It can be set in the QAbstractView * popup () element. My attempt:
const QAbstractItemView * popup = _completer->popup();
int padding = popup->width() - popup->viewport()->width();
int scrollbarWidth = qApp->style()->pixelMetric(QStyle::PM_ScrollBarExtent);
int frameWidth = popup->frameWidth();
int textWidth = popup->fontMetrics().boundingRect(QLocale().toString(string)).width();
int desiredWidth = textWidth + 4 * padding + 2 * frameWidth + scrollbarWidth;
_completer->popup()->setMinimumWidth(desiredWidth);
Result (tested on Windows 7, increase 2x):

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