Stability of global entanglement in thermal states of spin chains

We investigate the entanglement properties of a one dimensional chain of spin qubits coupled via nearest neighbor interactions. The entanglement measure used is the n-concurrence, which is distinct from other measures on spin chains such as bipartite entanglement in that it can quantify "global" entanglement across the spin chain. Specifically, it computes the overlap of a quantum state with its time-reversed state. As such this measure is well suited to study ground states of spin chain Hamiltonians that are intrinsically time reversal symmetric. We study the robustness of n-concurrence of ground states when the interaction is subject to a time reversal antisymmetric magnetic field perturbation. The n-concurrence in the ground state of the isotropic XX model is computed and it is shown that there is a critical magnetic field strength at which the entanglement experiences a jump discontinuity from the maximum value to zero. The n-concurrence for thermal mixed states is derived and a threshold temperature is computed below which the system has non zero entanglement.Comment: 13 pages, 3 figures. v.2 includes minor corrections and an added section treating the quantum XX model with open boundarie

Synthesis of Quantum Logic Circuits

We discuss efficient quantum logic circuits which perform two tasks: (i) implementing generic quantum computations and (ii) initializing quantum registers. In contrast to conventional computing, the latter task is nontrivial because the state-space of an n-qubit register is not finite and contains exponential superpositions of classical bit strings. Our proposed circuits are asymptotically optimal for respective tasks and improve published results by at least a factor of two. The circuits for generic quantum computation constructed by our algorithms are the most efficient known today in terms of the number of expensive gates (quantum controlled-NOTs). They are based on an analogue of the Shannon decomposition of Boolean functions and a new circuit block, quantum multiplexor, that generalizes several known constructions. A theoretical lower bound implies that our circuits cannot be improved by more than a factor of two. We additionally show how to accommodate the severe architectural limitation of using only nearest-neighbor gates that is representative of current implementation technologies. This increases the number of gates by almost an order of magnitude, but preserves the asymptotic optimality of gate counts.Comment: 18 pages; v5 fixes minor bugs; v4 is a complete rewrite of v3, with 6x more content, a theory of quantum multiplexors and Quantum Shannon Decomposition. A key result on generic circuit synthesis has been improved to ~23/48*4^n CNOTs for n qubit

Asymptotically Optimal Quantum Circuits for d-level Systems

As a qubit is a two-level quantum system whose state space is spanned by |0>, |1>, so a qudit is a d-level quantum system whose state space is spanned by |0>,...,|d-1>. Quantum computation has stimulated much recent interest in algorithms factoring unitary evolutions of an n-qubit state space into component two-particle unitary evolutions. In the absence of symmetry, Shende, Markov and Bullock use Sard's theorem to prove that at least C 4^n two-qubit unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa (VMS) use the QR matrix factorization and Gray codes in an optimal order construction involving two-particle evolutions. In this work, we note that Sard's theorem demands C d^{2n} two-qudit unitary evolutions to construct a generic (symmetry-less) n-qudit evolution. However, the VMS result applied to virtual-qubits only recovers optimal order in the case that d is a power of two. We further construct a QR decomposition for d-multi-level quantum logics, proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing the complexity question for all d-level systems (d finite.) Gray codes are not required, and the optimal Theta(d^{2n}) asymptotic also applies to gate libraries where two-qudit interactions are restricted by a choice of certain architectures.Comment: 18 pages, 5 figures (very detailed.) MatLab files for factoring qudit unitary into gates in MATLAB directory of source arxiv format. v2: minor change

Parallelism for Quantum Computation with Qudits

Robust quantum computation with d-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high fidelity two-qudit gates. We first describe how to implement parallel single qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the atomic alkalies $^{87}$Rb and $^{133}$Cs. Second, we provide a protocol for implementing parallelized non-local two-qudit gates using the assistance of entangled qubit pairs. Because the entangled qubits can be prepared non-deterministically, this offers the possibility of high fidelity two-qudit gates.Comment: 9 pages, 3 figure

Minimal Universal Two-qubit Quantum Circuits

We give quantum circuits that simulate an arbitrary two-qubit unitary operator up to global phase. For several quantum gate libraries we prove that gate counts are optimal in worst and average cases. Our lower and upper bounds compare favorably to previously published results. Temporary storage is not used because it tends to be expensive in physical implementations. For each gate library, best gate counts can be achieved by a single universal circuit. To compute gate parameters in universal circuits, we only use closed-form algebraic expressions, and in particular do not rely on matrix exponentials. Our algorithm has been coded in C++.Comment: 8 pages, 2 tables and 4 figures. v3 adds a discussion of asymetry between Rx, Ry and Rz gates and describes a subtle circuit design problem arising when Ry gates are not available. v2 sharpens one of the loose bounds in v1. Proof techniques in v2 are noticeably revamped: they now rely less on circuit identities and more on directly-computed invariants of two-qubit operators. This makes proofs more constructive and easier to interpret as algorithm