arXiv subjects

Quanlong Wang

Publications and source records attributed to Quanlong Wang.

36 records · Page 2Linked to original sources

arXiv2020

Towards a Minimal Stabilizer ZX-calculus

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: one can transform a stabilizer ZX-diagram into another one using the graphical rewrite rules if and only if these two diagrams represent the same quantum evolution or quantum state. We previously showed that the stabilizer ZX-calculus can be simplified by reducing the number of rewrite rules, without losing the property of completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show that most of the remaining rules of the language are indeed necessary. We do however leave as an open question the necessity of two rules. These include, surprisingly, the bialgebra rule, which is an axiomatisation of complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that a weaker ambient category -- a braided autonomous category instead of the usual compact closed category -- is sufficient to recover the meta rule 'only connectivity matters', even without assuming any symmetries of the generators.

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arXiv2020

Completeness of algebraic ZX-calculus over arbitrary commutative rings and semirings

ZX-calculus is a strict mathematical formalism for graphical quantum computing which is based on the field of complex numbers. In this paper, we extend its power by generalising ZX-calculus to such an extent that it is universal both in an arbitrary commutative ring and in an arbitrary commutative semiring. Furthermore, we follow the framework of arXiv:2007.13739 to prove respectively that the proposed ZX-calculus over an arbitrary commutative ring (semiring) is complete for matrices over the same ring (semiring), via a normal form inspired from matrix elementary operations such as row addition and row multiplication. This work could lead to various applications including doing elementary number theory in string diagrams.

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arXiv2020

Techniques to Reduce $π/4$-Parity-Phase Circuits, Motivated by the ZX Calculus

To approximate arbitrary unitary transformations on one or more qubits, one must perform transformations which are outside of the Clifford group. The gate most commonly considered for this purpose is the T = diag(1, exp(i π/4)) gate. As T gates are computationally expensive to perform fault-tolerantly in the most promising error-correction technologies, minimising the "T-count" (the number of T gates) required to realise a given unitary in a Clifford+T circuit is of great interest. We describe techniques to find circuits with reduced T-count in unitary circuits, which develop on the ideas of Heyfron and Campbell [arXiv:1712.01557] with the help of the ZX calculus. Following [arXiv:1712.01557], we reduce the problem to that of minimising the T count of a CNOT+T circuit. The ZX calculus motivates a further reduction to simplifying a product of commuting "π/4-parity-phase" operations: diagonal unitary transformations which induce a relative phase of exp(i π/4) depending on the outcome of a parity computation on the standard basis (which motivated Kissinger and van de Wetering [1903.10477] to introduce "phase gadgets"). For a number of standard benchmark circuits, we show that these techniques -- in some cases supplemented by the TODD subroutine of Heyfron and Campbell [arXiv:1712.01557] -- yield T-counts comparable to or better than the best previously known results.

quant-ph↗

arXiv2020

Fast and effective techniques for T-count reduction via spider nest identities

In fault-tolerant quantum computing systems, realising (approximately) universal quantum computation is usually described in terms of realising Clifford+T operations, which is to say a circuit of CNOT, Hadamard, and $π/2$-phase rotations, together with T operations ($π/4$-phase rotations). For many error correcting codes, fault-tolerant realisations of Clifford operations are significantly less resource-intensive than those of T gates, which motivates finding ways to realise the same transformation involving T-count (the number of T gates involved) which is as low as possible. Investigations into this problem [arXiv:1206.0758, 1303.2042, 1308.4134, 1601.07363, 1606.01904, 1701.00140] has led to observations that this problem is closely related to NP-hard tensor decomposition problems [arXiv:1712.01557] and is tantamount to the difficult problem of decoding exponentially long Reed-Muller codes [arXiv:1601.07363]. This problem then presents itself as one for which must be content in practise with approximate optimisation, in which one develops an array of tactics to be deployed through some pragmatic strategy. In this vein, we describe techniques to reduce the T-count, based on the effective application of "spider nest identities": easily recognised products of parity-phase operations which are equivalent to the identity operation. We demonstrate the effectiveness of such techniques by obtaining improvements in the T-counts of a number of circuits, in run-times which are typically less than the time required to make a fresh cup of coffee.

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arXiv2019

A Categorical Compositional Distributional Modelling for the Language of Life

The Categorical Compositional Distributional (DisCoCat) Model is a powerful mathematical model for composing the meaning of sentences in natural languages. Since we can think of biological sequences as the "language of life", it is attempting to apply the DisCoCat model on the language of life to see if we can obtain new insights and a better understanding of the latter. In this work, we took an initial step towards that direction. In particular, we choose to focus on proteins as the linguistic features of protein are the most prominent as compared with other macromolecules such as DNA or RNA. Concretely, we treat each protein as a sentence and its constituent domains as words. The meaning of a word or the sentence is just its biological function, and the arrangement of domains in a protein corresponds to the syntax. Putting all those into the DisCoCat framework, we can "compute" the function of a protein based on the functions of its domains with the grammar rules that combine them together. Since the functions of both the protein and its domains are represented in vector spaces, we provide a novel way to formalize the functional representation of proteins.

q-bio.QM↗

arXiv2018

Quantum-enhanced Logic-based Blockchain I: Quantum Honest-success Byzantine Agreement and Qulogicoin

We proposed a framework of quantum-enhanced logic-based blockchain, which improves the efficiency and power of quantum-secured blockchain. The efficiency is improved by using a new quantum honest-success Byzantine agreement protocol to replace the classical Byzantine agreement protocol, while the power is improved by incorporating quantum protection and quantum certificate into the syntax of transactions. Our quantum-secured logic-based blockchain can already be implemented by the current technology. The cryptocurrency created and transferred in our blockchain is called qulogicoin. Incorporating quantum protection and quantum certificates into blockchain makes it possible to use blockchain to overcome the limitations of some quantum cryptographic protocols. As an illustration, we show that a significant shortcoming of cheat-sensitive quantum bit commitment protocols can be overcome with the help of our blockchain and qulogicoin.

quant-ph↗

arXiv2018

ZX-Rules for 2-qubit Clifford+T Quantum Circuits

ZX-calculus is a high-level graphical formalism for qubit computation. In this paper we give the ZX-rules that enable one to derive all equations between 2-qubit Clifford+T quantum circuits. Our rule set is only a small extension of the rules of stabilizer ZX-calculus, and substantially less than those needed for the recently achieved universal completeness. One of our rules is new, and we expect it to also have other utilities. These ZX-rules are much simpler than the complete of set Clifford+T circuit equations due to Selinger and Bian, which indicates that ZX-calculus provides a more convenient arena for quantum circuit rewriting than restricting oneself to circuit equations. The reason for this is that ZX-calculus is not constrained by a fixed unitary gate set for performing intermediate computations.

quant-ph↗

arXiv2018

A Simple Voting Protocol on Quantum Blockchain

This paper proposes a simple voting protocol based on quantum blockchain. Besides being simple, our voting protocol is anonymous, binding, non-reusable, verifiable, eligible, fair and self-tallying. Our protocol is also realizable by the current technology.

quant-ph↗

arXiv2018

Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics

In this paper, we show that a qutrit version of ZX-calculus, with rules significantly different from that of the qubit version, is complete for pure qutrit stabilizer quantum mechanics, where state preparations and measurements are based on the three dimensional computational basis, and unitary operations are required to be in the generalized Clifford group. This means that any equation of diagrams that holds true under the standard interpretation in Hilbert spaces can be derived diagrammatically. In contrast to the qubit case, the situation here is more complicated due to the richer structure of this qutrit ZX-calculus.

quant-ph↗

arXiv2018

Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics

Recently, we gave a complete axiomatisation of the ZX-calculus for the overall pure qubit quantum mechanics. Based on this result, here we also obtain a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics by restricting the ring of complex numbers to its subring corresponding to the Clifford+T fragment resting on the completeness theorem of the ZW-calculus for arbitrary commutative ring. In contrast to the first complete axiomatisation of the ZX-calculus for the Clifford+T fragment, we have two new generators as features rather than novelties: the triangle can be employed as an essential component to construct a Toffoli gate in a very simple form, while the lambda box can be slightly extended to a generalised phase so that the generalised supplementarity (cyclotomic supplementarity ) is naturally seen as a special case of the generalised spider rule.

quant-ph↗

arXiv2017

A universal completion of the ZX-calculus

In this paper, we give a universal completion of the ZX-calculus for the whole of pure qubit quantum mechanics. This proof is based on the completeness of another graphical language: the ZW-calculus, with direct translations between these two graphical systems.

quant-ph↗

arXiv2017

ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.

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arXiv2017

Equivalence of Local Complementation and Euler Decomposition in the Qutrit ZX-calculus

In this paper, we give a modified version of the qutrit ZX-calculus, by which we represent qutrit graph states as diagrams and prove that the qutrit version of local complementation property is true if and only if the qutrit Hadamard gate $H$ has an Euler decomposition into $4π/3$-green and red rotations. This paves the way for studying the completeness of qutrit ZX-calculus for qutrit stabilizer quantum mechanics.

quant-ph↗

arXiv2017

A Simplified Stabilizer ZX-calculus

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics.The language is sound and complete: a stabilizer ZX-diagram can be transformed into another one if and only if these two diagrams represent the same quantum evolution or quantum state. We show that the stabilizer ZX-calculus can be simplified, removing unnecessary equations while keeping only the essential axioms which potentially capture fundamental structures of quantum mechanics. We thus give a significantly smaller set of axioms and prove that meta-rules like 'colour symmetry' and 'upside-down symmetry', which were considered as axioms in previous versions of the language, can in fact be derived. In particular, we show that the additional symbol and one of the rules which had been recently introduced to keep track of scalars (diagrams with no inputs or outputs) are not necessary.

quant-ph↗

arXiv2016

Supplementarity is Necessary for Quantum Diagram Reasoning

The ZX-calculus is a powerful diagrammatic language for quantum mechanics and quantum information processing. We prove that its π/4-fragment is not complete, in other words the ZX-calculus is not complete for the so called "Clifford+T quantum mechanics". The completeness of this fragment was one of the main open problems in categorical quantum mechanics, a programme initiated by Abramsky and Coecke. The ZX-calculus was known to be incomplete for quantum mechanics. On the other hand, its π/2-fragment is known to be complete, i.e. the ZX-calculus is complete for the so called "stabilizer quantum mechanics". Deciding whether its π/4-fragment is complete is a crucial step in the development of the ZX-calculus since this fragment is approximately universal for quantum mechanics, contrary to the π/2-fragment. To establish our incompleteness result, we consider a fairly simple property of quantum states called supplementarity. We show that supplementarity can be derived in the ZX-calculus if and only if the angles involved in this equation are multiples of π/2. In particular, the impossibility to derive supplementarity for π/4 implies the incompleteness of the ZX-calculus for Clifford+T quantum mechanics. As a consequence, we propose to add the supplementarity to the set of rules of the ZX-calculus. We also show that if a ZX-diagram involves antiphase twins, they can be merged when the ZX-calculus is augmented with the supplementarity rule. Merging antiphase twins makes diagrammatic reasoning much easier and provides a purely graphical meaning to the supplementarity rule.

quant-ph↗

arXiv2015

Generalised Compositional Theories and Diagrammatic Reasoning

This chapter provides an introduction to the use of diagrammatic language, or perhaps more accurately, diagrammatic calculus, in quantum information and quantum foundations. We illustrate the use of diagrammatic calculus in one particular case, namely the study of complementarity and non-locality, two fundamental concepts of quantum theory whose relationship we explore in later part of this chapter. The diagrammatic calculus that we are concerned with here is not merely an illustrative tool, but it has both (i) a conceptual physical backbone, which allows it to act as a foundation for diverse physical theories, and (ii) a genuine mathematical underpinning, permitting one to relate it to standard mathematical structures.

quant-ph↗

arXiv2014

Qutrit Dichromatic Calculus and Its Universality

We introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a counterexample to Ranchin's universality proof, and give another proof by Lie theory that the qudit ZX calculus contains all single qudit unitary transformations, which implies that qudit ZX calculus, with qutrit dichromatic calculus as a special case, is universal for quantum mechanics.

quant-ph↗

arXiv2012

Strong Complementarity and Non-locality in Categorical Quantum Mechanics

Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.

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