Strong algorithmic cooling in large star-topology quantum registers

Cooling the qubit into a pure initial state is crucial for realizing fault-tolerant quantum information processing. Here we envisage a star-topology arrangement of reset and computation qubits for this purpose. The reset qubits cool or purify the computation qubit by transferring its entropy to a heat bath with the help of a heat-bath algorithmic cooling procedure. By combining standard NMR methods with powerful quantum control techniques, we cool central qubits of two large star-topology systems, with 13 and 37 spins, respectively. We obtain polarization enhancements by a factor of over 24, and an associated reduction in the spin temperature from 298 K down to 12 K. Exploiting the enhanced polarization of computation qubit, we prepare combination coherences of orders up to 15. By benchmarking the decay of these coherences we investigate the underlying noise process. Further, we also cool a pair of computation qubits and subsequently prepare them in an effective pure state.

Figure 1

(a) Star topology with a central computation qubit and a set of reset qubits surrounded by a bath, (b) a four-star quantum register, and (c) HBAC procedure. The different steps of AC are illustrated in (d)–(f). In (d) the various levels are labeled as $|\alpha \rangle |N-j,j\rangle$, where $\alpha =\{0,1\}$ corresponds to the two states of the computation qubit and $j\in \{0,\cdots ,N\}$ denotes the number of reset qubits in state $|1\rangle$.

Figure 2

HBAC of (a) TMS and (b) TTSS. Shown in each case are molecular structure, ${}^1\mathrm{H}$-decoupled ${}^{29}\mathrm{Si}$ spectra before and after HBAC, and magnetization versus HBAC iteration number ($n$). Here the theoretical data points are calculated with ideal HBAC controls. In the right spectrum, the sidebands (indicated by stars) are due to ${}^{29}\mathrm{Si}$–${}^{13}\mathrm{C}$ J coupling.

Figure 3

(a) Quantum circuit for preparing combination coherences. Here due to the star symmetry, each cnot gate acts simultaneously on all the reset qubits. A pair of pulsed-field gradients ${\mathrm{G}}_1$ and ${\mathrm{G}}_2$ select a coherence of order $q$ which is converted into an observable single-quantum coherence by the second cnot [25]. (b)${}^{29}\mathrm{Si}$ reference spectrum of TTSS and characteristic spectra corresponding to various coherences of odd orders $q$. (c) The decay rates (${\mathrm{\Gamma }}_q$) versus $q^2$ measured in TTSS.

Figure 4

HBAC of a pair of computation qubits. (a) The reference spectra of ${}^{29}\mathrm{Si}$ corresponding to the thermal equilibrium magnetization. Spectra obtained after (b) cooling the left qubit, (c) transferring the polarization from left qubit to right qubit, (d) cooling the left qubit again, and (e) finally preparing the two ${}^{29}\mathrm{Si}$ qubits in the $|00\rangle$ pseudopure state.

Figure 5

Estimated swapping factors ${\eta }_j$ versus the transition number $j$ of the computation qubit in (a) TMS and (b) TTSS as indicated.

Figure 6

Simulated magnetizations after 15 iterations of HBAC versus star-size ($N$) for a set of relative gyromagnetic ratios $\left(\gamma \right)$. The HBAC parameters are shown in the inset.

Figure 7

(a) Simulated magnetization of TTSS versus the number of top levels swapped ($m$) during HBAC iterations. In the case of randomized ${\eta }_j$ values, ${\eta }_j=0$ and ${\eta }_j=1$ are replaced with distributions $[0,0.2]$ and $[0.8,1]$, respectively. (b) Magnetization versus the heat-bath delay.

Figure 8

(a) The inversion profile of the single-step grape pulse. To illustrate its robustness with respect to RF inhomogeneity, three inversion profiles with 0.8, 1.0, and 1.2 times the nominal RF amplitudes are displayed. (b) Performance of the single-step grape pulse experimentally observed by shifting a single-resonance peak. The reference spectrum of TTSS is shown at the bottom. First 15 lines of TTSS are inverted since $m=15$ (see main text).

Figure 9

Comparing the HBAC performance of the single-step grape pulse with IBURP-1 [34].

Figure 10

(a) The inversion profile of the double-step grape pulse. To illustrate its robustness with respect to RF inhomogeneity, three inversion profiles with 0.8, 1.0, and 1.2 times the nominal RF amplitudes are displayed. (b) Performance of the double-step grape pulse experimentally observed by shifting a single-resonance peak.