Sequential Implementation of Global Quantum Operations

We study the possibility for a global unitary applied on an arbitrary number of qubits to be decomposed in a sequential unitary procedure, where an ancillary system is allowed to interact only once with each qubit. We prove that sequential unitary decompositions are in general impossible for genuine entangling operations, even with an infinite-dimensional ancilla, being the controlled-NOT gate a paradigmatic example. Nevertheless, we find particular nontrivial operations in quantum information that can be performed in a sequential unitary manner, as is the case of quantum error correction and quantum cloning.

Figure 1

Sketch of the proposed sequential quantum factory of operations. When possible, it decomposes a global unitary operation $U$ acting on $N$ qubits into a sequence of local unitary steps where each qubit interacts only once with the ancilla $A$, and measurements are not allowed. When each qubit-ancilla interaction is finished, the corresponding qubit in the chain leaves the operation region, and the following qubit starts its interaction with $A$.

Figure 2

Trivial protocol for the sequential decomposition of a $1\to N$ isometry. First, the desired isometry is implemented locally between qubit 1 and $N-1$ qubits in the ancilla degrees of freedom. Second, each ancilla qubit is mapped locally to its corresponding blank chain qubits.

Figure 3

Diagram showing the sequential decomposition of a $1\to N$ isometry $U$, acting on an arbitrary initial state $|\psi ⟩$.