Quantum Information Processing with Delocalized Qubits under Global Control

Conventional quantum computing schemes are incompatible with nanometer-scale “hardware,” where the closely packed spins cannot be individually controlled. We report the first experimental demonstration of a global control paradigm: logical qubits delocalize along a spin chain and are addressed via the two terminal spins. Using NMR studies on a three-spin molecule, we implement a globally clocked quantum mirror that outperforms the equivalent swap network. We then extend the protocol to support dense qubit storage and demonstrate this experimentally via Deutsch and Deutsch-Jozsa algorithms.

Figure 1

A three-spin globally controlled quantum mirror, as recently proposed in 8, 9. The left-hand panel shows the “delocalize-and-revive” patterns for each qubit during three complete mirror cycles; in each cycle the upper and lower qubits exchange places. The right-hand panel shows the detailed evolution during one mirror cycle. Two types of global pulse are applied alternately: one to generate qubit-qubit phase gates, one to generate Hadamard gates. Lighter and darker colors correspond to the switch between $X$ and $Z$ bases after each pulse.

Figure 2

Quantum mirror performance. Upper panel: the structure of ${}^{13}\mathrm{C}$-labeled alanine in ${\mathrm{D}}_2\mathrm{O}$ together with the relevant NMR parameters. All frequencies are in hertz (Hz); following NMR conventions frequencies are measured from the rf reference frequency and increase from right to left. Left panel: spectrum for an initial mixed state $\mathbf{1}+p({\sigma }_z^A-{\sigma }_x^B+{\sigma }_x^C)$ with $p\sim {10}^{-5}$, and spectra after successive quantum mirroring operations [light gray (blue)] and equivalent conventional swap networks [dark gray (red), offset right]. We see precisely the predicted behavior: states on $A$ and $C$ exchange places, while states on $B$ delocalize over all spins and revive on B. Right panel: the results of large numbers of mirror and swap operations, with an initial state corresponding to observable signal on one of the three qubits; the intensity of the signal from the target qubit is plotted as a function of the number of cycles. Stars show experimental data points from mirror operations, and circles show data from conventional swap networks. Smooth lines are fitted curves as described in the text.

Figure 3

Our extension to the protocol in Ref. 9 to support higher storage densities. Here we depict blocks of $M=3$ qubits. (a) The evolution of six qubits, overlaid vertically to show that qubits initially within a block must always overlap with neighbors at the chain edge. (b) Intrablock operations can permute an internal qubit out to the block border [27].

Figure 4

A NMR implementation of interblock logic operations, performing Deutsch’s algorithm on a two-qubit computer with the central spin used as a buffer. The circuit for this algorithm is shown in (a), while (b) depicts the phase gate on logical qubits needed for $f_{01}$ and $f_{10}$: spin $A$ is inverted during $CZ$ operations isolating it from the mirror process and trapping it at the edge until spin $C$ is brought into contact. Experimental results are shown in (c); as expected the signal from spin $C$ is in emission, spin $B$ gives no signal, and spin $A$ is in absorption for the constant functions $f_{00}$ and $f_{11}$ and emission for the balanced functions $f_{01}$ and $f_{10}$.

Figure 5

A NMR implementation of intrablock logic on a three-spin system. Spectra in (b) show how a $z$ rotation can be performed on the central spin using global control methods as shown in (a); note that the final step (fixing the spurious local gates) can normally be omitted if swaps are performed in pairs. Spectra are shown for the pseudopure initial state $|0⟩|+⟩|0⟩$ (an absorption signal on spin $B$) which is converted to $|0⟩|-⟩|0⟩$ (an emission signal on spin $B$) by the $S_z$ gate. Spectra in (c) show complete implementations of the Deutsch-Jozsa algorithm (analyzing functions from 2 bits to 1 bit) for the constant function $f_{1111}$ and the balanced function $f_{0011}$ whose implementation requires a controlled-not operation between qubits $A$ and $C$ which are not directly coupled.