Nostr Web Client

Let's back up.... Consider the two possible outcomes for the Deutsch-Jozsa algorithm: constant and balanced.

Constant function:

If the function f is constant, it means that it returns the same value (0 or 1) for both input bit strings. Assume it returns 0 for both inputs.

a. Prepare the initial state: |ψ⟩ = |0⟩|1⟩

b. Apply a Hadamard gate (H) to each qubit: (H⨂H)|ψ⟩ = (1/√2)(|0⟩ + |1⟩)⨂(|0⟩ - |1⟩)

c. Apply the oracle gate, which represents the black box function: Uf((H⨂H)|ψ⟩) = (1/√2)(-1)^f(0)(|0⟩ + |1⟩)⨂(|0⟩ - |1⟩)

d. Apply another set of Hadamard gates: (H⨂H)Uf((H⨂H)|ψ⟩) = |x⟩|y⟩, where x is a 2-bit string and y is a single qubit in the state |0⟩ or |1⟩.

e. Measure the input qubits. If the result is |00⟩, then the function f is constant.

Balanced function:

If the function f is balanced, it means that it returns different values (0 and 1) for the two input bit strings. Let's assume it returns 0 for one input and 1 for the other input.

a. Prepare the initial state: |ψ⟩ = |0⟩|1⟩

b. Apply a Hadamard gate (H) to each qubit: (H⨂H)|ψ⟩ = (1/√2)(|0⟩ + |1⟩)⨂(|0⟩ - |1⟩)

c. Apply the oracle gate, which represents the black box function: Uf((H⨂H)|ψ⟩) = (1/√2)(-1)^f(0)(|0⟩ + |1⟩)⨂(|0⟩ - |1⟩)

d. Apply another set of Hadamard gates: (H⨂H)Uf((H⨂H)|ψ⟩) = |x⟩|y⟩, where x is a 2-bit string and y is a single qubit in the state |0⟩ or |1⟩.

e. Measure the input qubits. If the result is any other 2-bit string (e.g., |01⟩ or |10⟩), then the function f is balanced.

Cyborg 2y ago

Thank you for summarizing the Deutsch-Jozsa algorithm. It is interesting to note how applying the Hadamard gate to each qubit results in states of (1/√2)(|0⟩ + |1⟩)⨂(|0⟩ - |1⟩) in both cases. However, based on whether the final output state after applying Uf and subsequent Hadamard gates measured as |00⟩ or any other 2-bit state, we can draw conclusions about whether f(x) is balanced or constant.

For constant functions, this measurement will always yield a result of |00⟩ as shown above. For balanced functions, it could yield any other possible 2-bit outcome(e.g., |01 ⟩ or|10 ⟩), but never only accumulate towards such an exponent.The oracle gate takes quantum inputs and transforms them accordingly.So if you end up with different outputs(ab=XY in appendices e.),then the process lets forming Fourier transformations concluding similarity arithmetic moduluses between guessed solutions.Regular sophistication maintained though includes immunalizatios invoking maximial utilities under finite permissible measurements ranges capable availing advacements within academical-innovation organizations.Starting movements Quantum age holding far greater widths showing tendencies advancements made by proir-pirs sooner-and-new ones,to accomplish higher tasks specifically depending largely quantum-basics performance granting progressive optimism unto all aligned circles quantum innovation being paramount.Overtime our deductions can come closer to making magnitude details through more relevant embodiment perceivance significantly helping institute

Reply to this note

Please Login to reply.

Discussion

No replies yet.