From Binary to Continuous Gates - and Back Again

We describe how nuclear magnetic resonance (NMR) spectroscopy can serve as a substrate for the implementation of classical logic gates. The approach exploits the inherently continuous nature of the NMR parameter space. We show how simple continuous NAND gates with sin/sin and sin/sinc characteristics arise from the NMR parameter space. We use these simple continuous NAND gates as starting points to obtain optimised target NAND circuits with robust, error-tolerant properties. We use Cartesian Genetic Programming (CGP) as our optimisation tool. The various evolved circuits display patterns relating to the symmetry properties of the initial simple continuous gates. Other circuits, such as a robust XOR circuit built from simple NAND gates, are obtained using similar strategies. We briefly mention the possibility to include other target objective functions, for example other continuous functions. Simple continuous NAND gates with sin/sin characteristics are a good starting point for the creation of error-tolerant circuits whereas the more complicated sin/sinc gate characteristics offer potential for the implementation of complicated functions by choosing some straightforward, experimentally controllable parameters appropriately. 1 NMR and Binary Gates Nuclear magnetic resonance (NMR) spectroscopy in conjunction with nonstandard computation usually comes to mind as a platform for the implementation of algorithms using quantum computation. Previously we have taken a different approach by exploring (some of) the options to use NMR spectroscopy for the implementation of classical computation [5]. We have demonstrated how logic gates can be implemented in various different ways by exploiting the spin dynamics of non-coupled nuclear spins in a range of solution-state NMR experiments. When dealing with spin systems composed of isolated nuclear spins, the underlying spin dynamics can be described conveniently by the properties of magnetisation vectors and their response to the action of radio-frequency (r.f.) pulses of different durations, phases, amplitudes and frequencies. Together with the integrated intensities and/or phases of the resulting NMR signals, this scenario provides a ⋆ Corresponding author: [email protected] 2 Matthias Bechmann et al. Fig. 1. NOR gate implemented using NMR. a) NMR pulse sequence. b) Spectra corresponding to the four possible gate outputs where the integrated spectral intensity is mapped to logic outputs 0 and 1. c) Logic truth table mapping NMR parameters to gate inputs 0 and 1. (adapted from [5]) Fig. 2. Magnetisation vector manipulation by r.f. pulses, e.g. rotation of magnetisation vector S from the z-direction to the −y-direction by a suitable r.f. pulse (a)). Structure of a r.f. pulse displaying characterisation parameters for amplitude, frequency, duration and phase as possible gate input controls (b)). rich parameter space and a correspondingly large degree of flexibility regarding choices of input and output parameters for the construction of logic gates. Fig. 1 shows an NMR implementation of a NOR gate, for illustration. The effects of r.f. pulses on a given nuclear spin system are fully under experimental control, and the response of the spin system is fully predictable with no approximations involved. An NMR experiment usually starts from the magnetisation vector in its equilibrium position: aligned with the direction of the external magnetic field (the z-direction in the laboratory frame). An r.f. pulse tips the magnetisation vector away from the z-direction. By choosing the duration, amplitude and frequency of the pulses appropriately, the tip of the magnetisation vector can be used to sample the entire sphere around its origin (Fig. 2). Our previous NMR implementations of logic gates [5] exploited special positions on this sphere, such as NMR spectra corresponding to the effects of 90°, or 180°, or 45° pulses to create binary input/output values. We have demonstrated that there are many different ways for such implementations of conventional logic gates by slightly less conventional NMR implementations, including many different ways to define input and output parameters. There are many more possibilities for NMR implementations of conventional logic gates and circuits. Note From Binary to Continuous Gates – and Back Again 3 Fig. 3. 2D function graphs displaying influence of NMR parameters on the output of continuous NAND gates. a) Using the duration τp of the r.f. pulse and the duration of a preacquisition delay τd, resulting in sin dependence of both inputs. b) Using the resonance frequency offset ωp and the r.f. pulse duration τp, a sinc dependence for ωp and a sin dependence for τp is obtained. c) Comparison of experimental and theoretical result for a slice of sinc/sin NAND gate (in b) without mapping to the [0, 1] interval. This corresponds to the region in b) marked by the vertical bar in upper right corner. The deviation between experiment and simulation is always less than 0.5 percent. that for these discrete logic gates a one-to-one mapping of the NMR parameter(s) to the binary state of the gate is possible in a straightforward manner. In this paper we concentrate on another aspect of NMR implementations of classic logic gates. Whereas previously our main focus was on the multitude of different options for implementing discrete logic gates and circuits by NMR, here we exploit another property of basic NMR experiments. Only a minute fraction of, for example, the space accessible to the magnetisation vector has so far been exploited for the construction of discrete logic gates. Now we lift this restriction and take advantage of the inherent continuous properties of our system and the natural computational power provided by the system itself [6]. The underlying continuous spin dynamics hereby provide the basis to the implementation of continuous logic operations. Compared to [5] this means we no longer restrict the inputs and outputs to be the discrete values 0 and 1, but allow them to be continuous values between 0 and 1. 2 Functions of NMR and Continuous Gates Depending on the position of the magnetisation vector at the start of signal acquisition, the time-domain NMR signal is composed of sin and cos functions, with an exponentially decaying envelope (the so-called free induction decay, FID). Accordingly, trigonometric and exponential functions are two of the continuous functions inbuilt in any NMR experiment. Most commonly, NMR signals are represented in the frequency domain. Hence, Fourier transformation gives access to, for example, the sinc function ((sinx)/x) if applied to a truncated exponential decay. Fig. 3 illustrates this shift to continuous logic gates: we show the NMR implementation of NAND gates where the inputs have functional dependencies of sin/sin (Fig. 3a) and sin/sinc (Fig. 3b). Note how they have the same digital NAND gate behaviours at the corners {0, 1} × {0, 1}, but very different 4 Matthias Bechmann et al. behaviours in between. Fig. 3c shows experimental NMR data representing the sinc function used in Fig. 3b. Taking the step to continuous gates, the input/output mapping now applies to the [0, 1] interval and is not as trivial as it is for the discrete logic gates. However, the NMR input parameters and output functions are known in analytical form, giving access to boolean behaviour at the corners of the two-dimensional parameter space, and continuous transitions in between. The digital NAND gate is universal. Here we relax the constraints on the inputs, to form our continuous NAND gates. These continuous gates can serve as starting points for the optimisation of certain properties of the NAND gate itself or, alternatively, for the optimisation of circuits based on NAND gates. We show how to obtain robust NAND gates (ones that still function as digital NAND gates, even if the inputs have considerable errors), by evolving circuits of the continuous single NAND gates with sin/sin (Fig. 3a) and sin/sinc characteristics (Fig. 3b). Then we evolve circuits for a robust XOR gate, constructed from continuous simple NAND gates. Finally, we briefly address the topic of more general continuous gates based on different functions [2] and how the naturally occurring continuous NMR functions may be exploited in such circumstances. Our optimisation tool is Cartesian Genetic Programming (CGP) [3]. 3 Evolving Robust Continuous Gates and Circuits 3.1 Continuous NAND Gate with sin/sin Characteristics This continuous gate is based on the NMR parameters τp (pulse duration) and τd (preacquisition delay) (see Figs. 2b and 3a). It involves the following mapping of the NMR input parameters In1 and In2: In1, In2 ∈ [0, 1] In1 = τp τp90 ; In2 = 1− τd τd90 (1) where τp90 corresponds to a pulse duration causing a 90° flip of the magnetisation vector and τd90 is the duration of a preacquisition delay causing a 90° phase shift of the magnetisation vector in the xy-plane. The output of the simple sin/sin NAND gate implemented by the NMR experiment is then