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Cloud computing services offer time on quantum computers, but users are forced to each use the entire quantum computer to run their programs as there is no way to multiplex a quantum computer among multiple programs at the same time. We present HyperQ, a system that introduces virtual machines for quantum computers to provide fault isolation, better resource utilization, and lower latency for quantum cloud computing. A quantum virtual machine is defined in terms of quantum computer hardware, specifically its quantum gates and qubits arranged in a hardware-specific topology. HyperQ enables quantum virtual machines to be simultaneously executed together on a quantum computer by multiplexing them in time and space on the hardware and ensuring that they are isolated from one another. HyperQ works with existing quantum programs and compiler frameworks; programs are simply compiled to run in virtual machines without the programs or compilers needing to know what else might be executed at the same time. We have implemented HyperQ for the IBM quantum computing service, the largest quantum computing fleet in the world. Our experimental results running quantum programs in virtual machines using the IBM service demonstrate that HyperQ can increase utilization and throughput while reducing program latency, by up to an order of magnitude, without sacrificing, and in some cases improving, fidelity in the results of quantum program execution.