Algorithm 1: Improved quantum algorithm for solving the quantum LWE problem.

Quantum oracle: $|\mathit{u}\rangle |y\rangle \to |\mathit{u}\rangle |\mathit{u}·\mathbf{sk}+e_{\mathit{u}}+y\rangle$

1: Set the initial state to ${|0\rangle }^{\otimes n}|1\rangle$

2: Apply a quantum Fourier transform on the all registers

and obtain ${\displaystyle \frac{1}{\sqrt{q^n}}\sum _{\mathbf{u}\in Z_q^n}|\mathbf{u}\rangle \frac{1}{\sqrt{q}}\sum _{x\in Z_q}{\omega }^x|x\rangle }$

3: Apply a quantum oracle query and obtain

${\displaystyle \frac{1}{\sqrt{q^n}}\sum _{\mathbf{u}\in Z_q^n}{\omega }^{-\mathbf{u}·\mathbf{sk}-e_{\mathbf{u}}}|\mathbf{u}\rangle \frac{1}{\sqrt{q}}\sum _{x\in Z_q}{\omega }^x|x\rangle }$

4: Apply a quantum Fourier transform on the first register

and obtain ${\displaystyle \frac{1}{q^n}\sum _{\mathbf{u},\mathbf{y}\in Z_q^n}{\omega }^{-e_{\mathbf{u}}}|\mathbf{y}\rangle \frac{1}{\sqrt{q}}\sum _{x\in Z_q}{\omega }^x|x\rangle }$

5: Discard the second register and measure the first register

6: Output $\mathbf{sk}$