CF 1001A - Generate plus state or minus state

Rating: 1100
Tags: *special
Solve time: 58s
Verified: yes

Solution

Problem Understanding

We are working in a quantum programming setting where a single qubit is already in a known initial basis state, and we are asked to transform it into one of two target basis states depending on the value of an integer parameter called sign. Conceptually, you can think of the qubit as a two-level system that currently encodes a fixed computational basis state, and we are allowed to apply quantum gates to force it into a required target state.

The task is not about measurement or returning a classical result. Instead, the function is an operation that mutates the quantum state in place. After the operation finishes, the qubit must be left in a specific state: one state if sign = 1, and the opposite phase-flipped state if sign = -1.

The constraints are trivial in the classical sense because there is only a single qubit and a constant amount of work per test case. This immediately rules out any need for simulation or decomposition logic. The only meaningful reasoning is how to map a classical control value into a quantum phase transformation.

The main subtle edge case is that quantum state transformations are not arbitrary assignments. A careless approach might try to “set” the state directly, which is not how quantum computation works. Instead, we must apply valid unitary gates that achieve the desired transformation. Another potential mistake is ignoring the case sign = -1, which requires introducing a phase flip rather than a bit flip. Since both target states differ only by a global phase or a sign change in amplitude, the correct operation must respect phase behavior rather than classical negation intuition.

Approaches

A brute-force mental model would be to think in terms of reconstructing the full state vector of the qubit and then overwriting it with the target vector. That would correspond to explicitly representing amplitudes and assigning them to match the required state. While this is conceptually straightforward, it is not a valid quantum operation in a real system because it ignores the requirement that transformations must be unitary. Even if we imagined a simulator, constructing and normalizing state vectors per operation would be unnecessary overhead.

The key observation is that the two possible target states differ only by a sign controlled by the integer sign. In quantum computing, this is exactly the role of a phase flip operation. The Pauli-Z gate leaves the computational basis state |0⟩ unchanged and flips the phase of |1⟩. In the context of this problem’s encoding, the effect we need can be reduced to conditionally applying a Z gate depending on whether we want the positive or negative variant.

The initial state is fixed and known, so there is no need to reconstruct or inspect it. The entire problem reduces to applying a conditional unitary transformation: do nothing when sign = 1, and apply a phase flip when sign = -1.

This reduces the solution to a constant-time decision followed by at most one quantum gate application.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force state reconstruction	O(1) conceptual, invalid quantum operation	O(1)	Not valid in quantum model
Apply conditional phase flip	O(1)	O(1)	Accepted

Algorithm Walkthrough

Read the integer sign. This value determines whether we keep the state unchanged or introduce a phase flip. The qubit itself is already passed by reference, so we do not construct or return anything.
Check whether sign equals -1. This is the only condition that requires action because sign = 1 corresponds to the identity transformation.
If sign = -1, apply a Pauli-Z gate to the qubit. This flips the phase of the |1⟩ component while preserving computational basis probabilities, producing the required negated state.
If sign = 1, do nothing. The identity operation already preserves the required state.

Why it works

The correctness relies on the fact that the two required outputs differ only by a phase change that is exactly representable by a Pauli-Z operation. Since quantum operations must be unitary, any valid transformation between these two states must preserve amplitudes up to phase. The identity and Z gates form a complete decision set for the two cases, and no intermediate transformations are required. The invariant is that after each step, the qubit remains in a valid quantum state and matches the required phase condition implied by sign.

Python Solution

The actual implementation is in Q# style, but we present the logic in a Python-like structure for clarity of control flow. In real submission, this corresponds to applying Z from the quantum library.

import sys
input = sys.stdin.readline

def solve(q, sign):
    if sign == -1:
        # apply phase flip
        q.apply_Z()
    return

The key implementation decision is the conditional application of the Z gate. There is no branching complexity beyond this check, and no state inspection is required. The function is pure mutation on the qubit reference.

The most common mistake in implementation is attempting to modify amplitudes directly or treating the qubit as a classical bit. Another subtle mistake is applying a bit flip (X gate) instead of a phase flip (Z gate), which would change measurement outcomes instead of phase, producing an incorrect quantum state.

Worked Examples

Consider two cases: one where sign = 1 and one where sign = -1.

For sign = 1, the operation does nothing.

Step	sign	Operation	Qubit State
1	1	None	Original state

This confirms that identity behavior preserves the initial state exactly, which matches the required target for sign = 1.

For sign = -1, we apply a Z gate.

Step	sign	Operation	Qubit State
1	-1	Apply Z	Phase-flipped state

This shows that only the relative phase changes while the measurement probabilities remain unchanged. That matches the requirement for the negative variant of the target state.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(1)	Only a single conditional check and at most one quantum gate
Space	O(1)	No auxiliary structures are created

The solution trivially satisfies the constraints because it performs a constant amount of work regardless of input values.

Test Cases

import sys, io

def run(inp: str) -> str:
    sys.stdin = io.StringIO(inp)

    # Since we cannot simulate Q# here, we only test decision logic
    # We mock qubit as a dict for illustration
    class Qubit:
        def __init__(self):
            self.ops = []
        def apply_Z(self):
            self.ops.append("Z")

    def solve(q, sign):
        if sign == -1:
            q.apply_Z()

    # test 1: sign = 1
    q1 = Qubit()
    solve(q1, 1)
    assert q1.ops == []

    # test 2: sign = -1
    q2 = Qubit()
    solve(q2, -1)
    assert q2.ops == ["Z"]

    return "OK"

assert run("") == "OK"

# custom cases
# minimum behavior
q = type("Q", (), {"ops": [], "apply_Z": lambda self: self.ops.append("Z")})()
q.ops = []
if -1 == -1:
    q.apply_Z()
assert q.ops == ["Z"]

q = type("Q", (), {"ops": [], "apply_Z": lambda self: self.ops.append("Z")})()
q.ops = []
if 1 == 1:
    pass
assert q.ops == []

Test input	Expected output	What it validates
sign = 1	no operation	identity case
sign = -1	Z applied	phase flip case
empty/mock	OK	structural correctness

Edge Cases

For sign = 1, the algorithm performs no operation. Starting from the initial qubit state, the check fails the condition sign == -1, so execution falls through without applying any gate. The qubit remains unchanged, which is exactly the required outcome.

For sign = -1, the condition triggers and the Z gate is applied once. The qubit transitions from its initial state to a phase-inverted state. Since no other transformations occur, there is no risk of compounding errors or incorrect basis changes.