CF 1001B - Generate Bell state

Rating: 1400
Tags: *special
Solve time: 1m 2s
Verified: yes

Solution

Problem Understanding

We are working in a quantum setting with two qubits that initially form the computational basis state corresponding to both being zero. The task is to transform these two qubits in-place into one of four specific entangled states, chosen by an integer index from 0 to 3.

These target states are the four Bell states, which differ by how probability mass is distributed across the basis states |00⟩, |01⟩, |10⟩, and |11⟩, and by the relative phase between their components. Instead of returning a value, the operation must directly modify the quantum state of the provided qubits using quantum gates.

The input consists of a fixed-size register of exactly two qubits and an integer index. The output is not printed or returned in a classical sense, but is the resulting quantum state after applying a sequence of unitary transformations.

The constraints are minimal in the classical sense because the input size is constant. The important constraint is conceptual rather than computational: all transformations must be implemented using valid quantum operations, and the solution must avoid any classical simulation or branching over quantum states.

The main subtlety is that different Bell states differ only by simple local transformations applied after a common preparation procedure. A naive approach might try to separately construct each state from scratch using ad hoc reasoning, but that risks redundancy and mistakes in phase handling.

Edge cases are limited but still worth considering. If one incorrectly assumes that applying bit flips before entangling is equivalent to applying them after, they may produce incorrect phase structure. For example, swapping X and Z order incorrectly can change the sign of amplitudes in a way that breaks the intended Bell state. Another potential pitfall is forgetting that phase flips affect only basis states where the qubit is |1⟩, which becomes relevant when distinguishing |Φ+⟩ from |Φ−⟩.

Approaches

A brute-force way to think about the problem is to directly construct each Bell state independently. One could attempt to reason from the definition of each state and manually design a circuit that maps |00⟩ to the desired superposition. For example, to produce (|00⟩ + |11⟩)/√2, one might try to explicitly enforce equal amplitude between these two basis states while suppressing others.

This approach is correct in principle, but it quickly becomes error-prone. Each state would require re-deriving a different circuit, and keeping track of amplitude signs under different gate sequences is non-trivial. Even worse, independently designing four circuits hides the shared structure among all Bell states.

The key observation is that all four Bell states share a common backbone. First, we always create a single entangled pair using a Hadamard on the first qubit followed by a controlled-NOT from the first qubit to the second. This produces the canonical |Φ+⟩ state. From this single state, the remaining three Bell states can be obtained using only single-qubit Pauli operations, which either flip computational basis states or adjust relative phases.

This reduces the problem from designing four circuits to designing one base circuit plus a small conditional post-processing step.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force separate constructions	O(1)	O(1)	Too error-prone
Shared Bell construction + corrections	O(1)	O(1)	Accepted

Algorithm Walkthrough

Apply a Hadamard gate to the first qubit. This creates a uniform superposition between |0⟩ and |1⟩ on that qubit, which is the necessary starting point for entanglement.
Apply a CNOT gate with the first qubit as control and the second as target. This correlates the two qubits so that their values match, producing the entangled state |Φ+⟩ = (|00⟩ + |11⟩) / √2.
If the index is 0, do nothing further because |Φ+⟩ is already the desired state.
If the index is 1, apply a Z gate to the first qubit. This introduces a phase flip only on the |1⟩ component of the first qubit, which transforms |11⟩ into −|11⟩ while leaving |00⟩ unchanged, producing |Φ−⟩.
If the index is 2, apply an X gate to the second qubit. This flips the second qubit in both basis components, mapping |00⟩ to |01⟩ and |11⟩ to |10⟩, which yields |Ψ+⟩.
If the index is 3, apply an X gate to the second qubit followed by a Z gate on the first qubit. The X gate swaps the computational basis states into the |01⟩ and |10⟩ pair, and the Z gate introduces a relative phase flip on the component where the first qubit is 1, producing |Ψ−⟩.

Why it works comes from the fact that the initial Hadamard and CNOT generate a fixed entangled basis state, and the Pauli X and Z gates act as symmetry operations within the Bell basis. These operations permute or phase-shift Bell states without destroying entanglement, so every target state can be reached from |Φ+⟩ by a unique combination of these local transformations.

Python Solution

import sys
input = sys.stdin.readline

def Solve(qs, index):
    from Microsoft.Quantum.Canon import H, CNOT, X, Z  # conceptual Q# style imports

    H(qs[0])
    CNOT(qs[0], qs[1])

    if index == 1:
        Z(qs[0])
    elif index == 2:
        X(qs[1])
    elif index == 3:
        X(qs[1])
        Z(qs[0])

In an actual Q# implementation, the gates come from the Quantum.Canon namespace, and the operation mutates the qubit register directly. The structure of the solution is intentionally linear: first establish entanglement, then apply a minimal correction based on the index.

A subtle point is that the order of X and Z in the index 3 case does not affect correctness here because they act on different qubits and therefore commute. However, preserving a consistent order helps avoid mistakes when reasoning about more complex circuits where commutation does not hold.

Worked Examples

Consider index 1, which should produce |Φ−⟩.

Step	Q0 state action	Q1 state action	Resulting structure
Start
After H	superposition		(
After CNOT	correlated	copied from Q0	(
After Z on Q0	phase flip on	1⟩ branch	(

This confirms that a single phase operation is sufficient to distinguish the two Φ states.

Now consider index 2, producing |Ψ+⟩.

Step	Q0 state action	Q1 state action	Resulting structure
Start
After H	superposition		(
After CNOT	entangled	copied	(
After X on Q1	unchanged	bit flip	(

This shows that bit flips move between Φ and Ψ families while preserving phase.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(1)	A constant number of quantum gates is applied regardless of input
Space	O(1)	Only two qubits are used and no auxiliary storage is needed

The algorithm is optimal under the model because quantum gate sequences are fixed-length transformations on a constant-size register.

Test Cases

import sys, io

def run(inp: str) -> str:
    sys.stdin = io.StringIO(inp)
    # placeholder: in real setting this would call Solve via Q# harness
    return "ok"

# provided samples (conceptual placeholders)
# assert run("0") == "expected1"
# assert run("1") == "expected2"

# custom cases
assert run("0") == "ok", "identity case"
assert run("1") == "ok", "phase flip case"
assert run("2") == "ok", "bit flip case"
assert run("3") == "ok", "combined transformation case"

Test input	Expected output	What it validates
index = 0		Φ+⟩
index = 1		Φ−⟩
index = 2		Ψ+⟩
index = 3		Ψ−⟩

Edge Cases

One subtle edge case is when transformations are applied in a different order than intended. For example, applying X then Z versus Z then X can matter in general, but here it is safe because the gates act on different qubits. The input:

index = 3

Another potential issue is forgetting that Z does not swap amplitudes but only modifies phases. A mistaken intuition might assume it affects both basis components equally, which would incorrectly break the distinction between Φ+ and Φ−.