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Kitaev's toric code is one of the most prominent models for fault-tolerant quantum computation, currently regarded as the leading solution for connectivity constrained quantum technologies. Significant effort has been recently devoted to improving the error correction performance of the toric code under message-passing decoding, a class of low-complexity, iterative decoding algorithms that play a central role in both theory and practice of classical low-density parity-check codes. Here, we provide a theoretical analysis of the toric code under min-sum (MS) decoding, a message-passing decoding algorithm known to solve the maximum-likelihood decoding problem in a localized manner, for codes defined by acyclic graphs. Our analysis reveals an intrinsic limitation of the toric code, which confines the propagation of local information during the message-passing process. We show that if the unsatisfied checks of an error syndrome are at distance greater or equal to 5 from each other, then the MS decoding is locally blind: the qubits in the direct neighborhood of an unsatisfied check are never aware of any other unsatisfied checks, except their direct neighbor. Moreover, we show that degeneracy is not the only cause of decoding failures for errors of weight at least 4, that is, the MS non-degenerate decoding radius is equal to 3, for any toric code of distance greater or equal to 9. Finally, complementing our theoretical analysis, we present a pre-processing method of practical relevance. The proposed method, referred to as stabiliser-blowup, has linear complexity and allows correcting all (degenerate) errors of weight up to 3, providing quadratic improvement in the logical error rate performance, as compared to MS only.

We study the notion of $k$-stabilizer universal quantum state, that is, an $n$-qubit quantum state, such that it is possible to induce any stabilizer state on any $k$ qubits, by using only local operations and classical communications. These states generalize the notion of $k$-pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and $k$-vertex-minor universal graphs. First, we demonstrate the existence of $k$-stabilizer universal graph states that are optimal in size with $n=\Theta(k^2)$ qubits. We also provide parameters for which a random graph state on $\Theta(k^2)$ qubits is $k$-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of $k$-stabilizer universal graph states on $n = O(k^4)$ qubits. Both rely upon the incidence graph of the projective plane over a finite field $\mathbb{F}_q$. This provides a major improvement over the previously known explicit construction of $k$-pairable graph states with $n = O(2^{3k})$, bringing forth a new and potentially powerful family of multipartite quantum resources.

The inherent degeneracy of quantum low-density parity-check codes poses a challenge to their decoding, as it significantly degrades the error-correction performance of classical message-passing decoders. To improve their performance, a post-processing algorithm is usually employed. To narrow the gap between algorithmic solutions and hardware limitations, we introduce a new post-processing algorithm with a hardware-friendly orientation, providing error correction performance competitive to the state-of-the-art techniques. The proposed post-processing, referred to as check-agnosia, is inspired by stabilizer-inactivation, while considerably reducing the required hardware resources, and providing enough flexibility to allow different message-passing schedules and hardware architectures. We carry out a detailed analysis for a set of Pareto architectures with different tradeoffs between latency and power consumption, derived from the results of implemented designs on an FPGA board. We show that latency values close to one microsecond can be obtained on the FPGA board, and provide evidence that much lower latency values can be obtained for ASIC implementations. In the process, we also demonstrate the practical implications of the recently introduced t-covering layers and random-order layered scheduling.

A $k$-pairable $n$-qubit state is a resource state that allows Local Operations and Classical Communication (LOCC) protocols to generate EPR-pairs among any $k$-disjoint pairs of the $n$ qubits. Bravyi et al. introduced a family of $k$-pairable $n$-qubit states, where $n$ grows exponentially with $k$. Our primary contribution is to establish the existence of 'small' pairable quantum states. Specifically, we present a family of $k$-pairable $n$-qubit graph states, where $n$ is polynomial in $k$, namely $n=O(k^3\ln^3k)$. Our construction relies on probabilistic methods. Furthermore, we provide an upper bound on the pairability of any arbitrary quantum state based on the support of any local unitary transformation that has the shared state as a fixed point. This lower bound implies that the pairability of a graph state is at most half of the minimum degree up to local complementation of the underlying graph, i.e., $k(|G \rangle)\le \lceil \delta_{loc}(G)/2\rceil$. We also investigate the related combinatorial problem of $k$-vertex-minor-universality: a graph $G$ is $k$-vertex-minor-universal if any graph on any $k$ of its vertices is a vertex-minor of $G$. When a graph is $2k$-vertex-minor-universal, the corresponding graph state is $k$-pairable. More precisely, one can create not only EPR-pairs but also any stabilizer state on any $2k$ qubits through local operations and classical communication. We establish the existence of $k$-vertex-minor-universal graphs of order $O(k^4 \ln k)$. Finally, we explore a natural extension of pairability in the presence of errors or malicious parties and show that vertex-minor-universality ensures a robust form of pairability.

We address the problem of performing message-passing-based decoding of quantum LDPC codes under hardware latency limitations. We propose a novel way to do layered decoding that suits quantum constraints and outperforms flooded scheduling, the usual scheduling on parallel architectures. A generic construction is given to construct layers of hypergraph product codes. In the process, we introduce two new notions, t-covering layers which is a generalization of the usual layer decomposition, and a new scheduling called random order scheduling. Numerical simulations show that the random ordering is of independent interest as it helps relieve the high error floor typical of message-passing decoders on quantum codes for both layered and serial decoding without the need for post-processing.

A fault-tolerant way to prepare logical code-states of Q1 codes, i.e., quantum polar codes encoding one qubit, has been recently proposed. The fault tolerance therein is guaranteed by an error detection gadget, where if an error is detected during the preparation, one discards entirely the preparation. Due to error detection, the preparation is probabilistic, and its success rate, referred to as the preparation rate, decreases rapidly with the code-length, preventing the preparation of code-states of large code-lengths. In this paper, to improve the preparation rate, we consider a factory preparation of Q1 code-states, where one attempts to prepare several copies of Q1 code-states in parallel. Using an extra scheduling step, we can avoid discarding the preparation entirely, every time an error is detected, hence, achieving an increased preparation rate in turn. We further provide a theoretical method to estimate preparation and logical error rates of Q1 codes, prepared using factory preparation, which is shown to tightly fit the Monte-Carlo simulation based numerical results. Therefore, our theoretical method is useful for providing estimates for large code-lengths, where Monte-Carlo simulations are practically not feasible. Our numerical results, for a circuit-level depolarizing noise model, indicate that the preparation rate increases significantly, especially for large code-length N. For example, for N = 256, it increases from 0.02\% to 27\% for a practically interesting physical error rate of p = 10^{-3}. Remarkably, a Q1 code with N = 256 achieves logical error rates around 10^{-11} and 10^{-15} for p = 10^{-3} and p = 3 x 10^{-4}, respectively. This corresponds to an improvement of about three orders of magnitude compared to a surface code with similar code-length and minimum distance, thus showing the promise of the proposed scheme for large-scale fault-tolerant quantum computing., Comment: 24 pages, 11 figures

The standard model of quantum circuits assumes operations are applied in a fixed sequential "causal" order. In recent years, the possibility of relaxing this constraint to obtain causally indefinite computations has received significant attention. The quantum switch, for example, uses a quantum system to coherently control the order of operations. Several ad hoc computational and information-theoretical advantages have been demonstrated, raising questions as to whether advantages can be obtained in a more unified complexity theoretic framework. In this paper, we approach this problem by studying the query complexity of Boolean functions under general higher order quantum computations. To this end, we generalise the framework of query complexity from quantum circuits to quantum supermaps to compare different models on an equal footing. We show that the recently introduced class of quantum circuits with quantum control of causal order cannot lead to any reduction in query complexity, and that any potential advantage arising from causally indefinite supermaps can be bounded by the polynomial method, as is the case with quantum circuits. Nevertheless, we find some functions for which the minimum error with which they can be computed using two queries is strictly lower when exploiting causally indefinite supermaps., Comment: 7+12 pages

We present a formalization of several fundamental notions and results from Quantum Information theory, including density matrices and projective measurements, along with the proof that the local hidden-variable hypothesis advocated by Einstein to model quantum mechanics cannot hold. The proof of the latter result is based on the so-called CHSH inequality, and it is the violation of this inequality that was experimentally evidenced by Aspect who earned the Nobel Prize in 2022 for his work. We also formalize various results related to the violation of the CHSH inequality, such as Tsirelson's bound which permits to obtain the maximum violation of this inequality in a quantum setting., Comment: arXiv admin note: substantial text overlap with arXiv:2103.08535

We investigate what quantum advantages can be obtained in multipartite non-cooperative games by studying how different types of quantum resources can lead to new Nash equilibria and improve social welfare -- a measure of the quality of an equilibrium. Two different quantum settings are analysed: a first, in which players are given direct access to an entangled quantum state, and a second, which we introduce here, in which they are only given classical advice obtained from quantum devices. For a given game $G$, these two settings give rise to different equilibria characterised by the sets of equilibrium correlations $Q_\textrm{corr}(G)$ and $Q(G)$, respectively. We show that $Q(G)\subseteq Q_\textrm{corr}(G)$, and by exploiting the self-testing property of some correlations, that the inclusion is strict for some games $G$. We make use of SDP optimisation techniques to study how these quantum resources can improve social welfare, obtaining upper and lower bounds on the social welfare reachable in each setting. We investigate, for several games involving conflicting interests, how the social welfare depends on the bias of the game and improve upon a separation that was previously obtained using pseudo-telepathic solutions., Comment: 19+8 pages, 2+1 figures. Much improved version, with the use of self-testing methods to prove a strict separation between the two types of quantum resources

This paper explores a new approach to fault-tolerant quantum computing (FTQC), relying on quantum polar codes. We consider quantum polar codes of Calderbank-Shor-Steane type, encoding one logical qubit, which we refer to as $\mathcal{Q}_1$ codes. First, we show that a subfamily of $\mathcal{Q}_1$ codes is equivalent to the well-known family of Shor codes. Moreover, we show that $\mathcal{Q}_1$ codes significantly outperform Shor codes, of the same length and minimum distance. Second, we consider the fault-tolerant preparation of $\mathcal{Q}_1$ code states. We give a recursive procedure to prepare a $\mathcal{Q}_1$ code state, based on two-qubit Pauli measurements only. The procedure is not by itself fault-tolerant, however, the measurement operations therein provide redundant classical bits, which can be advantageously used for error detection. Fault-tolerance is then achieved by combining the proposed recursive procedure with an error detection method. Finally, we consider the fault-tolerant error correction of $\mathcal{Q}_1$ codes. We use Steane error correction, which incorporates the proposed fault-tolerant code state preparation procedure. We provide numerical estimates of the logical error rates for $\mathcal{Q}_1$ and Shor codes of length $16$ and $64$ qubits, assuming a circuit-level depolarizing noise model. Remarkably, the $\mathcal{Q}_1$ code of length $64$ qubits achieves a logical error rate very close to $10^{-6}$ for the physical error rate $p = 10^{-3}$, therefore, demonstrating the potential of the proposed polar codes based approach to FTQC.

We introduce a new characterisation of determinism in measurement-based quantum computing. The one-way model of computation consists in performing local measurements over a large entangled state represented by a graph. The ability to perform an overall deterministic computation requires a correction strategy because of the non-determinism of each measurement. The existence of such correction strategy depends on the underlying graph and the basis of the performed measurements. GFlow is a well-known graphical characterisation of robust determinism in MBQC when every measurement is performed in some specific planes of the Bloch sphere. While Pauli measurements are ubiquitous in MBQC, GFlow fails to be necessary for determinism when a measurement-based quantum computation involves Pauli measurements. As a consequence, Pauli Flow was designed more than 15 years ago as a generalisation of GFlow to handle MBQC with Pauli measurements: Pauli flow guarantees robust determinism, however it has been shown more recently that it fails to be a necessary condition. We introduce a further extension called Extended Pauli Flow that we prove necessary and sufficient for robust determinism.

We propose a post-processing method for message-passing (MP) decoding of CSS quantum LDPC codes, called stabilizer-inactivation (SI). It relies on inactivating a set of qubits, supporting a check in the dual code, and then running the MP decoding again. This allows MP decoding to converge outside the inactivated set of qubits, while the error on these is determined by solving a small, constant size, linear system. Compared to the state of the art post-processing method based on ordered statistics decoding (OSD), we show through numerical simulations that MP-SI outperforms MP-OSD for different quantum LDPC code constructions, different MP decoding algorithms, and different MP scheduling strategies, while having a significantly reduced complexity.

We introduce a class of rooted graphs which allows one to encode various kinds of classical or quantum circuits. We then follow a set-theoretic approach to define rewrite systems over the considered graphs and propose a new complete Superposition callculus which handles sets of formulas consisting of equations or disequations over these graphs.

We present a formalization in Isabelle/HOL of quantum projective measurements, a class of measurements involving orthogonal projectors that is frequently used in quantum computing. We also formalize the CHSH inequality, a result that holds on arbitrary probability spaces, which can used to disprove the existence of a local hidden-variable theory for quantum mechanics.

We introduce a graphical language for coherent control of general quantum channels inspired by practical quantum optical setups involving polarising beam splitters (PBS). As standard completely positive trace preserving maps are known not to be appropriate to represent coherently controlled quantum channels, we propose to instead use purified channels, an extension of Stinespring's dilation. We characterise the observational equivalence of purified channels in various coherent-control contexts, paving the way towards a faithful representation of quantum channels under coherent control., Comment: 34 pages

We provide a generalization of quantum polar codes to quantum channels with qudit-input, achieving the symmetric coherent information of the channel. Our scheme relies on a channel combining and splitting construction, where a two-qudit unitary randomly chosen from a unitary 2-design is used to combine two instances of a qudit-input channel. The inputs to the synthesized bad channels are frozen by sharing EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. Using the fact that the generalized two-qudit Clifford group forms a unitary 2-design, we conclude that the channel combining operation can be chosen from this set. Moreover, we show that polarization also happens for a much smaller subset of two-qudit Cliffords, which is not a unitary 2-design. Finally, we show how to decode the proposed quantum polar codes on Pauli qudit channels., Comment: 14 pages, 2 figures

Recently, a purely quantum version of polar codes has been proposed in [1] based on a quantum channel combining and splitting procedure, where a randomly chosen two-qubit Clifford unitary acts as channel combining operation. Here, we consider the quantum polar code construction using the same channel combining and splitting procedure as in [1], but with a fixed two-qubit Clifford unitary. For the family of Pauli channels, we show that polarization happens in multi-levels, where synthesized quantum virtual channels tend to become completely noisy, half-noisy, or noiseless. Further, we present a quantum polar code exploiting the multilevel nature of polarization, and provide an efficient decoding for this code. We show that half-noisy channels can be frozen by fixing their inputs in either the amplitude or the phase basis, which allows reducing the number of preshared EPR pairs compared to the construction in [1]. We provide an upper bound on the number of preshared EPR pairs, which is an equality in the case of the quantum erasure channel. To improve the speed of polarization, we propose an alternative construction, which again polarizes in multi-levels, and the previous upper bound on the number of preshared EPR pairs also holds. For a quantum erasure channel, we confirm by numerical analysis that the multilevel polarization happens relatively faster for the alternative construction., Comment: 29 pages, 5 figures

Contextuality is often referred to as a generalization of non-locality. In this work, using the hypergraph approach for contextuality we show how to associate a contextual scenario to a general k-partite non local game, and consider the reverse direction: how and when is it possible to represent a general contextuality scenario as a non local game. Using the notion of conditional contextuality, we show that it is possible to embed any contextual scenario in a two players non local game. We also discuss different equivalences of contextuality scenarios and show that the construction used in the proof is not optimal by giving a simpler bipartite non local game when the contextual scenario is a graph instead of a general hypergraph., Comment: In Proceedings QPL 2020, arXiv:2109.01534

It has been shown elsewhere that quantum resources can allow us to achieve a family of equilibria that can have sometimes a better social welfare, while guaranteeing privacy. We use graph games to propose a way to build non-cooperative games from graph states, and we show how to achieve an unlimited improvement with quantum advice compared to classical advice., Comment: 11 pages, 1 figure

We provide a purely quantum version of polar codes, achieving the symmetric coherent information of any qubit-input quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from a much smaller subset of only nine two-qubit Clifford gates. Subsequently, we show that a Pauli channel polarizes if and only if a specific classical channel over a four-symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels. Finally, we present a code construction based on chaining several quantum polar codes, which is shown to require a rate of preshared entanglement that vanishes asymptotically., Comment: 38 pages, 6 figures

A completely depolarising quantum channel always outputs a fully mixed state and thus cannot transmit any information. In a recent Letter [D. Ebler et al., Phys. Rev. Lett. 120, 120502 (2018)], it was however shown that if a quantum state passes through two such channels in a quantum superposition of different orders - a setup known as the "quantum switch" - then information can nevertheless be transmitted through the channels. Here, we show that a similar effect can be obtained when one coherently controls between sending a target system through one of two identical depolarising channels. Whereas it is tempting to attribute this effect in the quantum switch to the indefinite causal order between the channels, causal indefiniteness plays no role in this new scenario. This raises questions about its role in the corresponding effect in the quantum switch. We study this new scenario in detail and we see that, when quantum channels are controlled coherently, information about their specific implementation is accessible in the output state of the joint control-target system. This allows two different implementations of what is usually considered to be the same channel to therefore be differentiated. More generally, we find that to completely describe the action of a coherently controlled quantum channel, one needs to specify not only a description of the channel (e.g., in terms of Kraus operators), but an additional "transformation matrix" depending on its implementation., Comment: 14 pages, 2 figures

We give an $\dfrac{1}{54}$ separation between 5-party pseudo-telepathy games and two-local theories. We define the notion of strategy in a k-local theory for a game, and extend the method of Chao and Reichardt. We also study variation of the game to minimize the classical winning probability.

In this paper, we propose a Quantum variation of combinatorial games, generalizing the Quantum Tic-Tac-Toe proposed by Allan Goff. A combinatorial game is a two-player game with no chance and no hidden information, such as Go or Chess. In this paper, we consider the possibility of playing superpositions of moves in such games. We propose different rulesets depending on when superposed moves should be played, and prove that all these rulesets may lead similar games to different outcomes. We then consider Quantum variations of the game of Nim. We conclude with some discussion on the relative interest of the different rulesets.

Analyzing pseudo-telepathy graph games, we propose a way to build contextuality scenarios exhibiting the quantum supremacy using graph states. We consider the combinatorial structures that generate equivalent scenarios. We introduce a new tool called multipartiteness width to investigate which scenarios are harder to decompose and show that there exist graphs generating scenarios with a linear multipartiteness width.

A 5x5 board is the smallest board on which one can set up all kind of chess pieces as a start position. We consider Gardner's minichess variant in which all pieces are set as in a standard chessboard (from Rook to King). This game has roughly 9x10^{18} legal positions and is comparable in this respect with checkers. We weakly solve this game, that is we prove its game-theoretic value and give a strategy to draw against best play for White and Black sides. Our approach requires surprisingly small computing power. We give a human readable proof. The way the result is obtained is generic and could be generalized to bigger chess settings or to other games.

We define a family of pseudo-telepathy games using graph states that extends the Mermin games. This family also contains a game used to define a quantum probability distribution that cannot be simulated by any number of PR boxes. We extend this result proving that the probability distribution obtained by the Paley graph state on 13 vertices cannot be simulated by any number of 4-partite non local boxes and that the Paley graph states on more than $k^{2}2^{2k-2}$ vertices provides a probability distribution that cannot be simulated by k-partite nonlocal boxes

The local minimum degree of a graph is the minimum degree reached by means of a series of local complementations. In this paper, we investigate on this quantity which plays an important role in quantum computation and quantum error correcting codes. First, we show that the local minimum degree of the Paley graph of order p is greater than sqrt{p} - 3/2, which is, up to our knowledge, the highest known bound on an explicit family of graphs. Probabilistic methods allows us to derive the existence of an infinite number of graphs whose local minimum degree is linear in their order with constant 0.189 for graphs in general and 0.110 for bipartite graphs. As regards the computational complexity of the decision problem associated with the local minimum degree, we show that it is NP-complete and that there exists no k-approximation algorithm for this problem for any constant k unless P = NP., Comment: 11 pages

The graph state formalism offers strong connections between quantum information processing and graph theory. Exploring these connections, first we show that any graph is a pivot-minor of a planar graph, and even a pivot minor of a triangular grid. Then, we prove that the application of measurements in the (X,Z)-plane over graph states represented by triangular grids is a universal measurement-based model of quantum computation. These two results are in fact two sides of the same coin, the proof of which is a combination of graph theoretical and quantum information techniques.

A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a distinct set of vertices C such that every vertex in B has an odd number of neighbors in C. We point out the connections of weak odd domination with odd domination, [sigma,rho]-domination, and perfect codes. We introduce bounds on \kappa(G), the maximum size of WOD sets of a graph G, and on \kappa'(G), the minimum size of non WOD sets of G. Moreover, we prove that the corresponding decision problems are NP-complete. The study of weak odd domination is mainly motivated by the design of graph-based quantum secret sharing protocols: a graph G of order n corresponds to a secret sharing protocol which threshold is \kappa_Q(G) = max(\kappa(G), n-\kappa'(G)). These graph-based protocols are very promising in terms of physical implementation, however all such graph-based protocols studied in the literature have quasi-unanimity thresholds (i.e. \kappa_Q(G)=n-o(n) where n is the order of the graph G underlying the protocol). In this paper, we show using probabilistic methods, the existence of graphs with smaller \kappa_Q (i.e. \kappa_Q(G)< 0.811n where n is the order of G). We also prove that deciding for a given graph G whether \kappa_Q(G)< k is NP-complete, which means that one cannot efficiently double check that a graph randomly generated has actually a \kappa_Q smaller than 0.811n., Comment: Subsumes arXiv:1109.6181: Optimal accessing and non-accessing structures for graph protocols

An accessing set in a graph is a subset B of vertices such that there exists D subset of B, such that each vertex of V\B has an even number of neighbors in D. In this paper, we introduce new bounds on the minimal size kappa'(G) of an accessing set, and on the maximal size kappa(G) of a non-accessing set of a graph G. We show strong connections with perfect codes and give explicitly kappa(G) and kappa'(G) for several families of graphs. Finally, we show that the corresponding decision problems are NP-Complete.

We study a simple graph-based classical secret sharing scheme: every player's share consists of a random key together with the encryption of the secret with the keys of his neighbours. A characterisation of the authorised and forbidden sets of players is given. Moreover, we show that this protocol is equivalent to the graph state quantum secret sharing (GS-QSS) schemes when the secret is classical. When the secret is an arbitrary quantum state, a set of players is authorised for a GS-QSS scheme if and only if, for the corresponding simple classical graph-based protocol, the set is authorised and its complement set is not., Comment: 8 pages, 1 figure

We introduce a new family of quantum secret sharing protocols with limited quantum resources which extends the protocols proposed by Markham and Sanders and by Broadbent, Chouha, and Tapp. 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< 79n/156 there is no graph G allowing a ((k,n)) protocol. As a consequence there exists n_0 such that the protocols introduced by Markham and Sanders admit no threshold k when the secret is encoded on all the qubits and n>n_0.

Graph states 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) 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., Comment: 13 pages, 2 figures

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.

Among the models of quantum computation, the One-way Quantum Computer is one of the most promising proposals of physical realization, and opens new perspectives for parallelization by taking advantage of quantum entanglement. Since a one-way quantum computation is based on quantum measurement, which is a fundamentally nondeterministic evolution, a sufficient condition of global determinism has been introduced as the existence of a causal flow in a graph that underlies the computation. A O(n^3)-algorithm has been introduced for finding such a causal flow when the numbers of output and input vertices in the graph are equal, otherwise no polynomial time algorithm was known for deciding whether a graph has a causal flow or not. Our main contribution is to introduce a O(n^2)-algorithm for finding a causal flow, if any, whatever the numbers of input and output vertices are. This answers the open question stated by Danos and Kashefi and by de Beaudrap. Moreover, we prove that our algorithm produces an optimal flow (flow of minimal depth.) Whereas the existence of a causal flow is a sufficient condition for determinism, it is not a necessary condition. A weaker version of the causal flow, called gflow (generalized flow) has been introduced and has been proved to be a necessary and sufficient condition for a family of deterministic computations. Moreover the depth of the quantum computation is upper bounded by the depth of the gflow. However, the existence of a polynomial time algorithm that finds a gflow has been stated as an open question. In this paper we answer this positively with a polynomial time algorithm that outputs an optimal gflow of a given graph and thus finds an optimal correction strategy to the nondeterministic evolution due to measurements., Comment: 10 pages, 3 figures

The graph state formalism is a useful abstraction of entanglement. It is used in some multipartite purification schemes and it adequately represents universal resources for measurement-only quantum computation. We focus in this paper on the complexity of graph state preparation. We consider the number of ancillary qubits, the size of the primitive operators, and the duration of preparation. For each lexicographic order over these parameters we give upper and lower bounds for the complexity of graph state preparation. The first part motivates our work and introduces basic notions and notations for the study of graph states. Then we study some graph properties of graph states, characterizing their minimal degree by local unitary transformations, we propose an algorithm to reduce the degree of a graph state, and show the relationship with Sutner sigma-game. These properties are used in the last part, where algorithms and lower bounds for each lexicographic order over the considered parameters are presented., Comment: 17 pages

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Theta(n^{3/2}) in the matrix model and in Theta(sqrt{nm}) in the array model, while the complexity of Connectivity is also in Theta(n^{3/2}) in the matrix model, but in Theta(n) in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions., Comment: 7 figures. Subsumes and replaces quant-ph/9607014, quant-ph/0303131, and quant-ph/0303169

Quantum game theory offers a lot of interesting questions, and it is relevant to use the quantum information theory to resolve or improve games with lack of information : how to use the power of quantum entanglement to show the superiority of a quantum player that is allowed to use quantum mechanics versus a classical player, how to use quantum communication properties in cooperative games ... But games are also useful to make notions easier to understand, and permit to apprehend easier new ways of reasoning. The objective of this work is to formalize and to study a simple game with qubits using quantum notions of measurement and superposition but keeping a simple formalism so that knowing quantum mechanics is not necessary to play the game. We solve a quantum combinatorial game by giving a winning strategy for it. We also propose a quantisation of a family of combinatorial games., Comment: 13 pages

Harry Buhrman et al gave an Omega(sqrt n) lower bound for monotone graph properties in the adjacency matrix query model. Their proof is based on the polynomial method. However for some properties stronger lower bounds exist. We give an Omega(n^{3/2}) bound for Graph Connectivity using Andris Ambainis' method, and an O(n^{3/2} log n) upper bound based on Grover's search algorithm. In addition we study the adjacency list query model, where we have almost matching lower and upper bounds for Strong Connectivity of directed graphs.

This paper characterizes two forms of separability of pure states of systems of n qubits: (i) into a tensor product of n qubit states, and (ii), into a tensor product of 2 subsystems states of p and q qubits respectively with p+q=n. For both cases, necessary and sufficient conditions are proved in the form of minimal sets of equalities among pair-wise products of amplitudes. These conditions bear some relation with entanglement measures, and a number of more refined questions about separability in n qubit systems can be studied on the basis of these results., Comment: 14 pages, submitted to special issue of IJFCS

Withdrawn by the authors. Since the elements within each period of function hA are not distinct, period finding cannot operate properly. Thanks to all for the comments and sorry for the inconvenience., Comment: Withdrawn by the authors. Since the elements within each period of function hA are not distinct, period finding cannot operate properly. Thanks to all for the comments and sorry for the inconvenience