Experimental Bayesian Quantum Phase Estimation on a Silicon Photonic Chip

Quantum phase estimation is a fundamental subroutine in many quantum algorithms, including Shor’s factorization algorithm and quantum simulation. However, so far results have cast doubt on its practicability for near-term, nonfault tolerant, quantum devices. Here we report experimental results demonstrating that this intuition need not be true. We implement a recently proposed adaptive Bayesian approach to quantum phase estimation and use it to simulate molecular energies on a silicon quantum photonic device. The approach is verified to be well suited for prethreshold quantum processors by investigating its superior robustness to noise and decoherence compared to the iterative phase estimation algorithm. This shows a promising route to unlock the power of quantum phase estimation much sooner than previously believed.

Figure 1

(a) Quantum circuit for standard iterative and Bayesian phase estimation algorithms. (b) Experimental setup and the integrated silicon quantum photonic device. The quantum chip can perform any controlled-$U(2)$ operation and any single-qubit state preparation and analysis. Photons are produced and guided in the silicon waveguides (black wires) and reconfigurably controlled by thermo-optical phase shifters. Coherent light was used to generate photons and superconducting nanowire detectors were used for the detection, both coupled to the chip through lensed-single mode fibers. The implementation of the algorithms was achieved by interfacing the quantum device with a classical CPU.

Figure 2

(a) Convergence of RFPE to the true phase value $2\pi {\varphi }_0=4.8741\mathrm{rad}$ related to the energy of the dissociated ${\mathrm{H}}_2$ molecule. The initial prior distribution is $\mathcal{N}(\pi ,{\pi }^2)$. Data points show an exponential shrink of the error, compatible with simulations (blue line) of the device performance averaged over 1000 runs of the RFPE algorithm (shaded area: 67.5% credible interval). The dashed black line denotes the convergence of the standard IPEA using the parameters discussed in the main text. Inset: convergence of the phase estimation to ${\varphi }_0$ (red line), where errors are evaluated using the s.d. of the prior distribution. Error bars are obtained from the standard deviation of the median. (b) Bonding energies of the ${\mathrm{H}}_2$ molecule for various atomic distances using RFPE with 50 steps. Energy estimations are achieved within chemical accuracy. Errors are smaller than the markers and neglected in the plot for more clarity. The dashed line represents theoretical energies.

Figure 3

Effects of different experimental noises on phase estimation strategies. (a) Infidelity of quantum operation. Each of the correct phases ${\overline{\phi }}_i$ for the phase gates is synthetically replaced with a Gaussian distributed ${\phi }_i\sim \mathcal{N}({\overline{\phi }}_i,{\sigma }_{\text{phase}})$, where ${\sigma }_{\text{phase}}$ represents a noise in the phases. (b) Decoherence. For IPEA data, experiments were repeated 10 times with 16-bit accuracy to evaluate median error and error bars, while the RFPE data were collected from a single run, after 100 measurements, and directly used to evaluate the error and uncertainty within the algorithm. Error bars for the estimated phase represent in both plots either a 67.5% credible region for RFPE, either a 67.5% confidence interval for IPEA. In the cases where error bars are smaller than the markers they have been omitted for clarity. Points are experimental data and dashed lines are simulations averaged over 1000 runs. The simulations take into account the characterized residual phase noise in the device.

Figure 4

Experimental data (points) and simulation (lines) of the behavior of RFPE under different decoherence times $T_2$. Shaded areas represent a 67.5% credible interval from data simulated averaging over 500 runs of RFPE.