Grover Speedup from Many Forms of the Zeno Effect

TL;DR

This paper shows that other manifestations of the Zeno effect can support an optimal speedup, whether due to measurement, decoherence, or even decay of the excited state into a computationally useless state, and suggest a wide variety of methods to realise speedup which do not rely on Zeno behaviour.

Abstract

It has previously been established that adiabatic quantum computation, operating based on a continuous Zeno effect due to dynamical phases between eigenstates, is able to realise an optimal Grover-like quantum speedup. In other words, is able to solve an unstructured search problem with the same $\sqrt{N}$ scaling as Grover's original algorithm. A natural question is whether other manifestations of the Zeno effect can also support an optimal speedup in a physically realistic model (through direct analogue application rather than indirectly by supporting a universal gateset). In this paper we show that they can support such a speedup, whether due to measurement, decoherence, or even decay of the excited state into a computationally useless state. Our results also suggest a wide variety of methods to realise speedup which do not rely on Zeno behaviour. We group these algorithms into three families to facilitate a structured understanding of how speedups can be obtained: one based on phase kicks, containing adiabatic computation and continuous-time quantum walks; one based on dephasing and measurement; and finally one based on destruction of the amplitude within the excited state, for which we are not aware of any previous results. These results suggest that there may be exciting opportunities for new paradigms of analog quantum computing based on these effects.

Figures (10)

• Figure 1: Illustration of how an optimal annealing schedule can be used to decide parameters to perform discrete operations (measurements, for example). In this example, seven evenly spaced values of $\tau$ (placed at $1/8$ increments in a way which excludes $0$ and $1$) are given to an optimal schedule taking the form of equation \ref{['eq:opt_sched']}. These values of $f(\tau)$ then determine the values, as the top figure shows this leads to concentration around the smallest gap (emphasised with faint green line).
• Figure 2: Probability of finding the ground state for a multi-stage quantum walk against $m_\mathrm{stage}$. Points and dotted lines are colour (or grayscale) coded based on different values of $t_\mathrm{scale}$. Dashed lines represent the adiabatic limit which each set of points is approaching, computed by setting $m_\mathrm{stage}=10,000$. The fact that the dashed lines begin at $m_\mathrm{stage}=50$ is an aesthetic choice and has no underlying mathematical significance.
• Figure 3: Marked state probability of an adiabatic protocol, which can be defined mathematically as the limit of a multi-stage quantum walk when $m_\mathrm{stage}\rightarrow \infty$ versus $t_\mathrm{scale}$. Approximated numerically by setting $m_\mathrm{stage}=10,000$. Points represent the times used in figure \ref{['fig:msqw_mult_angle']} color coded to match that plot.
• Figure 4: Marked state probability illustrating approach of Zeno limit by performing multiple phase flipping operations (x-axis), each with a fixed rotation angle (colour coding) applied at points defined by an optimal adiabatic schedule. Dotted lines are a guide to the eye. Gold circles indicate exact phase inversion, which as the figure shows is equivalent to a rotation angle of $\pi$.
• Figure 5: Marked state probability versus number of operations (either dephasing or destruction) performed (with placement determined by the optimal adiabatic schedule, as depicted in figure \ref{['fig:sched_illustration']}) for a variety of operations. Filled circles correspond to performing a number of projective measurements in the instantaneous energy eigenbasis of eq. \ref{['eq:H_ac']} defined on the x-axis, while filled squares correspond to the same but partially destructive measurements. Unfilled symbols correspond to models (based on decoherence and dissipation, respectively) where the total rotation angles add to the fixed value for a single full operation. The filled symbols on the other hand correspond to versions where each operation corresponds to full dephasing (in the case of decoherence) or full removal of amplitude in the excited state (in the case of dissipation).