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

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 $p^w(p^w-1)(p^w-p)...(p^w-p^{w-1}) \ll \left( p^w \right)!$

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 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

A technology based complexity model for reversible Cuccaro ripple-carry adder

Reversible logic provides an alternative to classical computing, that may overcome many of the power dissipation problems. The paper presents a simple complexity model, from the study of a cascade of Cuccaro adders processed in standard 0.35 micrometer CMOS technology

Test of a majority-based reversible (quantum) 4 bits ripple-carry adder in adiabatic calculation

Quantum computing and circuits are of growing interest and so is reversible logic as it plays an important role in the synthesis of circuits dedicated to quantum computation. Moreover, reversible logic provides an alternative to classical computing machines, that may overcome many of the power dissipation problems in the near future. As a proof of concept we designed and tested a reversible 4 bits ripple-carry adder based on a do-spy-undo structure. This paper presents some performances obtained with such a chip processed in standard 0.35 μm CMOS technology and used in real reversible calculation (in this study, computations are performed in both directions such that addition and subtraction are made reversibly with the same chip). We also discuss the superiority of using adiabatic signals over classical rectangular pulses when using dual-line pass-transistor logic gates. Adiabatic signals allow the signal energy stored on the various capacitances of the circuit to be redistributed rather than being dissipated as heat. Finally, we show that adiabatic signals allow to avoid calculation errors introduced by the use of conventional rectangular pulses and allow to drastically reduce the number of pulse resynchronization in large circuits. Index Terms—reversible computation, design, implementation, pass-transistor logic, ripple-carry adder, Spectre simulation, quantum computation, adiabatic signal, test and measuremen