# ControlledOnBitString function

Namespace: Microsoft.Quantum.Canon

Package: Microsoft.Quantum.Standard

Returns a unitary operation that applies an oracle on the target register if the control register state corresponds to a specified bit mask.

function ControlledOnBitString<'T> (bits : Bool[], oracle : ('T => Unit is Adj + Ctl)) : ((Qubit[], 'T) => Unit is Adj + Ctl)


## Description

The output of this function is an operation that can be represented by a unitary transformation $U$ such that \begin{align} U \ket{b_0 b_1 \cdots b_{n - 1}} \ket{\psi} = \ket{b_0 b_1 \cdots b_{n-1}} \otimes \begin{cases} V \ket{\psi} & \textrm{if} (b_0 b_1 \cdots b_{n - 1}) = \texttt{bits} \\ \ket{\psi} & \textrm{otherwise} \end{cases}, \end{align} where $V$ is a unitary transformation that represents the action of the oracle operation.

## Input

### bits : Bool[]

The bit string to control the given unitary operation on.

### oracle : 'T => Unit is Adj + Ctl

The unitary operation to be applied on the target register.

## Output : (Qubit[],'T) => Unit is Adj + Ctl

A unitary operation that applies oracle on the target register if the control register state corresponds to the bit mask bits.

## Example

The following code snippets are equivalent:

(ControlledOnBitString(bits, oracle))(controlRegister, targetRegister);


and

within {
ApplyPauliFromBitString(PauliX, false, bits, controlRegister);
} apply {
Controlled oracle(controlRegister, targetRegister);
}


The following code prepares a state $\frac{1}{2}(\ket{00} - \ket{01} + \ket{10} + \ket{11})$:

using (register = Qubit[2]) {
ApplyToEach(H, register);
(ControlledOnBitString([false], Z))(register[0..0], register[1]);
}


## Remarks

The pattern given by bits may be shorter than controlRegister, in which case additional control qubits are ignored (that is, neither controlled on $\ket{0}$ nor $\ket{1}$). If bits is longer than controlRegister, an error is raised.

Given a Boolean array bits and a unitary operation oracle, the output of this function is an operation that performs the following steps:

• apply an X operation to each qubit of the control register that corresponds to false element of the bits;
• apply Controlled oracle to the control and target registers;
• apply an X operation to each qubit of the control register that corresponds to false element of the bits again to return the control register to the original state.

The output of the Controlled functor is a special case of ControlledOnBitString where bits is equal to [true, ..., true].