Computations with Greater Quantum Depth Are Strictly More Powerful (Relative to an Oracle)
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A conjecture of Jozsa states that any polynomial-time quantum computation can be simulated by polynomial-depth classical computation interleaved with logarithmic-depth quantum computation. This conjecture is a remarkable statement about the unresolved potential of combining classical and quantum computation. As a counterpoint we show that the Welded Tree Problem, which is an oracle problem that can be solved in quantum polynomial time as shown by Childs et al., cannot be solved in the class which Jozsa describes. Therefore, even when interleaved with arbitrary polynomial-time classical computation, greater "Quantum Depth" leads to strictly greater computational ability in this relativized setting.
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