Optimal purification of thermal graph states

In this paper, a purification protocol is presented and its performance is proven to be optimal when applied to a particular subset of graph states that are subject to local Z-noise. Such mixed states can be produced by bringing a system into thermal equilibrium, when it is described by a Hamiltonian which has a particular graph state as its unique ground state. From this protocol, we derive the exact value of the critical temperature above which purification is impossible, as well as the related optimal purification rates. A possible simulation of graph Hamiltonians is proposed, which requires only bipartite interactions and local magnetic fields, enabling the tuning of the system temperature.

Introduction. Any quantum technological implementation is plagued by environmental noise. The possibility to purify quantum states, or to use error correcting algorithms to stabilize quantum operations, is therefore a necessary step towards reaping the benefits of quantum technologies. Much attention has been focused lately on the purification [1,2,3,4,5,6,7] of a large class of multipartite states called graph states [8]. After the initial restriction to two-colourable graph states [4,5], the ideas have been extended to all graph states [7], and generalized to other stabilizer states [9]. The variety of different protocols trade off between a large tolerance to noise [4,5] and the rate of purification [6]. Graph states have proven important for realizing a variety of quantum information tasks such as performing quantum computation [10], quantum communication [11] and as a means for efficiently approximating other quantum states, such as the ground states of strongly correlated systems [12]. While some bounds have previously been proven on the ability to purify multipartite states [13,14], optimal results only exist for two-qubit states [15,16].
Here we concentrate on the purification of graph states that are subject to the physically motivated independent Z-noise. From the technological perspective, large-scale quantum computation is still too difficult to implement. However, graph states can be made and manipulated in the laboratory, e.g. by controlled collisions of alkali atoms trapped in optical lattices [17,18]. To enable the controlled collisions, it is necessary to employ two magnetically sensitive hyperfine levels of the atoms to form a qubit. This sensitivity means that the states are subject to decoherence from stray magnetic fields, in addition to any uncontrolled collisions that may occur. These errors are described by Z-errors, and as experiments improve, one can expect them to become more localised. An alternative approach to preparing a graph state involves implementing a Hamiltonian, known as a graph Hamil-tonian, which has the desired state as its ground state. As it is impossible to cool the system to absolute zero, the resulting equilibrium state will always be a thermal state. For graph Hamiltonians, the thermal noise corresponds to local Z-noise on each qubit.
In this paper, we consider a certain purification protocol applied to arbitrary graph states in the presence of independent Z-errors. Whilst this multipartite protocol is not novel or sophisticated [1,13,19], it has the advantage of being analytically tractable. Most importantly, we prove optimality of this protocol for these types of error, both in the sense of the level of noise that can be tolerated as well as the scaling of the purification rate, for a specific subset of states that includes the cluster and Greenberger Horne Zeilinger (GHZ) states. For up to seven qubits, this subset can be shown to be isomorphic to arbitrary graphs under local operations. We propose a method for simulating the graph Hamiltonians that is comprised of only two-body collisions and local magnetic fields, even though the resulting Hamiltonian has at least 3-qubit interactions [20,21]. The critical purification temperature related to these models can easily be made to lie above the typical temperatures given, e.g. from optical lattice realizations of such Hamiltonians.
Graph Hamiltonians and Graph States. Let us introduce a graph G, which is defined by a set of vertices V G , and a set of edges E G , describing the connections between the vertices. To each vertex of this graph, we attach a spin-1 2 particle (qubit), and define a graph state to be the ground-state of the following Hamiltonian, The B i 's are the coupling strengths, which we henceforth take to have equal magnitude, B i = B, and we assume B > 0. The interaction terms K i commute with each other, [K i , K j ] = 0, and hence each term individually stabilizes the eigenstates of H. A local Pauli Z-rotation on qubit i, Z i , commutes with all the K j where j = i, and anti-commutes with K i . Hence, the excitations of the Hamiltonian are given by local Z-rotations applied to the ground state.
A constructive way to produce graph states is found by close analogy with the cluster states [8,21]. We can produce a general graph state by creating the |+ = (|0 + |1 )/ √ 2 state on each vertex of G, and applying controlled-phase gates along the edges of the graph. The action of measurements on the cluster state also applies to the graph states, such that when a Z-measurement is performed, all nearest-neighbour bonds are severed. This means that graph states can be cut into sections using Z-measurements, and can be combined together using controlled-phase gates.
The key to what follows is the realization that Z-errors commute with both of these operations, and consequently remain as Z-errors.
Examples of graph states that are of particular interest in quantum computation and communication are the cluster [22] and GHZ [23] states. The first state corresponds to a graph that is given by a d-dimensional cube, while the latter state corresponds to a graph that is locally equivalent to a single qubit connected with all other qubits.
Purification of Thermal Graph States. The thermal state of the Hamiltonian in Eqn.
where we set the Boltzmann constant k B equal to unity. This density matrix can be written in terms of local Z- is the probability of a Z-error occurring at a certain site due to the non-zero temperature T . The graph state |ψ is the unique ground state of Hamiltonian (1). Our aim is to purify towards the state |ψ using many copies of ρ(p).
We consider that each vertex of the graph is controlled by a different party and that operations such as measurements and controlled-phase gates are only allowed locally, but involving the many copies. This restriction corresponds to the scenario of quantum repeaters [24], where the different parties are physically separated, and also serves to illustrate the entanglement properties of the system. We do not envisage implementation of the protocol in other scenarios, since it is generally cheaper in terms of resources to create the state directly -our aim is to prove optimality, providing a benchmark for all other protocols.
The purification procedure consists of breaking down the graph states into smaller blocks, purifying them, and then recombining them. We refer to it as the Divide and Rebuild Purification Protocol (DRPP). The smaller blocks that we choose to use are two-qubit states, for which there are analytic purification results, returning maximally entangled states. As already specified, the splitting of the graph state is readily achieved with Zmeasurements. This leaves us with a two-qubit mixed state ρ 2 (p), which can be purified to a maximally entangled state |ψ 2 provided Once we have generated a maximally entangled state for each of the edges of the graph, each local party (i) holds a number of qubits equal to the number of nearestneighbours |E i G |. These can be reduced to a single qubit that contains all these links by applying controlled-phase gates and performing X-measurements, as described in Fig. 1.
The condition for purification of the two-qubit state between neighbouring sites is readily found to be This is known to be necessary and sufficient for purifying states that are Bell diagonal, like the states considered here [15,16,24]. We can hence calculate that the maximum temperature at which the DRPP can purify graph states is given by Optimality and Rates. While the DRPP can be applied to any graph state (with minor modifications), we can prove its optimality for a specific subset of graphs. For clarity, we shall restrict to only cluster states (of arbitrary dimension) and GHZ states, all of which fall into this classification. The optimality that we prove is with respect to both the level of noise that can be tolerated such that purification is still possible, and with respect to the number of copies of the initial, noisy, state required to form a single pure copy. In particular, one can prove that states with higher levels of noise than the one dictated by Eqn. (4) can never be purified with any protocol. We do this by considering a purification protocol for a two-qubit state. The two parties sharing this state are allowed to introduce extra qubits and the local operations that they apply subsume multipartite considerations for the additional qubits.
If two parties, Alice and Bob, hold several copies of the noisy two-qubit state ρ 2 (p) [25], they can locally recreate the initial thermal state, ρ(p). For linear graphs (Fig. 2), this simply corresponds to Alice and Bob locally creating their own thermal cluster states, and connecting them to ρ 2 (p) with controlled-phase gates. In the case of a more general graph, we require some additional connections between Alice and Bob. These are achieved by using multiple copies of ρ 2 (p) (Fig. 3(b)). At this stage, we assume that purification of ρ(p) is possible, yielding |ψ . From there, Alice and Bob can measure out all the qubits that they added, leaving a pure two-qubit state. Hence, if ρ(p) can be purified, ρ 2 (p) can always be purified. However, we know that ρ 2 (p) cannot be purified if (1 − p) 2 ≤ 1 2 . Hence, under this condition, our assumption must be false i.e. the multipartite state ρ(p) cannot be purified, whatever the protocol. The DRPP saturates this bound and hence is optimal for independent Z-noise. While analysis of the protocol of [4,5] is a difficult problem, numerical results indicate that it also saturates the bound, and hence is optimal for this type of noise.
The rate of purification, R ψ , of the graph state |ψ can be calculated in terms of the yield of a Bell state, R 2 . We take the standard definition of rate, Copies of |ψ produced Copies of ρ consumed If we can purify ρ into |ψ ψ| at a rate R ψ (i.e. we require 1/R ψ copies of ρ to create |ψ ), then we can create a Bell state between any linked pair just by performing Zmeasurements on excess qubits. There could be a more efficient way to generate this Bell pair, requiring fewer copies, so R ψ ≤ R 2 .
Similarly, if we can purify Bell pairs, then we can generate |ψ . For that we take a Bell pair between each nearest-neighbour and the local parties perform the reconstruction as specified by the DRPP (e.g. controlledphase between local states and X-measurements). Most of the Bell pairs can be purified in parallel -we only require N geo copies of ρ to purify enough copies, where the geometric factor N geo depends only on the local degree of the graph, |E i G |, and is otherwise independent of the number of qubits in the graph. This is because the Zmeasurements commute with the errors, and divide the state ρ into separate blocks. As a result, R ψ ≥ R2 Ngeo . Combining the two results, For d-dimensional cluster states, we can readily evaluate N geo , since we must measure out all qubits connected directly to the pair that we are interested in isolating. Starting from the edge of the lattice, this uses up 3d − 2 qubits (we include in this number some single qubits that, while we might not intend to measure them, become isolated), plus the two for the state ρ 2 . If our lattice extends to N qubits in each direction, then we can generate on average (N −1)N d−1 /(3d) copies of ρ 2 from a single copy of ρ. We need d(N − 1)N d−1 different copies of ρ 2 , and hence N geo = 3d 2 . As demonstrated in Fig. 4, this corresponds to N geo = 3 for d = 1. The resulting rate is independent of the number of qubits in the system, only depending on R 2 , which is optimal up to a small numerical factor. For N -qubit GHZ states, N geo = N − 1.
Physical Implementation. A significant achievement in recent quantum engineering experiments is the construction of cluster states with optical lattices [17]. They are produced with a single operational step independent of the size of the system. It is natural to consider this setup for studying the purification properties of thermal cluster states. This requires the implementation of both the cluster Hamiltonian and a purification protocol in a physical setup. In the following, we will present a simple way of simulating the Hamiltonian (1) for cluster states, using proven experimental techniques in optical lattices.
When an entangled state decoheres, there is a characteristic lifetime that determines when the state becomes separable. On the other hand, if it is possible to obtain an interaction described by the Hamiltonian that has this entangled state as a ground state, then provided the energy gap is large enough in comparison to the decoherence rate, entanglement can survive indefinitely in the system, e.g. in the form of purifiable mixed states, as we have already seen.
Simulating Hamiltonian (1) for a general graph is a relatively straightforward task. The method we adopt here consists of a unitary operation, U † G [U G ], applied before [after] the evolution with respect to a local Hamiltonian. This evolution is generated by applying a uniform magnetic field in the x-direction. When U G corresponds to controlled phase gates between all pairs of qubits connected in the graph, then it is easy to show that the resulting effective Hamiltonian is of the form (1) [26]. The spectrum of the Hamiltonian corresponding to the magnetic field B i X i is the same as that of the Hamiltonian H, as they are related by an isospectral transformation. Hence, the generated thermal state directly corresponds to the one of H.
In optical lattices, the unitary U G can be realized in cubic lattices by controlled collisions between nearestneighbours. This is precisely the operation which is experimentally employed in [17] for the generation of a cluster state. Meanwhile, the local magnetic fields are implemented by globally applied Raman transitions between the hyperfine states that encodes a qubit. Thus, the realization of the graph Hamiltonian is readily achieved. The stationary state of this system is the thermal state of Eq. (2). Further, this system gives us the ability to vary the temperature. In previous experiments [29] the recorded temperature after performing optical cool-ing was given by the relation T ≈ 0.1U 0 , where U 0 is light shift potential created by the optical lattice. From the value of the critical temperature given in (5), one deduces that by employing moderate local magnetic fields with amplitude B 0.1U 0 , one can bring the system into the purifiable regime.
Once we are able to implement a purification protocol in optical lattices, whether it be the DRPP or any other, this provides us with the perfect test-bed to probe the maximum temperature that still allows purification, and verify the critical temperature given in Eqn. 5. The potential implementation of such schemes has been described extensively in, for example, [5]. The main drawback is the requirement of local addressability, which can be circumvented with the help of superlattices [18,27,28] by breaking of the translational invariance of the lattice.
Conclusions. Here, a purification protocol has been proven to be optimal when applied to a sub-class of graph states, including the cluster and GHZ states, subject to Z-errors. Although we have restricted ourselves to this particular form of noise in this paper, we emphasize that thermal states arising from graph Hamiltonians are precisely of this form, rendering the considered types of states interesting also from an experimental and practical point of view.
While the optimality proof cannot be applied to certain types of graphs (e.g. the icosahedron), and we are aware of examples of noise (e.g. local or global white noise) where the proposed protocol is not optimal, several extensions of our results are possible. Two of the authors have shown [30] that the optimality proof can be extended to a wider class of states and to different forms of noise. For instance, it can be shown that all graph states of up to seven qubits can be brought by local unitary operations to a form where the optimality proof can be applied [8]. Thus, all these graphs have a critical temperature given by Eqn. (5). In addition, it shown in [30] how the optimality proof presented in this paper can be extended to other types of noise, providing an upper-bound to the error probability that can be purified. Further extensions taking into account non-graph states and noisy local operations during the purification protocol will also be examined.