In such a way, no matter which request agent receives the request, the associated reveal agent receives two qutrits to retrieve the original qubit. In this subsection, we begin by introducing the Pauli group and the representation of Pauli operators using binary vectors. We explain the parameters characterizing a quantum error correcting code and the definition of erasure errors. Then we explain stabilizer codes [ 19 ], specially CSS codes [ 15 , 16 ]. Finally, we show the encoding of a stabilizer code. An n -qubit Pauli group [ 20 ] is.

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The Pauli group module sign is isomorphic to a Cartesian product of binary vector spaces according to. Two Pauli operators represented as binary vectors, and , mutually commute if and only if [ 20 ].

The distance of a quantum error correcting code is the minimum weight of a Pauli operator P such that. A stabilizer code [ 19 ] is the simultaneous eigenspace of all the elements of an Abelian subgroup of with eigenvalue one. Now we introduce CSS codes, which are a type of stabilizer codes. A CSS code [ 15 , 16 ] is specified by two classical linear codes and , where is a subcode of , i.

A CSS code is a stabilizer code whose stabilizer generators are either tensor products of X operators and identities, or tensor products of Z operators and identities [ 19 ]. Here we explain the encoding of a stabilizer code with stabilizer. The Pauli operators that preserve the stabilizer code space but act nontrivially on the encoded state are the logical Pauli operators on the encoded state [ 19 ]. The encoded logical states are [ 19 ]. We have discussed quantum error correction, especially stabilizer codes in this subsection.

Next we study the close relation between a graph and a binary vector, which provides a powerful tool to construct the CSS stabilizer code. In this subsection, we begin by defining graphs and the binary linear space. After explaining these two concepts, we describe an isomorphism from sets of edges of an n -vertex graph to binary vectors with length [ 17 ].

Our approach is inspired by homology theory to construct quantum error correcting codes [ 3 , 18 ]. Finally, we present the examples of triangle graphs and star graphs. A graph [ 17 ]. The power set of E K is denoted. The set of edges. Given any , an isomorphism is. The isomorphic mappings of the vector addition and the scalar multiplication are shown in table 1. Table 1.

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The mapping of the vector addition and the scalar multiplication on to those operations on. These examples are used later for the construction of the CSS stabilizer code. Figure 2. For , a the triangle graphs representing , b the star graphs representing and c the graphs representing.

### 1. Introduction

This subsection has shown that the power set of the edge set of an n -vertex complete graph forms a binary linear space, isomorphic to. Hence, we have constructed a graph representation of any -binary vector. The examples given in this subsection are useful to construct the CSS code for summoning. In section 4. An n , k CWS code [ 14 ] encodes a k -dimensional Hilbert space to n qubits. The word stabilizer specifies a unique such that ,. Under local Clifford operations, any CWS code is equivalent to its standard form [ 14 ], whose word stabilizer is a graph-state stabilizer [ 24 ], and whose word operators contain only Z operators and identities.

This CWS code can be used to summon one qubit in four causal diamonds [ 2 ] by employing twelve qubits. Figure 3. Each is adjacent to and , where and or k. This section has introduced quantum summoning as a quantum information processing task in spacetime and explained the conditions on the configurations to make quantum summoning feasible. We have briefly reviewed the background knowledge on quantum error correction, especially stabilizer codes and CSS codes, which are used for quantum summoning in this paper.

We have explained the connection between graphs and binary vectors, which is useful for the construction of the CSS code for summoning. Finally, we have reviewed the CWS code used for quantum summoning [ 2 ], with which we compare our CSS code in section 4. In this section, we mathematically define both classical and quantum summoning. We begin by formalizing the notions of past and future light cones and causal diamonds. We than establish notations for a configuration of causal diamonds and the sets of request and reveal points.

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Subsequently, we give a careful definition of both classical and quantum summoning, and when these tasks are trivial. The future light cone for x is. The past light cone for x is. A causal diamond for a pair of points satisfying is.

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The set of request points is. A starting point is such that , where there is a starting agent S.

We now formalize Kent's classical summoning protocol [ 1 ] by making each object mathematically well defined. Remark 1. Classical summoning is trivial because S broadcasts m to all [ 1 ]. The notion of quantum summoning [ 1 , 2 ] builds on the concept of classical summoning, which we formalize as follows.

Remark 2. Remark 3. The protocol is trivial in this case because quantum information can simply be sent along this causal curve. When quantum information arrives at , decides whether to send it to z j or to send it to the next causal diamond depending on whether she receives the request or not. Here we present a protocol for Alice to summon quantum information. S sends q ij to if. If receives the summoning request, she sends all the qubits in her possession to. Otherwise, she sends each qubit q ij in her possession to.

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Later, we prove that , the qubits. Figure 4 shows how the qubits are assigned to the request agents for a configuration of four causal diamonds. Figure 4. Three request points are placed at the base vertices of a equilateral triangular prism and a fourth y 4 is placed at the centroid of base vertices. The reveal points are placed at the midpoints of the top vertices and the centroid of the top vertices. The volume of the diamond is not shown for visual clarity. The black arrows represent causal connections between points. For the CSS code the qubit q ij is assigned to edge e ij.

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This new vertex can be seen as ficticious causal diamond causally related with every causal diamond, but the summoning request is never sent to this causal diamond. For any , the qubits. In this modified task, the set of causal diamonds are causally ordered and the request may be sent to multiple request agents but only one of them needs to comply with the summon. If the causal diamonds are causally ordered, then in the subset of request agents to whom the requests are sent,. This subsection has explained our protocol for quantum summoning.

Section 4. In this subsection, we propose a stabilizer code with each qubit assigned to an edge of. We show that it is an CSS code, which can be used to summon a qubit by following the protocol in section 4. Theorem 1. The stabilizer code, specified by a. Lemma 2. The first stabilizer generators contain only Z operators and identities and the other stabilizer generators contain only X operators and identities. Thus, the stabilizer code is a CSS code. In , the vectors representing the Z-type stabilizers and the X-type stabilizers span 28 and 30 for , respectively.

Lemma 3. For any , the CSS code in theorem 1 can correct erasure errors at qubits for. The proof of lemma 3 is in appendix A. The proof is a modified version of that for the continuous-variable code [ 3 ] with the infinite-dimensional field replaced by the finite-dimensional field.

One side effect of this modification is that has to be even. Lemma 3 is no longer true if is odd. To see this, we consider an example of a three-qubit code with stabilizer generators.