Demonstration of Shor’s factoring algorithm for N $$=$$ 21 on IBM quantum processors

We report a proof-of-concept demonstration of a quantum order-finding algorithm for factoring the integer 21. Our demonstration involves the use of a compiled version of the quantum phase estimation routine, and builds upon a previous demonstration. We go beyond this work by using a configuration of approximate Toffoli gates with residual phase shifts, which preserves the functional correctness and allows us to achieve a complete factoring of $$N=21$$ N = 21 . We implemented the algorithm on IBM quantum processors using only five qubits and successfully verified the presence of entanglement between the control and work register qubits, which is a necessary condition for the algorithm’s speedup in general. The techniques we employ may be useful in carrying out Shor’s algorithm for larger integers, or other algorithms in systems with a limited number of noisy qubits.

Order-by-disorder from bond-dependent exchange and intensity signature of nodal quasiparticles in a honeycomb cobaltate

Recent theoretical proposals have argued that cobaltates with edge-sharing octahedral coordination can have significant bond-dependent exchange couplings thus offering a platform in 3d ions for such physics beyond the much-explored realisations in 4d and 5d materials. Here we present high-resolution inelastic neutron scattering data within the magnetically ordered phase of the stacked honeycomb magnet CoTiO3 revealing the presence of a finite energy gap and demonstrate that this implies the presence of bond-dependent anisotropic couplings. We also show through an extensive theoretical analysis that the gap further implies the existence of a quantum order-by-disorder mechanism that, in this material, crucially involves virtual crystal field fluctuations. Our data also provide an experimental observation of a universal winding of the scattering intensity in angular scans around linear band-touching points for both magnons and dispersive spin-orbit excitons, which is directly related to the non-trivial topology of the quasiparticle wavefunction in momentum space near nodal points.

Atomic-scale insights into quantum-order parameters in bismuth-doped iron garnet

Bismuth and rare earth elements have been identified as effective substituent elements in the iron garnet structure, allowing an enhancement in magneto-optical response by several orders of magnitude in the visible and near-infrared region. Various mechanisms have been proposed to account for such enhancement, but testing of these ideas is hampered by a lack of suitable experimental data, where information is required not only regarding the lattice sites where substituent atoms are located but also how these atoms affect various order parameters. Here, we show for a Bi-substituted lutetium iron garnet how a suite of advanced electron microscopy techniques, combined with theoretical calculations, can be used to determine the interactions between a range of quantum-order parameters, including lattice, charge, spin, orbital, and crystal field splitting energy. In particular, we determine how the Bi distribution results in lattice distortions that are coupled with changes in electronic structure at certain lattice sites. These results reveal that these lattice distortions result in a decrease in the crystal-field splitting energies at Fe sites and in a lifted orbital degeneracy at octahedral sites, while the antiferromagnetic spin order remains preserved, thereby contributing to enhanced magneto-optical response in bismuth-substituted iron garnet. The combination of subangstrom imaging techniques and atomic-scale spectroscopy opens up possibilities for revealing insights into hidden coupling effects between multiple quantum-order parameters, thereby further guiding research and development for a wide range of complex functional materials.

Quantum phase transition phenomena in odd-A nuclei

Quantum phase transitions in even-even nuclei been extensively studied both in theory and experiment in the recent years. Odd-even nuclei can be considered as examples of mixed Bose-Fermi systems with a single fermion coupled to an even-even core. The eÆect of a fermion with angular momentum j on quantum phase transi- tions of a bosonic system comprised of s (L=0) and d (L=2) bosons is investigated. The analysis is based on an Interacting Boson Fermion Model (IBFM) Hamiltonian and its classical limit and it demostrates the changes in the energy level structure as well as the classical and quantum order parameters involved. Some experimental evidence is also presented and compared with theoretical calculations

Gauging permutation symmetries as a route to non-Abelian fractons

We discuss the procedure for gauging on-site Z_2Z2 global symmetries of three-dimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories. We then apply this general procedure to lattice models of several well known fracton phases: two copies of the X-Cube model, two copies of Haah’s cubic code, and the checkerboard model. Where the former two models possess an on-site Z_2Z2 layer exchange symmetry, that of the latter is generated by the Hadamard gate. For each of these models, upon gauging, we find non-Abelian subdimensional excitations, including non-Abelian fractons, as well as non-Abelian looplike excitations and Abelian fully mobile pointlike excitations. By showing that the looplike excitations braid non-trivially with the subdimensional excitations, we thus discover a novel gapped quantum order in 3D, which we term a ‘panoptic" fracton order. This points to the existence of parent states in 3D from which both topological quantum field theories and fracton states may descend via quasi-particle condensation. The gauged cubic code model represents the first example of a gapped 3D phase supporting (inextricably) non-Abelian fractons that are created at the corners of fractal operators.

A quantum primality test with order finding

Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer N, the test tries to find an element of the multiplicative group of integers modulo N with order N-1. If one is found, the number is known to be prime. During the test, we can also show most of the times N is composite with certainty (and a witness) or, after \log\log N unsuccessful attempts to find an element of order N-1, declare it composite with high probability. The algorithm requires O((\log n)^2 n^3) operations for a number N with n bits, which can be reduced to O(\log\log n (\log n)^3 n^2) operations in the asymptotic limit if we use fast multiplication.