Quantum Circuits for Incompletely Specified Two-Qubit Operators

While the question ``how many CNOT gates are needed to simulate an arbitrary two-qubit operator'' has been conclusively answered -- three are necessary and sufficient -- previous work on this topic assumes that one wants to simulate a given unitary operator up to global phase. However, in many practical cases additional degrees of freedom are allowed. For example, if the computation is to be followed by a given projective measurement, many dissimilar operators achieve the same output distributions on all input states. Alternatively, if it is known that the input state is |0>, the action of the given operator on all orthogonal states is immaterial. In such cases, we say that the unitary operator is incompletely specified; in this work, we take up the practical challenge of satisfying a given specification with the smallest possible circuit. In particular, we identify cases in which such operators can be implemented using fewer quantum gates than are required for generic completely specified operators.Comment: 15 page

An Arbitrary Two-qubit Computation In 23 Elementary Gates

Quantum circuits currently constitute a dominant model for quantum computation. Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and as small a number of elementary quantum gates as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worst-case upper bound of 23 elementary gates, of which at most 4 (CNOT) entail multi-qubit interactions. Our analysis shows that synthesis algorithms suggested in previous work, although more general, entail much larger quantum circuits than ours in the special case of two qubits. One such algorithm has a worst case of 61 gates of which 18 may be CNOTs. Our techniques rely on the KAK decomposition from Lie theory as well as the polar and spectral (symmetric Shur) matrix decompositions from numerical analysis and operator theory. They are related to the canonical decomposition of a two-qubit gate with respect to the ``magic basis'' of phase-shifted Bell states, published previously. We further extend this decomposition in terms of elementary gates for quantum computation.Comment: 18 pages, 7 figures. Version 2 gives correct credits for the GQC "quantum compiler". Version 3 adds justification for our choice of elementary gates and adds a comparison with classical library-less logic synthesis. It adds acknowledgements and a new reference, adds full details about the 8-gate decomposition of topC-V and stealthily fixes several minor inaccuracies. NOTE: Using a new technique, we recently improved the lower bound to 18 gates and (tada!) found a circuit decomposition that requires 18 gates or less. This work will appear as a separate manuscrip

Optimal Quantum Circuits for General Two-Qubit Gates

In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requires at most 2 CNOTs and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, y-rotation, z-rotation, and phase gates.Comment: 6 pages, 8 figures, new title, final journal versio

Balanced Tripartite Entanglement, the Alternating Group A4 and the Lie Algebra sl(3,C)⊕u(1)sl(3,C) \oplus u(1)

We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group $W(E_8)$ in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group dipoles and the enactment of Weyl/Coxeter group $W(E_8)$ by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar "condensation" of $W(E_8)$ into the four-letter alternating group $A_4$, obtained from a chain of maximal subgroups. Group $A_4$ is realized from two B-type generators and found to correspond to the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$. Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.Comment: 14 page