The Control Data Corp. yesterday filed a $1.3-million claim against the U.S. government, in effect asking the taxpayers to pay for a Soviet art exhibit the U.S. public never saw.

The ill-fated exhibition, "Art from the Hermitage Museum in Leningrad," was to have opened in Washington last May at the National Gallery of Art. In January, however, its U.S. tour was canceled following the Soviet Invasion of Afghanistan. The Minneapolis computer firm -- which blames the cancellation on "U.S. government action" -- is seeking to recover the costs that it incurred in sponsoring the show.

The Hermitage exhibit was to have surveyed all of the departments of that huge museum. Prehistoric carvings, Scythian gold, suits of armor, coins, medals and Old Master paintings were to have been displayed. Control Data had planned to publish an exhibition catalogue with 400 color plates, and had spent considerable sums of related projects before the show canceled.

"Before 1980, our business with the Soviets was some $20 million a year," Robert D. Schmidt, Control Data's Executive vice president, said yesterday. "Most of the equipment we sold them was from the bottom end of our large-scale computer line. It was used for seismic data processing -- to make underground maps for oil exploration. But since January 1980, we haven't done any business to speak of with the Russians."

Control Data's claim was filed with Comptroller General Elmer R. Staats in accordance with the Meritorious Claims Act of 1928. That act provides for governmental payments "in cases involving . . . circumstances of an unusual nature which are unlikely to constitute a recurring problem." But even if Staats finds the claim meritorious, a special act of Congress would have to be passed before the firm is paid.

The exhibition was canceled when the International Communications Agency, determing that the show was not in the national interest, declined to grant the usually pro forma "waiver of judicial seizure" without which the Soviets would not let it proceed.

"The funds to prepare the exhibition were spent in good faith at a time when increased cultural interchange with the Soviet Union was being promoted as part of national policy," said Lawrence Perlman, vice president and general counsel of Control Data. "In fairness, we believe the U.S. government should repay the costs associated with setting up the tour."

The government's decision to postpone U.S.-Soviet cultural exchanges also led to the U.S. boycott of the Moscow Olympics. NBC-TV, which paid $87 million for the rights to televise the games -- and partially insured its investment -- recently collected $85 million from its insurance companies. Control Data's investment was, however, not insured. "We didn't think anything could go wrong," said one Control Data official.