Murray back in quarter-finals for first time in six months
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Murray won the opening set against his Kazakh opponent but had to come from 5-2 down in the second to clinch the win.
The former number one said his confidence is improving after defeating Bublik in one and three-quarter hours.
"It's nice to be back at this stage," said Murray who will take on top seed Stefanos Tsitsipas for a place in the semi-finals.
"This was a solid win and I'll try and push through to the semis.
"When I was down in the second set it was frustrating, I was thinking of what I could do differently.
"He hit some great shots - but some that you wouldn't teach. The multiple drop shots were unexpected."
Murray's last appearance in the quarter-finals was in January when he reached the final in Sydney.
A win over Tsitsipas on Friday would push Murray closer to a return to the top 50 for the first time in more than four years.