Scaling a unitary matrix

The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called 'scaled matrix' which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all line sums equal to 1. We conjecture that a similar procedure exists, which allows, starting from an arbitrary unitary matrix, to find a scaled matrix which is unitary and has all line sums equal to 1. The existence of such algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.Comment: A proof of the conjecture is now provided by Idel & Wolf (http://arxiv.org/abs/1408.5728

The Birkhoff theorem for unitary matrices of prime-power dimension

The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity. If the dimension~$n$ of the unitary matrix equals a power of a prime $p$, i.e.\ if $n=p^w$, then the Birkhoff decomposition does not need all $n!$ possible permutation matrices, as the epicirculant permutation matrices suffice. This group of permutation matrices is isomorphic to the general affine group GA($w,p$) of order only $p^w(p^w-1)(p^w-p)...(p^w-p^{w-1}) \ll \left( p^w \right)!$

The decomposition of an arbitrary 2w×2w2^w\times 2^w unitary matrix into signed permutation matrices

Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of~2 (say $2^w$) deserve special attention, as they represent quantum qubit circuits. We investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix. It turns out to be a matrix group isomorphic to the extraspecial group {\bf E}$_{2^{2w+1}}^+$ of order $2^{2w+1}$. An associated projective group of order $2^{2w}$ equally suffices.Comment: 4th paper in a series of Birkhoff decompositions for unitary matrices [(1) arXiv:1509.08626; (2) arXiv:1606.08642; (3) arXiv:1812.08833

Logics between classical reversible logic and quantum logic

Classical reversible logic and quantum computing share the common feature that all computations are reversible, each result of a computation can be brought back to the initial state without loss of information

The Birkhoff theorem for unitary matrices of prime dimension

The Birkhoff's theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. It can be proven that a similar theorem holds for unitary matrices with equal line sums for prime dimensions

The block-ZXZ synthesis of an arbitrary quantum circuit

Given an arbitrary $2^w \times 2^w$ unitary matrix $U$, a powerful matrix decomposition can be applied, leading to four different syntheses of a $w$-qubit quantum circuit performing the unitary transformation. The demonstration is based on a recent theorem by F\"uhr and Rzeszotnik, generalizing the scaling of single-bit unitary gates ($w=1$) to gates with arbitrary value of~$w$. The synthesized circuit consists of controlled 1-qubit gates, such as NEGATOR gates and PHASOR gates. Interestingly, the approach reduces to a known synthesis method for classical logic circuits consisting of controlled NOT gates, in the case that $U$ is a permutation matrix.Comment: Improved (non-sinkhorn) algorithm to obtain the proposed circui

Read-Green resonances in a topological superconductor coupled to a bath

We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modelled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchange, which can be observed through resonances in low-energy and -momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.Comment: 7 pages, 4 figure

On two subgroups of U(n), important for quantum computing

ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n X n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n X n unitary diagonal matrices with first entry equal to 1