Bernstein-Vazirani算法

B-V算法

(1) 问题描述

给定布尔函数 f : { 0 , 1 } n → 0 , 1 f:{\left\{ {0,1} \right\}^n} \to{0,1} f:{0,1}n→0,1, 函数 f f f的值是由输入比特串 x x x和确定的比特串 s s s做模2意义下的内积： f ( x ) = x ⋅ s ( m o d 2 ) , f\left( x \right) = x \cdot s\left( {\bmod 2} \right), f(x)=x⋅s(mod2),其中 x ⋅ s = ∑ i ( x i ⊕ s i ) x \cdot s = \sum\limits_i {\left( {{x_i} \oplus {s_i}} \right)} x⋅s=i∑(xi⊕si)
前提：可以调用访问函数 f f f的黑盒
问题：计算出比特串 s s s

经典意义下：
依次输入比特串 x x x:
00...00 00...01 00...10 . . . 01...00 10...00 \begin{array}{l} 00...00\\ 00...01\\ 00...10\\ ...\\ 01...00\\ 10...00 \end{array} 00...0000...0100...10...01...0010...00
对于第 i i i次输入：
000100...00 → x ⋅ s ( m o d 2 ) = s i 000100...00 \to x \cdot s\left( {\bmod 2} \right) = {s_i} 000100...00→x⋅s(mod2)=si
重复该流程 n n n次，即可确定比特串 s s s，上述方法的查询复杂度为 O ( n ) O\left( n \right) O(n)

(2) 量子算法核心思路：

基础知识： H ⊗ n ∣ s ⟩ = 1 2 n 2 ∑ x ( − 1 ) s ⋅ x ∣ x ⟩ H^{\otimes n}|s\rangle=\frac{1}{2^{\frac{n}{2}}} \sum_{x}(-1)^{s \cdot x}|x\rangle H⊗n∣s⟩=22n1∑x(−1)s⋅x∣x⟩

Step1：制备初始量子比特 ∣ Φ 0 ⟩ = ∣ 0 ⟩ ⊗ n \left| {{\Phi _0}} \right\rangle ={\left| 0 \right\rangle ^{ \otimes n}} ∣Φ0⟩=∣0⟩⊗n
Step2：作用 H ⊗ n {H^{ \otimes n}} H⊗n，得到量子态 ∣ Φ 0 ⟩ = 1 2 n 2 ∑ x ∣ x ⟩ \left| {{\Phi _0}} \right\rangle = \frac{1}{{{2^{\frac{n}{2}}}}}\sum\limits_x {|x\rangle } ∣Φ0⟩=22n1x∑∣x⟩
Step3：作用量子黑盒 O f {O_f} Of， O f : ∣ x ⟩ → ( − 1 ) x ⋅ s ∣ x ⟩ {O_f}:\left| x \right\rangle \to {\left( { - 1} \right)^{x \cdot s}}\left| x \right\rangle Of:∣x⟩→(−1)x⋅s∣x⟩，此时系统状态为 ∣ Φ 1 ⟩ = 1 2 n 2 ∑ x ( − 1 ) s ⋅ x ∣ x ⟩ \left| {{\Phi _1}} \right\rangle = \frac{1}{{{2^{\frac{n}{2}}}}}\sum\limits_x {{{\left( { - 1} \right)}^{s \cdot x}}|x\rangle } ∣Φ1⟩=22n1x∑(−1)s⋅x∣x⟩
Step4：作用 H ⊗ n {H^{ \otimes n}} H⊗n，系统状态变为 ∣ s ⟩ |s\rangle ∣s⟩
此时测量量子系统即可得到比特串 s s s，该算法的查询复杂为 O ( 1 ) O(1) O(1)

备注：上述量子黑盒 O f O_f Of的实现方法与Deutsh算法相似，具体方法如下

(1) 制备量子态 ∣ Ψ 0 ⟩ = ∣ 0 ⟩ n ∣ 1 ⟩ \left| {{\Psi _0}} \right\rangle = {\left| 0 \right\rangle ^n}\left| 1 \right\rangle ∣Ψ0⟩=∣0⟩n∣1⟩
(2) 作用 H ⊗ n {H^{ \otimes n}} H⊗n，量子系统变为 ∣ Ψ 1 ⟩ = 1 2 n 2 ∑ x ∣ x ⟩ ∣ − ⟩ \left| {{\Psi _1}} \right\rangle = \frac{1}{{{2^{\frac{n}{2}}}}}\sum\limits_x {|x\rangle } \left| - \right\rangle ∣Ψ1⟩=22n1x∑∣x⟩∣−⟩
(3) 作用 U f ： ∣ x ⟩ ∣ y ⟩ → ∣ x ⟩ ∣ y ⊕ f ( x ) ⟩ U_f：\left|x\right\rangle\left|y\right\rangle \to\left|x\right\rangle\left|y\oplus f\left( x \right)\right\rangle Uf：∣x⟩∣y⟩→∣x⟩∣y⊕f(x)⟩，量子系统演变为 ∣ Ψ 2 ⟩ = 1 2 n 2 ∑ x ∣ x ⟩ 1 2 1 2 ( ∣ 0 ⊕ f ( x ) ⟩ − ∣ 1 ⊕ f ( x ) ⟩ ) \left| {{\Psi _2}} \right\rangle = \frac{1}{{{2^{\frac{n}{2}}}}}\sum\limits_x {|x\rangle } \frac{1}{{{2^{\frac{1}{2}}}}}\left( {\left| {0 \oplus f\left( x \right)} \right\rangle - \left| {1 \oplus f\left( x \right)} \right\rangle } \right) ∣Ψ2⟩=22n1x∑∣x⟩2211(∣0⊕f(x)⟩−∣1⊕f(x)⟩)
当 f ( x ) = 0 {f\left( x \right)}=0 f(x)=0时， ∣ x ⟩ 1 2 1 2 ( ∣ 0 ⊕ f ( x ) ⟩ − ∣ 1 ⊕ f ( x ) ⟩ ) = ∣ x ⟩ 1 2 1 2 ( ∣ 0 ⟩ − ∣ 1 ⟩ ) = ∣ x ⟩ ∣ − ⟩ \left|x\right\rangle \frac{1}{{{2^{\frac{1}{2}}}}}\left( {\left| {0 \oplus f\left( x \right)} \right\rangle - \left| {1 \oplus f\left( x \right)} \right\rangle } \right) = |x\rangle \frac{1}{{{2^{\frac{1}{2}}}}}\left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right) = |x\rangle \left| - \right\rangle ∣x⟩2211(∣0⊕f(x)⟩−∣1⊕f(x)⟩)=∣x⟩2211(∣0⟩−∣1⟩)=∣x⟩∣−⟩
当 f ( x ) = 1 {f\left( x \right)}=1 f(x)=1时， ∣ x ⟩ 1 2 1 2 ( ∣ 0 ⊕ f ( x ) ⟩ − ∣ 1 ⊕ f ( x ) ⟩ ) = ∣ x ⟩ 1 2 1 2 ( ∣ 0 ⟩ − ∣ 1 ⟩ ) = − ∣ x ⟩ ∣ − ⟩ \left|x\right\rangle \frac{1}{{{2^{\frac{1}{2}}}}}\left( {\left| {0 \oplus f\left( x \right)} \right\rangle - \left| {1 \oplus f\left( x \right)} \right\rangle } \right) = |x\rangle \frac{1}{{{2^{\frac{1}{2}}}}}\left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right) = -|x\rangle \left| - \right\rangle ∣x⟩2211(∣0⊕f(x)⟩−∣1⊕f(x)⟩)=∣x⟩2211(∣0⟩−∣1⟩)=−∣x⟩∣−⟩
不难发现 U f U_f Uf的作用为： ∣ x ⟩ ∣ − ⟩ → ( − 1 ) f ( x ) ∣ x ⟩ ∣ − ⟩ = ( − 1 ) s ⋅ x ∣ x ⟩ ∣ − ⟩ |x\rangle \left| - \right\rangle \to {\left( { - 1} \right)^{f\left( x \right)}}|x\rangle \left| - \right\rangle={\left( { - 1} \right)^{s \cdot x}}|x\rangle \left| - \right\rangle ∣x⟩∣−⟩→(−1)f(x)∣x⟩∣−⟩=(−1)s⋅x∣x⟩∣−⟩
舍弃掉最后一个量子比特（辅助比特） ∣ − ⟩ \left| - \right\rangle ∣−⟩，即实现了Step3中的黑盒 O f O_f Of

参考资料
[1] Bernstein-Vazirani Algorithm 学习笔记
[2] 量子计算【算法篇】第7章 Deutsch-Josza算法及实现
(3) 由 Fourier Sampling 到 Deutsch-Jozsa Algorithm & Bernstein-Vazirani Algorithm