Exponential Speedup with a Single Bit of Quantum Information: Measuring the Average Fidelity Decay

We present an efficient quantum algorithm to measure the average fidelity decay of a quantum map under perturbation using a single bit of quantum information. Our algorithm scales only as the complexity of the map under investigation. Thus for those maps admitting an efficient gate decomposition, it provides an exponential speedup over known classical procedures. Fidelity decay is important in the study of complex dynamical systems, where it is conjectured to be a signature of eigenvector statistics. Our result also illustrates the role of chaos in the process of decoherence.

Figure 1

Quantum circuit evaluating the average fidelity $\overline{F_n(\psi )}$ between the perturbed and unperturbed maps $U$ and $U_p=UP$. The gates $R_k$ are $\pi /2$ rotation in the Bloch sphere around axis $k=x$ or $y$. When $k$ is set to $x$, we get the real part of $\mathrm{Tr}\{(U^n{)}^{†}{U}_p^n\}/N$ while $k=y$ yields the imaginary part. The unitary operator $P$ is applied conditionally: when the probe qubit is in state $|1⟩$, the unitary $P$ is applied to the lower register while no transformation is performed when the state of the probe qubit is $|0⟩$.

Figure 2

Fidelity decay $F_n(\psi )$ averaged over 50 initial computational basis states for $U_{\mathrm{QKT}}$ in a regular regime ($k=1$, squares) and chaotic regime ($k=12$, circles). The dashed lines represent the exact average Eq. (5) and the full line shows the exponential decay at the Fermi golden rule rate $|V^2|{\delta }^2$.