# ApplyAnd operation

Namespace: Microsoft.Quantum.Canon

Package: Microsoft.Quantum.Standard

Inverts a given target qubit if and only if both control qubits are in the 1 state, using measurement to perform the adjoint operation.

operation ApplyAnd (control1 : Qubit, control2 : Qubit, target : Qubit) : Unit is Adj + Ctl


## Description

Inverts target if and only if both controls are 1, but assumes that target is in state 0. The operation has T-count 4, T-depth 2 and requires no helper qubit, and may therefore be preferable to a CCNOT operation, if target is known to be 0. The adjoint of this operation is measurement based and requires no T gates.

The controlled application of this operation requires no helper qubit, 2^c Rz gates and is not optimized for depth, where c is the number of overall control qubits including the two controls from the ApplyAnd operation. The adjoint controlled application requires 2^c - 1 Rz gates (with an angle twice the size of the non-adjoint operation), no helper qubit and is not optimized for depth.

## Input

### control1 : Qubit

First control qubit

### control2 : Qubit

Second control qubit

### target : Qubit

Target auxiliary qubit; must be in state 0

## References

• Cody Jones: "Novel constructions for the fault-tolerant Toffoli gate", Phys. Rev. A 87, 022328, 2013 arXiv:1212.5069 doi:10.1103/PhysRevA.87.022328
• Craig Gidney: "Halving the cost of quantum addition", Quantum 2, page 74, 2018 arXiv:1709.06648 doi:10.1103/PhysRevA.85.044302
• Mathias Soeken: "Quantum Oracle Circuits and the Christmas Tree Pattern", Blog article from December 19, 2019 (note: explains the multiple controlled construction)