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We say that f (n) =Ω (g (n)) if g (n) =O (f (n)), and f (n) =Θ (g (n)) if f (n) =O (g (n)) and f (n) =Ω (g (n)).
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We say that f (n) =Ω (g (n)) if g (n) =O (f (n)), and f (n) =Θ (g (n)) if f (n) =O (g (n)) and f (n) =Ω (g (n)).
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In fact this is not quite true: Kendon and Tregenna conjectured this for the discrete-time Hadamard walk. We prove it for both walks.
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In fact this is not quite true: Kendon and Tregenna conjectured this for the discrete-time Hadamard walk. We prove it for both walks.
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27
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34948862219
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Alagic and Russell exhibit a decoherent quantum walk on the hypercube which converges to the uniform distribution and preserves the quantum mixing speedup proven by Moore and Russell, but in contrast to our work, it is their single (carefully chosen) final measurement that forces uniform convergence, not the decoherence itself (which only adds "noise" that if small enough does not destroy the speedup).
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Alagic and Russell exhibit a decoherent quantum walk on the hypercube which converges to the uniform distribution and preserves the quantum mixing speedup proven by Moore and Russell, but in contrast to our work, it is their single (carefully chosen) final measurement that forces uniform convergence, not the decoherence itself (which only adds "noise" that if small enough does not destroy the speedup).
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31
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34948817718
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Though repeatedly measured quantum walks generate mixed states, our analysis is simplest without the introduction of density matrices. We refer the reader lacking sufficient background in quantum computing to the excellent text of Kitaev, Shen, and Vyalyi.
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Though repeatedly measured quantum walks generate mixed states, our analysis is simplest without the introduction of density matrices. We refer the reader lacking sufficient background in quantum computing to the excellent text of Kitaev, Shen, and Vyalyi.
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Similarly, we can retrieve a good approximation to π by generating and then measuring a good approximation to the ground state |π = xS π (x) |x of the Hamiltonian H=DP D-1.
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Similarly, we can retrieve a good approximation to π by generating and then measuring a good approximation to the ground state |π = xS π (x) |x of the Hamiltonian H=DP D-1.
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37
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Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA
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A. Childs, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA 2004.
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Childs, A.1
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34948825430
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For a walk on the symmetric group Sn, Gerhardt and Watrous showed that 1 2 Π-u 1† 1 1 n - 1 nn! (2n-2 n-1); for a walk on the hypercube Z2n, Moore and Russell showed that there exists an >0 such that 1 2 Π-u 1† 1
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For a walk on the symmetric group Sn, Gerhardt and Watrous showed that 1 2 Π-u 1† 1 1 n - 1 nn! (2n-2 n-1); for a walk on the hypercube Z2n, Moore and Russell showed that there exists an >0 such that 1 2 Π-u 1† 1 =.
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40
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34948863899
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Π is also positive semidefinite: it is the Gram matrix of { fs } with fs (kl): = s| k l |s if λk = λl, 0 otherwise.
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Π is also positive semidefinite: it is the Gram matrix of { fs } with fs (kl): = s| k l |s if λk = λl, 0 otherwise.
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41
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34948842462
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For d=1 it does not; the Grover walk on Zn with measurement ω= ν̄ mixes perfectly in T=n time steps.
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For d=1 it does not; the Grover walk on Zn with measurement ω= ν̄ mixes perfectly in T=n time steps.
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42
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0004342981
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AMS, Providence, RI
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A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation (AMS, Providence, RI 2002).
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Kitaev, A.1
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Vyalyi, M.3
|