A Birkhoff connection between quantum circuits and linear classical reversible circuits

Birkhoff's theorem tells how any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. Similar theorems on unitary matrices reveal a connection between quantum circuits and linear classical reversible circuits. It triggers the question whether a quantum computer can be regarded as a superposition of classical reversible computers

The Birkhoff theorem for unitary matrices of arbitrary dimensions

It was shown recently that Birkhoff's theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension.Comment: This manuscript presents a proof for the general unitary birkhoff theorem, conjectured in arXiv:1509.0862

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

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

Some Initial Guidelines for Building Reusable Quantum Oracles

The evolution of quantum hardware is highlighting the need for advances in quantum software engineering that help developers create quantum software with good quality attributes. Specifically, reusability has been traditionally considered an important quality attribute in terms of efficiency of cost and effort. Increasing the reusability of quantum software will help developers create more complex solutions, by reusing simpler components, with better quality attributes, as long as the reused components have also these attributes. This work focuses on the reusability of oracles, a well-known pattern of quantum algorithms that can be used to perform functions used as input by other algorithms. In particular, in this work, we present several guidelines for making reusable quantum oracles. These guidelines include three different levels for oracle reuse: the ideas inspiring the oracle, the function which creates the oracle, and the oracle itself. To demonstrate these guidelines, two different implementations of a range of integers oracle have been built by reusing simpler oracles. The quality of these implementations is evaluated in terms of functionality and quantum circuit depth. Then, we provide an example of documentation following the proposed guidelines for both implementations to foster reuse of the provided oracles. This work aims to be a first point of discussion towards quantum software reusability. Additional work is needed to establish more specific criteria for quantum software reusability.Comment: 10 page

Approximating Fractional Time Quantum Evolution

An algorithm is presented for approximating arbitrary powers of a black box unitary operation, $\mathcal{U}^t$, where $t$ is a real number, and $\mathcal{U}$ is a black box implementing an unknown unitary. The complexity of this algorithm is calculated in terms of the number of calls to the black box, the errors in the approximation, and a certain `gap' parameter. For general $\mathcal{U}$ and large $t$, one should apply $\mathcal{U}$ a total of $\lfloor t \rfloor$ times followed by our procedure for approximating the fractional power $\mathcal{U}^{t-\lfloor t \rfloor}$. An example is also given where for large integers $t$ this method is more efficient than direct application of $t$ copies of $\mathcal{U}$. Further applications and related algorithms are also discussed.Comment: 13 pages, 2 figure

Operating with Quantum Integers: an Efficient 'Multiples of' Oracle

Quantum algorithms are a very promising field. However, creating and manipulating these kind of algorithms is a very complex task, specially for software engineers used to work at higher abstraction levels. The work presented here is part of a broader research focused on providing operations of a higher abstraction level to manipulate integers codified as a superposition. These operations are designed to be composable and efficient, so quantum software developers can reuse them to create more complex solutions. Specifically, in this paper we present a 'multiples of' operation. To validate this operation we show several examples of quantum circuits and their simulations, including its composition possibilities. A theoretical analysis proves that both the complexity of the required classical calculations and the depth of the circuit scale linearly with the number of qubits. Hence, the 'multiples of' oracle is efficient in terms of complexity and depth. Finally, an empirical study of the circuit depth is conducted to further reinforce the theoretical analysis.Comment: 19 pages, 18 figures, preprint submitted to SummerSOC 202