Two examples of discrete-time quantum walks taking continuous steps

This note introduces some examples of quantum random walks in R and proves the weak convergence of their rescaled n-step densities. One of the examples is called the Plancherel quantum walk because the “quantum coin flip” is the Fourier Integral (or Plancherel) Transform. The other examples are the Birkhoff quantum walks, so named because the coin flips are effected by means of measure preserving transformations to which the Birkhoff’s Ergodic Theorem is applied. Quantum walks of the type we consider in this note were introduced in [1], which defined and analyzed the Hadamard quantum walk on Z, and a “new type of convergence theorem” for such quantum walks on Z was discovered by Konno [4, 5]. A much simpler proof of Konno’s theorem has recently appeared in [3], allowing the theorem to be generalized to quantum walks in Zd. Inspired by the technique of [3], I have proven that Konno’s theorem also holds for an analog of the quantum walk that takes steps in Rd instead of Zd. In this note, I describe a couple of quantum walks that take steps in Rd: the Birkhoff quantum walk and the Plancherel quantum walk. These are analogs of the Hadamard quantum walk of [1], which is reviewed next. The Hadamard random walker steps along the lattice Z, carrying with her a “quantum coin.” Formally, the Walker&Coin state is specified by a unit vector in l2(Z)⊗C2; the standard basis vectors of the auxilliary “coin space” C2 will be denoted |H〉 for “Heads” and |T 〉 for “Tails.” A complete measurement of the walker’s position would find her at j ∈ Z with probability P (j;ψ) = ∣∣ 〈 (j ⊗H) ∣ψ 〉∣∣ 2 + ∣∣ 〈 (j ⊗ T ) ∣ψ 〉∣∣ 2 (1) if the state of the Walker&Coin is ψ ∈ l2(Z)⊗C2. But the walker walks unobserved, and her position will become entangled with the state of her coin. To take a step, the quantum walker flips her coin by a Hadamard transform |H〉 7−→ 1 √ 2 (|H〉+ |T 〉) |T 〉 7−→ 1 √ 2 (|H〉 − |T 〉) (2) and takes one step to the left or right depending on the outcome. This conditional step is implemented by the unitary operator S on l2(Z)⊗ C2 defined by S(|j〉 ⊗ |H〉) = |j + 1〉 ⊗ |H〉 S(|j〉 ⊗ |T 〉) = |j − 1〉 ⊗ |T 〉 ,