mikeevmm / quantum-tptp

Equational Theory for Quantum Circuits - TPTP formalisation

Formalisation of an Incomplete Equational Theory for Quantum Circuits

This repository contains a first-order formalisation of the $QC_0$ portion of the complete axiomatisation for ancilla-free quantum circuits presented in [1]. Axioms for $QC_0$ are given in the TPTP format [2], as well as axioms for arithmetic modulo $n$, and there are also tools to write circuits in a convenient notation and convert it to TPTP. There are also example problems in the $QC_0$ theory, such as some of the corollaries in [1] as well as other true and false conjectures.

Contents

• Axioms/QUA001-1.axq: Axiomatisation of $QC_0$.
• Axioms/gen-ARI001-0.py: Generator for an axiomatisation of arithmetic modulo $n$.
• Problems/QUA0xx-1.pq: Corollaries in [1].
• Problems/QUA1xx-1.pq: Other true conjectures.
• Problems/QUA2xx-1.pq: Other false conjectures.
• Problems/QUAxxx-1.min.pq: Variants where only the necessary axioms are given, rather than the whole $QC_0$.
• Solutions/QUAxxx-1.n.sq: Step n of a proof of problem QUAxxx-1.
• Solutions/QUAxxx-1.n.min.sq: Variants where only the necessary axioms are given, rather than the whole $QC_0$.
• Tools/preprocess.py: Tool to parse the $(…) macros containing custom notation.

These files use custom notation inside $(…) macros (see below). Use Tools/preprocess.py to preprocess them into valid tptp files to feed into any theorem prover of your choosing (the script reads from stdin and writes to stdout).

Build

Run make to preprocess all files in the repo (.axq → .ax, .pq → .p, .sq → .s) and generate an ARI001 instance.

Custom notation

The encoding of quantum circuits is cumbersome and hard to read and write directly, so we provide a small tool that parses notation inside $(…) into that encoding.

A gate over n qubits, applied to qubits q1, …, qn is notated <gate> <q1> … <qn>. A sequence of gates is notated by a - separated pipeline, where the first element is the initial state and the rest are the gates in order of application. A global phase is notated on the initial state with o(<phase>,<state>).

When notating formulas over many-qubit quantum circuits, we need to assert that the qubits are distinct. When using this notation this is automatic; simply wrap your entire formula in another $(…)

Examples:

Notation	Circuit	TPTP
$(S - h Q)		app1(h,Q,S)
$(S - h Q1 - cnot Q1 Q2)		app2(cnot,Q1,Q2,app1(h,Q1,S))
cnf(example, plain, $(S - h Q - h Q) = S ).		cnf(example, plain, app1(h,Q,app1(h,Q,S)) ).
cnf(example, plain,$( $(S - p(φ) Q1 - cnot Q1 Q2) = $(S - cnot Q1 Q2 - p(φ) Q1) )).		cnf(example, plain, app2(cnot,Q1,Q2,app1(p(φ),Q1,S)) = app1(p(φ),Q1,app2(cnot,Q1,Q2,S)) | Q1=Q2 ).

References

Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix, Benoit Valiron. “A Complete Equational Theory for Quantum Circuits”. [1]