43 research outputs found
Verifying the Steane code with Quantomatic
In this paper we give a partially mechanized proof of the correctness of
Steane's 7-qubit error correcting code, using the tool Quantomatic. To the best
of our knowledge, this represents the largest and most complicated verification
task yet carried out using Quantomatic.Comment: In Proceedings QPL 2013, arXiv:1412.791
Graphical Structures for Design and Verification of Quantum Error Correction
We introduce a high-level graphical framework for designing and analysing
quantum error correcting codes, centred on what we term the coherent parity
check (CPC). The graphical formulation is based on the diagrammatic tools of
the zx-calculus of quantum observables. The resulting framework leads to a
construction for stabilizer codes that allows us to design and verify a broad
range of quantum codes based on classical ones, and that gives a means of
discovering large classes of codes using both analytical and numerical methods.
We focus in particular on the smaller codes that will be the first used by
near-term devices. We show how CSS codes form a subset of CPC codes and, more
generally, how to compute stabilizers for a CPC code. As an explicit example of
this framework, we give a method for turning almost any pair of classical
[n,k,3] codes into a [[2n - k + 2, k, 3]] CPC code. Further, we give a simple
technique for machine search which yields thousands of potential codes, and
demonstrate its operation for distance 3 and 5 codes. Finally, we use the
graphical tools to demonstrate how Clifford computation can be performed within
CPC codes. As our framework gives a new tool for constructing small- to
medium-sized codes with relatively high code rates, it provides a new source
for codes that could be suitable for emerging devices, while its zx-calculus
foundations enable natural integration of error correction with graphical
compiler toolchains. It also provides a powerful framework for reasoning about
all stabilizer quantum error correction codes of any size.Comment: Computer code associated with this paper may be found at
https://doi.org/10.15128/r1bn999672
Graphical CSS Code Transformation Using ZX Calculus
In this work, we present a generic approach to transform CSS codes by
building upon their equivalence to phase-free ZX diagrams. Using the ZX
calculus, we demonstrate diagrammatic transformations between encoding maps
associated with different codes. As a motivating example, we give explicit
transformations between the Steane code and the quantum Reed-Muller code, since
by switching between these two codes, one can obtain a fault-tolerant universal
gate set. To this end, we propose a bidirectional rewrite rule to find a (not
necessarily transversal) physical implementation for any logical ZX diagram in
any CSS code.
Then we focus on two code transformation techniques: code morphing, a
procedure that transforms a code while retaining its fault-tolerant gates, and
gauge fixing, where complimentary codes can be obtained from a common subsystem
code (e.g., the Steane and the quantum Reed-Muller codes from the [[15,1,3,3]]
code). We provide explicit graphical derivations for these techniques and show
how ZX and graphical encoder maps relate several equivalent perspectives on
these code-transforming operations.Comment: In Proceedings QPL 2023, arXiv:2308.1548
Verifying the smallest interesting colour code with quantomatic
In this paper we present a Quantomatic case study, verifying the basic properties of the Smallest Interesting Colour Code error detection code
A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics
We introduce the first complete and approximatively universal diagrammatic
language for quantum mechanics. We make the ZX-Calculus, a diagrammatic
language introduced by Coecke and Duncan, complete for the so-called Clifford+T
quantum mechanics by adding four new axioms to the language. The completeness
of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open
questions in categorical quantum mechanics. We prove the completeness of the
Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus,
a calculus dealing with integer matrices. We also prove that the Clifford+T
fragment of the ZX-Calculus represents exactly all the matrices over some
finite dimensional extension of the ring of dyadic rationals
A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond
We consider a ZX-calculus augmented with triangle nodes which is well-suited
to reason on the so-called Toffoli-Hadamard fragment of quantum mechanics. We
precisely show the form of the matrices it represents, and we provide an
axiomatisation which makes the language complete for the Toffoli-Hadamard
quantum mechanics. We extend the language with arbitrary angles and show that
any true equation involving linear diagrams which constant angles are multiple
of Pi are derivable. We show that a single axiom is then necessary and
sufficient to make the language equivalent to the ZX-calculus which is known to
be complete for Clifford+T quantum mechanics. As a by-product, it leads to a
new and simple complete axiomatisation for Clifford+T quantum mechanics.Comment: In Proceedings QPL 2018, arXiv:1901.09476. Contains Appendi
Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
We give a scheme for interpreting shaded tangles as quantum programs, with
the property that isotopic tangles yield equivalent programs. We analyze many
known quantum programs in this way -- including entanglement manipulation and
error correction -- and in each case present a fully-topological formal
verification, yielding in several cases substantial new insight into how the
program works. We also use our methods to identify several new or generalized
procedures.Comment: In Proceedings QPL 2017, arXiv:1802.0973
The ZX-calculus is incomplete for quantum mechanics
We prove that the ZX-calculus is incomplete for quantum mechanics. We suggest
the addition of a new 'color-swap' rule, of which currently no analytical
formulation is known and which we suspect may be necessary, but not sufficient
to make the ZX-calculus complete.Comment: In Proceedings QPL 2014, arXiv:1412.810