He made the remarks at an event where he was discussing AI with Professor Brian Cox during BBC’s Tomorrow’s World Live at London’s Science Museum.
One audience member asked Mr Schmidt: “(Vladimir) Putin said last year that whoever wins this race or in fact leads in AI will rule the world.
“And we also had China saying that by 2024 they want to lead this race. What are your views on that?”
Mr Schmidt responded: “I’m very concerned about this.
“I think that both the Russian and the Chinese leaders have recognised the value of this, not just for their commercial aspirations, but also their military aspirations.”
The former Google CEO also said the US and Europe need more funding to combat the problem.
He added: “It is very, very important that the incredible engines that exist in Europe, and Britain, wherever, United States etc. get more funding for basic research, ethics and so forth.
“I’d like us to deal with the Chinese and Russian competition not by copying their approaches, but being more like us, more of the incredible intellects that have been produced here in this country.
“Think about the tradition of science represented by the Science Museum here and the history. I literally am incredibly in debt to the last 150 years, as you all are of the British scientists.
“Let’s have more of them. Let’s celebrate them. Let’s get as many people to come here from other countries to do the same thing. Let’s outrun them with our own intelligence, rather than any other outcome.”
His words have come after his warning last year that China will overtake the US in AI by 2025.
Mr Schmidt last year also blasted Donald Trump’s administration for falling behind China’s government when it came to AI.
Speaking to the Centre for New American Security’s Paul Scharre at the Artificial Intelligence & Global Security Summit on Wednesday, he said: “I’m assuming our [US] lead will continue over the next five years and then that China will catch up extremely quickly.
“We need to get our act together, as a country… This is the moment when the [US] Government collectively, and private industry, needs to say, ‘these technologies are important.”
In July last year, China revealed its national plan for the future of artificial intelligence.
Donald Trump’s 2018 budget request has reduced funds for basic science and research by $4.3 billion (£3 billion), roughly 13 per cent compared to 2016.