Elementary quantum gates in different bases

Sergey A. Podoshvedov

We introduce transformation matrix connecting sets of the displaced states with different displacement amplitudes. Arbitrary pure one-mode state can be represented in new basis of the displaced number (Fock) states ( representation) by multiplying the transposed transformation matrix on a column vector of initial state. Analytical expressions of the representation of superposition of vacuum and single photon and two-mode squeezed vacuum (TMSV) are obtained. On the basis of the developed mathematical formalism, we consider the mechanism of interaction between qubits which is based on their displaced properties. Superposed coherent states deterministically displace target state on equal modulo but opposite on sign values. Registration of the single photon in auxiliary mode (probabilistic operation) results in constructive interference and gives birth to entangled hybrid state corresponding to outcome of elementary quantum gates. The method requires minimal number of resource and works in realistic scenario

Wolfgang Scherer

Daniel Grier, Luke Schaeffer

We present a complete classification of quantum stabilizer gates in terms of the functions they generate assuming the ability to swap qubits and use ancillary workspace. Because we view these stabilizer circuits as subroutines of some general quantum computation, we insist that any ancilla qubits used during the computation must not change in an input-dependent manner. This is the first attempt at a quantum extension of the classification of reversible classical gates introduced by Aaronson et al., another part of an ambitious program to classify all quantum gate sets. The classification uses, at its center, a reinterpretation of the tableau representation of stabilizer gates to give circuit decompositions, from which elementary generators can easily be extracted. There are a total of 57 different stabilizer classes generated in this way, 30 of which arise from the single-qubit subgroups of the Clifford group. At a high level, the remaining classes are arranged according to the bases they preserve. For instance, the CNOT gate preserves the X and Z bases because it maps X-basis elements to X-basis elements and Z-basis elements to Z-basis elements. The remaining classes are characterized by more subtle tableau invariants; for instance, the T_4 and phase gate generate a proper subclass of Z-preserving gates.

Sergey A. Podoshvedov, Jaewan Kim, Kisik Kim

Daniel Grier, Luke Schaeffer

We examine the following problem: given a collection of Clifford gates, describe the set of unitaries generated by circuits composed of those gates. Specifically, we allow the standard circuit operations of composition and tensor product, as well as ancillary workspace qubits as long as they start and end in states uncorrelated with the input, which rule out common "magic state injection" techniques that make Clifford circuits universal. We show that there are exactly 57 classes of Clifford unitaries and present a full classification characterizing the gate sets which generate them. This is the first attempt at a quantum extension of the classification of reversible classical gates introduced by Aaronson et al., another part of an ambitious program to classify all quantum gate sets. The classification uses, at its center, a reinterpretation of the tableau representation of Clifford gates to give circuit decompositions, from which elementary generators can easily be extracted. The 57 different classes are generated in this way, 30 of which arise from the single-qubit subgroups of the Clifford group. At a high level, the remaining classes are arranged according to the bases they preserve. For instance, the CNOT gate preserves the X and Z bases because it maps X-basis elements to X-basis elements and Z-basis elements to Z-basis elements. The remaining classes are characterized by more subtle tableau invariants; for instance, the T_4 and phase gate generate a proper subclass of Z-preserving gates.