Detector tomography on IBM quantum computers and mitigation of an imperfect measurement

We use quantum detector tomography to characterize the qubit readout in terms of measurement positive operator-valued measures (POVMs) on IBM quantum computers IBM Q 5 Tenerife and IBM Q 5 Yorktown. Our results suggest that the characterized detector model deviates from the ideal projectors, ranging from 10 to 40%. This is mostly dominated by classical errors, evident from the shrinkage of arrows from the poles in the corresponding Bloch-vector representations. There are also small deviations that are not “classical,” of order 3% or less, represented by the tilt of the arrows from the $z$ axis. Further improvement on this characterization can be made by adopting two- or more-qubit detector models instead of independent single-qubit detectors for all the qubits in one device. We also find evidence indicating correlations in the detector behavior, i.e., the detector characterization is slightly altered (to a few percent) when other qubits and their detectors are in operation. Such peculiar behavior is consistent with characterization from the more sophisticated approach of the gate set tomography. We also discuss how the characterized detectors' POVMs, despite deviation from the ideal projectors, can be used to estimate the ideal detection distribution.

Figure 1

Detector spheres for qubits 0 to 4 of (a, b) IBM Q 5 Tenerife and (c, d) IBM Q 5 Yorktown. The arrow represents the vector $(a_1,a_2,a_3)/a_0$ from measurement ${\tilde{\mathrm{\Pi }}}^{(n=0,1)}={\vec{a}}^{\left(n\right)}·\vec{\sigma }$, where the north pole and the south pole correspond to ideal $|0\rangle \langle 0|$ and $|1\rangle \langle 1|$, respectively. Positivity is reflected by the length of the arrow being smaller than 1. The width of the arrow represents the weight $a_0$ in the corresponding POVM element, for which the ideal case is $1/2$. These data are in Tables 5 and 6. For example, individual measurement of qubit 4 of IBM Q 5 Tenerife produces the last sphere in (a), which corresponds to ${\mathrm{\Pi }}^{\left(0\right)}=0.521\left(1\right)\mathbb{1}-0.012\left(2\right){\sigma }_x-0.0122\left(4\right){\sigma }_y+0.3798\left(4\right){\sigma }_z$ and ${\mathrm{\Pi }}^{\left(1\right)}=0.479\left(1\right)\mathbb{1}+0.012\left(2\right){\sigma }_x+0.0122\left(4\right){\sigma }_y-0.3798\left(4\right){\sigma }_z$.

Figure 2

The left panels explain the comparison scheme: the single-qubit detector on the left is obtained by tracing out another qubit in a two-qubit detector while that on the right is from individual measurement. The tables contain the distances between these two: the entry in the $i\mathrm{th}$ row and $j\mathrm{th}$ column is the distance between the single-qubit detector of qubit $i$ conditioned on qubit $j$ and that for qubit $i$ obtained from individual measurement. Values between 0.01 and 0.1 are highlighted in blue and those above 0.1 are highlighted in green. The right panels are schematic influence diagrams where each arrow points from the qubit traced out to the qubit of interest and the thickness is proportional to the distance. This comparison is shown for IBM Q 5 Tenerife (upper panels) and IBM Q 5 Yorktown (bottom panels), respectively.

Figure 3

The left panels explain the comparison scheme: the single-qubit detector on the left is obtained by tracing out another qubit in a two-qubit detector while that on the right is from parallel measurement. The tables contain the distances between these two: the entry in the $i\mathrm{th}$ row and $j\mathrm{th}$ column is the distance between the single-qubit detector of qubit $i$ conditioned on qubit $j$ and that for qubit $i$ obtained from parallel measurement. Values between 0.01 and 0.1 are highlighted in blue and those above 0.1 are highlighted in green. The right panels are schematic influence diagrams where each arrow points from the qubit traced out to the qubit of interest and the thickness is proportional to the distance. This comparison is shown for IBM Q 5 Tenerife (upper panels) and IBM Q 5 Yorktown (bottom panels), respectively.

Figure 4

The left panels explain the comparison scheme: the single-qubit detector on the left is obtained by tracing out another qubit in a two-qubit detector while that on the right is from pairwise parallel measurement. The tables contain the distances between these two: the entry in the $i\mathrm{th}$ row and $j\mathrm{th}$ column is the distance between the single-qubit detector of qubit $i$ conditioned on qubit $j$ and that for qubit $i$ obtained from pairwise parallel measurement. Values between 0.01 and 0.1 are highlighted in blue and those above 0.1 are highlighted in green. The right panels are schematic influence diagrams where each arrow points from the qubit traced out to the qubit of interest and the thickness is proportional to the distance. This comparison is shown for IBM Q 5 Tenerife (upper panels) and IBM Q 5 Yorktown (bottom panels), respectively.

Figure 5

Visualizing the single-qubit detector parameters for the five qubits in IBM Q 5 Tenerife obtained by (a) individual measurement and (b) parallel measurement. The parameters $a{r}_{x,y,z}=a_{1,2,3}$. These are displayed for both ${\mathrm{\Pi }}^{\left(1\right)}$ and ${\mathrm{\Pi }}^{\left(0\right)}$.

Figure 6

Visualizing the single-qubit detector parameters for the five qubits in IBM Q 5 Yorktown obtained by (a) individual measurement and (b) parallel measurement. The parameters $a{r}_{x,y,z}=a_{1,2,3}$. These are displayed for both ${\mathrm{\Pi }}^{\left(1\right)}$ and ${\mathrm{\Pi }}^{\left(0\right)}$.

Figure 7

Detector spheres for qubits 0 to 13 of IBM Q 16 Melbourne (from left to right, the rows corresponding to qubits 0 to 3, 4 to 7, 8 to 11, and 12 and 13, respectively).

Figure 8

Qubit relaxation: probability $p$ of measuring $|1\rangle$ vs a time $t$, calculated using the identity gate from (a) best GST estimate without imposing TP or CPTP condition, (b) projecting the identity gate from (a) to be TP, and (c) projecting the identity gate from (a) to be CPTP.