Scalable spider nests
(...or how to graphically grok transversal non-Clifford gates)
Aleks Kissinger & John van de Wetering
+
ZX-diagrams
...are the language of the ZX calculus.
They are made of spiders:
 |
$:=\ \ |0...0\rangle\langle 0...0| + e^{i \alpha} |1...1\rangle\langle 1...1|$ |
 |
$:=\ \ |{+}...{+}\rangle\langle {+}...{+}| + e^{i \alpha} |{-}...{-}\rangle\langle {-}...{-}|$ |
We love ZX diagrams because
- They can represent any linear map $(\mathbb C^2)^{\otimes m} \to (\mathbb C^2)^{\otimes n}$
- They readily encode universal gatesets, e.g.
$\textit{CNOT} :=$
$\sqrt{X} :=$ 
$Z_\alpha :=$ 
- Complete for circuit equality with only a few rules
...But there's a catch
Working with full-powered ZX diagrams is (exponentially) hard.
Clifford ZX calculus
A complete set of equations for qubit Clifford QC
efficient synthesis, equality checking, classical simulation, ...
"graphical stabiliser theory and stabiliser codes"
Phase-free ZX calculus
A complete set of equations for qubit phase-free QC
efficient synthesis, equality checking, classical simulation, ...
"graphical $\mathbb F_2$-linear algebra and CSS codes"
Phase-free ZX diagrams
...are made of spiders with $\alpha = 0$:
 |
\[ :=\ \ |0...0\rangle\langle 0...0| + |1...1\rangle\langle 1...1| \] |
 |
\[ :=\ \ |{+}...{+}\rangle\langle {+}...{+}| + |{-}...{-}\rangle\langle {-}...{-}| \] |
| \[= \ \ N \sum_{\oplus_i b_i = 0} |b_1...b_n\rangle\langle b_{n+1}...b_{n+m}|\] |
Simplification
- Apply (sp) and (id) as much as possible.
- Apply (sc) once where
is not an input and
is not an output.
- Repeat.
Normal forms
Phase-free ZX diagrams can be reduced efficiently to (pseudo-)normal form:
Special case: unitaries
Unitary $\implies$ $m = n ,\ \ j = k = 0$
Special case: isometries
Isometry $\quad \implies \quad m \leq n ,\ \ j = 0$
Q: What structures are between "full-stength" ZX and efficient (Clifford/PP) ZX?
$\ \ ::\ \ |x_1 \ldots x_n \rangle \mapsto e^{i \alpha \cdot x_1 \oplus \ldots \oplus x_n} |x_1 \ldots x_n\rangle$
Phase gadgets
- $\frac\pi 4$ phase gadgets generate all diagonal gates on the 3rd level of the Clifford hierarchy $\mathcal D_3$, e.g.
$\textit{T} \ = \ $
$\textit{CCZ} \ = \ $ 
- simple T count optimisation using gadget fusion, which follows from Clifford rules:
For general $\alpha \in \mathbb R$, that's pretty much the whole story.
But for $\alpha = \pi/4$, there are (infinitely) many non-trivial rules, e.g.
These are called spider nest identities.
Q: Is there a nice way to classify all spider nests and work with them effectively?
A: Yes, but we need some new tools.
Scalable notation
Scalable notation
Scalable notation
Scalable notation
All spider nest identities
All spider nest identities
Theorem. The Clifford ZX calculus plus the S4 rule:
is diagonally complete for Clifford+T ZX diagrams, i.e. any true equation of the form:
is provable.
All spider nest identities
...where:
All spider nest identities
Then:
(Uses equivalence between triorthonality and degree $\leq n-4$ polynomials/Reed-Muller codewords, cf. Nezami & Haah. Classification of small triorthogonal codes. 2022)
Quantum error correction
...is done by encoding some space of logical qubits into a bigger space of physical qubits:
- $E$ (or just $\textrm{Im}(E)$) is called a quantum error correcting code
- Stabiliser code $:= E$ is a Clifford ZX diagram
- Calderbank-Shor-Steane code $:= E$ is a phase-free ZX diagram
Fault-tolerant computation
...is done by implementing logical operations $f$
with physical operations $F$:
Transversal diagonal gates
$\forall i . |P^i| = 1$
Transversal diagonal gates
$M = \begin{pmatrix} H & 0 \\ PL & PS \end{pmatrix}$
$D_M$ vanishes $\iff M$ is triorthogonal
Theorem. A CSS code with generator matrix $(L|S)$ admits a transversal implementation of a gate $D_H^\dagger \in \mathcal D_3$ if and only if there exists a matrix $P$ whose rows have Hamming weight $1$ such that the matrix
\[ M = \begin{pmatrix} H & 0 \\ PL & PS \end{pmatrix} \]
is triorthogonal.
Example
\[
M =
\footnotesize
\left(
\begin{array}{c|ccccccccccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline
0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1
\end{array}
\right)^T
\]
- $(L|S)$ is a $[[15, 1, 3]]$ quantum Reed-Muller code
- $P = I$, $D_P = T^{\otimes 15}$
- $D_H^\dagger = T^\dagger$
Example
\[
M =
\footnotesize
\left(
\begin{array}{cccccccc|cccccccc}
0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1
\end{array}
\right)^T
\]
- $(L|S)$ is the $[[8, 3, 2]]$ "smallest interesting colour code"
- $P = I$, $D_P = T^{\otimes 8}$
- $D_H^\dagger = \textrm{CCZ}$
Conclusions
- Scalable notation tames the complexity of spider nests
$\qquad \leadsto \qquad$ 
- Scalable diagrams represent Clifford and non-Clifford parts together, allowing $\mathbb F_2$-linear structures to interact
- Open Qs:
- Can Clifford+S4 be completed?
- Does this help with code search, distillation protocols, etc?
- Application to other kinds of FT operations?
- Beyond CSS/stabilser codes (e.g. Floquet codes)?