Robust logic gates and realistic quantum computation

The composite rotation approach has been used to develop a range of robust quantum logic gates, including single qubit gates and two qubit gates, which are resistant to systematic errors in their implementation. Single qubit gates based on the BB1 family of composite rotations have been experimentally demonstrated in a variety of systems, but little study has been made of their application in extended computations, and there has been no experimental study of the corresponding robust two qubit gates to date. Here we describe an application of robust gates to nuclear magnetic resonance studies of approximate quantum counting. We find that the BB1 family of robust gates is indeed useful, but that the related NB1, PB1, B4, and P4 families of tailored logic gates are less useful than initially expected.

Figure 1

A quantum circuit for implementing quantum counting on a two qubit NMR quantum computer; the central sequence of gates, surrounded by brackets, is applied $r$ times. A similar circuit can be constructed for a larger search space by replacing the (lower) target bit by a register and replacing gates applied to the target by multibit versions. Gates marked $h$ implement the NMR pseudo-Hadamard operation, while those marked $h^{-1}$ implement the inverse operation.

Figure 2

The effects of pulse length errors on a single qubit pseudo-Hadamard gate (a ${90}_y^{°}$ rotation). The figures show the effect of applying this gate to a spin in the thermal equilibrium state, which is equivalent to a pseudopure state ∣0⟩, followed by observation of the NMR spectrum, effectively measuring the off-diagonal components of the single-spin density matrix. For further details see the main text.

Figure 3

An implementation of approximate quantum counting with naive and BB1 single qubit gates. The use of BB1 single qubit gates greatly reduces the apparent decoherence rate, indicating that much of the signal loss actually arises from pulse length errors, and that BB1 is effective in correcting for this in a complex sequence.

Figure 4

Pulse sequence for a BB1 robust coupling gate to implement a controlled-NOT gate in a two spin $\left(IS\right)$ system. Boxes correspond to single qubit rotations with rotation angles of $\psi =\arccos (-1∕8)\approx 97.2°$ applied along the $\pm y$ axes as indicated; time periods correspond to free evolution under the scalar coupling, $\pi J2I_z{S}_z$, for multiples of the time $t=1∕4J$. The naive coupling gate corresponds to free evolution for a time $2t$.

Figure 5

Implementation of an coupling gate in the presence of extraneous couplings. For details see the main text. The ten ${}^{13}\mathrm{C}$ spectra in each subfigure correspond to increasing periods of evolution under the coupling, and the desired pattern is a simple alternation in signal intensity between $+1$ and $-1$; deviations in the naive implementation arise from evolution under the extraneous couplings to the methyl protons. It might be expected that evolution under this small coupling would be effectively suppressed by the NB1 sequence, but this performs very poorly, and the best results are seen from the BB1 sequence.