Practical and reliable error bars for quantum process tomography

Current techniques in quantum process tomography typically return a single point estimate of an unknown process based on a finite albeit large amount of measurement data. Due to statistical fluctuations, however, other processes close to the point estimate can also produce the observed data with near certainty. Unless appropriate error bars can be constructed, the point estimate does not carry any sound operational interpretation. Here, we provide a solution to this problem by constructing a confidence region estimator for quantum processes. Our method enables reliable estimation of essentially any figure of merit for quantum processes on few qubits, including the diamond distance to a specific noise model, the entanglement fidelity, and the worst-case entanglement fidelity, by identifying error regions which contain the true state with high probability. We also provide a software package, QPtomographer, implementing our estimator for the diamond norm and the worst-case entanglement fidelity. We illustrate its usage and performance with several simulated examples. Our tools can be used to reliably certify the performance of, e.g., error correction codes, implementations of unitary gates, or more generally any noise process affecting a quantum system.

Figure 1

The workflow of rigorous process tomography. Our data analysis QPtomographer supports both prepare-and-measure and ancilla-assisted experimental schemes. The conclusion is guaranteed without any prior information on the unknown quantum process.

Figure 2

Illustrations of (a) prepare-and-measure and (b) ancilla-assisted tomographic schemes for an unknown channel ${\mathrm{\Lambda }}_{A\to B}$. In prepare and measure, one can only prepare input state ${\sigma }_A^j$ which is fed into an unknown channel whose output state is measured by some POVM with elements $E_B^{\ell }$. In ancilla assisted, one can prepare entangled input state with some reference system $P$, and measure jointly the output using some POVM with elements $E_{BP}^{\ell }.$

Figure 3

Typical output from QPtomographer. The (blue) dots form the estimated distribution of values of the diamond norm distance to the identity channel as determined by the Metropolis-Hastings random walk. These are well fitted to the (red) curve, which is compactly described by the triple of numbers $(v_0,\mathrm{\Delta },\gamma )$ which we called quantum error bars. Here, $v_0$ is the position of the peak, $\mathrm{\Delta }$ is half-width at relative height $1/e$, and $\gamma$ is a measure of skewness. These data encode information about the performance of the quantum memory, and enable us to construct confidence intervals certifying its quality.

Figure 4

Distribution of the figures of merit for the single-qubit noisy quantum memory example relevant to construct confidence regions. This figure is obtained by the channel-space sampling method run over simulated data from a known true state; vertical dashed lines are true figure-of-merit values of the channel. The data points are the histogram resulting from our Metropolis-Hastings integration method, and the solid lines a corresponding fit to the empirical model of [22] (see main text). Left: the diamond norm is chosen as figure of merit. Right: the figure of merit is chosen to be either the worst-case entanglement fidelity [blue (dark gray)] or the average entanglement fidelity [red (light gray)]. These plots should be understood as tools to construct confidence regions: Given a threshold on the $x$ axis, one may easily calculate from these curves the confidence with which one may ascertain the true figure of merit (see main text). The present example is included in our software package [33].

Figure 5

Distribution of the diamond norm distance for the two-qubit noisy quantum memory example. Results from other jump distributions of the same sampling method (not shown), namely, bipartite state or channel space, coincide with plotted curves. The difference between the two methods for the entanglement fidelity is not a contradiction, rather, in this situation the channel-space method [blue (dark gray)] gives better tomographic results compared to the bipartite-state method [red (light gray)]. The reason is due to the additional use of prior information about the input state in the channel-space method. This figure can be reproduced in its entirety from example 2qubit-noisy-identity in our software package [33].

Figure 6

Distribution of the figures of merit for the two-qubit noisy quantum memory example in log-scale. This figure is obtained by all combinations of sampling methods and jump distributions. Left: log-plot of histogram bins with vertical error bars for diamond norm distance. Bipartite-state methods are plotted as the two right dotted bins [standard as red (light gray), optimized as blue (dark gray)], while channel-space method are two left dotted bins [elr as magenta (light gray), eiH as green (darker gray)]. Different jump distributions for each method, bipartite state and channel space, give consistent results. Right: log-plot of worst-case entanglement fidelity and bipartite-state sampling. The default fit model (with legend fit1) does not match the histogram well, which is fixed by another fit model (legend fit2). In any case, however, the exponential decay of the fit function ensures that the quantum error bars provide a meaningful estimation of the estimation error. The present example is included in our software package [33].

Figure 7

Convergence of quantum error bars to the true value of figure of merit in the limit of many measurements, for a noisy identity process on a qutrit. Measurements using Gell-Mann matrices as observables on the input and the output systems were simulated with ${10}^6$ outcomes per setting, providing the reference experiment (labeled “100%”). Analyses as in Fig. 4 were then carried out after artificially rescaling the measured frequency counts by various factors (percentage labels), allowing us to compare regimes with different number of measurements while still keeping the estimated expectation values of the measured observables constant to facilitate comparison. As the number of measurements increases, the distribution of $f_{\diamond }$, the diamond norm to the identity channel, peaks to the known true value of $3.556\times {10}^{-2}$. Data points display the numerical histogram (bipartite-state sampling method: blue-green; channel-space method: red-yellow) which are fit to our model #1. Inset: the quantum error bars ($v_0,\mathrm{\Delta },\gamma$) obtained from the fit [22] (channel-space method only) are plotted against the number of measurements relative to the reference experiment; markers represent $v_0$ with an error bar representing $[v_0-(\mathrm{\Delta }-\gamma ),v_0+\mathrm{\Delta }+\gamma ]$ for each analysis instance. The dotted line indicates the known true value of $f_{\diamond }$.

Figure 8

One of the quantum error bars, $\mathrm{\Delta }$, is observed to scale approximately as $1/\sqrt{N}$, where $N$ indicates the number of measurements, as expected in standard (nonadaptive) quantum tomography. The setting is the same as in Fig. 7. By choosing more sophisticated measurement operators, for instance by adapting the measurement settings based on earlier outcomes, the scaling could be improved [13, 50, 51, 52].

Figure 9

An intermediate tomographic scheme. Scenario (c) comes from restricting $E_k^{\ell }$ acting on $BP$ of Fig. 2 to be a tensor product measurement. The measurement $E_P^j$ on half of an entangled state in (c) can be seen as a probabilistic state preparation similar to Fig. 2.