Scalable quantum detector tomography by high-performance computing

TL;DR

This work performs quantum tomography on a megascale quantum photonic detector covering a Hilbert space of 106 and enables the reconstruction of large-scale quantum sources, processes and detectors used in computation and sampling tasks, which may be necessary to prove their nonclassical character or quantum computational advantage.

Abstract

At large scales, quantum systems may become advantageous over their classical counterparts at performing certain tasks. Developing tools to analyse these systems at the relevant scales, in a manner consistent with quantum mechanics, is therefore critical to benchmarking performance and characterising their operation. While classical computational approaches cannot perform like-for-like computations of quantum systems beyond a certain scale, classical high-performance computing (HPC) may nevertheless be useful for precisely these characterisation and certification tasks. By developing open-source customised algorithms using high-performance computing, we perform quantum tomography on a megascale quantum photonic detector covering a Hilbert space of $10^6$. This requires finding $10^8$ elements of the matrix corresponding to the positive operator valued measure (POVM), the quantum description of the detector, and is achieved in minutes of computation time. Moreover, by exploiting the structure of the problem, we achieve highly efficient parallel scaling, paving the way for quantum objects up to a system size of $10^{12}$ elements to be reconstructed using this method. In general, this shows that a consistent quantum mechanical description of quantum phenomena is applicable at everyday scales. More concretely, this enables the reconstruction of large-scale quantum sources, processes and detectors used in computation and sampling tasks, which may be necessary to prove their nonclassical character or quantum computational advantage.

Figures (8)

• Figure 1: (a) Experimental setup to perform high dynamic range quantum detector tomography. The coherent states from a picosecond pulsed laser can be attenuated with variable optical attenuators, to control their mean photon numbers. Polarisation controllers before the fiber beam splitter loop (blue line) and the SNSPD are used to optimize the performance of the devices. The beam splitter loop creates sub-pulses with a temporal separation of $\tau=156~\mathrm{ns}$ and has an adaptable out-coupling $R$ and loop-efficiency $\eta_{\mathrm{loop}}$. A time tagger records raw time tags of the electrical output signal of the SNSPD. (b) Poisson distributions for mean photon numbers of $\bar{n}=1$, $\bar{n}=100$ and $\bar{n}=10^6$, which are set by the variable optical attenuator. (c) Schematic matrix representation of the matrices $\boldsymbol{F}_{D\times M}$ containing the coherent input states, $\boldsymbol{\Pi}_{M\times N}$ containing the (unknown) POVMs of the detector and $\boldsymbol{P}_{D\times N}$ containing the measured outcomes of the detector. (d) Probability distributions that a certain number of time-bins of the detector click, i.e., the outcomes of the detector for mean photon numbers of $\bar{n}=1$, $\bar{n}=100$ and $\bar{n}=10^6$.
• Figure 2: (a-c) Reconstructed POVMs from the experimentally measured data for $M=1210581$, $N=151$ outcomes, and $D=1076$. Due to the span of the input space, the first $\sim$50 outcomes are occupied and shown. (a) Does not include regularisation (smoothing) of the POVMs, (b) uses the standard nearest-neighbour smoothing with a regularisation parameter of $\gamma=10^{-5}$ and (c) in addition to the nearest-neighbour smoothing, utilises a novel long-range approach to the regularisation of the POVMs (see Methods, Regularisation and smoothing, for further detail). (d) Shows the analytical POVMs of the detector. The inset shows the infidelity Eq. \ref{['eq:fidelities']} between the three regularisation approaches and the analytical model for all occupied outcomes of the detector.
• Figure 3: Wigner functions of the reconstructed POVMs using the long-range smoothing (shown in Fig. \ref{['fig:results']}(c)) and the analytical POVMs (shown in Fig. \ref{['fig:results']}(d)) for a subset of outcomes (a) $n=0$, (b) $n=1$, (c) $n=2$ and (d) $n=40$. Clear negativity in (b) shows the non-classical nature of the corresponding POVM $\boldsymbol{\pi}_{n=1}$. The inset shows that the general overlap of the Wigner functions based on the experimental (colored lines) and analytical (black dashed lines) POVMs is good, however, some noise appears in the Wigner functions for the experimental POVMs at larger outcomes.
• Figure 4: Runtimes (a) and memory usage (b) of the CVXPY-MOSEK-solver of Liu et al. liu2023optimized (orange dots) and the solver proposed in this work (blue crosses, 8 CPU cores; blue pluses, 128 CPU cores) for the detector setting of Liu et al. for different numbers of outcomes $N$. The orange diamond symbol show the estimated runtime and memory usage based on linear extrapolation for the solver of Liu et al. for $N=1000$.
• Figure 5: Scaling of the reconstruction wall time for a simulated detector with $N=151$ and $M=(D-1)^2$ up to one full compute node of the Noctua 2 cluster with 256 GiB of main memory ($N\cdot M\leq 4.4\cdot 10^9$). Beyond $N\cdot M\approx 4.4\cdot 10^9$ the wall time scaling for $N\approx 150$, $D=\sqrt{M}+1$ is estimated from the measurements of the scalability of the underlying operations like the gradient combined with the estimated numbers of operations described in Sec. \ref{['sec:scalability_large']} assuming that the lowest possible number of compute nodes is used for the reconstruction, i.e., filling the main memory of the nodes. The blue cross represents the runtime for reconstructing the experimental POVMs using one compute node for comparison.