Construction of optimal resources for concatenated quantum protocols
We consider the explicit construction of resource states for measurement-based quantum information processing. We concentrate on special-purpose resource states that are capable to perform a certain operation or task, where we consider unitary Clifford circuits as well as non-trace preserving completely positive maps, more specifically probabilistic operations including Clifford operations and Pauli measurements. We concentrate on and operations, i.e. operations that map one input qubit to output qubits or vice versa. Examples of such operations include encoding and decoding in quantum error correction, entanglement purification or entanglement swapping. We provide a general framework to construct optimal resource states for complex tasks that are combinations of these elementary building blocks. All resource states only contain input and output qubits, and are hence of minimal size. We obtain a stabilizer description of the resulting resource states, which we also translate into a circuit pattern to experimentally generate these states. In particular, we derive recurrence relations at the level of stabilizers as key analytical tool to generate explicit (graph-) descriptions of families of resource states. This allows us to explicitly construct resource states for encoding, decoding and syndrome readout for concatenated quantum error correction codes, code switchers, multiple rounds of entanglement purification, quantum repeaters and combinations thereof (such as resource states for entanglement purification of encoded states).
pacs:03.67.Hk, 03.67.Lx, 03.67.Pp
Measurement-based quantum computation Briegel et al. (2009); Raussendorf and Briegel (2001); Gottesman and Chuang (1999) enables one to perform any quantum computation via a sequence of single-qubit measurements on a large, highly entangled resource state, the 2D cluster state Briegel and Raussendorf (2001). Hence this specific state is universal for quantum computation. As the 2D cluster state might be very large in terms of the number of qubits, in many situations one is interested in resource states of minimal size for implementing a specific quantum task Raussendorf et al. (2003).
Such a measurement-based approach to quantum information processing has been discussed in various contexts, including quantum computation Briegel et al. (2009) and quantum communication Zwerger et al. (2016a) and was in some cases experimentally demonstrated Lanyon et al. (2013); Barz et al. (2014). In contrast to a circuit-based approach, no coherent manipulation of quantum information via the application of single- and two-qubit gates is required. One rather needs to prepare certain resource states, which are then manipulated by means of measurements, e.g. by coupling input qubits via Bell measurements to the resource state, similarly as in a teleportation process where however in this case a specific operation, determined by the resource state, is performed. The only sources of errors in such a scheme are imperfect preparation of resource states, and imperfect measurements.
Two main advantages of such a measurement-based approach have been identified Zwerger et al. (2016a, 2014, 2013): On the one hand, one finds that the acceptable error rates are very high - depending on the task, 10% noise per particle or more can be tolerated when assuming a noise model where each qubit of the resource state is subjected to single-qubit depolarizing noise. On the other hand, such a measurement-based approach allows for various ways to prepare resource states, including probabilistic (heralded) schemes. This opens the possibility to use probabilistic processes such as parametric-down conversion, error detection, entanglement purification or even cooling to a ground state for resource state preparation.
Depending on the task at hand, different resource states need to be prepared. However, obtaining the explicit resource state or even an efficient preparation procedure for a given task is not straightforward, and explicit constructions are often limited to small system sizes. In turn, any experimental realization requires knowledge of the exact form of resource states and how to prepare them. Also from a theoretical side, knowing the explicit resource state for a specific task allows one to investigate its entanglement features, stability under noise and imperfections and to design optimized ways to generate them with high fidelity, e.g. by means of entanglement purification
Here we provide an explicit construction of resource states for complex tasks that correspond to a composition of elementary building blocks. These building blocks consist of unitary and non-unitary or operations, including e.g. encoding and decoding in quantum error correction, entanglement swapping and entanglement purification which is a probabilistic process. We develop a general framework for concatenating resource states for different quantum operations according to their stabilizer description and obtain resource states of minimal size which consist only of input and output particles (or only input particles for tasks where there is no quantum output). We provide an efficient and explicit description of families of resource states via recurrence relations in terms of stabilizers for complex quantum operations, and also obtain a representation of these stabilizer states in terms of graph states. This leads directly to an efficient quantum circuit that prepares these states using at most commuting two qubit gates for any task with input and output systems. Furthermore, this allows one to use several of the methods and techniques developed for graph states Hein et al. (2006, 2004), including entanglement purification Dür and Briegel (2007); Dür et al. (2003), or to analyze their stability under noise and decoherence. Entanglement purification is of particular importance in this context, as this does not only allow one to prepare resource states with high fidelity, but also yields states which can be well described by a local noise model Wallnöfer and Dür (2017), thereby confirming above mentioned local error model that was used e.g. in Zwerger et al. (2016a, 2014, 2013, 2012).
The examples we provide include resource states for multiple steps of entanglement purification using a recurrence protocol Bennett et al. (1996); Deutsch et al. (1996); Dür and Briegel (2007), encoding, decoding and error syndrome readout for concatenated quantum error correction Nielsen and Chuang (2010), quantum code switchers that allow one to change between different error correction codes (e.g. for storage and data processing), entanglement purification of encoded states or quantum repeater stations for long-distance quantum communication Briegel et al. (1998); Sangouard et al. (2011).
This paper is organized as follows. In Sec. II we relate our findings to earlier works in the field and highlight the novelty of our approach. In Sec. III we provide some background information on stabilizer states, graph states and Clifford circuits, and give a brief introduction to measurement-based quantum computation and the Choi-Jamiolkowksi isomorphism Jamiołkowski (1972). We also settle the notation we use throughout the article in this section. In Sec. IV we describe the general framework of concatenated quantum tasks, and present our main technical results to efficiently construct a stabilizer description of corresponding resource states. In Sec. V we present several applications of our method. We provide resource states for multiple steps of entanglement purification using the recurrence protocol of Deutsch et al. (1996), concatenated quantum error correction using a generalized Shor code, codes switchers and entanglement purification of encoded states. We summarize and discuss our results in Sec. VI.
Ii Relation to prior work
Measurement-based quantum computation Briegel et al. (2009); Raussendorf and Briegel (2001); Gottesman and Chuang (1999); Raussendorf et al. (2003); Nielsen (2006); Aliferis and Leung (2004); Childs et al. (2005); Gross and Eisert (2007) is a paradigm where quantum information is processed by measurements only. Certain states, e.g. the 2D cluster state Briegel and Raussendorf (2001), serve as universal resources Mantri et al. (2017). That is, by performing only single qubit measurements, an arbitrary quantum computation can be performed, or equivalently an arbitrary quantum state can be generated. For an introduction and review on measurement-based quantum computation we also refer to Browne and Briegel (2006); Jozsa (2005); Campbell and Fitzsimons (2010) and for their implementation in specific systems to Kwek et al. (2012); Kyaw et al. (2014); Benjamin et al. (2009). Despite the probabilistic character of measurements, the desired state is generated deterministically up to local Pauli corrections. There are also special purpose resource states Raussendorf et al. (2003); Zwerger et al. (2016a, 2014, 2012) that allow one to perform a specific task or operation. Typically, these special purpose resource states are smaller in size. Such resource states can be constructed in two different ways: First, one may start with a 2D cluster state, and perform all measurements corresponding to Clifford operations. This leaves one with a state of reduced size, which can in principle be determined using the stabilizer formalism Gottesman (1997) or graph-state formalism Hein et al. (2006). On the other hand, for circuits that only include Clifford operations and Pauli measurements, one can construct the resource state via the Jamiolokowski isomorphism Jamiołkowski (1972), i.e. by applying the circuit to part of a maximally entangled state (see e.g. Raussendorf et al. (2003); Zwerger et al. (2016a, 2014, 2012)). This equivalence is also apparent from Theorem 1 in Raussendorf et al. (2003). In both cases, the stabilizer formalism allows one in principle to efficiently obtain the description of the resource state in terms of its stabilizers. However, taking care of local correction operations and stabilizer update rules is a tedious task, making such a direct approach difficult for complex tasks and operations, especially for operations that act on many qubits and consist of many Clifford operations and measurements.
Measurement-based quantum information processing using special purpose resource states has been investigated in different contexts. In Raussendorf et al. (2003) explicit resources for quantum adder and the quantum fourier transform have been constructed. Quantum error correction codes associated with graph states have been proposed in Schlingemann and Werner (2001). This type of quantum error correction codes especially fit in a measurement-based setting as their resource state corresponds to a graph. Quantum error correction codes where the codewords correspond to graph states were studied e.g. in Schlingemann (2002); Grassl et al. (2007); Hein et al. (2006, 2004). The concatenation of quantum error correction codes is a standard way to obtain efficient codes for fault-tolerant quantum computation, see e.g. Gottesman (1997). The measurement-based implementations of quantum error correction codes were studied in Zwerger et al. (2014, 2016a); Barz et al. (2014); Lanyon et al. (2013) where explicit resource states for the repetition code and cluster-ring code were provided. In Dawson et al. (2006a, b) the practicality of optical cluster-state quantum computation Browne and Rudolph (2005); Nielsen (2004) using quantum error correction codes was numerically investigated and it was found that scalable optical quantum computing is possible. Furthermore it was also investigated in fault-tolerant quantum computation on cluster-states in Nielsen and Dawson (2005); Raussendorf et al. (2007, 2006). In Zwerger et al. (2012) explicit resource states for entanglement purification and entanglement swapping, as well as combinations thereof were constructed. In particular, resource states for one and two rounds of entanglement purification using the protocol of Deutsch et al. (1996) and entanglement purification followed by entanglement swapping. The latter is a building block for a measurement-based quantum repeater, which allows for long-distance quantum communication.
All examples mentioned so far have in common that they construct a resource state for one specific task. Even though these resource states do implement a particular quantum operation, they still lack of a composable description. It is a non-trivial task to concatenate those elementary quantum operations at the level of resource states as already easy examples like two rounds of entanglement purification show. Here we address this problem and provide an explicit construction of resource states for concatenated tasks that are combinations of elementary building blocks. Rather than calculating the required resource state directly, we develop a method to combine and concatenate small building blocks in terms of their stabilizer description via a set of recurrence relations. We would like to emphasize that our construction is applicable not only to unitary Clifford circuits, but also to circuits that contain Pauli measurements. In particular, also probabilistic operations such as entanglement purification can be treated in this way. Pauli measurements can be done beforehand, thereby obtaining resource states of smaller (minimal) size. Furthermore, the result of the Pauli measurement (which determines whether the overall operation is successful and the output states should be kept) can be determined from the results of the incoupling Bell measurements Zwerger et al. (2016a, 2013, 2012, 2014). This ensures full functionality of circuits, including post-selection based on measurement outcomes, or application of correction operations depending on the encountered error syndrome.
Consider for example a resource state for syndrome read-out and error correction for a concatenated five-qubit code with four concatenation levels. This corresponds to an error correction code of qubits, i.e. a resource states of 1250 qubits. The circuit to implement the required error correction operation thus contains several thousand gates, and the direct computation of the resource state from the 2D cluster state or via the Jamiolkowski isomorphism is difficult. With our approach, we make use of the recursive and concatenated structure of the states, and can easily construct the required resource states for decoding and encoding for such concatenated codes in terms of their stabilizers. The combination of decoding (with syndrome readout) and encoding allows then to obtain the resource state for syndrome readout and error correction, or alternatively a code switcher between different codes. In a similar way, one can combine different kinds of elementary building blocks, and obtain resource states for multiple rounds of entanglement purification, entanglement purification of encoded states or quantum repeater stations for encoded quantum information. This allows for full flexibility, and for a broad applicability of our findings. Following our approach one immediately obtains a stabilizer description of a concatenated quantum operation rather than computing its implementing resource state from scratch. Furthermore, as the construction scheme relies on stabilizers, this description turns out to be especially suited for studying scaling and stability properties of resource states, crucial for experimental implementations. For all relevant examples, we also provide an explicit description of the resulting resource states as graph states. This has the advantage that one obtains directly an efficient way to prepare these states via elementary two-qubit operations. In addition, as entanglement purification protocols for all graph states exist Kruszynska et al. (2006), one also obtains a way to generate all these states with high-fidelity from multiple copies via entanglement purification.
Iii Background and Notation
In the following we recall some basic notations and results concerning stabilizer states, graph states and measurement-based quantum computation which will be used throughout the paper.
iii.1 Stabilizer states, Clifford group and Graph states
Let denote the qubit Pauli group, i.e. is the group consisting of all fold tensor products of the Pauli operators and as well as the identity. We call an qubit state a stabilizer state if it is stabilized by elements of . More precisely, there exist such that for Nielsen and Chuang (2010). The stabilizers of form a subgroup of .
Graph states Hein et al. (2006, 2004) are specific stabilizer states. Given a mathematical graph , where denotes the set of vertices and the set of edges, the associated graph state is stabilized by the operators
where the superscripts in brackets indicate on which Hilbert space the Pauli operator acts. Hence the graph state is the common eigenstate of the family of operators . Alternatively, the graph state can be generated via
where is a controlled gate. Notice that this implies an efficient preparation procedure for all graph states, i.e. knowing the graph state description of a state provides one with a way to prepare the state with at most quadratically many commuting two-qubit gates.
We call two graph states and local unitary equivalent (LU equivalent) if there exist unitaries such that .
An important group of unitaries is the so-called Clifford group. The Clifford group is the set of all qubit unitaries such that , or, in other words, the Clifford group is the normalizer of the Pauli group. An important result which we will use here frequently is that any stabilizer state is local Clifford equivalent (LC equivalent) to a graph state Van den Nest et al. (2004) and that those local Clifford operations can be determined efficiently. This equivalence can be most easily derived in terms of the binary representation of stabilizers of a stabilizer state.
Finally, we denote the four Bell-basis states by
iii.2 Measurement-based quantum computation and the Jamiolkowski isomorphism
In measurement-based quantum computation a specific quantum operation is realized via a sequence of single-qubit measurements on an entangled state. It has been shown in Raussendorf and Briegel (2001) that the 2D cluster state Briegel and Raussendorf (2001) is universal for quantum computation Mantri et al. (2017). The 2D cluster state is a graph state associated with a two-dimensional square lattice.
This enables a correspondence between a quantum operation and a specific sequence of measurements on the 2D cluster, as those measurements implement the quantum operation in a measurement-based way. However, the outcomes of measurements are random, leading to Pauli correction operations. For general quantum circuits, this implies that measurements need to be modified depending on previous measurement results, and quantum information processing takes place in a sequential way. Nevertheless, by using a sufficiently large resource state, one can deterministically implement an arbitrary quantum circuit following this approach. Determinism in measurement-based quantum computation was addressed in more depth in Danos and Kashefi (2006); Browne et al. (2007).
The 2D cluster state is a universal resource and can thus be used to simulate any quantum circuit. However, there can be a large overhead in terms of the number of auxiliary systems. So if one is interested in a specific quantum task, it might be beneficial to consider a special-purpose resource state that can be used to realize a specific operation, and find a state of minimal size or complexity Raussendorf et al. (2003). The Jamiolkowski isomorphism Jamiołkowski (1972) establishes a one-to-one correspondence between completely positive maps and quantum states. More precisely, for every quantum operation there exists a mixed state (which we will refer to as resource state) such that allows one to probabilistically implement the operation via Bell-measurements at the input qubits of the resource state, see Fig. 1.
For unitary operations , or unitary operations followed by projective measurements on some of the output qubits, the corresponding resource state is pure, . We only consider such situations in the following.
In this case, the Jamiolkowski state for the quantum operation is given by
(see also Theorem 1 in Raussendorf et al. (2003)), where denotes the number of input qubits of , and and denote input and output qubits respectively. For an operation, i.e. an operation acting on input qubits and producing output qubits, the resource state is of size . The processing of quantum information now takes place by coupling the qubits of the input states to the input qubits of the resource state via Bell measurements, similarly as in teleportation. Depending on the outcome of the Bell measurements, the output state is then given by a state, where first some Pauli operations (determined by the outcomes of the Bell measurements) act, followed by the application of the desired operation . In general, the Pauli operations and do not commute, resulting in a probabilistic implementation of . In particular, if all outcomes of the Bell measurements are given by (which happens with probability ), the desired operation is applied.
For specific operations , including all Clifford circuits (unitary operations from the Clifford group and Pauli measurements), the Pauli operations can be corrected, and one obtains a deterministic implementation of the map in this case. All circuits we consider throughout the paper are of this form.
It is also straightforward to concatenate different quantum tasks, as the quantum computation is done via Bell-measurements at the input qubits. In particular, one can combine resource states via Bell-measurements on the respective inputs and outputs, see Fig. 2.
Because those Bell-measurements might, in general, yield other outcomes than one would need to deal in sequential implementations of concatenated quantum tasks with Pauli byproduct operators. We emphasize that within the framework proposed here these coupling measurements are done virtually only, thus enabling deterministic generation of resource states for complex quantum tasks. That is, the intermediate output-input qubits only appear virtually, and are not part of the resulting resource state that contains input and output qubits of the general task. This is similar as for qubits which are measured in the Pauli basis in a Clifford circuit: also in this case, the measurement can be done beforehand on qubits in the resource state, thereby leading to a state of reduced size that only contains input and output qubits. The result of Pauli measurements on beforehand measured (virtual) systems is determined by the outcomes of the incoupling Bell measurements see e.g. Zwerger et al. (2012) for entanglement purification. This leads to a possible reinterpretation of the outcomes at the read-in.
In the following we will use at some points of the paper the notation:
We denote by that the quantum operation maps input qubits to output qubits. Furthermore, all resource states we are concerned with here are stabilizer states.
In this section we present a general framework to construct the stabilizers of resource states for concatenated quantum tasks.
iv.1 Basic observation
Suppose we implement the quantum operation via its Jamiolkowski state in a measurement-based way, for example encoding quantum information in a quantum error correcting code. We rewrite as
where and denote the eigenstates of , the input qubit and the system of output qubits. Observe that the states and are not normalized. Furthermore assume there exist two classes of operators, which we call auxiliary operators and , which satisfy the equations:
The operator serves as logical and the operator as a logical operator for the states . We denote by and the sets of all operators satisfying (10) and (11) respectively. Using (10) and (11) we immediately observe that the operators
stabilize , i.e. and for and . So we infer that some stabilizers of the resource state are given by .
According to the following argument the stabilizers can always be brought to this form: If there are at least two generators of the stabilizer group with different Pauli operators on the first qubit we can construct a set of generators that are of form (12) and (13) by picking different elements of the stabilizer group. This can be done by multiplying two of the known stabilizers together to get a new one. Furthermore, all meaningful (i.e. entangled, so they actually can implement an operation) resource states have at least two stabilizers with different Pauli operators as every set of stabilizers is local Clifford equivalent to a set of graph state stabilizers Van den Nest et al. (2004). So, in the graph state picture there is an edge from the first qubit to at least one of the other qubits. That means the stabilizers can always be brought to the required form if the input qubit is entangled with the other qubits.
In the following we will construct the stabilizers of a resource state implementing a concatenated quantum task, such as concatenated quantum error correction or entanglement purification at a logical level. Suppose we want to concatenate the quantum operations and , where the latter acts on each output particles of the former, i.e. the resulting quantum operation is , and both operations are implemented in a measurement-based way
Because we are interested in the construction of the resource state for implementing the composite rather than their sequential execution we assume that the (virtual) coupling Bell-measurements reveal simultaneously the outcome. It is straightforward to define connecting functions
for where . To summarize, the resource state for implementing the quantum operation by denoting is given by
where . We observe that (17) is again of the form (9). In the rest of this section we elaborate on the construction of the auxiliary operators of in terms of the auxiliary operators of . We emphasize that the same techniques apply to the concatenation of quantum operations with a single qubit output.
iv.2 Main results
In the following we denote concrete auxiliary operators of by and , i.e. and . The theorem below relates the auxiliary operator of to auxiliary operators and of .
Theorem 1 (Recurrence relation for ).
Let with , and be the type operator of .
Then there exist , and where such that the auxiliary operator of is given by where , , and for all and .
We sketch the proof of Theorem 1 as follows: First one observes that the new auxiliary operator must satisfy (10), i.e. . The application of to leads to . From the definition of the connecting functions (16), the expansion of in the Pauli basis and the well-known relationship follows the claim. We provide a detailed proof in Appendix A.
Theorem 1 establishes a recurrence relation for auxiliary operators of type for concatenated quantum tasks. We provide a similar theorem for the auxiliary operators of type .
Theorem 2 (Recurrence relation for ).
Let with , and be the type operator of .
Then there exist , and where such that the auxiliary operator of is given by where , , and for all and .
The proof is similar to the proof of Theorem 1: must satisfy (11), i.e. . Hence applying to implies the condition . Again, from the definition of (16), and the expansion of the claim follows. The detailed proof is provided in Appendix A.
Thus, Theorem 1 and 2 show that the sets and are given by
where the families , and , denote the decompositions in the Pauli basis of the initial auxiliary operators in and respectively.
The sets and enable us to provide a complete set of recurrence relations for auxiliary operators via (18) and (19). Recall that the auxiliary operators immediately translate to stabilizers via (12) and (13). In general the sets and will contain too many auxiliary operators depending on and and and . This is rather obvious as all initial auxiliary operators in and enable the application of Theorem 1 and 2. The new stabilizers uniquely describe the resulting state because we can choose a sufficient number (i.e. the number of qubits in the resulting state) of linear independent stabilizers of the form (12) and (13) from the sets and . For the examples we are concerned with in the following sections this follows immediately from the construction of the new stabilizers using Theorem 1 and 2. We address this issue in detail for those cases in Appendix B. Furthermore, we show that this method always provides a full set of independent stabilizers in Appendix C.
We conclude the technical results by providing a theorem concerning the concatenation of single qubit output with single qubit input quantum tasks, crucial for the measurement-based implementation of code switchers and quantum repeaters Briegel et al. (1998). The setting is illustrated in Fig. 4.
Theorem 3 (Coupling of resource states).
Suppose we are given two resource states, and , implementing a single-qubit input and single-qubit output quantum task respectively. Furthermore assume that we are provided the sets of auxiliary operators and , i.e.
where and . Then the stabilizers of the concatenated quantum task, i.e. the resource state obtained by connecting and through a Bell-measurement between and , are given by
First we observe that . Hence we find for the state after connecting and via the Bell-measurement
It follows for and that
which shows that and stabilize as claimed. ∎
iv.3 Dealing with byproduct operators
Finally we have to deal with Pauli byproduct operators due to the Bell-measurements at the read-in of the resource state. Because all quantum operations we are concerned with here belong to the Clifford group, the Pauli byproduct operators at the input qubits propagate through the resource state leading to (possibly different) Pauli operations at the output qubits, see Sec. III.1. Thus it is left to determine how those Pauli errors propagate through a concatenated resource state which we discuss below.
If the resource states and implementing the quantum operations and are concatenated to obtain the resource state implementing , the Pauli byproduct operators due to the read-in Bell-measurements need to be propagated through . This is done as follows: the Pauli byproduct operator at the read-in of translates to an effective Pauli error on the output of . Because and were connected via a virtual Bell-measurement with outcome, the Pauli error at the output of is also the effective Pauli error on the input of . Hence one obtains the resulting Pauli error up to a global phase on the output of by propagating through leading to a deterministic correctable Pauli error at the output of .
In Sec. IV we showed theorems which allow one to construct auxiliary operators for concatenated quantum tasks, and determine the stabizers of the resulting resource states. Now we provide applications of those theorems to different tasks in quantum communication and computation. In particular, we use Theorems 1-3 to construct stabilizers and graph state representation of resource states for measurement-based implementations of multiple rounds of entanglement purification protocols, for quantum error correction including encoding, decoding and syndrome readout for concatenated quantum codes, as well as for code switchers. Finally we construct resource states for the measurement-based implementation of entanglement purification protocols at a logical level.
v.1 Bipartite entanglement purification protocols
Bipartite entanglement purification protocols are used to distill high-fidelity entangled states, ideally a perfect Bell-pair, from a set of noisy copies by means of local operations and classical communication. Entanglement purification is an important primitive in quantum information processing Dür and Briegel (2007), and constitutes a possible way to prepare states with high fidelity, both locally where they have the role of resource states to perform certain operations or tasks Wallnöfer and Dür (2017), as well as in a distributed, non-local way, where entanglement purification is used as a central element in long-distance quantum communication protocols, the quantum repeater Briegel et al. (1998). Entanglement purification protocols exist for all graph states Kruszynska et al. (2006), and the methods we develop here are also applicable to such protocols. That is, one can construct the corresponding resource states to perform multiple rounds of multi-party entanglement purification in a similar way as outlined below for bipartite states.
Several different protocols for bipartite entanglement purification have been proposed Deutsch et al. (1996); Bennett et al. (1996); Dür and Briegel (2007); Aschauer (2004). Here we provide a detailed analysis of a measurement-based implementation of the entanglement purification protocol of Deutsch et al. (1996), which we refer to as DEJMPS protocol in the following. The DEJMPS protocol is a recurrence-type entanglement purification protocol that operates on two noisy pairs and produces probabilistically one pair with improved fidelity. This is achieved by applying at each site, referred to as Alice and Bob, a certain two-qubit operation between the two pairs, followed by a local measurement of the second pair. Depending on the measurement outcome, the remaining pair is either kept or discarded, a step for which two-way classical communication is necessary. This constitutes the basic purification step, where only Clifford operations and Pauli measurements are involved and hence a three-qubit resource state with two input and one output particle can be found for a measurement-based implementation Zwerger et al. (2013, 2012). This basic purification step (also referred to as purification round) may be applied in an iterative manner, using the output pairs of the previous round as input pairs for the next round. One may combine of these basic purification steps, thereby obtaining a protocol that operates on input pairs which produces (probabilistically) one output pair. The corresponding resource state for a measurement-based implementation at each site is of size . This setting has been studied for in Zwerger et al. (2012). Notice that the resource state for performing several purification rounds in one steps contains fewer qubits than the resource states to perform the same task in a sequential fashion. For two rounds, one needs at each site three three-qubit states, i.e. nine qubits, when performing the protocol in a sequential fashion, while a resource state of five qubits (four input plus one output) suffices for the overall task. This reduction in size of resource states seems to be the crucial feature that leads to very high error thresholds in a measurement-based implementation, where a threshold of more than noise per qubit was found Zwerger et al. (2013). In turn, performing multiple rounds in one step leads to a smaller success probability. Obtaining the explicit form of resource states that allow one to perform entanglement purification with such a high robustness and tolerance against noise and imperfections is highly relevant, and will be the subject of the remainder of this section.
We emphasize that our results go beyond the analysis provided in Zwerger et al. (2013, 2012), as we offer a framework for constructing analytically the concrete stabilizers of the resource state for an arbitrary number of rounds of entanglement purification explicitly rather than considering only a small number of entanglement purification rounds. Fig. 5 shows the measurement-based implementation of two rounds of the DEJMPS protocol.
Construction of resource state for several rounds of the DEJMPS protocol
The resource state for one basic purification step of the DEJMPS protocol at Alice’s side Zwerger et al. (2012) is
Hence we have that and . The resource state at Bobs side differs in the sign of .
One easily verifies that the initial auxiliary operators of are , and . Hence and . Thus we obtain via Theorem 1 that and where for both cases. Furthermore Theorem 2 implies , and . To summarize, the sets of auxiliary operators and are given by
The stabilizers of the resource state for performing purification steps of the DEJMPS protocol are, according to (12) and (13), and . One easily verifies that those stabilizers are linear independent for all by construction.
Graph state representation of resource states
In the following we transform the resource states of the DEJMPS protocol via local unitaries to graph states and show that there exists a construction scheme solely based on simple graph rules for concatenating the DEJMPS protocol. For that purpose we observe from (28) and (29) that
where for depend on and . One computes in a similar fashion for that for and . To summarize, the stabilizers of the resource state for rounds of purification are
where and for all . Multiplying (34), (36) and (37) with appropriate choices of and , yields the stabilizer , where we have used that Pauli operators either commute or anti-commute. Thus we obtain as stabilizers of the resource state for rounds of the DEJMPS
where for and for all . From and we deduce
which implies for all . Now we show that the stabilizers (38)-(42) correspond to a graph state up to local Clifford operations.
We observe from (33)-(36) that if the operators in (odd number of purification rounds) or (even number of purification rounds) contain exactly one or respectively then all subsequent will also contain exactly one or . (43) implies that the output particle will be connected to all input particles and that the stabilizers describe a valid graph state.
Furthermore we find from (43) and (38) - (41) that the resource state for rounds of purification is a tensor product corresponding to two disjoint subgraphs, each described by (38)-(39) and (40)-(41). Both subgraphs are of the form
Remove the output qubit from the resource state. This results in two disjoint graphs and .
Duplicate the disjoint graphs of item (1) and label them with and .
Connect and according to the following rules:
Each particle of with each particle of
Each particle of with each particle of
Each particle of with each particle of
Each particle of with each particle of
We denote the resulting graph .
Duplicate the graph of (3) and denote it .
Insert the output qubit and connect it to all particles in and