fix: proofread Quantum Algorithms part

[vEnhance]

Proofreading pass over the Quantum Algorithms part (tex/quantum/vectors.tex, circuits.tex, shor.tex), part of the proofreading campaign (#315).

Most fixes are minor typos/grammar; two are small calculation/logic errors worth highlighting.

vectors.tex

• Singlet-state measurement (logic error). In the problem solution for measuring $\sigma_x^A \otimes \id_B$ on $\ket{\Psi_-}$, the text read "$+1$ gives $\xdown_B$ and $-1$ gives $\xdown_A$". The question asks for the state of qubit $B$, and from $\ket{\Psi_-} = -\tfrac{1}{\sqrt2}(\xup_A\otimes\xdown_B - \xdown_A\otimes\xup_B)$, the $-1$ outcome ($\xdown_A$) leaves $B$ in $\xup_B$. Fixed to "$-1$ gives $\xup_B$".
• Eigenspace-projection collapse sum: index range $0 \le i \le n$ → $0 \le i \le n-1$ (basis is $\ket0,\dots,\ket{n-1}$), and removed a redundant nested \substack.

circuits.tex

• Grammar: "output some more bits for input" → "for output".
• "such gates must take the same number of input and output gates" → "bits".
• Spelling: "Tofolli" → "Toffoli"; "Fredkind" → "Fredkin".
• Deutsch–Jozsa problem solution mixed the qubit count $m$ and $n$ within the same formulas (e.g. $H^{\otimes m+1}$ alongside $\sum_{x_1,\dots,x_n}$). Unified everything to $n$ to match the problem statement, and made the $C(y_1,\dots,y_n)$ notation consistent.

shor.tex

• Grammar: "This is the exactly the same definition" → "This is exactly the same definition".
• Worked example $M=77$ (calculation error). With $x=2$, the second register holds $x^k \bmod 77$, a residue $< 77$, so the measured state cannot be $\ket{128}$. Since $2^7 = 128 \equiv 51 \pmod{77}$, the measured value (and the tensor factor in the collapsed state) should be $\ket{51}$; corrected in all three places it appears. ($k_0 = 7$ and the later continued-fraction computation are unaffected and check out.)

No large mathematical errors found; the arguments are otherwise sound.

🤖 Generated with Claude Code

---

Generated by Claude Code