Algorithmic cooling
A qubit (or quantum bit) is a unit of information that can be in a superposition of two states, denoted as formula_1 and formula_2. The general superposition can be written as formula_3where formula_4 and formula_5. If one measures the state of the qubit in the orthonormal basis composed of formula_1 and formula_2, one gets the result formula_1 with probability formula_9 and the result formula_2 with probability formula_11.
Algorithmic cooling
The algorithm can be written using quantum operations on qubits, as opposed to the classical treatment. In particular, the C-NOT and C-SWAP steps can be replaced by a single unitary quantum operator that operates on the 3 qubits. Although this operation changes qubits formula_50 in a different manner than the two classical steps, it yields the same final bias for qubit formula_51. The operator formula_107 can be uniquely defined by its action on the computational basis of the Hilbert space of 3 qubits:
In matrix form, this operator is the identity matrix of size 8, except that the 4th and 5th rows are swapped. The result of this operation can be obtained by writing the product state of the 3 qubits, formula_116, and applying formula_107 on it. Afterwards, the bias of qubit formula_51 can be calculated by projecting its state on the state formula_1 (without projecting qubits formula_50) and taking the trace of the result (see density matrix measurement):