QCQI notes: Discrete sets of universal operations

In fact, many physical realizations of quantum computers can not perform arbitrary unitary operations, but can only perform the standard set of universal gates which consists of the Hadamard, phase, controlled-NOT, and $\pi/8$ gates.

References

[1] Nielsen MA, Chuang I. Quantum computation and quantum information.

Error evaluation

We use a simple formula to evaluate the error of the approximation of U using V,

$E(U,V)=\max_{|\psi\rangle}||(U-V)|\psi\rangle||$

here $ |\psi\rangle$ is a normalized vector. Its definition is interesting, obviously, the input state $|\psi\rangle$ does affect the result of error value.

The standard set

As we said, the standard contains 4 quantum gates, Hadamard, phase, controlled-NOT, and $\pi/8$ gates can approximate arbitrary rotation gate.

T had HTH gates

$T$ and $HTH$ gates can approximate arbitrary rotation gate, firstly, let’s take a look at their matrix forms, the Exercise 4.14 in QCQI is about the transformation of those two gates,

$T=\exp(i\pi/8) \left[\begin{array}{ll}{e^{-i\pi/8}} & {0} \\ {0} & {e^{i\pi/8}}\end{array}\right]=e^{i\pi/8}R_z(\pi/4),$

if we ignore the global phase $\exp(i\pi/8)$, $ T$ gate is actually a rotation gate aroung $z$ axis $R_z(\pi/4)$. Let us see the $HTH$ gate,

$$HTH=\frac{1}{2}\left[\begin{array}{ll}{1+e^{i\pi/4}} & {1-e^{i\pi/4}} \\ {1-e^{i\pi/4}} & {1+e^{i\pi/4}}\end{array}\right]=e^{i\pi/8}R_x(\pi/4),$$

in this example, the first thing that we can see is that the $ H$ gates can switch the roration axis from $X$ to $latex Z$, or from $Z$ to $ X$, which is also mentioned in Exercise 4.13 in QCQI.

Rz and Rx gates

A important combination is $R_z(\pi/4)R_x(\pi/4)$,

$$\exp \left(-i \frac{\pi}{8} Z\right) \exp \left(-i \frac{\pi}{8} X\right) =\left[\cos \frac{\pi}{8} I-i \sin \frac{\pi}{8} Z\right]\left[\cos \frac{\pi}{8} I-i \sin \frac{\pi}{8} X\right] $$

$$=\cos ^{2} \frac{\pi}{8} I-i\left[\cos \frac{\pi}{8}(X+Z)+\sin \frac{\pi}{8} Y\right] \sin \frac{\pi}{8}$$

The detailed derivation is, note that the construction of $Y$ gate,

This is a rotation of Bloch sphere along $\vec{n}=(\cos\frac{\pi}{8}, \sin\frac{\pi}{8}, \cos\frac{\pi}{8})$, $\vec{n}$ is not a unit vector, its normalized vector $\hat{n}$, and the rotation angle is $latex \theta$, such that $\cos(\theta/2)=\cos^2(\pi/8)$. In fact, it is not so easy to understand this, but we can now use this conclusion, in one sentence, $ THTH$ gate is a rotation by $2\arccos(\cos^2\frac{\pi}{8})$ about axis $\vec{n}=(\cos\frac{\pi}{8}, \sin\frac{\pi}{8}, \cos\frac{\pi}{8})$.

Here we discuss how should we deal with the above rotation opeartor,

$\exp \left(-i \frac{\pi}{8} Z\right) \exp \left(-i \frac{\pi}{8} X\right)=\exp \left[-i \frac{\pi}{8} (Z+X)\right] $

$ =\cos(\pi/8)I-i\sin(\pi/8)(Z+X)$

Why? In math principle, it looks resonable, however, Pauli operators do not commute with each other, hence we cannot write is as $\cos(\pi/8)I-i\sin(\pi/8)(Z+X)$, instead, we need to decompose these two gates independently and product them, like we did in previous paragraphs.

Repeat T and HTH

We define $R_{\hat{n}}(\theta)$ as $ THTH$ gate, where $latex \theta$ satisfies $ \cos(\theta/2)=\cos^2(\pi/8)$, and $R_{\hat{n}}(\alpha)$ is the target unitary operator we want to approximate about $ \hat{n}$ axis.

The main idea is repeating the $R_{\hat{n}}(\theta)$ for finite time. As we said before, $\theta$ is an irrational number, that’s why we can we can approximate and $ \alpha$ with $k$ times, the approximating rotation angle is $l(\theta_k-\theta_j)$, where

$$ \theta_k=k\theta\mod 2\pi$$,

it is same to $\theta_j$, and $l$ is an integer. we can prove that by using pigeonhole principle, $|\theta_k-\theta_j|$ can approximate to an small number $\delta>2\pi/N$, where $j,k \in \{1,2,\dots,N\}$. Because $|\theta_k-\theta_j|$ is small enough, so that we can obtain a sequence fills up $[0,2\pi]$ by timing an integer $l$.

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