Creating Entanglement

Intent

Enforce a strong correlation between qubits by entangling them.

Context

Entanglement is a unique characteristic of quantum mechanics and one of the causes for the power of quantum algorithms (Bruß and Macchiavello 2011). Although entanglement is not necessarily needed for a powerful quantum algorithm (Biham et al. 2004), it is required to achieve an exponential speedup over classical algorithms (Jozsa and Linden 2003).
Consequently, a quantum register is often entangled for further processing after the Initialization.

Forces

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Solution

There exist numerous approaches for the creation of an entangled state. For example, entanglement can be created by a Boolean function $f:\{0, 1\}^n \rightarrow \{0, 1\}^m$, a corresponding unitary operation

$$ \newcommand{\colVec}[1]{% inline column vector \bigl( \begin{smallmatrix}#1\end{smallmatrix}\bigr) } \newcommand{\bigColVec}[1]{% inline column vector \left( \begin{matrix}#1\end{matrix}\right) } \newcommand{\state}[1]{{\left| #1 \right>}}U_f: \{0, 1\}^{n+m} \rightarrow \{0, 1\}^{n+m}, U_f \left(\left| x,y \right> \right) = \left| x,y\oplus f(x) \right> $$

and a Uniform Superposition in the first $n$ qubits of the quantum register:

$$ \newcommand{\colVec}[1]{% inline column vector \bigl( \begin{smallmatrix}#1\end{smallmatrix}\bigr) } \newcommand{\bigColVec}[1]{% inline column vector \left( \begin{matrix}#1\end{matrix}\right) } \newcommand{\state}[1]{{\left| #1 \right>}}U_f \left(H^{\otimes n} \otimes I^{\otimes m} \right) \left( \left| 0 \right>^{\otimes n} \otimes \left|0 \right> ^{\otimes m} \right) $$

$$ \newcommand{\colVec}[1]{% inline column vector \bigl( \begin{smallmatrix}#1\end{smallmatrix}\bigr) } \newcommand{\bigColVec}[1]{% inline column vector \left( \begin{matrix}#1\end{matrix}\right) } \newcommand{\state}[1]{{\left| #1 \right>}}= U_f \left( \dfrac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n-1} \left|x \right> \otimes \state{0}^{\otimes m} \right) $$

The resulting state is entangled. For $f = id$, $U_{f}=CNOT$ and consequently $CNOT \left(H \otimes I \right) \left( \left| 0 \right> \otimes \left|0 \right> \right)$ is entangled.

Result

Unitary transformations that create entanglement must include multi-qubit operators such as CNOT.
Such operators typically have a lower gate fidelity than single qubit operators and therefore increase the overall gate error.

Examples

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Related Patterns

A quantum register must make use of Initialization before entanglement can be created. Applying $U_{f}$ as described above computes a Function Table and makes use of Uniform Superposition.

Known Uses

Most algorithms use entangled states. A Function Table may create an entangled state, by making use of the above-defined unitary operation $U_f$ of a Boolean function $f$.

Quantum Phase Estimation (PlanQK) Grover's Algorithm (PlanQK) Deutsch algorithm (PlanQK)

Extended Quantum Computing Patterns