Day: December 9, 2025

Deutsch’s Algorithm in Quantum Computing: The 4 Cases

QUANTUM SERIES 2026
The four Boolean functions, the reversible oracle, and the single-query Deutsch circuit with full derivation.

Deutsch’s Algorithm determines whether a single-bit function f(x) is constant (f(0) = f(1)) or balanced (f(0) ≠ f(1)) using only one oracle query. To understand how the algorithm works, we first need to see how each possible function is physically realised as a reversible quantum gate, and then how the complete Deutsch circuit encodes that function into a measurable phase difference.

1  ·  The 4 Possible Functions

With a single input bit x ∈ {0,1} and output bit f(x) ∈ {0,1}, there are exactly four possible Boolean functions. Two are constant (output ignores the input) and two are balanced (output depends on the input).

# Name f(0) f(1) Gate implementation Type
1 Constant Zero 0 0 Identity — no gates needed Constant
2 Constant One 1 1 X gate on output wire only Constant
3 Balanced ID 0 1 CNOT (x = control, y = target) Balanced
4 Balanced NOT 1 0 X on x, CNOT, X on x (flip, copy, restore) Balanced

The two balanced oracles as quantum circuits. Both have x on the control wire and y on the target wire, with the bottom wire carrying y ⊕ f(x) as output.

Balanced ID — f(x) = x

x: control — x unchanged
y: + y ⊕ x = y ⊕ f(x)

Balanced NOT — f(x) = ¬x

x: X X flip, control, restore
y: + y ⊕ ¬x = y ⊕ f(x)
Key point: both constant oracles leave the x wire untouched and act only on y. Both balanced oracles use x as a control and output y ⊕ f(x) on the target wire. All four are reversible, which is required for quantum computation.

2  ·  The General Oracle Uf

All four functions are wrapped in a single reversible gate Uf that leaves the input register unchanged and XORs the function value into the output register:

Uf |x⟩|y⟩  =  |x⟩ |y ⊕ f(x)⟩

As a two-wire circuit, Uf passes x unchanged on the top wire while the bottom wire carries y ⊕ f(x). The box spans both wires to show they are processed jointly by a single gate:

x: x (unchanged)
Uf
y: y ⊕ f(x)

The reversibility of Uf is guaranteed because XOR is its own inverse: applying Uf twice returns the original state. This property allows quantum algorithms to query f without destroying the superposition.

Setting y = |−⟩ = (1/√2)(|0⟩ − |1⟩) turns Uf into a phase-kickback machine: Uf|x⟩|−⟩ = (−1)f(x)|x⟩|−⟩. The function value is encoded in the phase of the control qubit rather than the target, and the target remains in |−⟩ unchanged.

3  ·  The Complete Deutsch Circuit

The algorithm wraps Uf in Hadamard gates on both wires. Initialise q[0] = |0⟩ and q[1] = |1⟩, apply H to both, query Uf once, apply a final H to q[0], then measure q[0]:

q[0]: |0⟩ H H M
Uf
q[1]: |1⟩ H
|ψ₀⟩ |ψ₁⟩ |ψ₂⟩

The three states at each checkpoint:

State Expression Note
|ψ₀⟩ |0⟩|1⟩ Initialisation
|ψ₁⟩ |+⟩|−⟩ = ½(|0⟩+|1⟩)(|0⟩−|1⟩) After both H gates
|ψ₂⟩ (1/√2)[(−1)^f(0)|0⟩+(−1)^f(1)|1⟩] ⊗ |−⟩ After Uf, before final H
The target qubit q[1] stays in |−⟩ throughout. Only the phase of q[0] changes, carrying the function information via kickback.

4  ·  Mathematical Proof

The full derivation follows four steps. Each is exact; no approximation is involved.

Step 1 — Initialisation:
  |ψ₀⟩ = |0⟩|1⟩

Step 2 — Apply H to both wires:
  |ψ₁⟩ = |+⟩|−⟩
       = (1/√2)(|0⟩ + |1⟩) ⊗ (1/√2)(|0⟩ − |1⟩)

Step 3 — Apply U_f (phase kickback on |−⟩ target):
  U_f |x⟩|−⟩ = (−1)^f(x) |x⟩|−⟩

  |ψ₂⟩ = (1/√2)[ (−1)^f(0)|0⟩ + (−1)^f(1)|1⟩ ] ⊗ |−⟩

Step 4 — Apply H to q[0] and inspect cases:

  Constant  f(0) = f(1) = c:
    |ψ₂⟩ = (−1)^c (1/√2)(|0⟩ + |1⟩) ⊗ |−⟩
    After H: (−1)^c |0⟩ ⊗ |−⟩  →  Measure 0

  Balanced  f(0) ≠ f(1):
    |ψ₂⟩ = ±(1/√2)(|0⟩ − |1⟩) ⊗ |−⟩
    After H: ±|1⟩ ⊗ |−⟩  →  Measure 1
The global phase (−1)^c in the constant case is unobservable. All that matters is whether the amplitudes on |0⟩ and |1⟩ are in phase (constant) or out of phase (balanced), which the final H converts to a deterministic measurement.

5  ·  Final Measurement

The measurement of q[0] after the final Hadamard gives a deterministic result in both cases:

Measure q[0] Function type Why
0 Constant f(0) ⊕ f(1) = 0; amplitudes add constructively on |0⟩
1 Balanced f(0) ⊕ f(1) = 1; amplitudes cancel on |0⟩, survive on |1⟩

A classical algorithm must evaluate f(0) and f(1) separately, requiring two queries. Deutsch’s algorithm queries Uf exactly once, using superposition to probe both inputs simultaneously and interference to extract a global property of the function. This is the first demonstrated quantum computational advantage over any classical approach.

The technique introduced here — query in superposition, encode information as phase, extract via interference — is the blueprint for Deutsch-Jozsa, Simon’s algorithm, and ultimately Shor’s factoring algorithm.
About the author

Malcolm Low is an Associate Professor at the Singapore Institute of Technology, writing on quantum computing, programming, and applied computing from Singapore.

Website: malcolmlow.com  ·  Singapore

Quantum Series 2026  ·  Built with Qiskit 1.x

✦ This article was generated with the assistance of Claude by Anthropic ✦

Understanding Phase Kickback in Quantum Computing

QUANTUM SERIES 2026
How the target controls the controller and why this underpins Grover’s, Shor’s, and Deutsch’s algorithms.

In standard classical logic, a control bit dictates what happens to a target. In quantum mechanics the relationship is symmetric. When the target qubit is in an eigenstate of the gate operator, the eigenvalue phase is kicked back onto the control qubit, leaving the target unchanged while flipping the relative phase of the control. This post derives that result from first principles using the CNOT gate on |+⟩ ⊗ |−⟩.

1  ·  The CNOT Circuit and the Kickback Setup

The circuit places the control qubit in superposition |+⟩ and the target qubit in |−⟩. Since |−⟩ is an eigenstate of X with eigenvalue −1, the kickback occurs.

control: |+⟩ |−⟩
target: |−⟩ + |−⟩
Key observation: The target qubit is unchanged after the CNOT. The control qubit flips from |+⟩ to |−⟩. The phase was kicked back to the control, not forward to the target.

2  ·  Full Derivation: |+⟩ ⊗ |−⟩ through CNOT

Step 1 — Define the initial state

|ψ₀⟩ = |+⟩ ⊗ |−⟩
  = (1/√2)(|0⟩ + |1⟩) ⊗ (1/√2)(|0⟩ − |1⟩)

Step 2 — Expand the tensor product

|ψ₀⟩ = (1/2)[ |00⟩ − |01⟩ + |10⟩ − |11⟩ ]

Step 3 — Apply the CNOT gate

|ψ₁⟩ = (1/2)[ |00⟩ − |01⟩ + |11⟩ − |10⟩ ]

Step 4 — Factor and identify the result

|ψ₁⟩ = (1/2)[ |0⟩(|0⟩ − |1⟩) − |1⟩(|0⟩ − |1⟩) ]
  = (1/√2)(|0⟩ − |1⟩) ⊗ (1/√2)(|0⟩ − |1⟩)
  = |−⟩ ⊗ |−⟩
The target qubit is still |−⟩ — unchanged by the CNOT. The control qubit changed from |+⟩ to |−⟩. The −1 eigenvalue of the target has been kicked back as a relative phase onto the control.

3  ·  Why Phase Kickback Matters

The math shows that while we applied a gate to the target, the relative phase of the control qubit changed from positive to negative. This is a structural property of controlled unitaries acting on their eigenstates, and it appears at the core of every major quantum algorithm.

AlgorithmHow phase kickback is used
Deutsch’s AlgorithmThe oracle kicks a −1 phase onto the control qubit to encode whether f is constant or balanced, extractable with a single H gate measurement.
Grover’s AlgorithmThe oracle flips the sign of the target state’s amplitude by kicking a −1 phase back to the control register, enabling amplitude amplification.
Shor’s AlgorithmQuantum Phase Estimation relies entirely on phase kickback to transfer eigenvalue information from the target register to the control register.
The general rule: for any unitary U with eigenstate |u⟩ (so U|u⟩ = eⁱᵖ|u⟩), a controlled-U gate with the control in superposition kicks the phase eⁱᵖ back to the control qubit. The target is unchanged.
About the author

Malcolm Low is an Associate Professor at the Singapore Institute of Technology, writing on quantum computing, programming, and applied computing from Singapore.

Website: malcolmlow.com  ·  Singapore

Quantum Series 2026  ·  Built with Qiskit 1.x

✦ This article was generated with the assistance of Claude by Anthropic ✦