Is there a non-Clifford gate preserving both $X$ and $Z$ errors?

Question

I would like to know if there exists an $n$-qubit (for $n \geq 2$) quantum gate $G_n$ that preserves both $X$ and $Z$ errors and that is additionnally non-Clifford.

In other words, I would like that $G_n$ satisfy

$$ \forall P^X_n, \exists E^X_n | \ G_n P^X_n = E^X_n G_n $$ $$ \forall P^Z_n, \exists E^Z_n | \ G_n P^Z_n = E^Z_n G_n $$

where

• $P^X_n$ is any $n$-qubit Pauli matrix composed of $X$ and $I$ terms only.
• $E_n^X$ is an operator that once decomposed on the Pauli matrices can be expressed as a linear combination composed of terms involving pauli $X$ and identity. The coefficients in this linear combination don't have to be identical.

Similarly, for $Z$ errors.

To be a little bit clearer, $P_2^X$ could be $X_1 X_2$ or $I_1 X_2$ but not $Z_1 X_2$. $E_2^X$ could be $X_1 X_2 + I_1 X_2$, $X_1 X_2 + \frac{1}{3}I_1 X_2$ but not $X_1 X_2 + Z_1 X_2$ (the example I give here might not work strictly speaking as $G_n$ is unitary, but this is just to say that any linear combination is allowed as long as it only involves I and $X$ operators, I do not ask all the coefficients in the linear combination to be equal, even up to a sign). The fact that I am asking the gate to be non-Clifford implies that there exist at least one $E_n^X$ or one $E_n^Z$ that is not a Pauli operator.

As examples of gates that fulfill some but not all of my properties there is:

• CNOT gate. It preserves both $X$ and $Z$ errors but the $E_2^X$ and $E_2^Z$ will always be tensor product of Pauli (the gate is Clifford).
• Toffoli gate. It satisfies the properties for $Z$ errors (and one $E_n^Z$ will be a sum of Paulis but not a Pauli itself: the gate is then non-Clifford). However, it can convert a $X$ error into a $Z$ one.

Do gates satisfying the properties I am looking for exist? If so what are some examples?

Answer 1

TL;DR: It turns out that $G_n$ acts as a phased permutation of the computational basis, sending $|x\rangle$ to $(-1)^{s(x)}|\sigma(x)\rangle$ . Similarly, $H^{\otimes n}G_nH^{\otimes n}$ sends $|y\rangle$ to $(-1)^{t(y)}|\tau(y)\rangle$. Moreover, $$ t(y)+x\cdot\tau(y)=y\cdot\sigma^{-1}(x) + s(\sigma^{-1}(x)) $$ for any two $n$-bit strings $x,y\in\mathbb{F}_2^n$. But then $s$ and $\sigma$ are affine functions on $\mathbb{F}_2^n$, so $G_n$ is Clifford.

$G_n$ is a phased permutation

We begin with the observation that $P_n^Z$ and $E_n^Z$ are diagonal in the computational basis. Moreover, all operators $P_n^Z$ form a basis of the real vector space of diagonal Hermitian matrices. Thus, conjugation by $G_n$ maps diagonal matrices to diagonal matrices. Let $\rho$ be the density matrix of a computational basis state. In other words, $\rho$ is a diagonal matrix with a single $1$ on diagonal and all other elements equal to zero. Then $G_n\rho G_n^\dagger$ is also diagonal with a single element equal $1$. Thus, $G_n$ maps computational basis states to computational basis states, i.e. $G_n$ is a "phased permutation gate" $$ G_n|x\rangle = e^{i\theta(x)}|\sigma(x)\rangle\tag1 $$ where $x\in\mathbb{F}_2^n$ is an $n$-bit string, $e^{i\theta(x)}$ is a phase factor and $\sigma\in S_{2^n}$ is a permutation of the computational basis.

$G_n$ is real

Now, consider the matrix $2^nG_n|+\rangle\langle+|G_n^\dagger$. On one hand, its elements are equal to $e^{i\theta(x_i)-i\theta(x_j)}$ for all pairs of $n$-bit strings $x_i$ and $x_j$. On the other hand, the matrix $2^nG_n|+\rangle\langle+|G_n^\dagger=2^n E_n^X$ is a real symmetric matrix. Consequently, the phase factors $e^{i\theta(x)}$ are $\pm 1$. Thus, we can rewrite $(1)$ as $$ G_n|x\rangle = (-1)^{s(x)}|\sigma(x)\rangle\tag{1'} $$ where $s:\mathbb{F}_2^n\to\mathbb{F}_2$. Incidentally, this implies that $G_1$ is a Pauli gate.

$H^{\otimes n}G_nH^{\otimes n}$ is another phased permutation

If $G_n$ preserves $X$- and $Z$-errors then so does $H^{\otimes n}G_nH^{\otimes n}$. Therefore, $$ H^{\otimes n}G_nH^{\otimes n}|y\rangle = (-1)^{t(y)}|\tau(y)\rangle\tag{2} $$ where $t:\mathbb{F}_2^n\to\mathbb{F}_2$ and $\tau\in S_{2^n}$ is a permutation of the computational basis. But then $$ \begin{align} G_nH^{\otimes n}|y\rangle &= (-1)^{t(y)}H^{\otimes n}|\tau(y)\rangle\tag{3.1}\\ G_n\sum_{x\in\mathbb{F}_2^n}(-1)^{x\cdot y}|x\rangle &= (-1)^{t(y)}\sum_{x\in\mathbb{F}_2^n}(-1)^{x\cdot\tau(y)}|x\rangle\tag{3.2}\\ \sum_{x\in\mathbb{F}_2^n}(-1)^{s(x)+x\cdot y}|\sigma(x)\rangle &= \sum_{x\in\mathbb{F}_2^n}(-1)^{t(y)+x\cdot\tau(y)}|x\rangle\tag{3.3}\\ \sum_{x\in\mathbb{F}_2^n}(-1)^{s(\sigma^{-1}(x))+\sigma^{-1}(x)\cdot y}|x\rangle &= \sum_{x\in\mathbb{F}_2^n}(-1)^{t(y)+x\cdot\tau(y)}|x\rangle\tag{3.4}\\ \end{align} $$ where $\cdot$ denotes the dot product in $\mathbb{F}_2^n$. By linear independence we have $$ s(\sigma^{-1}(x))+\sigma^{-1}(x)\cdot y = t(y)+x\cdot\tau(y)\tag{4} $$ for all $x,y\in\mathbb{F}_2^n$.

$G_n$ is Clifford

Setting $u(x)=s(\sigma^{-1}(x))$ and $\upsilon=\sigma^{-1}$, we can rewrite $(4)$ as $$ u(x)+y\cdot\upsilon(x) = t(y)+x\cdot\tau(y).\tag{5} $$ More explicitly, $(5)$ may be written as $$ u(x_1,\dots,x_n)+\sum_{i=1}^n y_i\upsilon_i(x_1,\dots,x_n)= t(y_1,\dots,y_n)+\sum_{i=1}^n x_i\tau_i(y_1,\dots,y_n).\tag{6} $$ By setting all $y_i=0$, we see that $u:\mathbb{F}_2^n\to\mathbb{F}_2$ is affine. By setting a single $y_i=1$ and all others to zero, we see that each $\upsilon_i:\mathbb{F}_2^n\to\mathbb{F}_2$ is affine. Therefore, $\upsilon:\mathbb{F}_2^n\to\mathbb{F}_2^n$ is affine. Hence, $s$ and $\sigma$ are affine. Therefore, $G_n$ is Clifford.