The Queen has said her "heart goes out" to those affected by the Virginia Tech shootings last month, in a speech on her first US trip for 16 years.
The national anthem was played for the Queen's US arrival
She and the Duke of Edinburgh arrived earlier at Virginia's Richmond Airport, for the six-day East Coast tour.
The state visit will include two days in Washington DC with the President.
The royal couple will visit Jamestown on Friday to celebrate the 400th anniversary of the US's first permanent English settlement.
There have already been calls for her to apologise for the introduction of slavery and slaughter of American Indians.
The programme is aimed at forging bonds between Britain and the US and focusing on their shared future.
The Queen's last visit in 1991 was when President George Bush's father was in power.
In a speech to the state's general assembly she talked of how the bonds between the US and the UK were far stronger than any "temporary differences of opinion".
"Friendship is a complex concept," she said.
The Queen also extended her sympathies to the students and families and friends of those killed at the Virginia tech college in April, when a gunman killed 32 people.
"As a state and as a nation you are still coming to terms with the dreadful events at Virginia Tech," she said in the speech.
"My heart goes out to the students, friends and families of all those killed, and to the many others who have been affected, some of whom I shall be meeting shortly.
"On behalf of the people of the United Kingdom I extend my deepest sympathies at this time of such grief and sorrow," she said.
As the Queen arrived at Richmond's Capitol building 11 military cadets from Virginia Tech stood guard in uniform.
While there should have been 12, one space was left empty in memory of Matthew La Porte - a member of the university's cadet force who died in the shootings.
Workers' day off
When the Queen and Prince Philip arrived in the US there was some delay in the royal party disembarking their British Airways aircraft as officials and workers struggled to get the steps to reach the plane door and line up the red carpet.
A crowd of an estimated 5,000 people in Richmond's Capitol Square were told through a PA system: "The Queen has landed".
All state workers in Virginia have been given the day off for the royal visit at a cost of almost £6m ($11.9m).
But officials said the visit's benefits in terms of tourism and publicity would outweigh the cost.
Crowds turned out for the monarch's visit
The Queen will fulfil a lifelong dream during the trip, by attending the Kentucky Derby at the Churchill Downs racetrack at the weekend.
The finale will be two days in Washington DC, where she will be entertained by President Bush, visit a Nasa space centre and meet some GI brides.
It will be the fourth state visit to the US by the Queen and the Duke, following tours in 1957, 1976, for the US Bicentennial, and 1991.
They also visited Ronald Reagan in 1983.
It was revealed on Wednesday that Mr Reagan wrote in his diaries how the Prince was, "horror of horrors", served a cup of tea with the teabag left in it.
The Queen's carbon footprint will be calculated and offset for the first time for a state visit.