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Adiabatic Quantum State Generation and Statistical Zero Knowledge
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The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem, by studying the problem of 'quantum state generation'. This approach provides intriguing links between many different areas: quantum computation, adiabatic evolution, analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing Markov chains, the complexity class statistical zero knowledge, quantum random walks, and more. We first show that many natural candidates for quantum algorithms can be cast as a state generation problem. We define a paradigm for state generation, called 'adiabatic state generation' and develop tools for adiabatic state generation which include methods for implementing very general Hamiltonians and ways to guarantee non negligible spectral gaps. We use our tools to prove that adiabatic state generation is equivalent to state generation in the standard quantum computing model, and finally we show how to apply our techniques to generate interesting superpositions related to Markov chains.
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