# Physical Computing: Unifying Real Number Computation to Enable Energy Efficient Computing

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## Abstract

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## 1. Introducing Physical Computing and Physical Turing Machine Modeling

## 2. Algorithmic Complexity of Real-Valued Computation

**R**), representations, and timescales. Analog computing can compute over continuous-valued voltage or current state variables (e.g., 1 V to 3 V) as well as computing over continuous-valued timesteps. Neuromorphic computing, which includes neuromorphic biological or electronic computing, involves similar capabilities to analog computing, at least over limited regions, as well as continuous-spatial regions at least over limited regions (e.g., dendritic cables). Optical computing also involves continuous amplitudes operating over continuous-valued timesteps with two-dimensional (2D) and three-dimensional (3D) continuous spatial dimensions. Quantum computing operates using a continuous probabilities modeling continuous-time and continuous 3D space, as well as continuous superposition mixtures of two or more states. All of these techniques compute effectively utilizing at least one real-valued dimension.

**R**) between 0 and 1 is infinitely larger than all integers (

**Z**) between 0 and ∞. The size of

**Z**between 0 and ∞ is represented as ${\aleph}_{0}$, and the size bf R between 0 and 1 is represented at ${\aleph}_{1}$. Two

**R**dimensions can fit into a single

**R**dimension. Working along the diagonals of a 2D

**R**map, one can recount the 2D space into a single one-dimensionsal (1D) space (Figure 2) as the same order of infinity (${\aleph}_{1}$). More available dimensions opens up additional implementation opportunities and resulting efficiencies, although these dimensions do not affect the system complexity.

**R**values and timesteps, similar to a Digital Turing Machine models computing over

**Z**values and timesteps (Figure 1 and Figure 3). A Digital Turing machine computes over a

**Z**set of input and output alphabets with

**Z**internal variables and

**Z**size tape operating over

**Z**timesteps. The theoretical Turing Machine model does not require the computation follows the same approach, and yet, digital computing often resembles parts of the Turing Machine techniques. To fully model the continuous-time (CT) computation over

**R**values, a Physical Turing Machine Model computes over a

**R**set of input and output alphabets with

**R**internal variables and

**R**size tape (internal memory) operating over

**R**timesteps. Utilizing

**Z**size input and output alphabets for a Physical Computation is still modeled by the Physical Turing Machine. Physical computing operates over ${\aleph}_{1}$ (∞ for

**R**) as opposed to synchronous digital computing operating over ${\aleph}_{0}$ (∞ for

**Z**). Model allows directly extending known properties and theorems for

**Z**-valued Turing machines to these

**R**-valued Turing machines. Although one could consider a physical implementation specific model (e.g., [22,23]), it likely misses the wider computational space and becomes harder to generalize across all

**R**-valued computing, as well as requiring to build an entirely new theoretical infrastructure.

where he states a“Every finitely realizible physical system can be perfectly simulated by a universal model computing machine operating by finite means”(p. 99) [24].

**Z**-valued Turing machine cannot perfectly simulate a classical dynamical system, as well as he recognizes the impact of non-decreasing entropy (e.g., loss) impacts these computations. Deutsch only imagines

**Z**-valued input and output alphabets, while he opens the possibility of computing over a continuum of values, which is more concretely defined as computation over

**R**-values, and then moves that a quantum dynamics provides a means towards reaching these opportunities, while mostly missing this opportunity in

**R**-valued systems. He mathematically attempts to show non-dynamical quantum operations could be operating over a continuum, inspired by the introduction of quantum computing by Feynman [25,26]. A model that computes with

**R**-valued alphabets also simplifies to computing with

**Z**-valued input and output alphabets. The approach in this discussion generalizes

**R**-valued computation for the continuum of classical and quantum physics, heavily based in decades of physical (e.g., analog, neuromorphic) computing.

with the explicit restatement of the Church-Turing thesis for-“Every ‘function which would naturally be regarded as computable’ can be computed by the universal (Z-valued) Turing machine”—Turing [1]

**R**-valued computation:

where the“Every finitely realizible ‘function which would naturally be regarded as computable’ can be computed by the universal (R-valued) Turing machine”

**Z**-valued Turing machine ⊂

**R**-valued Turing machine. Finite means, which includes finite resources as well as finite amount of time, is essential to any practical physical computation.

**R**versus

**Z**Turing capabilities (Figure 3). Algorithm complexity (Figure 3) considers whether a particular computing structure can compute certain algorithms in Polynomial (P) time, or scales by some other function, such as exponential time (EXPTIME). One can consider the class of polynomial-time physical (P${}_{A}$) and digital (P${}_{D}$) algorithms, the class of digital NP problems (NP${}_{D}$) and similar class of NP problems for analog computation (NP${}_{A}$), as well as the class of exponential-time analog (EXPTIME${}_{A}$) and digital (EXPTIME${}_{D}$) algorithms (Figure 3). One must consider a product of time and resources, although if one considers only polynomial resources, time complexity is sufficient. P${}_{A}$ completely overlaps with P${}_{D}$ as one can make digital gates from analog blocks (Figure 3).

**Z**) input and output alphabets. This interpretation is far too restrictive of physical computing approaches.

**R**verses

**Z**computation. Transformations from

**Z**to

**R**are straight-forward, and one expects a physical system to implement computations over integer (e.g., digital) values. Transformations from

**R**to

**Z**computation require interpolations to, or numeric approximations from

**R**to

**Z**(Figure 4). Synchronous digital simulation results of physical computation must always be approached cautiously, as digital computation is more limited than the resulting physical substrate (Figure 4). Physical ODEs are solved in ${\aleph}_{1}$, with

**R**-valued timesteps compared to

**Z**-valued timesteps by digital emulation. Multiple-timescale nonlinear ODEs (Figure 4) inherently utilize

**R**-valued timesteps to directly handle exponentially fast moving nonlinear physical system dynamics. Analog WTA physical system uses two (or more) competitively strong nonlinearities, resulting in consistent analog solutions while creating a difficult ODEs to solve numerically (e.g., [37]). The error of digital ODE solutions are limited as O(f${}^{k+1}$(·)) for a kth order solver (RK45 is 4th order) [38], so high derivatives, like exponential functions destroy the accuracy and convergence of digital ODE solutions. Attempting to validate physical computing systems (e.g., systems of ODEs) through digital numerical approximations creates the unnecessary concern over having exponentially fast moving nonlinear results for a physical system that are very real for synchronous digital computation. Physical algorithms must be developed and verified only through physical hardware, and discrete simulation or analysis of physical algorithms cannot invalidate potential results.

**Z**-based computing. For example, a Winner-take-all (WTA) circuit (Figure 5) operating with subthreshold currents utilizes competing exponential functions modeled by the transistor current-voltage relationships:

**Z**-based computation, resulting in a significant computational issue (e.g., [16]), as the I${}_{bias}$ and input currents I${}_{1}$, I${}_{2}$ can be orders of magnitude different from each other.

**Z**-based simulation. To eliminate the stiff ODE computation, ${I}_{s0}={I}_{s01}$ resulting in ${\tau}_{1}=\tau $. The resulting ODE system where time is normalized by $\tau $ (time becomes unitless):

## 3. The Continuous-Valued Physical World

#### 3.1. Noise Does Not Negate $\mathit{R}$-Valued Computation

#### 3.2. The Physical World Is $\mathit{R}$-Valued in Space and Time

**R**-valued space and time gives infinitely more steps, and therefore resources, between each

**Z**-value (Figure 7b).

## 4. Connecting Quantum Computing and Analog Computing Applications

**Z**-valued algorithms cannot fully simulate a quantum computer [48], and an initial demonstration through a discrete bench-top analog circuit for a small quantum system (q-bits) utilizing sinusoidal input and output signals [49,50,51]. Typical quantum computing tends to be performed using fixed devices, such as qubits, and assume that the computation is instantaneous, effectively reaching its steady-state rapidly in the measurement timescales. Therefore, the transformation between

**Z**-valued inputs to the measured outputs through these fixed-position qubits,

**R**-valued computation described through a Unitary matrix and nonlinear measurement operations. With increased CMOS scaling, analog integrated circuit techniques use more quantum concepts in their fundamental devices, providing another bridge between these techniques.

**R**or

**Z**) that allows for a number of signals or waveforms to simultaneously exist in an overlapping space. Superposition is taught from the first analog circuit analysis class. Superposition enables simultanious quantum wavefunctions or analog states including for these examples.

## 5. Relationships between Physical Computing Approaches

**R**-valued computing has equivalents to other

**R**-valued techniques. These concepts build on the existence and properties of a continuous-valued environment. Optical, Neuromorphic, Quantum, and Analog computing compute over

**R**amplitude and

**R**time with various spatial forms (Figure 12). All four physical systems described show properties of superposition within their linear operating region, allowing for a number of signals or waveforms to simultaneously exist in the same representation. Optical communication systems extensively make use of superposition to have multiple frequency or wavelength carriers communicate on the same fiber. All four physical systems may have infinite time and/or spatial responses. Optimization problems are routinely solved in analog, quantum and neuromorphic techniques, resulting from coupled ODEs or PDEs propagating energy down the established energy surfaces, either to a global minimum or local minimas. Quantum computing relaxation and annealing as in Grover’s algorithm [36], find similar concepts within as well as analog energy relaxation techniques (e.g., [59]) and neuromorphic L${}_{1}$ norm minimization [60]. A good recent review shows the different physical computing techniques for similar energy and power surface minimization [61].

**R**-valued computational mediums (Figure 12) computing over

**R**-valued amplitude, space, and time (Table 1). Optical computing utilizes waves, described through a second-order time and space PDE formulation of Maxwell’s equations, computing through continuous time, amplitude, and 1- to 3-dimensional space utilizing a number of spatial filters, lenses, and a variety of tunable light modulators & mirrors can modify an input optical signal. Quantum computing uses complex probability wavefunctions ($\mathsf{\Psi}$) governed through PDEs (Schrodinger’s equation) computing through continuous time, amplitude, and 1- to 3-dimensional space. Typical implementations compute multiple quantum wells, potential energy barriers, and connected states governed by these PDEs. Neuromorphic computing (e.g., [10,62]), including Neural Networks (e.g., [30,31]), uses physical computing devices, typically of an analog nature in Si or hybrid system, modeling part of a neurobiological computations (e.g., neurons) using continuous amplitude, time, with at least 1-dimensional space PDEs in neuron computation (e.g., dendritic systems) [63]. Neurons are spatio-temporal computing elements with hundreds if not thousands of inputs, modeling voltages governed by diffusing and wave-propagating PDEs (Figure 12). Analog neuron implementations have demonstrated hundreds of inputs (e.g., [64]) as well as wave-propagating PDEs (e.g., [63]). The dendritic PDEs have a significant linear operating range within the overall biological structure.

**R**-value in space, but practically the parameters change in particular points with a finite granularity of parameter resolution setting, as well as output measurement capability. The PDE could be coupled transistors (e.g., resistive networks [59]) or transistor circuits (e.g., ladder filters [65]). Inputs, outputs and boundary conditions are set through additional analog circuitry. Analog techniques provide the mode advanced physical implementation capabilities, including programmable and reconfigurable techniques in standard CMOS processes [18].

**R**-valued translations (Section 4). The translation between optical and quantum computing goes through their similar hyperbolic wave-propagating PDEs and similar mathematical formulations, where both heavily utilize the temporal and spatial duality between real values and Fourier transformed values. Analog techniques involve a wide range of ODE and PDE techniques that include the range of linear and nonlinear PDE systems. Analog techniques can compute the same PDEs for Maxwell’s equations as optical systems (e.g., [65]), although with a polynomially larger complexity in many cases. Neuromorphic systems utilize parabolic PDEs (e.g., diffusion), and yet these networks can be approximate waveguiding systems by altering spatial parameters [63] as well as through local (e.g., active channels) or active network (e.g., synfire chain [66]) properties. The translation between analog and neuromorphic is straight-forward as most neurormophic models are built with some analog circuit modeling (e.g., [10,62]). The spatial steady-state solutions typically form elliptic PDE problems (Poison’s equation). Each approach has a linear region, although nonlinear operations are harder in some domains (e.g., Optical), possible in some cases (e.g., Quantum), and easy or too easy in other cases (e.g., Analog or Neuromorphic). In each case, even and odd nonlinearities are possible and a linear region are possible where some systems are more optimal in some places than others. One expects translations between other physical systems not identified, all systems computing with multiple

**R**-valued operations and representations.

## 6. Computing Opportunities for Physical Computation

**R**versus

**Z**), as well as showing the transformations between these techniques, the discussion moves to considering the opportunities for physical computing. At one level, physical computing approaches often have far higher computational efficiency and lower computational area [7,8,9,18,19], as mentioned for analog systems (1000×, 100×) at the beginning of this discussion. Neuromorphic approaches have demonstrated even greater computational efficiencies with roadmaps possible for both improved analog and neuromorphic cases [10]. These opportunities alone are sufficient to explore these techniques, particularly in an ever energy-constrained environment. The availability of large-scale programmable and configurable analog techniques has enabled a realistic analysis of physical computing techniques.

**R**versus

**Z**likely has additional benefits including addressing the relationship between NP${}_{D}$ and EXPTIME${}_{D}$ to P${}_{A}$ and NP${}_{A}$ (Figure 13). Some examples in P${}_{A}$ appear to be beyond P${}_{D}$. such as Shor’s quantum factorization algorithm [35]. Multiple coupled ODEs systems have been proposed to solve NP problems, such at the 3-SAT problem [33,34,70,71,72], that could be implemented on a configurable continuous-time platform [19], although to date it has not been verified through physical computing approaches. The potential overlap of EXPTIME${}_{D}$ (e.g., Shor Factorization) with P${}_{A}$ and NP${}_{A}$ gives motivation for deeper explorations. Throughout this process, synchronous digital simulation results of physical computation must always be approached cautiously, as digital computation is more limited than the resulting physical substrate (Figure 4).

**R**-valued computing platforms. It remains an open question to connect the Traveling salesman Problem (TSP) or Max cut problem to this

**R**-valued optimal path planning. This discussion sits at a place where we have the framework to explore if NP${}_{D}$ is part of P${}_{A}$, looking for a constructive approach based on some potential examples.

## 7. Summary and Discussion: Implications and Opportunities of Physical Computation

**R**-values as compared to traditional digital computing over

**Z**-values.

**R**-valued computation and its

**R**-valued Turing Machine, and there is starting theory of

**R**-valued numerical analysis, architectures, and abstraction primarily coming from analog implementations, one still requires a constructive framework for engineers to design physical computing systems for an application. Although there are significant guideposts for physical computation so we no longer require a miracle to occur (Figure 1) for such implementations, one still does not find the well-traveled paths typical of linear digital design.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Digital Computation builds from the framework of Turing Machines, setting up capability of computer architectures, computer algorithms, and resulting numerical analysis, being the basis for our day to day digital computing. Classical Analog Computation is perceived to have little computational modeling, as well as architectures and algorithms, seeming to be bottom-up artwork rather than top-down digital computing design. Configurable Analog Computation, a Physical Computing real-valued computing technique originally FPAA enabled, builds on recent framework in architectures, algorithms, abstraction, and numerical analysis. This approach enables a Physical Turing Machine model unifying real-value computation.

**Figure 2.**One can map a two dimensional infinity, whether countable or real, to a one-dimensional infinity of the same order (countable or real, respectively) by projecting the successive diagonals of the two-dimensional space into the one-dimensional space. Each box is as small a region possible (real or countable), and they project into small regions into the one-dimensional space. The difference between real or countable is the size of these regions.

**Figure 3.**Physical (Real-Valued,

**R**) and Digital (Integer-Valued,

**Z**) computation comparison. Physical computing includes quantum, optical, analog and neuromorphic approaches. Polynomial time (P) digital algorithms are part of P physical algorithm space. Non-polynomial time (NP, EXPTIME) digital algorithms might be part of P physical algorithm space.

**Figure 4.**Equivalences between physical computing techniques (real variables) and the challenges equating these approaches digital (integer variables). An integer-valued computational framework can approximate a real valued computation, and may work within reasonable bounds for a range of cases, but will not be successful within a reasonable amount of time (e.g., exponential algorithm scaling) in other cases.

**Figure 5.**Two-input Winner-Take-All (WTA) circuit dynamics are set through competing nonlinear functions described by ${y}_{1}$, ${y}_{2}$, and y. Time is normalized by $\tau $ that is a function of the bias current (I${}_{bias}$) and C. When simulating ODEs using

**Z**-valued computation, these competing exponential functions result in large derivatives, resulting in significant errors in the computation, in addition to linear timeconstants that are potentially a different order of magnitude.

**R**-valued computation benefits from the

**R**-valued timesteps verses

**Z**-valued timesteps available for

**Z**-valued simulation, as well as

**R**-valued computation does not have accumulation errors typical of

**Z**-valued integration [16]. In a

**Z**-valued timesteps (h) are an ∞ of

**R**-valued points. The numerical solution (e.g., RK45) of this two-input WTA circuit results in different stability for a small change in starting conditions. ODE numerical solution for a step from an initial starting condition to ${x}_{1}={x}_{2}=2$ for two initial condition cases (Case I: [${y}_{1}$${y}_{2}$y] = [−4 −3 2], Case II: [${y}_{1}$${y}_{2}$y] = [−8 −6 4]). Case I converged, while Case II was numerically unstable.

**Figure 6.**Do physical systems operate over continuous or discrete spaces? Difference between continuous-values with noise versus resolution. One abstraction for real variables with noise and uncertainty is to quantize the number of levels to the extent of that uncertainty. Resolution is this abstraction. In analog systems, this abstraction is computed using an Analog-to-Digital Converter (ADC). This abstraction, while useful for some engineering systems, only approximates the real world, and in no way precludes the continuous-value nature of the resulting computation. The uncertainty would be part of the resulting numerical analysis of the physical computation.

**Figure 7.**Do particles indicate nature operates over discrete quantities? (

**a**) The abstraction of a particle, rather than a wave, effectively takes a first and second moment (mean and approximate extent) of the resulting wave and resulting wavefunction. The position of the particle in space and its position in time are continuous values. (

**b**) The computational implication of operating in a

**R**-valued world over a

**Z**-valued world allows an infinitely more time, space, and amplitude

**R**-valued steps between the minimum

**Z**-valued step.

**Figure 8.**Translation between Qubit-based quantum computation and analog computation. Both Physical computing approaches are computations over real values. (

**a**) Although one starts with individual states (<0 or <1), typically one has simultaneous probability of <0 and a probability of <1 (superposition). (

**b**) Discrete valued inputs, represented as two quantum or voltage states, through a transformation (

**c**) Real-valued inputs, represented as a mixture of two quantum states or an analog voltage, moves through a transformation to another mixture of two quantum states or an analog voltage. (

**d**) Real-valued inputs, represented as a mixture of two quantum states or an analog voltage, moves through a transformation and then moves to a discrete value through measurement. For quantum computation, this measure operation typically is the basic nonlinear operation element. This measurement operation for analog computation is a comparison operation of some form.

**Figure 9.**The operation of one or several qubits is a linear operation over complex numbers. Measurement provides a non-linear operation effectively thresholding the resulting probability. A combination of linear, non-time dependent measurements results in a complex VMM where the measurement operation looks like a threshold or a noisy threshold operation. This structure is related to a one-layer Perceptron network, directly implementable by analog computation, with complex weights and real or complex inputs.

**Figure 10.**Floating-Gate (FG) based circuit for a complex signal (V${}_{inR}$ + j V${}_{inI}$) multiplied by a complex weight (W${}_{R}$ + j W${}_{I}$). The differential complex input terms (V${}_{inR}^{+}$, V${}_{inR}^{-}$, V${}_{inI}^{+}$, V${}_{inI}^{-}$) to differential complex output terms (Real and Imaginary Output Components), require several partial products to compute the full four-quadrant complex multiplication. V${}_{tun}$ is used to erase the FG elements through the tunneling capacitors, and otherwise is held at a fixed potential through this operation.

**Figure 11.**Potential Continuous-Time Computation Architecture for Qubit computation. Source of M1, M2 might be biased higher to keep everything in subthreshold. The complex VMM operation could be either gate or source coupled structure.

**Figure 12.**Example computational medium for Physical Computing. All cases are described by linear or nonlinear PDEs capable of diffusion and wave propagation, where they operate over space as well as one continuous variable as well as continuous time. We show representative PDEs for potential computations for each case; the diagrams do not capture all possible phenomena. (

**a**) Optical Computing: Highly efficient and linear computation using light in multiple spatial dimensions, described through second-order wave equations. (

**b**) Quantum Computing: Wave-based physics coupling particles to all other particles. Typical implementations tend to use multiple quantum wells, energy barriers (modeled by V), and connected states implementing computation through the resulting wavefunctions ($\mathsf{\Psi}$). (

**c**) Neuromorphic Computing: Physical computing inspired by computing in animal nervous systems. Looking along a single axis of a single neuron, the physics of the membrane voltage (V) is a combination of diffusion and waveguiding behavior. (

**d**) Analog Computing: An example utilizing analog devices computing in voltage (V) a temporal-spatial PDE including second-order space and time dynamics. These systems have some continuous spatial behavior through the spatial dynamics in individual dynamics, whether or not those intermediate results are used.

**Figure 13.**Mapping the transformation between

**Z**computations and

**R**computations remains an open question. Some algorithms that solve in P${}_{D}$, such as energy surface relaxation applications, directly translate to algorithms in P${}_{A}$, even if they sometimes give good enough solutions for NP${}_{D}$, NP${}_{A}$. Some algorithms might provide an opportunity to bridge between NP${}_{D}$ and P${}_{A}$.

R | Core | ||||
---|---|---|---|---|---|

Time | Amplitude | Space | Values | PDEs | |

Quantum | 1 | 1 | 1–3 | >=3 | 1st/2nd-Order Hyperbolic |

Analog | 1 | 1 | 0–3? | >=2 | 1st & 2nd-Order Space & Time |

Neuromorphic | 1 | 1 | 1 | >=3 | 2nd-Order Parabolic |

Optical | 1 | 1 | 1–3 | <=3 | 2nd-Order Hyperbolic |

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## Share and Cite

**MDPI and ACS Style**

Hasler, J.; Black, E.
Physical Computing: Unifying Real Number Computation to Enable Energy Efficient Computing. *J. Low Power Electron. Appl.* **2021**, *11*, 14.
https://doi.org/10.3390/jlpea11020014

**AMA Style**

Hasler J, Black E.
Physical Computing: Unifying Real Number Computation to Enable Energy Efficient Computing. *Journal of Low Power Electronics and Applications*. 2021; 11(2):14.
https://doi.org/10.3390/jlpea11020014

**Chicago/Turabian Style**

Hasler, Jennifer, and Eric Black.
2021. "Physical Computing: Unifying Real Number Computation to Enable Energy Efficient Computing" *Journal of Low Power Electronics and Applications* 11, no. 2: 14.
https://doi.org/10.3390/jlpea11020014