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Quantum Circuits for Maximally Entangled States

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Quantum Circuits for Maximally Entangled States Quantum circuits for maximally entangled states Alba Cervera-Lierta,1, 2, ∗ José Ignacio Latorre,2, 3, 4 and Dardo Goyeneche5 1Barcelona Supercomputing Center (BSC). 2Dept. Física Quàntica i Astrofísica, Universitat de Barcelona, Barcelona, Spain. 3Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands. 4Center for Quantum Technologies, National University of Singapore, Singapore. 5Dept. Física, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile. (Dated: May 16, 2019) We design a series of quantum circuits that generate absolute maximally entangled (AME) states to benchmark a quantum computer. A relation between graph states and AME states can be exploited to optimize the structure of the circuits and minimize their depth. Furthermore, we find that most of the provided circuits obey majorization relations for every partition of the system and every step of the algorithm. The main goal of the work consists in testing efficiency of quantum computers when requiring the maximal amount of genuine multipartite entanglement allowed by quantum mechanics, which can be used to efficiently implement multipartite quantum protocols. I. INTRODUCTION nique, e.g. tensor networks [9], is considered. We believe that item (i) is fully doable with the current state of the There is a need to set up a thorough benchmarking art of quantum computers, at least for a small number strategy for quantum computers. Devices that operate of qubits. On the other hand, item (ii) is much more in very different platforms are often characterized by the challenging, as classical computers efficiently work with number of qubits they offer, their coherent time and the a large number of bits. fidelities of one- and two-qubit gates. This is somewhat Quantum correlations depend on the delicate balance misleading as the performance of circuits are far below of the coefficients of the wave function. It is natural to the expected when the amount of genuine multipartite expect that quantum computers will have to be very re- entanglement contained in state is relatively high. fined to achieve such a good description of multipartite There exist several figures of merit that try to quantify correlations along the successive action of gates. Entan- the success performance of a quantum device. Methods glement is at the heart of quantum efficiency [10]. Again, such as randomized benchmarking [1], state and process if a quantum computer is not able to generate faithful tomography [2] and gateset tomography [3, 4] are used to large entanglement, it will remain inefficient. quantify gate fidelities. However, they are only useful for A fundamental factor in quantum computing is the few-qubit experiments and fail when used to evaluate the ability to generate large entangled states, such as area performance of greater circuits [5, 6]. In that sense, IBM law violating states [11]. However, such ability has to proposed a metric to be used in arbitrary large quantum be accomplished by a sufficiently large coherence time circuits called quantum volume [7]. This figure takes into for such multipartite maximally entangled states. Note account several circuit variables like number of qubits, that GHZ-like states are highly entangled and useful to connectivity and gate fidelities. The core of the protocol test violate qubit Bell inequalities [12], but even more is the construction of arbitrary circuits formed by one- entangled are the Absolute Maximally Entangled (AME) and two-qubit gates that are complex enough to repro- states [13–15], which are maximally entangled in every duce a generic n-qubit operation. One can expect to bipartition of the system. generate high entanglement in this kind of circuits. Even Following this line of thought, along this work we ex- though, we should certify that this amount of entangle- plore techniques to efficiently construct quantum circuits ment will be large enough to perform some specific tasks for genuinelly multipartite maximally entangled states. arXiv:1904.07955v2 [quant-ph] 15 May 2019 that, precisely, demand high entanglement. A further Some preliminar results reflect the difficulty to deal with relevant reference concerns to volumetric framework for quantum computers. For instance, the amount of Bell quantum computer benchmarks [8]. inequality violation rapidly decreases with the number Two reasonable ways to tests for quantum computers of qubits considered [16].Also, the exact simulation of an are the following: (i) implement a protocol based on max- analytical solvable model in a quantum computer signif- imally entangled states, (ii) solve a problem that is hard icantly differs from the expected values, even when con- for a classical computer. These two ways are linked by sidering four qubits and less than thirty basic quantum the fact that quantum advantage requires large amounts gates [17]. These examples illustrate the difference be- of entanglement, so that classical computers are unable tween gate fidelity and circuit fidelity, being the second to carry the demanded task even when a sofisticated tech- one much harder to improve. We shall present quantum circuits required to build multipartite maximally entangled states. This pro- posal differs from bosonic sampling method, where large ∗ [email protected] amounts of entanglement are faithfully reproduced by 2 classical simulations [18]. AME states find applications in of multipartite entanglement. That is, when the average multipartite teleportation [19] and quantum secret shar- Von Neumann entropy S(ρ) = Tr[ρ log ρ], taken over all ing [15, 19]. Along our work, maximally entangled states reductions to bn=2c parties, achieves the global maximum will be exclusively used to test the strength of multipar- value S(ρ) = bn=2c, where logarithm is taken in basis d. tite correlations in quantum computers. Another possi- For instance, Bell states and GHZ states are AME states ble use, not discussed by us, could be generate quantum for bipartite and three partite systems, respectively, for advantage with respect to a classical computer or, ideally, any number of internal levels d. to achieve quantum supremacy. We describe a benchmark suit of quantum circuits, where each one should deliver an AME state. The cir- cuits provided were designed to minimize the number of The existence of AME states for n qudit systems com- required gates under the presence of restricted connectiv- posed by d levels each, denoted AME(n; d), is a hard ity of qubits. Some of them consider individual systems open problem in general [20]. This problem is fully solved composed by more than two internal levels each, which for any number of qubits: an AME(n; 2) exists only for sometimes can be effectively reduced to qubits. In gen- n = 2; 3; 5; 6 [21–24]. Among all existing AME states, eral, we consider simple and compact circuits, illustrating there is one special class composed by minimal support the way in which multipartite entanglement is generate states. These states are defined as follows: an AME(n; d) step by step along the circuit. We have also pay atten- state has minimal support if it can be written as the tion to a simple criteria of majorization of the entropy superposition of dbn=2c fully separable orthogonal pure of reductions, which basically implies that multipartite states. Here, we consider superposition at the level of entanglement, quantified by the averaged entropy of re- vectors, in such a way that the linear combination of pure ductions, monotonically increase for all partitions. states always produce another pure state. For example, This work is organized as follows: In Section II, we generalized Bell states for two-qudit systems and gener- review the basic properties of AME states and show ex- alized GHZ states for three-qudit systems have minimal plicit examples. In Section III, we present the quantum support. It is simple to show that all coefficients of ev- circuits that generate AME states by using the proper- ery AME state having minimal support can be chosen to ties of graph states. We also propose the simulation of be identically equal to d−bn=2c=2, i.e. identical positive AME states having local dimension larger than 2 by us- numbers. By contrast, AME states having non-minimal ing qubits instead of qudits. In Section IV, we analyze support require to be composed by non-trivial phases in the entanglement majorization criteria in the proposed its entries in order to have all reduced density matrices circuits and find further optimal circuits for experimen- proportional to the identity. In other words, non-minimal tal implementation, by imposing a majorization arrow in support AME states require destructive interference. terms of entanglement. In Section V, we implement GHZ and AME states for five qubit systems in IBM quantum computers, quantify the state preparation quality by test- ing maximal violation of suitably chosen Bell inequalities. AME states connect to several mathematical tools. It Finally, in Section VI we discuss and summarize the main is known that AME states composed by n parties and results of the paper. having minimal support, e.g. AME(2,2), AME(3,2) and AME(4,3), are one-to-one related to a special class of maximum distance separable (MDS) codes [25], index II. REVIEW OF AME STATES unity orthogonal arrays [26], permutation multi-unitary matrices when n is even [14] and to a set of m = n−bn=2c mutually orthogonal Latin hypercubes of size d defined The study of AME states have become an intensive in dimension bn=2c [27]. On the other hand, AME states area of research along the last years due to both the- inequivalent to minimal support states, e.g. AME(5,2) oretical foundations and practical applications. In this or AME(6,2), are equivalent to quantum error correc- section we briefly review the current state of the art of tion codes [21], quantum orthogonal arrays [27], non- the field.
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