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quantum-dummy-text [June 13, 2026 at 00:58] – created Ivan Janevskiquantum-dummy-text [June 13, 2026 at 03:13] (current) – external edit 127.0.0.1
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 The six cardinal points of the Bloch sphere are the most commonly used single-qubit states. The $\pm z$ poles are $\lvert 0\rangle$ and $\lvert 1\rangle$. The $\pm x$ equatorial points are $\lvert +\rangle$ and $\lvert -\rangle$, which differ only in the relative sign of their amplitudes. The $\pm y$ equatorial points are $\lvert +i\rangle$ and $\lvert -i\rangle$, which carry a complex relative phase of $\pm i$. These six states are the eigenstates of the three Pauli operators $X$, $Y$, $Z$, and come up in measurement bases, gate decompositions, and state preparation constantly. The six cardinal points of the Bloch sphere are the most commonly used single-qubit states. The $\pm z$ poles are $\lvert 0\rangle$ and $\lvert 1\rangle$. The $\pm x$ equatorial points are $\lvert +\rangle$ and $\lvert -\rangle$, which differ only in the relative sign of their amplitudes. The $\pm y$ equatorial points are $\lvert +i\rangle$ and $\lvert -i\rangle$, which carry a complex relative phase of $\pm i$. These six states are the eigenstates of the three Pauli operators $X$, $Y$, $Z$, and come up in measurement bases, gate decompositions, and state preparation constantly.
  
-Multi-qubit entangled states do not fit on a single Bloch sphere — the state space is too large for that picture. The [[bell-states|Bell states]] are the four maximally entangled two-qubit states and are the building block of quantum teleportation, superdense coding, and Bell inequality tests. Three-qubit entanglement splits into two distinct classes: the [[ghz-state|GHZ state]] is a fragile "all or nothing" superposition that collapses to a fully unentangled product state if you lose even one qubit, while the [[w-state|W state]] is robust — tracing out any one qubit still leaves the other two entangled.+Multi-qubit entangled states do not fit on a single Bloch sphere — the state space is too large for that picture. The [[bell-states|Bell states]] are the four maximally entangled two-qubit states and are the building block of quantum teleportation, superdense coding, and Bell inequality tests. Three-qubit entanglement splits into two distinct classes: the [[ket-ghz|GHZ state]] is a fragile "all or nothing" superposition that collapses to a fully unentangled product state if you lose even one qubit, while the [[ket-w|W state]] is robust — tracing out any one qubit still leaves the other two entangled.
  
  
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 Single-qubit gates correspond to rotations on the [[bloch-sphere|Bloch sphere]]. The [[x-gate|X gate]] rotates $180°$ around the $x$-axis, swapping $\lvert 0\rangle \leftrightarrow \lvert 1\rangle$ like a classical NOT. The [[h-gate|Hadamard gate]] rotates the north pole to the $+x$ equatorial point, turning $\lvert 0\rangle$ into equal superposition $\lvert +\rangle$. The [[z-gate|Z gate]] leaves $\lvert 0\rangle$ unchanged and flips the sign of $\lvert 1\rangle$. The [[rotation-gates|rotation gates]] $R_x$, $R_y$, $R_z$ generalize these to arbitrary angles, and the [[u-gate|U gate]] parameterizes every possible single-qubit rotation in three angles $(\theta, \phi, \lambda)$. Single-qubit gates correspond to rotations on the [[bloch-sphere|Bloch sphere]]. The [[x-gate|X gate]] rotates $180°$ around the $x$-axis, swapping $\lvert 0\rangle \leftrightarrow \lvert 1\rangle$ like a classical NOT. The [[h-gate|Hadamard gate]] rotates the north pole to the $+x$ equatorial point, turning $\lvert 0\rangle$ into equal superposition $\lvert +\rangle$. The [[z-gate|Z gate]] leaves $\lvert 0\rangle$ unchanged and flips the sign of $\lvert 1\rangle$. The [[rotation-gates|rotation gates]] $R_x$, $R_y$, $R_z$ generalize these to arbitrary angles, and the [[u-gate|U gate]] parameterizes every possible single-qubit rotation in three angles $(\theta, \phi, \lambda)$.
  
-Two-qubit gates are where entanglement enters the picture. The [[cnot-gate|CNOT gate]] flips a target qubit if and only if the control qubit is $\lvert 1\rangle$; applied to $\lvert +\rangle\lvert 0\rangle$ it produces a Bell state. Together with single-qubit gates, CNOT forms a **universal gate set** — any unitary on any number of qubits can be decomposed into CNOT plus single-qubit rotations. Designing a quantum algorithm then means choosing which gates to apply and in what order so that interference amplifies the probability of the right answer and kills off the wrong ones. That interference engineering is the central challenge of quantum programming.+Two-qubit gates are where entanglement enters the picture. The [[cx-gate|CX gate]] flips a target qubit if and only if the control qubit is $\lvert 1\rangle$; applied to $\lvert +\rangle\lvert 0\rangle$ it produces a Bell state. Together with single-qubit gates, CX forms a **universal gate set** — any unitary on any number of qubits can be decomposed into CX plus single-qubit rotations. Designing a quantum algorithm then means choosing which gates to apply and in what order so that interference amplifies the probability of the right answer and kills off the wrong ones. That interference engineering is the central challenge of quantum programming.
  
  
quantum-dummy-text.1781312298.txt.gz · Last modified: by Ivan Janevski