Scalable spider nests
(...or how to graphically grok transversal non-Clifford gates)
Aleks Kissinger & John van de Wetering
+
https://zxcalc.github.io/book
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 $\alpha = 0$:
 |
\[ :=\ \ |0...0\rangle\langle 0...0| + |1...1\rangle\langle 1...1| \] |
 |
\[ :=\ \ |{+}...{+}\rangle\langle {+}...{+}| + |{-}...{-}\rangle\langle {-}...{-}| \] |
| \[= \ \ \lambda \sum_{\oplus_i b_i = 0} |b_1...b_n\rangle\langle b_{n+1}...b_{n+m}|\] |
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 $\quad \implies \quad m \leq n ,\ \ j = 0$
Scalable notation
Scalable notation
Scalable notation
Scalable notation
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 $F$:
In general, it is easy to classify transversal Clifford gates, but transversal non-Clifford gates are less well-understood.
which are all generated by phase gadgets:
$\ \ ::\ \ |x_1 \ldots x_n \rangle \mapsto e^{i \alpha \cdot x_1 \oplus \ldots \oplus x_n} |x_1 \ldots x_n\rangle$
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, using scalable ZX.
All spider nest identities
All spider nest identities
- We can prove all of the spider nest identities graphically, adding just one rule to Clifford ZX.
- Idea: Start by defining the following inductive family:
- ...then add this one rule:
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
Proof sketch: Using the inductive definition of $s_n$:
...we immediately get:
$\implies$
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)
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/stabiliser codes (e.g. Floquet codes)?