The Exercise

The unitary of a quantum circuit is the product of the matrices of its gates. Several of the standard gates are a power of $1/\sqrt{2}$ times a matrix whose entries are $a + bi$ with $a$ and $b$ integers: the Hadamard gate is $1/\sqrt{2}$ times a matrix of entries $\pm 1$ , and the phase gate $\operatorname{diag}(1, i)$ needs no factor. The exercise is to show that this form survives multiplication, so that a circuit of such gates has a unitary $2^{-k/2} M$ for some natural number $k$ , with $M$ of the same kind, and that the $\pi/8$ gate, whose matrix holds $e^{i\pi/4}$ , does not have it. Circuits that use the $\pi/8$ gate keep the form once the entries of $M$ may also contain the powers of $e^{i\pi/4}$ .

Two Rings of Numbers

The Gaussian integers $\mathbb{Z}[i]$ are the complex numbers $a + bi$ with $a$ and $b$ integers.^✦ The primitive eighth root of unity is $\zeta_8 = e^{i\pi/4},$ (1)^✦ whose square is $i$ . The cyclotomic integers $\mathbb{Z}[\zeta_8]$ are the complex numbers $p + q\,\zeta_8, \qquad p, q \in \mathbb{Z}[i].$ (2)^✦

The $\pi/8$ Gate and the Scaled Form

The $\pi/8$ gate $T$ is the gate on one qubit whose matrix in the computational basis is $T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & \zeta_8 \end{pmatrix}.$ (3)^✦

Let $R$ be a subring of the complex numbers. A complex $m \times n$ matrix $U$ has a scaled form over $R$ if there are a natural number $k$ and an $m \times n$ matrix $M$ with entries in $R$ such that $U = \Bigl(\frac{1}{\sqrt{2}}\Bigr)^{k} M.$ (4)^✦

The Claim

The library states the exercise as three theorems.

Theorem 1. Let $U_1, \dots, U_n$ be square complex matrices of one size, each with a scaled form over $\mathbb{Z}[\zeta_8]$ . Then there are a natural number $k$ and a matrix $M$ of the same size, every entry of which is $a + b\zeta_8 + c\zeta_8^2 + d\zeta_8^3$ for some integers $a, b, c, d$ , such that $U_1 U_2 \cdots U_n = 2^{-k/2} M.$ (5)^✦

Theorem 2. Let $U_1, \dots, U_n$ be square complex matrices of one size, each with a scaled form over $\mathbb{Z}[i]$ . Then there are a natural number $k$ and a matrix $M$ of the same size, every entry of which is $a + bi$ for some integers $a$ and $b$ , such that $U_1 U_2 \cdots U_n = 2^{-k/2} M.$ (6)^✦

Theorem 3. The $\pi/8$ gate has no scaled form over $\mathbb{Z}[i]$ : whatever the natural number $k$ and the $2 \times 2$ matrix $M$ with entries in $\mathbb{Z}[i]$ , $T \neq \Bigl(\frac{1}{\sqrt{2}}\Bigr)^{k} M.$ (7)^✦

What Is Checked

The claims are stated over the scaled form (4) of a matrix over a subring of the complex numbers, taken over the Gaussian integers and over the cyclotomic integers (2), which are built on the eighth root of unity (1); the third is about the $\pi/8$ gate (3).

About these constructions the library proves that both sets of numbers are subrings of the complex numbers: each contains $0$ and $1$ and is closed under sums and negatives, and products stay in the set by $(a + bi)(c + di) = (ac - bd) + (ad + bc)\,i$ (8) for integers $a, b, c, d$ , and by $(p + q\,\zeta_8)(p' + q'\,\zeta_8) = (pp' + i\,qq') + (pq' + p'q)\,\zeta_8$ (9) for Gaussian integers $p, q, p', q'$ .