y}} & \targ & \rstick{\ket{x \oplus y }} \qw
}
Photo
Link
here
Bell states
Example of fully correlated states (maximally entangled),
Create Bell states
#+header: :headers '("\\usepackage{qcircuit}")
\Qcircuit @C=1em @R=1em {
\lstick{\ket{i}_A} & \gate{H} & \ctrl{1} & \qw \\
\lstick{\ket{j}_B} & \qw & \targ & \qw
}
Photo
Link
here
Teleportion
If Alise and Bob share the same bell states , Now Alias want to send stetas to Bob,
- Alice preforms a measurement in the Bell basis
- she send her classical output (i,j) to Bob
- Bob apply to get the orignal .
#+header: :headers '("\\usepackage{qcircuit}")
\Qcircuit @C=1em @R=.7em {
\lstick{\ket{\phi_s}_A} & \multigate{1}{Bell Meas} & \cw & \cw & \cwx[2] \\
\lstick{\ket{\varphi^{00}_A}} & \ghost{Bell Meas} & \cw & \cwx[1] \\
\lstick{\ket{\varphi^{00}_B}} & \qw & \qw & \gate{\sigma_x^j}& \gate{\sigma_x^j }& \qw & \rstick{\ket{\phi_s}_B}
}
Photo Link Here
Deutsche-Jose
Bit oracle
#+header: :headers '("\\usepackage{qcircuit}")
Photo
Link
here
, : phase oracle, which is independt of y, .
Hadamard on n qubits
for , ,
for for , , , ,
Deutsche Jose algorithm
#+header: :headers '("\\usepackage{qcircuit}")
\Qcircuit @C=1em @R=1em {
\lstick{\ket{0}} & \gate{H} & \multigate{2}{U_f} & \gate{H} & \meter & \cw & \rstick{\ket{y_0}} \\
\lstick{\ket{0}} & \gate{H} & \ghost{U_f} & \gate{H} & \meter & \cw & \rstick{\ket{y_1}} \\
\lstick{\ket{0}} & \gate{H} & \ghost{U_f} & \gate{H} & \meter & \cw & \rstick{\ket{y_n}}
}
Photo
Link
here
Proof:
because , so ,
The Probability to measure the Zero( ) string is: if f is constant, and if f is balanced