Uniform Superposition

Intent

Create a uniform superposition of all possible states of a quantum register

Context

The power of quantum algorithms partially originates in the ability to represent multiple states at once, the so-called quantum parallelism.
To realize quantum parallelism, some qubits of a quantum register must be brought into superposition.
Many algorithms start with obtaining an equally weighted superposition in (a part of) a quantum register. Therefore, for every state represented by (this part of) the quantum register, the measurement probability is the same.

Forces

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Solution

After initializing the quantum register as the unit vector $\left|0 \ldots 0\right>$, a Uniform Superposition is created via the Hadamard transformation:

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

The quantum register may also include ancilla bits that must not necessarily be brought into superposition. For example, if the last part of the quantum register is used for ancilla qubits, a different Initialization on the two parts of the register can be accomplished by using a tensor product operator $H^{\otimes n}\otimes U$ where the Hadamard transformation $H^{\otimes n}$ only operates on the first part and $U$ operates on the ancilla qubits. If the ancilla qubits are not brought into superposition, $U$ is the Identity (e.g., $U=I^{\otimes m}$):

$$ \newcommand{\state}[1]{{\left| #1 \right>}} H^{\otimes n} \otimes I^{\otimes m} \left( \left| 0 \right>^{\otimes n} \otimes \left| 0 \right>^{\otimes m} \right)= \frac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n-1} \left|x \right> \otimes \state{0}^{\otimes m} $$

Result

A Uniform Superposition can be achieved by performing a single parallel operation on each qubit, increasing the depth of the entire circuit by only one.

Examples

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Related Patterns

Uniform Superposition is a special case of Initialization.
After a register is brought into uniform superposition, Creating Entanglement may be used. A Function Tablecan be computed on a register in uniform superposition. An Oracle may require its input to be in uniform superposition.

Known Uses

Countless quantum algorithms start with a uniform superposition.

Grover's Algorithm (PlanQK) Dürr-Høyer Quantum Minimization Algorithm (PlanQK) Quantum Phase Estimation (PlanQK) Quantum approximate optimization algorithm (PlanQK)

Extended Quantum Computing Patterns