arXiv subjects

Salman Beigi

Publications and source records attributed to Salman Beigi.

62 records · Page 4Linked to original sources

arXiv2010

Graph Concatenation for Quantum Codes

Graphs are closely related to quantum error-correcting codes: every stabilizer code is locally equivalent to a graph code, and every codeword stabilized code can be described by a graph and a classical code. For the construction of good quantum codes of relatively large block length, concatenated quantum codes and their generalizations play an important role. We develop a systematic method for constructing concatenated quantum codes based on "graph concatenation", where graphs representing the inner and outer codes are concatenated via a simple graph operation called "generalized local complementation." Our method applies to both binary and non-binary concatenated quantum codes as well as their generalizations.

quant-ph↗

arXiv2009

C3, Semi-Clifford and Generalized Semi-Clifford Operations

Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called ${\mathcal{C}}_k$ hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). ${\mathcal{C}}_k$ gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev. A 77, 042313) raise the question of the relation between ${\mathcal{C}}_k$ hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any ${\mathcal{C}}_k$ gate is a generalized semi-Clifford operator. In this paper, we prove this conjecture for $k=3$. Using the techniques that we develop, we obtain more insight on how to characterize ${\mathcal{C}}_3$ gates. Indeed, the more we understand ${\mathcal{C}}_3$, the more intuition we have on ${\mathcal{C}}_k$, $k\geq 4$, and then we have a way of attacking the conjecture for larger $k$.

quant-ph↗

arXiv2009

NP vs QMA_log(2)

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve $\NP$-complete problems given a "short" quantum proof; more precisely, $\NP\subseteq \QMA_{\log}(2)$ where $\QMA_{\log}(2)$ denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion $\NP\subseteq \QMA_{\log}(2)$ has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap $\frac{1}{24n^6}$. Moreover, Aaronson {\it et al.} have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if $\QMA_{\log}(2)$ with a constant gap contains $\NP$. In this paper, we show that 3-SAT admits a $\QMA_{\log}(2)$ protocol with the gap $\frac{1}{n^{3+ε}}$ for every constant $ε>0$.

quant-ph↗

arXiv2008

The Power of Unentanglement

The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given ~O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, we show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all k>=2. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.

quant-ph↗

arXiv2008

On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels

One of the main problems in quantum complexity theory is that our understanding of the theory of QMA-completeness is not as rich as its classical analogue, the NP- completeness. In this paper we consider the clique problem in graphs, which is NP- complete, and try to find its quantum analogue. We show that, quantum clique problem can be defined as follows; Given a quantum channel, decide whether there are k states that are distinguishable, with no error, after passing through channel. This definition comes from reconsidering the clique problem in terms of the zero-error capacity of graphs, and then redefining it in quantum information theory. We prove that, quantum clique problem is QMA-complete. In the second part of paper, we consider the same problem for the Holevo capacity. We prove that computing the Holevo capacity as well as the minimum entropy of a quantum channel is NP-complete. Also, we show these results hold even if the set of quantum channels is restricted to entanglement breaking ones.

quant-ph↗

arXiv2007

Enumerating the Classes of Local Equivalency in Graphs

There are local operators on (labeled) graphs $G$ with labels $(g_{ij})$ coming from a finite field. If the filed is binary, in other words, if the graph is ordinary, the operation is just the local complementation. That is, to choose a vertex and complement the subgraph induced by its neighbors. But, in the general case, there are two different types of operators. The first type is the following. Let $v$ be a vertex of the graph and $a\in \mathbf{F}_q$, the finite field of $q$ elements. The operator is to obtain a graph with labels $g'_{ij}=g_{ij}+ag_{vi}g_{vj}$. For the second type of operators, let $0\neq b\in \mathbf{F}_q$ and the resulted graph is a graph with labels $g''_{vi}=bg_{vi}$ and $g''_{ij}=g_{ij}$, for $i,j$ unequal to $v$. The local complementation operator (binary case) has appeared in combinatorial theory, and its properties have studied in the literature. Recently, a profound relation between local operators on graphs and quantum stabilizer codes has been found, and it has become a natural question to recognize equivalency classes under these operators. In the present article, we show that the number of graphs locally equivalent to a given graph is at most $q^{2n+1}$, and consequently, the number of classes of local equivalency is $q^{\frac{n^2}{2}-o(n)}$.

math.CO↗

arXiv2007

An Efficient Algorithm to Recognize Locally Equivalent Graphs in Non-Binary Case

Let $v$ be a vertex of a graph $G$. By the local complementation of $G$ at $v$ we mean to complement the subgraph induced by the neighbors of $v$. This operator can be generalized as follows. Assume that, each edge of $G$ has a label in the finite field $\mathbf{F}_q$. Let $(g_{ij})$ be set of labels ($g_{ij}$ is the label of edge $ij$). We define two types of operators. For the first one, let $v$ be a vertex of $G$ and $a\in \mathbf{F}_q$, and obtain the graph with labels $g'_{ij}=g_{ij}+ag_{vi}g_{vj}$. For the second, if $0\neq b\in \mathbf{F}_q$ the resulted graph is a graph with labels $g''_{vi}=bg_{vi}$ and $g''_{ij}=g_{ij}$, for $i,j$ unequal to $v$. It is clear that if the field is binary, the operators are just local complementations that we described. The problem of whether two graphs are equivalent under local complementations has been studied, \cite{bouchalg}. Here we consider the general case and assuming that $q$ is odd, present the first known efficient algorithm to verify whether two graphs are locally equivalent or not.

cs.DS↗

arXiv2007

Graph States Under the Action of Local Clifford Group in Non-Binary Case

Graph states are well-entangled quantum states that are defined based on a graph. Of course, if two graphs are isomorphic their associated states are the same. Also, we know local operations do not change the entanglement of quantum states. Therefore, graph states that are either isomorphic or equivalent under the local Clifford group have the same properties. In this paper, we first establish a bound on the number of graph states which are neither isomorphic nor equivalent under the action of local Clifford group. Also, we study graph states in non-binary case. We translate the action of local Clifford group, as well as measurement of Pauli operators, into transformations on their associated graphs. Finally, we present an efficient algorithm to verify whether two graph states, in non-binary case, are locally equivalent or not.

quant-ph↗