Two-qubit Hamiltonian tomography by Bayesian analysis of noisy data

We present an empirical strategy to determine the Hamiltonian dynamics of a two-qubit system using only initialization and measurement in a single fixed basis. Signal parameters are estimated from measurement data using Bayesian methods from which the underlying Hamiltonian is reconstructed, up to three unobservable phase factors. We extend the method to achieve full control Hamiltonian tomography for controllable systems via a multistep approach. The technique is demonstrated and evaluated by analyzing data from simulated experiments including projection noise.

Figure 1

Manufactured qubit system. A pair of horizontally aligned double quantum dots (center) can act as two qubits. A qubit can be defined in each double quantum dot by two different charging states, e.g., a single excess electron located on the left or the right dot of each pair. Electrodes (top and bottom) control the potentials and electron tunneling rates. Single-electron transistors (left and right) measure the locations of the excess electrons which defines the measurement basis or logical states of the qubit. Due to finite manufacturing precision, the placement of the control and the measurement structures may not be exactly as calculated; hence, the Hamiltonian dependence upon control signals will have to be determined empirically [6].

Figure 2

(Color online) Simulated measurements of system 1 with $2^{10}+1$ data points per trace sampled at $\Delta t=0.1$, signal length $T=102.4$ (arbitrary units), number of experiment repetitions per data point $N_e=250$. Each graph is the probability $p_{k\ell }\left(t\right)$ at time $t$ of detecting the system in state $|\ell ⟩$ ($\ell =1,2,3,4$ left to right) if initialized in state $|k⟩$ ($k=1,2,3,4$ top to bottom).

Figure 3

(Color online) Estimated and actual values of the coefficients $a_{k\ell ;m}$ with estimated error bars of system 1. When both $N$ and $N_e$ are large (top) the error bars are nearly invisible, and the estimated and the actual values are almost indistinguishable. When $N$ and $N_e$ are both small (bottom), the error bars are significantly larger, mainly due to increased noise variances ${\sigma }_{k\ell }^2$; yet the actual and the estimated values for the coefficients are still almost indistinguishable. This suggests that the estimated coefficients are in fact much more accurate than the uncertainty estimates suggest.

Figure 4

(Color online) Power spectrum $C\left(\omega \right)$ of system 1. Although the power spectrum is noisy, the logarithmic plot of $C\left(\omega \right)$ of the measured signals shown in Fig. 2 shows six well-defined peaks for ${\omega }_m>0$ in addition to the peak at $\omega =0$. The inset shows the filtered power spectrum $C\left(\omega \right)>C_0$, from which the six peaks ${\omega }_m$ can easily be identified using standard peak detection.

Figure 5

(Color online) The power spectrum of system 12. The power spectrum $C\left(\omega \right)$ has only five peaks ${\omega }_m>0$ in addition to the peak at $\omega =0$. This could mean that the system has only five distinct transition frequencies, or that the measured signals are not sufficient to resolve two (closely spaced) transition frequencies.

Figure 6

(Color online) Logarithmic likelihood on $I_1\times I_1$ for system 12 with five-peak power spectrum shown in Fig. 5 shows symmetry about $y=x$ as ${\text{log}}_{10}P\left[\left({\omega }_1,{\omega }_2,\dots \right)\mid {\mathbf{d}}_{k\ell }\right]={\text{log}}_{10}P\left[\left({\omega }_2,{\omega }_1,\dots \right)\mid {\mathbf{d}}_{k\ell }\right]$ with twin peaks for $y\ne x$ indicating that the most probable model on this subspace of the parameter space is a six-frequency model.

Figure 7

(Color online) The transition frequency diagram for each of the 100 test systems shows that the transition frequencies range from 0.3 to 7, and there are six systems (12, 22, 34, 38, 73, and 78) with two transition frequencies that differ by less than 0.01 (circled), which are difficult to resolve, including one system (78) with two such cases.

Figure 8

(Color online) Histogram of the relative % error for the test 100 systems for sampling numbers of (a) $N=1025$, $N_e=125$ and (b) $N=16385$, $N_e=1000$. Inset graphs magnify the region around the origin showing the general distribution of errors, which is roughly exponential. The numbers 66, 12, 78, 73, and 77 in (a) refer to outlier systems.

Figure 9

(Color online) Contour plots of the ${\text{log}}_{10}$ mean (relative) errors for the frequencies $\omega$ and coefficients $a$, $b$, and $c$. The frequencies show the smallest errors (down to ${10}^{-6}$), while the $a$ coefficients show errors up to a few percent $\left(\sim 10^{-2}\right)$ for the shortest signal lengths and greatest projection noise.

Figure 10

(Color online) Possible arrangements for a generic four-level system with six distinct transition frequencies. Not shown are the other five configurations that correspond to a reflection of the energies, which merely flips the above level structures.

Figure 11

(Color online) A scatter plot of the relative errors $\Vert \Delta H\Vert /\Vert H\Vert$ of the estimated Hamiltonian with $\Vert \Delta H\Vert$ as defined in Eq. (41) vs the maximum constraint violation (37) on a log-log scale for our 100 systems and 20 data sets per system (total of 2000 data points) shows a strong correlation, suggesting that the maximum constraint violation (prior to refinement) is a good predictor of the accuracy of the estimated Hamiltonian.

Figure 12

(Color online) Evolution of populations $\left|{\alpha }_j\left(t\right)\right|$ under reference Hamiltonian (system 5) and error ${\sum }_j\mathbf{\left|}{\left|{\alpha }_j\right|}^2-\frac{1}{2}\mathbf{\right|}$ from the ideal balanced initial state. We selected the second minimum (which is the global minimum for $0\le t\le 10$) at $t=5.34$ as initial evolution time $t_{\ast }$ for the $\delta$ estimation step.

Figure 13

(Color online) Median relative error of estimated Hamiltonian after $\delta$ estimation as a function of signal lengths $N$ and projection-noise level $N_e$ of the measured traces $p_{\ell }\left(t_k\right)$. In all cases the most accurate estimates for $\tilde{H}$ from the previous step, i.e., $N=16385$ and $N_e=1000$, were used.