16 research outputs found
Completeness of Graphical Languages for Mixed States Quantum Mechanics
There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics.
We address the question of the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics.
We introduce a new construction, the discard construction, which transforms any dagger-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal.
Using this construction we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However this construction fails for some fringe cases like the Clifford+T quantum mechanics, as the category does not have enough isometries
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
Phase gadget synthesis for shallow circuits
We give an overview of the circuit optimisation methods used by tket, a compiler system for quantum software developed by Cambridge Quantum Computing Ltd. We focus on a novel technique based around phase gadgets, a family of multi-qubit quantum operations which occur naturally in a wide range of quantum circuits of practical interest. The phase gadgets have a simple presentation in the ZX-calculus, which makes it easy to reason about them. Taking advantage of this, we present an efficient method to translate the phase gadgets back to CNOT gates and single qubit operations suitable for execution on a quantum computer with significant reductions in gate count and circuit depth. We demonstrate the effectiveness of these methods on a quantum chemistry benchmarking set based on variational circuits for ground state estimation of small molecules
Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
The ZX-calculus is a universal graphical language for qubit quantum
computation, meaning that every linear map between qubits can be expressed in
the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any
equation involving linear maps that is derivable in the Hilbert space formalism
for quantum theory can also be derived in the calculus by rewriting. It has
widespread usage within quantum industry and academia for a variety of tasks
such as quantum circuit optimisation, error-correction, and education.
The ZW-calculus is an alternative universal graphical language that is also
complete for qubit quantum computing. In fact, its completeness was used to
prove that the ZX-calculus is universally complete. This calculus has advanced
how quantum circuits are compiled into photonic hardware architectures in the
industry.
Recently, by combining these two calculi, a new calculus has emerged for
qubit quantum computation, the ZXW-calculus. Using this calculus,
graphical-differentiation, -integration, and -exponentiation were made
possible, thus enabling the development of novel techniques in the domains of
quantum machine learning and quantum chemistry.
Here, we generalise the ZXW-calculus to arbitrary finite dimensions, that is,
to qudits. Moreover, we prove that this graphical rewrite system is complete
for any finite dimension. This is the first completeness result for any
universal graphical language beyond qubits.Comment: 47 pages, lots of figure
A Generic Normal Form for ZX-Diagrams and Application to the Rational Angle Completeness
Recent completeness results on the ZX-Calculus used a third-party language,
namely the ZW-Calculus. As a consequence, these proofs are elegant, but sadly
non-constructive. We address this issue in the following. To do so, we first
describe a generic normal form for ZX-diagrams in any fragment that contains
Clifford+T quantum mechanics. We give sufficient conditions for an
axiomatisation to be complete, and an algorithm to reach the normal form.
Finally, we apply these results to the Clifford+T fragment and the general
ZX-Calculus -- for which we already know the completeness--, but also for any
fragment of rational angles: we show that the axiomatisation for Clifford+T is
also complete for any fragment of dyadic angles, and that a simple new rule
(called cancellation) is necessary and sufficient otherwise