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Analysis of quantum hypergraph states in the ZH-calculus
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Abstract
In this dissertation, we prove the known quantum hypergraph state manipulation rules, and introduce metarules for showing the correctness of measurement-based quantum computing (MBQC) resource states and measurement patterns in the ZH-calculus. In particular, we study the action of local Pauli operators on quantum hypergraph states, prove the generalisation of local complementation laws for hypergraph states and derive hypergraph transformation rules for local measurements, which have never been studied in the graphical calculus before. We then apply the derived metarules to study the two hypergraph-based MBQC states: the Union Jack state and the Takeuchi-Morimae-Hayashi state. Our examples show that the use of graphical calculus yields simple and intuitive proofs; in particular the metarules provide a formal and straightforward way of validating MBQC measurement patterns.
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Measurement-based quantum computing is one of the most promising quantum computing models. Although various universal resource states have been proposed so far, it was open whether only two Pauli bases are enough for both of universal measurement-based quantum computing and its verification. In this paper, we construct a universal hypergraph state that only requires X and Z-basis measurements for universal measurement-based quantum computing. We also show that universal measurement-based quantum computing on our hypergraph state can be verified in polynomial time using only X and Z-basis measurements. Furthermore, in order to demonstrate an advantage of our hypergraph state, we construct a verifiable blind quantum computing protocol that requires only X and Z-basis measurements for the client.
We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZX-calculus. Consequently, N-controlled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.
This paper has two tightly intertwined aims: (i) To introduce an intuitive
and universal graphical calculus for multi-qubit systems, the ZX-calculus,
which greatly simplifies derivations in the area of quantum computation and
information. (ii) To axiomatise complementarity of quantum observables within a
general framework for physical theories in terms of dagger symmetric monoidal
categories. We also axiomatize phase shifts within this framework.
Using the well-studied canonical correspondence between graphical calculi and
symmetric monoidal categories, our results provide a purely graphical
formalisation of complementarity for quantum observables. Each individual
observable, represented by a commutative special dagger Frobenius algebra,
gives rise to an abelian group of phase shifts, which we call the phase group.
We also identify a strong form of complementarity, satisfied by the Z and X
spin observables, which yields a scaled variant of a bialgebra.
We give a detailed account of the one-way quantum computer, a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. We prove its universality, describe why its underlying computational model is different from the network model of quantum computation, and relate quantum algorithms to mathematical graphs. Further we investigate the scaling of required resources and give a number of examples for circuits of practical interest such as the circuit for quantum Fourier transformation and for the quantum adder. Finally, we describe computation with clusters of finite size.
We present a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. The measurements are used to imprint a quantum logic circuit on the state, thereby destroying its entanglement at the same time. Cluster states are thus one-way quantum computers and the measurements form the program.
Particular focus in this paper is on quantum information protocols, which exploit quantum-mechanical effects in an essential way. The particular examples we shall use to illustrate our approach will be teleportation (Benett et al., 1993), logic-gate teleportation (Gottesman and Chuang,1999), and entanglement swapping (Zukowski et al., 1993). The ideas illustrated in these protocols form the basis for novel and potentially very important applications to secure and fault-tolerant communication and computation (2001,1999,2000).
Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer [Raussendorf and Briegel 2001] stands out as fundamental.
We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns , and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems.
Furthermore we formalize several other measurement-based models, for example, Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.
We study the entanglement properties of a class of $N$ qubit quantum states that are generated in arrays of qubits with an Ising-type interaction. These states contain a large amount of entanglement as given by their Schmidt measure. They have also a high {\em persistency of entanglement} which means that $\sim N/2$ qubits have to be measured to disentangle the state. These states can be regarded as an entanglement resource since one can generate a family of other multi-particle entangled states such as the generalized GHZ states of $<N/2$ qubits by simple measurements and classical communication (LOCC). Comment: 4 pages, 1 figure. Revised version puts more emphasis on the presentation of the cluster states as a novel class of N-qubit entangled states, and on the discussion of their entanglement properties in terms of the notions of persistency, maximal connectedness, and the Schmidt measure. Introduction has been completely rewritten. More space is now devoted to motivating the notions of persistency and maximal connectedness, see the paragraph after Eq.(3) and (4). Discussion of the Schmidt measure of the cluster states has been added. More technical discussions of the 2D and 3D generalisations of the cluster states have been shortened
Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for three-uniform hypergraph states.
Measurement-based quantum computation (MQC) is a paradigm for studying quantum computation using many-body entanglement and single-qubit measurements. Although MQC has inspired wide-ranging discoveries throughout quantum information, our understanding of the general principles underlying MQC seems to be biased by its historical reliance upon the archetypal 2D cluster state. Here we utilise recent advances in the subject of symmetry-protected topological order (SPTO) to introduce a novel MQC resource state, whose physical and computational behaviour differs fundamentally from that of the cluster state. We show that, in sharp contrast to the cluster state, our state enables universal quantum computation using only measurements of single-qubit Pauli X, Y, and Z operators. This novel computational feature is related to the ‘genuine’ 2D SPTO possessed by our state, and which is absent in the cluster state. Our concrete connection between the latent computational complexity of many-body systems and macroscopic quantum orders may find applications in quantum many-body simulation for benchmarking classically intractable complexity.
The theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines. Quantum gates are the generalization of classical logic gates. A single type of gate, the univeral quantum gate, together with quantum 'unit wires', is adequate for constructing networks with any possible quantum computational property.
Ionicioiu and Spiller [Phys. Rev. A 85, 062313 (2012)] have recently
presented an axiomatic framework for mapping graphs to quantum states of a
suitable physical system. Based on their study, we first extend the axiomatic
framework to hypergraphs by means of modifying its axioms and consistency
conditions. Then we use the axiomatic approach to encode hypergraphs into a new
family of quantum states, called the hypergraph states. Moreover, we also try
to do the followings: (i) to show that real equally weighted states, which
occur in Grover and Deutsch-Joza algorithms, are equivalent to hypergraph
states; (ii) to describe the relations among hypergraph states, graph states
and stabilizer states; (iii) to provide some transformation rules, stated in
purely hypergraph theoretical terms, which completely characterize the
evolution of hypergraph states under some local operations, including operators
in Pauli group and some special local Pauli measurements; and (iv) to
investigate some properties of multipartite entanglement of hypergraph states
by hypergraph theory.
Quantum computations are easily represented in the graphical notation known
as the ZX-calculus, a.k.a. the red-green calculus. We demonstrate its use in
reasoning about measurement-based quantum computing, where the graphical syntax
directly captures the structure of the entangled states used to represent
computations, and show that the notion of information flow within the entangled
states gives rise to rewriting strategies for proving the correctness of
quantum programs.
Coecke and Duncan recently introduced a categorical formalisation of the
interaction of complementary quantum observables. In this paper we use their
diagrammatic language to study graph states, a computationally interesting
class of quantum states. We give a graphical proof of the fixpoint property of
graph states. We then introduce a new equation, for the Euler decomposition of
the Hadamard gate, and demonstrate that Van den Nest's theorem--locally
equivalent graphs represent the same entanglement--is equivalent to this new
axiom. Finally we prove that the Euler decomposition equation is not derivable
from the existing axioms.
Categories and quantum informatics
- Chris Heunen
Chris Heunen. Categories and quantum informatics, January 2018.
Introduction to quantum computing
- Petros Wallden
Petros Wallden. Introduction to quantum computing, September 2018.
- Aleks Kissinger
- John Van De Wetering
Aleks Kissinger and John van de Wetering. Pyzx: Large scale automated diagrammatic reasoning. arXiv preprint arXiv:1904.04735, 2019.