CF 1002A3 - Generate superposition of two basis states

Rating: 1500
Tags: *special
Solve time: 1m 9s
Verified: yes

Solution

Problem Understanding

We are working in a quantum setting where an initial register of $N \le 8$ qubits starts in the all-zero computational basis state. Alongside this register, we are given two classical bitstrings of equal length, each describing a valid basis state on these qubits. The two bitstrings are guaranteed to differ in at least one position, meaning they represent two distinct basis vectors in the computational basis.

The task is to transform the all-zero quantum state into an equal quantum superposition of exactly those two basis states. In other words, after applying the operation, the system should be in a state where measuring it yields the first bitstring with probability $1/2$ and the second bitstring with probability $1/2$, with the appropriate phase consistency between the two amplitudes.

The important constraint here is that we are not allowed to compute amplitudes or simulate quantum states. We must construct the target transformation using elementary quantum gates such as X, H, and controlled-NOT operations.

Since $N$ is at most 8, any solution that is linear or quadratic in $N$ is trivially fast. The real challenge is not performance but constructing a correct circuit that produces exactly two computational basis states in superposition without introducing unintended extra states.

A subtle edge case arises when the two bitstrings differ in multiple positions. A naive approach might try to “flip bits independently into superposition,” but that would create a superposition of all $2^k$ combinations over differing positions, which is incorrect. For example, if we try to Hadamard all differing qubits, we would generate four or more states instead of two. The correct transformation must correlate qubits so that only two global configurations exist.

Another failure mode occurs if we ignore the initial basis alignment. If we try to directly build a superposition of the two given bitstrings from $|0^N\rangle$, we risk constructing the wrong phase structure unless we first normalize the problem to a canonical form.

Approaches

The brute-force mindset would be to explicitly construct a circuit that maps $|0^N\rangle$ to each target basis state and then somehow “add” them. In a classical simulation this would correspond to building the full state vector of size $2^N$, setting two entries to $1/\sqrt{2}$, and normalizing. That approach conceptually works but is not a valid quantum construction path because it assumes direct amplitude manipulation. Even if we interpret it as circuit synthesis, enumerating all basis states and enforcing amplitudes would require exponential reasoning over $2^N$ states, which is unnecessary given the structure of the target.

The key structural observation is that the two target states differ only by a fixed bitmask. If we denote the bitwise XOR of the two strings as $d$, then one state can be obtained from the other by flipping exactly the bits where $d$ has ones. This reduces the problem to preparing a superposition of $|0^N\rangle$ and $|d\rangle$, after first mapping one of the basis states to zero.

This is the central simplification: instead of thinking about two arbitrary bitstrings, we shift the coordinate system so that one of them becomes the origin. Then the problem becomes constructing a two-state superposition where one component is the all-zero state and the other is a known bitmask configuration. That structure can be implemented using a single Hadamard and a controlled propagation of X gates.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force State Construction	O(2^N)	O(2^N)	Too slow / invalid model
Bitmask Reduction + Circuit Synthesis	O(N)	O(1)	Accepted

Algorithm Walkthrough

We construct the desired quantum state by reducing it to a canonical superposition and then building a controlled entanglement pattern.

Identify all positions where the two bitstrings differ. This defines a bitmask $d$. This mask encodes exactly which qubits must flip to transform one basis state into the other.
Apply X gates to all qubits where the first bitstring has a 1. This maps the basis state corresponding to bits0 into the all-zero state. This step is crucial because it reduces the problem to a standard starting point.
Apply the same X gates to bits1 implicitly through the transformation: after step 2, the second state becomes the bitmask $d = bits0 \oplus bits1$.
Choose any index $i$ such that $d[i] = 1$. Such an index must exist because the two bitstrings differ.
Apply a Hadamard gate to qubit $i$. This creates a local superposition $(|0\rangle + |1\rangle)/\sqrt{2}$, which will act as the “control backbone” for generating exactly two global states.
For every other index $j \ne i$ where $d[j] = 1$, apply a CNOT gate controlled by qubit $i$ targeting qubit $j$. This propagates the excitation from qubit $i$ to all required positions, ensuring that either all these bits remain zero or all flip together.
Finally, the system is in the state $(|0^N\rangle + |d\rangle)/\sqrt{2}$ in the transformed basis. Applying the inverse of step 2 (the same X gates again) restores the original coordinate system, yielding the required superposition of the original two basis states.

Why it works

The construction enforces a strict binary correlation structure: qubit $i$ is the sole source of superposition, and all other differing qubits are deterministically tied to it via CNOTs. This guarantees that the computational basis support of the state contains exactly two configurations. The initial X-layer ensures that these configurations are precisely the original bitstrings rather than shifted versions. Since all operations are unitary and reversible, no extraneous amplitude is introduced outside the intended two-dimensional subspace.

Python Solution

Although the original problem is posed in Q#, the logical structure can be expressed in a Python-like quantum API style where primitive gates are assumed to be available.

import sys
input = sys.stdin.readline

# We assume quantum primitives exist:
# X(q), H(q), CNOT(control, target)

def Solve(qs, bits0, bits1):
    n = len(qs)

    # Step 1: compute difference mask
    diff = [bits0[i] ^ bits1[i] for i in range(n)]

    # Step 2: align bits0 to |0...0>
    for i in range(n):
        if bits0[i]:
            X(qs[i])

    # Step 3: find pivot qubit
    pivot = -1
    for i in range(n):
        if diff[i]:
            pivot = i
            break

    # Step 4: create superposition on pivot
    H(qs[pivot])

    # Step 5: propagate correlations
    for j in range(n):
        if j != pivot and diff[j]:
            CNOT(qs[pivot], qs[j])

    # implicit end of operation

The first loop is a basis alignment step that is easy to overlook. Without it, the construction would produce a superposition anchored at the wrong computational basis point.

The pivot selection is safe because the problem guarantees at least one differing bit. Any valid index works, but choosing the first keeps the circuit deterministic.

The controlled propagation step is what prevents state explosion. Without CNOT tying all differing bits to a single control, each Hadamard would independently double the number of basis states.

Worked Examples

Consider a simple case with three qubits.

Input bitstrings are bits0 = 000 and bits1 = 110.

After computing the difference mask, we get d = 110.

We then apply no X gates because bits0 is already zero. The pivot is chosen as index 0.

Step	Qubit 0	Qubit 1	Qubit 2	Explanation
Start				(
H(0)	superposed	0	0	((
CNOT 0→1	entangled	same as 0	0	((
End				desired state

This confirms that only two computational basis states appear, and the second one matches bits1.

Now consider a case where bits0 = 101 and bits1 = 011.

After XOR, d = 110. We first apply X gates to qubits where bits0 has 1s, mapping $|101\rangle$ to $|000\rangle$. The system is then treated identically to the previous example, producing $(|000\rangle + |110\rangle)/\sqrt{2}$. Applying X gates again restores the original coordinate system, yielding $(|101\rangle + |011\rangle)/\sqrt{2}$.

This second trace shows that the transformation is invariant under basis shifting and depends only on the XOR structure.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(N)	Each qubit is processed a constant number of times for X, H, or CNOT operations
Space	O(1)	Only a small number of indices and the diff mask are stored

The bound $N \le 8$ makes even naive implementations fast, but the solution is already optimal in terms of asymptotic structure and uses only linear circuit construction.

Test Cases

import sys, io

def run(inp: str) -> str:
    sys.stdin = io.StringIO(inp)
    # This is a placeholder; in real judge environment Solve is called by framework
    return "ok"

# minimal differing single bit
assert run("...") == "...", "sample 1"

# custom cases
assert run("n=1 case") == "...", "single qubit trivial superposition"
assert run("all ones vs all zeros") == "...", "max difference spread case"
assert run("alternating bits") == "...", "multi-bit XOR structure"
assert run("two-bit difference") == "...", "checks CNOT fan-out"

Test input	Expected output	What it validates
1-qubit flip	equal superposition	base correctness
all bits differ	uniform propagation	full entanglement handling
sparse difference	selective CNOTs	no over-activation

Edge Cases

A key edge case is when the two bitstrings differ in only one position. In that scenario, the XOR mask has exactly one set bit, so the CNOT propagation loop does nothing. The algorithm reduces to applying X gates followed by a single Hadamard, which directly produces the correct single-qubit superposition without accidental entanglement.

Another edge case is when the first bitstring is not all zeros. The initial X-layer is what guarantees correctness here. Without it, the pivot-based construction would generate a superposition around the wrong basis origin. By explicitly mapping bits0 to zero, we ensure that the subsequent Hadamard and CNOT structure operates in a consistent coordinate system, and the final reversal preserves the intended pair of states.