Quanlong Wang
Quanlong Wang
Doctor of Philosophy
ZX-calculus, Quantum computing, models of consciousness
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59
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Introduction
Quanlong does research in Quantum Computing and mathematical models of consciousness within the framework of ZX-calculus which is a fine-grained version of categorical process theory.
Skills and Expertise
Publications
Publications (59)
ZX-calculus has proved to be a useful tool for quantum technology with a wide range of successful applications. Most of these applications are of an algebraic nature. However, other tasks that involve differentiation and integration remain unreachable with current ZX techniques. Here we elevate ZX to an analytical perspective by realising different...
Elementary matrices play an important role in linear algebra applications. In this paper, we represent all the elementary matrices of size 2^m\times 2^m using algebraic ZX-calculus. Then we show their properties on inverses and transpose using rewriting rules of ZX-calculus. As a consequence, we are able to depict any matrices of size 2^m\times 2^n...
We cast aspects of consciousness in axiomatic mathematical terms, using the graphical calculus of general process theories (a.k.a symmetric monoidal categories and Frobenius algebras therein). This calculus exploits the ontological neutrality of process theories. A toy example using the axiomatic calculus is given to show the power of this approach...
ZW-calculus is a useful graphical language for pure qubit quantum computing. It is via the translation of the completeness of ZW-calculus that the first proof of completeness of ZX-calculus was obtained. A d-level generalisation of qubit ZW-calculus (anyonic qudit ZW-calculus) has been given in \cite{amar} which is universal for pure qudit quantum...
This paper is a ‘spiritual child’ of the 2005 lecture notes Kindergarten Quantum Mechanics [24], which showed how a simple, pictorial extension of Dirac notation allowed several quantum features to be easily expressed and derived, using language even a kindergartner can understand. Central to that approach was the use of pictures and pictorial tran...
We cast aspects of consciousness in axiomatic mathematical terms, using the graphical calculus of general process theories (a.k.a symmetric monoidal categories and Frobenius algebras therein). This calculus exploits the ontological neutrality of process theories. A toy example using the axiomatic calculus is given to show the power of this approach...
ZX-calculus is graphical language for quantum computing which usually focuses on qubits. In this paper, we generalise qubit ZX-calculus to qudit ZX-calculus in any finite dimension by introducing suitable generators, especially a carefully chosen triangle node. As a consequence we obtain a set of rewriting rules which can be seen as a direct genera...
Scientific studies of consciousness rely on objects whose existence is assumed to be independent of any consciousness. On the contrary, we assume consciousness to be fundamental, and that one of the main features of consciousness is characterized as being other-dependent. We set up a framework which naturally subsumes this feature by defining a com...
This paper is a `spiritual child' of the 2005 lecture notes Kindergarten Quantum Mechanics, which showed how a simple, pictorial extension of Dirac notation allowed several quantum features to be easily expressed and derived, using language even a kindergartner can understand. Central to that approach was the use of pictures and pictorial transform...
We give a complete axiomatisation of qubit ZX-calculus via elementary transformations which are basic operations in linear algebra. This formalism has two main advantages. First, all the operations of the phases are algebraic ones without trigonometry functions involved, thus paved the way for generalising complete axiomatisation of qubit ZX-calcul...
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 both in an arbitrary commutative ring and an arbitrary commutative semiring. We follow the framework of [14] to prove respectively tha...
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 both in an arbitrary commutative ring and an arbitrary commutative semiring. We follow the framework of [14] to prove that the propose...
Scientific studies of consciousness rely on objects whose existence is independent of any consciousness. This theoretical-assumption leads to the "hard problem" of consciousness. We avoid this problem by assuming consciousness to be fundamental, and the main feature of consciousness is characterized as being other-dependent. We set up a framework w...
In this paper we give a complete axiomatisation of qubit ZX-calculus via elementary transformations which are basic operations in linear algebra. This formalism has two main advantages. First, all the operations of the phases are algebraic ones without trigonometry functions involved, thus paved the way for generalis-ing complete axiomatisation of...
Slides for my invited talk at QPL2020.
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 $\pi/2$-phase rotations, together with T operations ($\pi/4$-phase rotations). For many error correcting codes, fault-tolerant realisa...
In this extended abstract we give an axiomatisation of ZX-calculus over arbitrary commutative rings and semirings respectively. By a normal form inspired from matrix elementary operations such as row addition and row multiplication, we can obtain that these versions of ZX-calculus are still universal and complete.
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 $T = \mathrm{diag}(1, \exp(i \pi/4))$ gate. As $T$ gates are computationally expensive to perform fault-tolerantly in the most promising error-cor...
In this paper we give an algebraic complete axiomatisation of ZX-calculus in the sense that there are only ring operations involved for phases, without any need of trigonometry functions such as sin and cos, in contrast to previous universally complete axiomatisations of ZX-calculus. With this algebraic axiomatisation of ZX-calculus, we are able to...
In this note we exploit the utility of the triangle symbol in ZX-calculus, and its role within the ZX-representation of AND-gates in particular. First, we derive a decomposition theorem for large phase gadgets, something that is of key importance to recent developments in quantum circuit optimisation and T-count reduction in particular. Then, using...
While Blockchain technology is universally considered as a significant technology for the near future, some of its pillars are under a threat of another thriving technology, Quantum Computing. In this paper, we propose important safeguard measures against this threat by developing a framework of a quantum-secured, permissioned blockchain called Log...
While Blockchain technology is universally considered as a significant technology for the near future, some of its pillars are under a threat of another thriving technology, Quantum Computing. In this paper, we propose important safeguard measures against this threat by developing a framework of a quantum-secured, permissioned blockchain called Log...
Proof of completeness of the ZX-calculus for the full pure qubit quantum mechanics, Clifford+T quantum mechanics, 2-qubit Clifford+T circuits, and the qutrit stabilizer quantum mechanics, with some applications highlighted by a triangle node as a new generator.
This paper proposes a lottery protocol based on quantum bit commitment and quantum blockchain. We first develop an arbitrarily binding quantum bit commitment protocol, then we use it to build a lottery protocol. The design and the verification of the correctness and the binding property of our bit commitment protocol is inspired by the graphical la...
This paper proposes a simple voting protocol based on Quantum Blockchain. Despite its simplicity, our protocol satisfies the most important properties of secure voting protocols: is anonymous, binding, non-reusable, verifiable, eligible, fair and self-tallying. The protocol could also be implemented using presently available technology.
By combining the logic for quantum programs (LQP) and categorical quantum mechanics (CQM), we construct a categorical logic for quantum programs (CLQP). The crucial point of our construction is to represent the constant symbols of LQP by morphisms in the ZX-calculus, a graphical calculus of CQM. Inherited from the universality of the ZX-calculus, C...
Categorical quantum mechanics places finite-dimensional quantum theory in the context of compact closed categories, with an emphasis on diagrammatic reasoning. In this framework, two equational diagrammatic calculi have been proposed for pure-state qubit quantum computing: the ZW calculus, developed by Coecke, Kissinger and the first author for the...
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.
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 protectio...
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 protectio...
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 com...
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 Cliff...
Recently, we gave a complete axiomatisation of the ZX-calculus [1] for the overall pure qubit quantum mechanics [4]. In this paper, we first simplify the rule of addition (AD) and show that some rules can be derived from other rules in [4]. Then we obtained a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics by restric...
This is the second paper of the series of papers dealing with access control problems in cloud computing by adopting quantum techniques. In this paper we study the application of quantum encryption and quantum key distribution in the access control problem. We formalize our encryption scheme and protocol for key distribution in the setting of categ...
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 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...
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.
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 $4\pi/3$-green and red rotations. This paves the way for studying the compl...
The ZX-Calculus is a powerful diagrammatic language for quantum mechanics and quantum information processing. The completeness of the $\pi$ 4-fragment is a main open problem in categorical quantum mechanics, a program initiated by Abramsky and Coecke. It has recently been proven that this fragment, also called Clifford+T quantum mechanics, was not...
The stabilizer ZX-calculus is a rigorous graphical language for reasoning about stabilizer quantum mechanics. This language has been proved to be complete in two steps: first in a setting where scalars (diagrams with no inputs or outputs) are ignored and then in a more general setting where a new symbol and three additional rules have been added to...
There are many cases where people need to be aware of the security status of their network in order to be able to respond effectively to security risks. Most security mechanisms are however too complex and most users find it difficult to understand the system and respond effectively. There is therefore a need to design understandable security mecha...
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 re...
We introduce a graphical calculus for multi-qutrit systems (the qutrit ZX-calculus) based on the framework of dagger symmetric monoidal categories. This graphical calculus consists of generators for building diagrams and rules for transforming diagrams, which is obviously different from the qubit ZX-calculus. As an application of the qutrit ZX-calc...
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 $d\times d$ unitary m...
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 general...
Controlled complementary measurements are key to quantum key distribution protocols, among many other things. We axiomatize controlled complementary measurements within symmetric monoidal categories, which provides them with a corresponding graphical calculus. We study the BB84 and Ekert91 protocols within this calculus, including the case where th...
It is shown that each rational approximant to (omega, omega(2))(tau) given by the Jacobi-Perron algorithm (JPA) or modified Jacobi-Perron algorithm (MJPA) is optimal, where omega is an algebraic function (a formal Laurent series over a finite field) satisfying omega(3) + k omega - 1 = 0 or omega(3) + kd omega - d = 0. A result similar to the main r...
We study $C_{a, b}$ curves and their applications to coding theory. Recently,
Joyner and Ksir have suggested a decoding algorithm based on the automorphisms
of the code. We show how $C_{a, b}$ curves can be used to construct MDS codes
and focus on some $C_{a, b}$ curves with extra automorphisms, namely
$y^3=x^4+1, y^3=x^4-x, y^3-y=x^4$. The automor...
In this paper, we present a method of implementing the multi-continued fraction algorithm on a class of infinite multi-sequences.
As applications of our implementing method, we get the linear complexity and minimal polynomial profiles of some non-periodic
multi-sequences.
Sequences with almost perfect linear complexity proflle are deflned by H. Niederre- iter(4). C.P. Xing and K.Y. Lam(5, 6) extended this concept from the case of single sequences to the case of multi-sequences and furthermore proposed the concept of d-perfect. In this paper, based on the technique of m-continued fractions due to Dai et al, we invest...
This paper presents the property of d-perfect m-dimensional multisequences based on the technique of m-continued fractions. The concept of perfect linear complexity profile (PLCP) to measure the unpredictability and randomness of pseudorandom sequences is presented. The properties of unpredictability and randomness are widely used in the fields of...
Projects
Projects (3)
The ultimate goal of this project is to design a framework of blockchain which has the following advantages over the existing blockchain:
1. Cheaper.
2. More efficient.
3. More Secure.
4. Smarter.
5. Easier-to-regulate.
We will use quantum resource to achieve cheapness, efficiency
and security and techniques from logic to achieve smartness and regulatability.
