Efficient compression of quantum information

We propose a scheme for an exact efficient transformation of a tensor product state of many identically prepared qubits into a state of a logarithmically small number of qubits. Using a quadratic number of elementary quantum gates we transform $N$ identically prepared qubits into a state, which is nontrivial only on the first $\lceil \log {}_2(N+1)\rceil$ qubits. This procedure might be useful for quantum memories, as only a small portion of the original qubits has to be stored. Another possible application is in communicating a direction encoded in a set of quantum states, as the compressed state provides a high-effective method for such an encoding.

Figure 1

Sequence of gates of compression transformation in the five-qubit example. Operation $V$ represents the starting two-qubit transformation Eq. (10) and operations $U$ represent relevant $U(a,b)$ operations Eq. (12).

Figure 2

Sequence of gates for the final step operation in the five-qubit example.

Figure 3

Fidelity of the global state after the action of noise with and without compression and decompression. The fidelity of the compressed state is clearly higher than the fidelity of the uncompressed one.

Figure 4

Fidelity of the one-qubit state after the action of noise with and without compression and decompression. For the compression scenario, results for noise acting around $x$, $y$, and $z$ axis, as well as for noise averaged through all axes are shown.