Simon Perdrix

Generalized flow and determinism in measurement-based quantum computation

We extend the notion of quantum information flow defined by Danos and Kashefi (2006 Phys. Rev. A 74 052310) for the one-way model (Raussendorf and Briegel 2001 Phys. Rev. Lett. 86 910) and present a necessary and sufficient condition for the stepwise uniformly deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the (X, Y), (X, Z) and (Y, Z) planes. We apply both measurement calculus and the stabiliser formalism to derive our main theorem which for the first time gives a full characterization of the stepwise uniformly deterministic computation in the one-way model. We present several examples to show how our result improves over the traditional notion of flow, such as geometries (entanglement graph with input and output) with no flow but having generalized flow and we discuss how they lead to an optimal implementation of the unitaries. More importantly one can also obtain a better quantum computation depth with the generalized flow rather than with flow. We believe our characterization result is particularly valuable for the study of the algorithms and complexity in the one-way model.

Rewriting measurement-based quantum computations with generalised flow

We present a method for verifying measurement-based quantum computations, by producing a quantum circuit equivalent to a given deterministic measurement pattern. We define a diagrammatic presentation of the pattern, and produce a circuit via a rewriting strategy based on the generalised flow of the pattern. Unlike other methods for translating measurement patterns with generalised flow to circuits, this method uses neither ancilla qubits nor acausal loops.

Graph States and the Necessity of Euler Decomposition

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 Nests 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.

Unifying quantum computation with projective measurements only and one-way quantum computation

Quantum measurement is universal for quantum computation. Two models for performing measurement-based quantum computation exist: the one-way quantum computater was introduced by Briegel and Raussendorf and quantum computation via projective measurements only by Nielsen. The more recent development of this second model is based on state transfers instead of teleportation. From this development a finite but approximate quantum universal family of observables is exhibited which includes only one two-qubit observable while others are one-qubit observables. In this article an infinite but exact quantum universal family of observables is proposed including also only one two-qubit observable. The rest of the paper is dedicated to compare these two models of measurement-based quantum computation i.e. one-way quantum computation and quantum computation via projective measurements only. From this comparison which was initiated by Cirac and Verstraete closer and more natural connections appear between these two models. These close connections lead to a unified view of measurement-based quantum computation.

Classically-controlled Quantum Computation

It is reasonable to assume that quantum computations take place under the control of the classical world. For modelling this standard situation, we introduce a Classically-controlled Quantum Turing Machine (CQTM) which is a Turing machine with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control. In CQTM, unitary transformations and quantum measurements are allowed. We show that any classical Turing machine is simulated by a CQTM without loss of efficiency. Furthermore, we show that any k-tape CQTM is simulated by a 2-tape CQTM with a quadratic loss of efficiency. The gap between classical and quantum computations which was already pointed out in the framework of measurement-based quantum computation (see [S. Perdrix, Ph. Jorrand, Measurement-Based Quantum Turing Machines and their Universality, arXiv, quant-ph/0404146, 2004]) is confirmed in the general case of classically-controlled quantum computation. In order to appreciate the similarity between programming classical Turing machines and programming CQTM, some examples of CQTM will be given in the full version of the paper. Proofs of lemmas and theorems are omitted in this extended abstract.

Pivoting makes the ZX-calculus complete for real stabilizers

We show that pivoting property of graph states cannot be derived from the axioms of the ZX-calculus, and that pivoting does not imply local complementation of graph states. Therefore the ZX-calculus augmented with pivoting is strictly weaker than the calculus augmented with the Euler decomposition of the Hadamard gate. We derive an angle-free version of the ZX-calculus and show that it is complete for real stabilizer quantum mechanics.

Information Flow in Secret Sharing Protocols

The entangled graph states have emerged as an elegant and powerful quantum resource, indeed almost all multiparty protocols can be written in terms of graph states including measurement based quantum computation (MBQC), error correction and secret sharing amongst others. In addition they are at the forefront in terms of implementations. As such they represent an excellent opportunity to move towards integrated protocols involving many of these elements. In this paper we look at expressing and extending graph state secret sharing and MBQC in a common framework and graphical language related to flow. We do so with two main contributions. First we express in entirely graphical terms which set of players can access which information in graph state secret sharing protocols. These succinct graphical descriptions of access allow us to take known results from graph theory to make statements on the generalisation of the previous schemes to present new secret sharing protocols. Second, we give a set of necessary conditions as to when a graph with flow, i.e. capable of performing a class of unitary operations, can be extended to include vertices which can be ignored, pointless measurements, and hence considered as unauthorised players in terms of secret sharing, or error qubits in terms of fault tolerance. This offers a way to extend existing MBQC patterns to secret sharing protocols. Our characterisation of pointless measurements is believed also to be a useful tool for further integrated measurement based schemes, for example in constructing fault tolerant MBQC schemes.

New Protocols and Lower Bounds for Quantum Secret Sharing with Graph States

We introduce a new family of quantum secret sharing protocols with limited quantum resources which extends the protocols proposed by Markham and Sanders [14] and Broadbent, Chouha, and Tapp [2]. Parametrized by a graph G and a subset of its vertices A, the protocol consists in: (i) encoding the quantum secret into the corresponding graph state by acting on the qubits in A; (ii) use a classical encoding to ensure the existence of a threshold. These new protocols realize ((k,n)) quantum secret sharing i.e., any set of at least k players among n can reconstruct the quantum secret, whereas any set of less than k players has no information about the secret. In the particular case where the secret is encoded on all the qubits, we explore the values of k for which there exists a graph such that the corresponding protocol realizes a ((k,n)) secret sharing. We show that for any threshold k ≥ n − n 0.68 there exists a graph allowing a ((k,n)) protocol. On the other hand, we prove that for any \(k n 0.

Which Graph States are Useful for Quantum Information Processing

Graph statesa[ 5 ] are an elegant and powerful quantum resource for measurement based quantum computation (MBQC). They are also used for many quantum protocols (error correction, secret sharing, etc.). The main focus of this paper is to provide a structural characterisation of the graph states that can be used for quantum information processing. The existence of a gflow (generalized flow) [ 8 ] is known to be a requirement for open graphs (graph, input set and output set) to perform uniformly and strongly deterministic computations. We weaken the gflow conditions to define two new more general kinds of MBQC: uniform equiprobability and constant probability. These classes can be useful from a cryptographic and information point of view because even though we cannot do a deterministic computation in general we can preserve the information and transfer it perfectly from the inputs to the outputs. We derive simple graph characterisations for these classes and prove that the deterministic and uniform equiprobability classes collapse when the cardinalities of inputs and outputs are the same. We also prove the reversibility of gflow in that case. The new graphical characterisations allow us to go from open graphs to graphs in general and to consider this question: given a graph with no inputs or outputs fixed, which vertices can be chosen as input and output for quantum information processing? We present a characterisation of the sets of possible inputs and ouputs for the equiprobability class, which is also valid for deterministic computations with inputs and ouputs of the same cardinality.

Quantum Patterns and Types for Entanglement and Separability

As a first step toward a notion of quantum data structures, we introduce a typing system for reflecting entanglement and separability. This is presented in the context of classically controlled quantum computation where a classical program controls a sequence of quantum operations, i.e. unitary transformations and measurements acting on a quantum memory. Abstract models for such quantum computations are the Quantum Random Access Machine (QRAM [E. Knill, Conventions for Quantum Pseudocode, LANL report LAUR-96-2724, 1996]) and the Classically-Controlled Quantum Turing Machine (CQTM [S. Perdrix, Ph. Jorrand, Classically-Controlled Quantum Computation, arXiv:quant-ph/0407008, to appear in Mathematical Structures in Computer Science]). Several quantum programming languages follow this model [S. Bettelli, T. Calarco, L. Serafini, Toward an architecture for quantum programming, Eur. Phys. J. D 25 (2) (2003) 181-200, S.J. Gay, R. Nagarajan, Communicating quantum processes, Proceedings of the 32nd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Jens Palsberg and Martin Abadi (Eds.), POPL 2005, January 12-14, 2005, pp. 145-157, M. Lalire, Ph. Jorrand, A process algebraic approach to concurrent and distributed quantum computation: operational semantics, Proceedings of the 2nd International Workshop on Quantum Programming Languages, 2004, pp. 109-126, J.W. Sanders, P. Zuliani, Quantum Programming, Mathematics of Program Construction, Springer LNCS 1837, 80-99, 2000, P. Selinger, Towards a Quantum Programming Language, Mathematical Structures in Computer Science 14 (4) (2004) 527-586]. Among them, the functional language defined by Valiron [B. Valiron, Quantum Typing, Proceedings of the 2nd International Workshop on Quantum Programming Languages, 2004, pp. 163-178] is the basis for the language developed in this paper. This is work in progress.