Polynomial Signals
Exercise PLAN-512
primary source nl only. Current content-bound mathematical second pass accepted. No new Lean verification is claimed for this NL view.
Assumptions and conventions
- theta lies strictly between zero and a right angle so cosine theta is a nondegenerate singular value
- R(theta) is the planar rotation with cosine theta on the diagonal
- The input reflection restricts to Pauli Z on the first qubit
Question
Take the smallest nontrivial subspace-encoding example: a four-dimensional space, the identity as the accessing unitary, an input projector that keeps the first qubit in its zero state, and an output projector obtained by rotating the input projector by a real angle theta on the first qubit (via the planar rotation with cosine theta on the diagonal and the usual off-diagonal signs). With the encoded operator read off as the overlap between the rotated and unrotated flagged bases, prove that the encoded operator equals cosine theta times the two-dimensional identity, and that the qubitization iterate (product of the two reflections, since the unitary is the identity) equals the planar rotation by twice theta on the first qubit tensored with the identity. Deduce the eigenphases of the iterate and relate them to the encoded singular value.
Hint: mathematical route
In the identity-unitary two-qubit example the encoded operator is cosine theta times the identity and the two-reflection iterate is a rotation by twice theta, whose eigenphases plus/minus 2 theta encode the singular value cosine theta.
- The input and output flagged bases are the first-qubit-zero states and their R(theta) rotations.
- Overlaps between the two bases give cosine theta on the diagonal, zero off-diagonal.
- Hence the encoded operator is cosine theta times the identity, singular value cosine theta.
- With identity accessing unitary the iterate is the product of the two reflections.
- On the first qubit this product is R(theta) Z R(theta)-adjoint Z = R(2 theta).
- Eigenphases plus/minus 2 theta satisfy cosine of the half-phase equals the singular value cosine theta.
Reference solution
The input projector keeps the first qubit at zero, so its range is spanned by the two states with first qubit zero; the unrotated flagged basis is these two states. The output projector is the input projector conjugated by the single-qubit rotation R(theta) on the first qubit, so its range is spanned by R(theta) applied to the same two states, giving the rotated flagged basis. The encoded operator is the matrix of overlaps between the rotated output basis vectors and the unrotated input basis vectors: each overlap is the inner product of a state with its R(theta)-rotation, which on the first qubit is cosine theta and on the second qubit is one; cross terms vanish by orthogonality of the second qubit. Hence the overlap matrix is cosine theta times the two-dimensional identity, so the encoded operator is cosine theta times the identity and its single distinct singular value is cosine theta. Because the accessing unitary is the identity, the qubitization iterate is just the product of the two reflections: (output reflection)(input reflection). On the first qubit the input reflection is the Pauli Z (plus one on zero, minus one on one), and the output reflection is R(theta) Z R(theta)-adjoint; their product R(theta) Z R(theta)-adjoint Z equals the planar rotation by twice theta. Tensoring with the untouched second qubit, the iterate is R(2 theta) on the first qubit tensored with the identity. A planar rotation by angle 2 theta has eigenvalues e to the plus and minus i 2 theta, each with multiplicity two here. These eigenphases plus and minus 2 theta determine the encoded singular value through cosine theta: the half of the eigenphase is theta, and its cosine is exactly the encoded singular value cosine theta. Thus the two-reflection product converts the geometric overlap cosine theta into a rotation whose eigenphase is twice the arccosine of that singular value.
Assessment rubric
- Identifies the unrotated and rotated flagged bases and computes the overlap matrix as cosine theta times the identity (1 weight)
- Concludes the encoded operator is cosine theta times the identity with singular value cosine theta (1 weight)
- Shows the two-reflection product equals R(theta) Z R(theta)-adjoint Z = rotation by twice theta on the first qubit tensored with identity (1 weight)
- Gives eigenphases plus/minus 2 theta and relates them to the singular value via cosine theta = cosine of the half eigenphase (1 weight)
Graph-grounded prerequisites
These are support/source anchors, not an automatically extracted theorem-dependency proof.
- Operator and semigroup theory (frozen curriculum anchor; see graph registry)
- Matrix analysis (frozen curriculum anchor; see graph registry)
- Block Encoding (frozen curriculum anchor; see graph registry)
Lean view and evidence boundary
No aligned Lean view has been admitted for this native core in this textbook checkpoint. This is not a claim that the underlying mathematics lacks a formal proof. Contributions must preserve this exact target contract and split.
Sources and provenance
- website/research/sources.json
lin-wiebe-2026-qasc
Source SHA-256:9f85f45f20ed5a46931ad0dec3b94ee09748bc56e39fae29b0aefe910440b0ae· MIT metadata only - Primary source
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Graph snapshot: qcb-snapshot-ffa644f0da3031a1ce754b9a
Authored support steps: 6. This is not measured graph distance or model difficulty.