Picturing Quantum Software

CH1: Intro

Quantum Software is the code that runs on a quantum computer

INIT 5 CNOT 1 0 H 2 Z 3 H 0 H 1 CNOT 4 2 ...

↔

ZX Calculus

• ...is a convenient graphical language for reasoning about quantum software
• E.g. ZX diagrams readily express quantum circuits:

  $\textit{CNOT} :=$     $\sqrt{X} :=$     $Z_\alpha :=$     ...

• Complete for circuit equality with only a few rules

  

CH4: CNOT Circuits & Phase-free ZX

$\textit{CNOT} :=$

• CNOT circuits correspond to parity maps, which send basis elements to XORs of basis elements

  

• Using this fact, CNOT circuits can be efficiently represented as $\mathbb F_2$ matrices and synthesised using Gaussian elimination

• The phase-free ZX calculus is very simple:

  

• This is complete for phase-free diagrams, and can efficiently reduce those diagrams to generalised parity form:

  

• When $j = k = 0$, this reduces to Parity Normal Form, which we describe compactly as matrix arrows:

            $|\vec x\rangle \ \mapsto \ |A\vec x\rangle$

  ...a preview of scalable notation for ZX diagrams

• Phase-free rules correspond to $\mathbb F_2$-linear algebra, e.g. strong complementarity is substitution:

  

  This is used to develop techniques used for CNOT circuits (and later for generalised circuits and QEC)

CH5: Clifford Circuits & Diagrams

The Clifford ZX calculus

A complete set of equations for qubit Clifford QC

efficient synthesis, equality checking, & strong classical simulation

• Clifford circuits are built from:

  $\textit{CNOT} :=$         $H :=$         $S :=$

• Using Clifford ZX rules, these can efficiently be reduced to normal form

  GSLC form		AP form

  
  
• $\Rightarrow$ Clifford circuit (re)synthesis and normal forms
• $\Rightarrow$ pure-ZX proofs of efficient equality checking and Gottesman-Knill

CH7: Universal Circuits

• Efficient representation of CNOT+phase circuits using phase polynomials/phase gadgets:

      $=$    

• Circuit optimisation via "phase folding"
• Representing/rewrite universal circuits via
  • Sum-over-paths (a.k.a. Feynman sums/discrete Feynman integrals)
  • Pauli exponentials

• Hamiltonian simulation via Trotterization:

  

CH9: MBQC and ZX Circuit Extraction

ZX Circuit Optimisation has 2 phases. First simplify...

$\Rightarrow$

...and extract:

Circuit extraction from ZX diagrams is #P-hard in general, but it can be done in poly-time in the presence of generalised flow.

• ZX diagrams as MBQC programs
• Generalised flow (gflow) and determinism
• Polynomial-time circuit extraction using gflow

CH12: Quantum Error Correction

• $E$ (or just $\textrm{Im}(E)$) is a called a quantum error correcting code
• QECCs are used to:
  • encode (and sometimes decode) logical states
  • measure physical qubits to detect/correct errors
  • do fault-tolerant computations on encoded qubits
  
  

• The idea: represent the encoder isometry $E$ directly as a ZX diagram

  $\quad = \quad$

• Fault-tolerant QC $:=$ pushing through the encoder

  

• For CSS codes, this is especially convenient, since the encoders correspond directly to logical/check structure, e.g.

  Steane code	[[8, 3, 2]] colour code	Surface code

• We can even work with check matrices directly using scalable notation:

  $E \ \ =$         $=$    

Application: Transversal Cliffords

Application: transversal non-Cliffords

Application: lattice surgery

...and a couple more things to mention before we go:

• Fault-tolerant syndrome extraction

  

• Magic state distillation