Quantum Circuits and Spin(3n) Groups

All quantum gates with one and two qubits may be described by elements of $Spin$ groups due to isomorphisms $Spin(3) \simeq SU(2)$ and $Spin(6) \simeq SU(4)$. However, the group of $n$-qubit gates $SU(2^n)$ for $n > 2$ has bigger dimension than $Spin(3n)$. A quantum circuit with one- and two-qubit gates may be used for construction of arbitrary unitary transformation $SU(2^n)$. Analogously, the `$Spin(3n)$ circuits' are introduced in this work as products of elements associated with one- and two-qubit gates with respect to the above-mentioned isomorphisms. The matrix tensor product implementation of the $Spin(3n)$ group together with relevant models by usual quantum circuits with $2n$ qubits are investigated in such a framework. A certain resemblance with well-known sets of non-universal quantum gates e.g., matchgates, noninteracting-fermion quantum circuits) related with $Spin(2n)$ may be found in presented approach. Finally, a possibility of the classical simulation of such circuits in polynomial time is discussed.Comment: v1. REVTeX 4-1, 2 columns, 10 pages, no figures, v3. extended, LaTeX2e, 1 col., 23+2 pages, v4. typos, accepted for publicatio

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

Clifford algebras, Spin groups and qubit trees

Representations of Spin groups and Clifford algebras derived from structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deleting of superfluous branches. Usual Jordan-Wigner construction also may be formally obtained in such approach by bringing the process up to trivial qubit chain ("trunk"). The methods can be also used for effective simulations of some quantum circuits corresponding to the binary tree structure. Modeling of more general qubit trees and relation with mapping used in Bravyi-Kitaev transformation are also briefly outlined.Comment: LaTeX 12pt, 36 pages, 9 figures; v5: updated, with two new appendices. Comments are welcom

Quantum robustness and phase transitions of the 3D Toric Code in a field

We study the robustness of 3D intrinsic topogical order under external perturbations by investigating the paradigmatic microscopic model, the 3D toric code in an external magnetic field. Exact dualities as well as variational calculations reveal a ground-state phase diagram with first and second-order quantum phase transitions. The variational approach can be applied without further approximations only for certain field directions. In the general field case, an approximative scheme based on an expansion of the variational energy in orders of the variational parameters is developed. For the breakdown of the 3D intrinsic topological order, it is found that the (im-)mobility of the quasiparticle excitations is crucial in contrast to their fractional statistics