Targeting molecular quantum memory with embedded error correction

The implementation of a quantum computer requires both to protect information from environmental noise and to implement quantum operations efficiently. Achieving this by a fully fault-tolerant platform, in which quantum gates are implemented within quantum-error corrected units, poses stringent requirements on the coherence and control of such hardware. A more feasible architecture could consist of connected memories, that support error-correction by enhancing coherence, and processing units, that ensure fast manipulations. We present here a supramolecular {Cr7Ni}–Cu system which could form the elementary unit of this platform, where the electronic spin 1/2 of {Cr7Ni} provides the processor and the naturally isolated nuclear spin 3/2 of the Cu ion is used to encode a logical unit with embedded quantum error-correction. We demonstrate by realistic simulations that microwave pulses allow us to rapidly implement gates on the processor and to swap information between the processor and the quantum memory. By combining the storage into the Cu nuclear spin with quantum error correction, information can be protected for times much longer than the processor coherence.


Organic Thread Synthesis S3
Heterometallic Rotaxane Synthesis S4 Crystallography Details S7 Table S1: Crystallographic Information for 2, 4 and 10 S8 Computational Quantum Simulation and Processing Figures S16 Figure S8: Simulation of the Hadamard gate on the processing unit S17 Figure S9: Duration of the iSWAP S17 Figure S10: Simulation of iSWAP S18 SI References S19 Experimental section:

Structures
General remarks: All starting reagents and materials were sourced from Sigma-Aldrich, Alfa and/or Fluorochem. Unless stated otherwise, all reagents and solvents were used without further purification. The syntheses of the hybrid organic-inorganic rotaxanes were carried out in Erlenmeyer Teflon® FEP flasks supplied by Fisher. Column chromatography was performed using either 40-63 µm silica from Sigma-Aldrich or a Grace Reverelis ® X2 Autocolumn with Grace Reverelis ® NP cartridges. Chemical shifts are reported in parts per million (ppm) from low to high frequency and referenced to the residual solvent resonance. ESI mass spectrometry and microanalysis were carried out by the services at The University of Manchester.

Organic Thread Synthesis (R,R'NH):
All threads were prepared using reductive amination -Schiff base condensation methods. S1 Thread C and Thread D were prepared as per previously published S2 1.1 Thread A (ImCH 2 NHCH 2 CH 2 C 6 H 5 ): A solution of phenethylamine (0.962 mL, 10 mmol) and 4-Imidazolecarboxaldehyde (1 g, 10 mmol) in methanol (30 mL) was refluxed for 5 hr under an N 2 atmosphere, then stirred at room temperature for 3 hr. Excess NaBH 4 (1.52 g, 40 mmol) was added and the reaction mixture was stirred for 2 hr. The reaction was then quenched with water (20 mL) and the residue was extracted with chloroform (3 x 25 mL

Crystallography details
Data Collection. X-Ray data for compound in 2, 4 and 10 were collected at a temperature of 100 K using a Rigaku FR-X with Cu-Kα radiation equipped with a HypixHE6000 detector, equipped with an Oxford Cryosystems nitrogen flow gas system. Data was measured using CrysAlisPro suite of programs.
Crystal structure determinations and refinements. X-Ray data were processed and reduced using CrysAlisPro suite of programmes. Absorption correction was performed using empirical methods (SCALE3 ABSPACK) based upon symmetry-equivalent reflections combined with measurements at different azimuthal angles. S3 The crystal structure was solved and refined against all F 2 values using the SHELXL and Olex 2 suite of programmes. S4 Despite the highly intense X-ray source, crystals of 2 and 10 present a diffraction limit of 1.09 Å, and 1 Å respectively.
All atoms in crystal structures 4 were refined anisotropically with the exception of the hydrogens atoms. In the crystal structure 2 and 10, only the non-disordered toms were refined anisotropically in order to keep the highest data/parameters ratio. Hydrogen atoms were placed in the calculated idealized positions for all crystal structures. The pivalate ligands, threads and hfac ligands in crystal structures 2 and 10 were disordered and modelled over two positions, using structural same distance (SADI) and distance fix (DFIX) Shelxl restraints commands. The atomic displacement parameters (adp) of the ligands have been restrained using similar Ueq and rigid bond (SIMU) and Similar Ueq (SIMU) restraints.
DCM molecules in compound 10 were also disordered and modelled over two positions.
Compounds 2 present large voids filled with featureless electron density. Squeeze software implemented in Platon shows an electron count of 264 electrons which correspond to one molecule of acetone in the asymmetric unit.
A number of A and B alerts were found, especially for structure 2 and 10 , due to the crystal poor resolution obtained for the two crystal structures. Unfortunately, this resolution is common in large molecules with large intermolecular spaces filled with disordered solvent molecules or large amount of disorder.
CCDC 2057528-2057530 contains the supplementary crystallographic data for this paper.  The crystal structures and refinement details for 6 and 8 have been previously published. S2

EPR details and measurements:
Continuous wave Q-band (~34 GHz) and X-band (~9.5 GHz) EPR spectra were recorded with a Bruker EMX580 spectrometer. The continuous wave data were collected on polycrystalline powders and a solution of 1:1 toluene / DCM at 5 K (unless otherwise stated) using liquid helium cooling. All continuous wave spectra were field corrected using a 'Strong Pitch' standard (g = 2.0028) and all powder samples were checked for any polycrystalline nature, by measuring multiple random rotations.
Spectral simulations were performed using the EasySpin 5.2.25 simulation software S5 unless stated otherwise.

Continuous wave EPR details and spectra for 10:
The CW EPR spectra of 10 show two isolated components, with well resolved features. As 10 has the longest separation of the Cu and {Cr 7 Ni} components, the CW EPR spectra appears as a simple superposition of the independent spectra of the components ( Figure S4).
Simulations S5 using a 2J = 0 can reproduce the resolution and sharpness expected for two components that do not overlap or have any exchange interaction between them. For a 1:1 components. There is good resolution of a well-defined quartet from the 63,65 Cu hyperfine (I = 3/2) interaction on the g z (Cu) component, with A z = 450 MHz. A 2% g-Strain was applied to the {Cr 7 Ni} components.
The powder sample for 10 (Fig. S4) shows the same features as per the solution samples, but with better resolution of the {Cr 7 Ni} g z component. This shows extra flexing of the {Cr 7 Ni} ring for 10 due to an additional CH 2 group between the stoppers, with the resolution averaged out in the solution spectra. A best fit of the simulation for the powder sample came by initial using the parameters of the solution sample, with a small adjustment to give the optimal g values: g x,y,z (Cu) = 2.060, 2.052, 2.310, and g x,y,z (Cr 7 Ni) = 1.787, 1.775, 1.738.   All T m measurements were performed with a sample concentration of 0.2 mM at 3 K. A π/2-τ-π-τ-echo sequence was used with 40 (π/2) and 80 (π) ns pulses with a tau of 300 ns. The echo decays were fit to an exponential decay with the form I(2τ) = I(2τ 0 )exp(-2τ/T m ). ) or vice-versa. Within this subspace, the phase reduces to a |1/2⟩ 1 + | -1/2⟩ 1 2 ⨂ | -1/2⟩ 2 single-qubit rotation and hence does not create any entanglement between memory and processor.  s. Points with low factorization (<0.9) between Cr 7 Ni and Cu states have been = 1 = 10 excluded.