Explicit lower bounds on strong quantum simulation
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We consider the problem of strong (amplitude-wise) simulation of n-qubit quantum circuits, and identify a subclass of simulators we call monotone. This subclass encompasses almost all prominent simulation techniques. We prove an unconditional (i.e. without relying on any complexity theoretic assumptions) and explicit (n-2)(2^n-3-1) lower bound on the running time of simulators within this subclass. Assuming the Strong Exponential Time Hypothesis (SETH), we further remark that a universal simulator computing any amplitude to precision 2^-n/2 must take at least 2^n - o(n) time. Finally, we compare strong simulators to existing SAT solvers, and identify the time-complexity below which a strong simulator would improve on state-of-the-art SAT solving.
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