Manuscript X-Enter@1 · X-Enter.tex · section · 484b71bc

A two-qubit program without pre-combination

Option B: conditional relational inputs › A two-qubit program without pre-combination · lines 863–930

X-Enter@1#eq:phaseencoding, X-Enter@1#eq:relprogram

Fix the phase scales

σ > 0, σ^{'} = \frac{σ r _{ε}}{c _{ε}} \neq = 0.

The second scale may be signed. Freeze the normalized input encoding

∣ ψ (a, r)⟩ = \frac{1}{2} (∣ 0 ⟩ + e^{ia / σ} ∣ 1 ⟩) \otimes (∣ 0 ⟩ + e^{i r / σ^{'}} ∣ 1 ⟩) .

Write

α = a / σ, β = r / σ^{'}, θ = α - β = \frac{I _{rel}}{σ r _{ε}} .

Gauge translation adds the same phase $λ c_{ε} / σ$ to both $α$ and $β$ .

Let

H = \frac{1}{2} (11 1 - 1) .

CNOT uses the first qubit as control and the second as target. Define the finite decoder

d (b, e) = if e then if b then ⟨ 1, 0 ⟩ else ⟨ 0, 1 ⟩ else if b then ⟨ 0, 0 ⟩ else ⟨ 0, 0 ⟩ .

Its reported codes are

F = (0, 0), + = (0, 1), - = (1, 0) .

Both input bits are consumed by guards, including in the failure branch.

The program is

P_{rel} = λ z : Q \otimes Q . let ⟨ q_{a}, q_{r} ⟩ = z in let ⟨ u, v ⟩ = CNOT (⟨ q_{a}, q_{r} ⟩) in let b = Enter (H (u)) in let e = Enter (v) in ⟨ d (b, e), * ⟩ .

Its type is

Q \otimes Q ⊸ Bit^{\otimes 2} \otimes 1 .

It uses two live qubits and the same four-qubit budget, but changes the input interface and is not a member of the original single-input class $P_{4}$ .

Displayed formulas in this passage

X-Enter@1#eq:phaseencoding · line 872
$∣ ψ (a, r)⟩ = \frac{1}{2} (∣ 0 ⟩ + e^{ia / σ} ∣ 1 ⟩) \otimes (∣ 0 ⟩ + e^{i r / σ^{'}} ∣ 1 ⟩) .$
X-Enter@1#eq:relprogram · line 910
$P_{rel} = λ z : Q \otimes Q . let ⟨ q_{a}, q_{r} ⟩ = z in let ⟨ u, v ⟩ = CNOT (⟨ q_{a}, q_{r} ⟩) in let b = Enter (H (u)) in let e = Enter (v) in ⟨ d (b, e), * ⟩ .$
unlabeled · line 866
$σ > 0, σ^{'} = \frac{σ r _{ε}}{c _{ε}} \neq = 0.$
unlabeled · line 882
$α = a / σ, β = r / σ^{'}, θ = α - β = \frac{I _{rel}}{σ r _{ε}} .$
unlabeled · line 890
$H = \frac{1}{2} (11 1 - 1) .$
unlabeled · line 896
$d (b, e) = if e then if b then ⟨ 1, 0 ⟩ else ⟨ 0, 1 ⟩ else if b then ⟨ 0, 0 ⟩ else ⟨ 0, 0 ⟩ .$
unlabeled · line 903
$F = (0, 0), + = (0, 1), - = (1, 0) .$
unlabeled · line 924
$Q \otimes Q ⊸ Bit^{\otimes 2} \otimes 1 .$

A two-qubit program without pre-combination

Displayed formulas in this passage

Nearby in this section