2108.09836v2

## Probabilistic computing with p-bits Jan Kaiser 1, a) and Supriyo Datta 1 Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47906 USA (Dated: 14 October 2021) Digital computers store information in the form of bits that can take on one of two values 0 and 1, while quantum computers are based on qubits that are described by a complex wavefunction whose squared magnitude gives the probability of measuring either a 0 or a 1. Here we make the case for a probabilistic computer based on p-bits which take on values 0 and 1 with controlled probabilities and can be implemented with specialized compact energy-efficient hardware. We propose a generic architecture for such p-computers and emulate systems with thousands of p-bits to show that they can significantly accelerate randomized algorithms used in a wide variety of applications including but not limited to Bayesian networks, optimization, Ising models and quantum Monte Carlo. ## I. INTRODUCTION Feynman 1 famously remarked 'Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical' . In the same spirit we could say 'Many real life problems are not deterministic, and if you want to simulate them, you'd better make it probabilistic' . But there is a difference. Quantum algorithms require quantum hardware and this has motivated a worldwide effort to develop a new appropriate technology. By contrast probabilistic algorithms can be and are implemented on existing deterministic hardware using pseudo RNG's (random number generators). Monte Carlo algorithms represent one of the top ten algorithms of the 20 th century 2 and are used in a broad range of problems including Bayesian learning, protein folding, optimization, stock option pricing, cryptography just to name a few. So why do we need a p-computer ? A key element in a Monte Carlo algorithm is the RNG which requires thousands of transistors to implement with deterministic elements, thus encouraging the use of architectures that time share a few RNG's. Our work has shown the possibility of high quality true RNG's using just three transistors 3 , prompting us to explore a different architecture that makes use of large numbers of controlled-RNG's or p-bits . Fig. 1 (a) 4 shows a generic vision for a probabilistic or a p-computer having two primary components: an N-bit random number generator (RNG) that generates N-bit samples and a Kernel that performs deterministic operations on them. Note that each RNG-Kernel unit could include multiple RNGKernel sub-units (not shown) for problems that can benefit from it. These sub-units could be connected in series as in Bayesian networks (Fig. 2 (a) or in parallel as done in parallel tempering 5,6 or for problems that allow graph coloring 7 . The parallel RNG-Kernel units shown in Fig. 1 (a) are intended to perform easily parallelizable operations like ensemble sums using a data collector unit to combine all outputs into a single consolidated output. a) kaiser32@purdue.edu Ideally the Kernel and data collector are pipelined so that they can continually accept new random numbers from the RNG 4 , which is assumed to be fast and available in numerous numbers. The p-computer can then provide N p f c samples per second, N p being the number of parallel units 9 , and f c the clock frequency. We argue that even with N p = 1, this throughput is well in excess of what is achieved with standard implementations on either CPU (central processing unit) or GPU (graphics processing unit) for a broad range of applications and algorithms including but not limited to those targeted by modern digital annealers or Ising solvers 8,10-17 . Interestingly, a p-computer also provides a conceptual bridge to quantum computing, sharing many characteristics that we associate with the latter 18 . Indeed it can implement algorithms intended for quantum computers, though the effectiveness of quantum Monte Carlo depends strongly on the extent of the so-called sign problem specific to the algorithm and our ability to 'tame' it 19 . ## II. IMPLEMENTATION Of the three elements in Fig. 1, two are deterministic . The Kernel is problem-specific ranging from simple operations like addition or multiplication to more elaborate operations that could justify special purpose chiplets 20 . Matrix multiplication for example could be implemented using analog options like resistive crossbars 16,21-23 . The data collector typically involves addition and could be implemented with adder trees. The third element is probabilistic , namely the N-bit RNG which is a collection of N 1-bit RNG's or p -bits . The behavior of each p -bit can be described by 24 $$s _ { i } = \Theta [ \, \sigma ( I _ { i } - r ) \, ] \quad \, ( 1 )$$ where s i is the binary p-bit output, Θ is the step function, σ is the sigmoid function, I i is the input to the p-bit and r is a uniform random number between 0 and 1. Eq. (1) is illustrated in Fig. 1 (b). While the p-bit output is always binary, the p-bit input I i influences the mean of the output sequence. With I i = 0, the output FIG. 1. Probabilistic computer: (a) Overall architecture combining a probabilistic element (N-bit RNG) with deterministic elements (kernel and data collector). The N-bit RNG block is a collection of N 1-bit RNG's, or p-bits. (b) p-bit: Desired input-output characteristic along with two possible implementations, one with CMOS technology using linear feedback shift registers (LFSR's) and lookup tables (LUT's) 8 and the other using three transistors and a stochastic magnetic tunnel junction (s-MTJ) 3 . The first is used to obtain all the results presented here, while the second is a nascent technology with many unanswered questions. <details> <summary>Image 1 Details</summary>  ### Visual Description ## Diagram: p-Computer Architecture and 1-bit RNG Implementation ### Overview The image presents two interconnected technical diagrams: 1. **(a) Architecture for a p-computer**: A system with N-bit RNGs feeding into kernels, which aggregate data into a central collector. 2. **(b) 1-bit RNG: p-bit**: A characteristic graph and hardware implementation of a pseudo-random number generator (RNG) producing binary outputs. --- ### Components/Axes #### (a) p-Computer Architecture - **Components**: - **N-bit RNG**: Blue boxes labeled "N-bit RNG" (top and bottom). - **Kernel**: Green boxes labeled "Kernel" (center). - **Data Collector**: Orange box labeled "Data Collector" (right). - **Flow**: - Arrows connect N-bit RNGs → Kernels → Data Collector. - Dashed lines indicate feedback loops from Kernels to N-bit RNGs. #### (b) 1-bit RNG: p-bit - **Graph**: - **X-axis**: "Input" (horizontal). - **Y-axis**: "Output" (vertical). - **Lines**: - **Blue**: "p-bit output" (step function). - **Green**: "tanh(input)" (smooth curve). - **Yellow**: "Moving Average" (smoothed step function). - **Legend**: Located on the right, with color-coded labels. - **Implementation**: - Circuit diagram with labeled components: - **LFSR** (Linear Feedback Shift Register). - **LUT** (Look-Up Table). - **3T + s-MTJ** (3-transistor + spintronic magnetoresistive junction). - **FM** (Ferromagnetic) and **LBM** (Local Bit Manipulation). - Voltage labels: **VDD**, **VIN**, **VSS**, **VOUT**. --- ### Detailed Analysis #### (a) p-Computer Architecture - **Structure**: - Two N-bit RNGs (top and bottom) generate random data. - Kernels process inputs from RNGs and feed results to the Data Collector. - Feedback loops suggest iterative refinement of RNG outputs. #### (b) 1-bit RNG: p-bit - **Graph Trends**: - **p-bit output**: Binary step function (blue line) with abrupt transitions between 0 and 1. - **tanh(input)**: Smooth sigmoid curve (green line), mapping input to [-1, 1]. - **Moving Average**: Yellow line smooths the p-bit steps, reducing noise. - **Implementation**: - **LFSR**: Generates pseudo-random bits. - **LUT**: Maps LFSR outputs to p-bit values. - **3T + s-MTJ**: Hardware implementation for energy-efficient bit manipulation. --- ### Key Observations 1. **p-bit Output**: Binary steps (blue line) indicate discrete 0/1 states, typical of RNGs. 2. **tanh(input)**: Smooth transition suggests input normalization for analog processing. 3. **Moving Average**: Yellow line demonstrates noise reduction in binary data. 4. **Hardware Components**: - **LFSR** and **LUT** enable deterministic pseudo-randomness. - **3T + s-MTJ** implies low-power, high-density spintronic logic. --- ### Interpretation - **p-Computer Architecture**: - The system uses multiple N-bit RNGs to generate diverse random data, processed by kernels for aggregation. Feedback loops may optimize RNG performance or adapt to data patterns. - **1-bit RNG**: - The p-bit output (blue line) is a binary signal, while the tanh(input) (green) provides a continuous analog representation. The moving average (yellow) filters noise, critical for reliable data collection. - **Hardware Implementation**: - The 3T + s-MTJ circuit highlights a focus on energy-efficient, spintronic-based computing, aligning with emerging non-volatile memory technologies. - **Significance**: - The architecture bridges probabilistic computing (p-computer) with hardware implementations, emphasizing scalability (N-bit RNGs) and noise resilience (moving average). </details> is distributed 50 -50 between 0 and 1 and this may be adequate for many algorithms. But in general a nonzero I i determined by the current sample is necessary to generate desired probability distributions from the N-bit RNG -block. One promising implementation of a p-bit is based on a stochastic magnetic tunnel junction (s-MTJ) as shown in Fig. 1 (b) whose resistance state fluctuates due to thermal noise. It is placed in series with a transistor, and the drain voltage is thresholded by an inverter 3 to obtain a random binary output bit whose average value can be tuned through the gate voltage V IN . It has been shown both theoretically 25,26 and experimentally 27,28 that sMTJ -based p-bits can be designed to generate new random numbers in times ∼ nanoseconds . The same circuit could also be used with other fluctuating resistors 29 , but one advantage of s-MTJ's is that they can be built by modifying magnetoresistive random access memory (MRAM) technology that has already reached gigabit levels of integration 30 . Note, however, that the examples presented here all use p-bits implemented with deterministic CMOS elements or pseudo-RNG's using linear feedback shift registers (LFSR's) combined with lookup tables (LUT's) and thresholding elements 8 as shown in Fig. 1 (b). Such random numbers are not truly random, but have a period that is longer than the time range of interest. The longer the period, the more registers are needed to implement it. Typically a p-bit requires ∼ 1000 transistors 30 , the actual number depending on the quality of the pseudo RNG that is desired. Thirty-two stage LFSR's require ∼ 1200 transistors, while a Xoshiro128+ 31 would require around four times as many. Physics-based approaches, like s-MTJ's , naturally generate true random numbers with infinite repetition period and ideally require only 3 transistors and 1 MTJ. A simple performance metric for p-computers is the ideal sampling rate N p f c mentioned above. The results presented here were all obtained with an fieldprogrammable gate array (FPGA) running on a 125 MHz clock, for which 1 /f c = 8 ns, which could be significantly shorter (even ∼ 0 . 1 ns 25 ) if implemented with s-MTJ's . Furthermore, s-MTJ's are compact and energy-efficient, allowing up to a factor of 100 larger N p for a given area and power budget. With an increase of f c and N p , a performance improvement by 2-3 orders of magnitude over the numbers presented here may be possible with s-MTJ's or other physics-based hardware. We should point out that such compact p-bit implementations are still in their infancy 30 and many questions remain. First is the inevitable variation in RNG characteristics that can be expected. Initial studies suggest that it may be possible to train the Kernel to compensate for at least some of these variations 32,33 . Second is the quality of randomness, as measured by statistical quality tests which may require additional circuitry as discussed for example in Ref. 27 . Certain applications like simple integration (Section III A) may not need high quality random numbers, while others like Bayesian correlations (Section III B) or Metropolis-Hastings methods that require a proposal distribution (Section III C) may have more stringent requirements. Third is the possible difficulty associated with reading sub-nanosecond fluctuations in the output and communicating them faithfully. Finally, we note that the input to a p-bit is an analog quantity requiring Digital-to-Analog Converters (DAC's) FIG. 2. Bayesian network for genetic relatedness mapped to a p-computer (a) with each node represented by one p-bit. With increasing N S , the correlations (b) between different nodes is obtained more accurately. <details> <summary>Image 2 Details</summary>  ### Visual Description ## Bayesian Network and Correlation Graph Analysis ### Overview The image contains two primary components: 1. **(a) Bayesian Network**: A computational model depicting data flow through RNGs (Random Number Generators), a Kernel, and correlation calculations. 2. **(b) Correlation Graph**: A log-log plot showing correlation decay across different data relationships (self, child, grandchild, etc.) as a function of sample size. --- ### Components/Axes #### Bayesian Network (a) - **Nodes**: - **Inputs**: `F1`, `F2`, `M1`, `M2` (feature/marker nodes). - **Outputs**: `G61`, `G62`, `G63`, `G64` (generated data nodes). - **Components**: - **RNG (Left)**: Contains `LUT` (Look-Up Table) and `LFSR` (Linear Feedback Shift Register) blocks. - **Kernel (Center)**: Processes data via `CPT LUT` (Conditional Probability Table Look-Up Table) and `Multiply & Accumulate` operations. - **RNG (Right)**: Mirrors the left RNG structure. - **Flow**: - Data flows from `data collector` → Left RNG → Kernel → Right RNG → `data collector`. - Arrows (`→`, `↑`, `↘`) indicate directional dependencies. #### Correlation Graph (b) - **Axes**: - **X-axis**: `Number of Samples, N_S` (log scale: 10¹ to 10⁶). - **Y-axis**: `Correlation` (log scale: 10⁻³ to 10⁰). - **Legend**: - **Lines**: - `Self` (solid blue): Self-correlation. - `Child` (dashed orange): Parent-child correlation. - `Grandchild` (dotted green): Grandparent-grandchild correlation. - `3rd generation` (dash-dot red): Three-generation correlation. - `4th generation` (dash-dot purple): Four-generation correlation. - `6th generation` (dotted pink): Six-generation correlation. - `Stranger` (gray): Unrelated data correlation. - `1/√N_S` (black dashed): Theoretical decay threshold. --- ### Detailed Analysis #### Bayesian Network - **RNG Blocks**: - Both RNGs use `LUT` and `LFSR` for pseudo-random number generation. - Left RNG feeds into the Kernel; Right RNG receives processed data. - **Kernel**: - Combines inputs via `CPT LUT` and arithmetic operations (`×`, `+`). - Outputs to the Right RNG for final data generation. #### Correlation Graph - **Trends**: - **Self (blue)**: Maintains near-constant correlation (~10⁰) across all sample sizes. - **Child (orange)**: Starts at ~10⁻¹, decays to ~10⁻² by 10⁵ samples. - **Grandchild (green)**: Begins at ~10⁻¹, stabilizes near ~10⁻². - **3rd/4th/6th generations**: Show gradual decay, with 6th generation dropping below 10⁻². - **Stranger (gray)**: Sharp initial drop to ~10⁻³, then plateaus. - **Theoretical (black)**: Follows `1/√N_S`, intersecting most lines at ~10³ samples. --- ### Key Observations 1. **Self-correlation** remains dominant, unaffected by sample size. 2. **Stranger correlation** decays fastest, aligning closely with the theoretical `1/√N_S` line. 3. **Generational correlations** (Child, Grandchild, etc.) decay slower than strangers but faster than self. 4. **6th generation** exhibits the weakest correlation, suggesting diminishing relatedness over generations. --- ### Interpretation 1. **Bayesian Network**: - Models a closed-loop system where data is processed through RNGs and a Kernel, likely simulating stochastic processes or cryptographic operations. - The dual RNGs and Kernel suggest a focus on data transformation and correlation analysis. 2. **Correlation Graph**: - Demonstrates that **self-similarity** (e.g., identical data) persists regardless of sample size, while **relatedness** (e.g., parent-child) decays predictably. - The `1/√N_S` line validates the expected decay rate for uncorrelated data, confirming that stranger correlations align with randomness. - **Generational decay** implies that shared patterns (e.g., genetic, behavioral) weaken over time, consistent with entropy principles. 3. **Anomalies**: - The `Stranger` line’s sharp initial drop suggests strong initial dissimilarity, possibly due to data preprocessing or inherent randomness. - The `6th generation` line’s divergence from the theoretical curve may indicate residual correlations in long-term data. --- ### Conclusion The Bayesian Network and correlation graph together illustrate a system where data relationships (self, generational, stranger) are quantified and modeled. The decay trends align with theoretical expectations, validating the network’s ability to simulate and analyze complex dependencies. This could apply to fields like cryptography, genetics, or machine learning, where understanding data similarity is critical. </details> unless the kernel itself is implemented with analog components. ## III. APPLICATIONS ## A. Simple integration Avariety of problems such as high dimensional integration can be viewed as the evaluation of a sum over a very large number N of terms. The basic idea of the Monte Carlo method is to estimate the desired sum from a limited number N s of samples drawn from configurations α generated with probability q α : $$M = \sum _ { \alpha = 1 } ^ { N } m _ { \alpha } \approx \frac { 1 } { N _ { S } } \sum _ { \alpha = 1 } ^ { N _ { S } } \frac { m _ { \alpha } } { q _ { \alpha } } \quad ( 2 ) \quad H o n$$ The distribution { q } can be uniform or could be cleverly chosen to minimize the standard deviation of the estimate 34 . In any case the standard deviation goes down as 1 / √ N s and all such applications could benefit from a p-computer to accelerate the collection of samples. ## B. Bayesian Network A little more complicated application of a p-computer is to problems where random numbers are generated not according to a fixed distribution, but by a distribution determined by the outputs from a previous set of RNG ′ s . Consider for example the question of genetic relatedness in a family tree 35,36 with each layer representing one generation. Each generation in the network in Fig. 2 (a) with N nodes can be mapped to a N-bit RNG -block feeding into a Kernel which stores the conditional probability table (CPT) relating it to the next generation. The correlation between different nodes in the network can be directly measured and an average over the samples computed to yield the correct genetic correlation as shown in Fig. 2 (b). Nodes separated by p generations have a correlation of 1 / 2 p . The measured absolute correlation between strangers goes down to zero as 1 / √ N s . This is characteristic of Monte Carlo algorithms, namely, to obtain results with accuracy ε we need N s = 1 /ε 2 samples. The p-computer allows us to collect samples at the rate of N p f c = 125 MSamples per second if N p = 1 and f c = 125 MHz. This is about two orders of magnitude faster than what we get running the same algorithm on a Intel Xeon CPU. How does it compare to deterministic algorithms run on CPU ? As Feynman noted in his seminal paper 1 , deterministic algorithms for problems of this type are very inefficient compared to probabilistic ones because of the need to integrate over all the unobserved nodes { x B } in order to calculate a property related to nodes { x A } $$P _ { A } ( x _ { A } ) = \int d x _ { B } P ( x _ { A } , x _ { B } ) \quad \quad ( 3 )$$ By contrast, a p-computer can ignore all the irrelevant nodes { x b } and simply look at the relevant nodes { x A } . We used the example of genetic correlations because it is easy to relate to. But it is representative of a wide class of everyday problems involving nodes with oneway causal relationships extending from 'parent' nodes to 'child' nodes 37-39 , all of which could benefit from a p-computer . FIG. 3. Example of MCMC - Knapsack problem: (a) Mapping to the general p-computer framework. (b) Performance of the p-computer compared to CPU implementation of the same probabilistic algorithm and along with two well-known deterministic approaches. A deterministic algorithm like Pisinger's 40 which is optimized specifically for the Knapsack problem can outperform MCMC. But for a given MCMC algorithm, the p-computer provides orders of magnitude improvement over the standard CPU implementation. <details> <summary>Image 3 Details</summary>  ### Visual Description ## Diagram and Chart Analysis: Knapsack Problem and Computational Efficiency ### Overview The image contains two components: 1. **Diagram (a)**: A flowchart titled "Knapsack problem" illustrating algorithmic steps. 2. **Chart (b)**: A line graph titled "Time to solution" comparing computational methods across problem sizes. --- ### Components/Axes #### Diagram (a) - **Blocks**: - **RNG**: Contains "LUT" and "LFSR" (repeated). - **Kernel**: Contains: - ΔWeight Calculation → Weight < Capacity - ΔValue Calculation → Value Function - Accept/Reject decision node. - **Flow**: - RNG → LUT/LFSR → Kernel (calculations) → Accept/Reject → Data collector. #### Chart (b) - **X-axis**: Problem size, *n* (log scale: 10² to 10⁴). - **Y-axis**: Time to solution (log scale: 10⁻⁵ to 10³ seconds). - **Legend**: - **Blue**: CPU (DP) - **Red**: GPU (DP) - **Purple (dashed)**: CPU (MCMC) - **Orange**: p-comp. emul. (MCMC) - **Brown**: CPU (Pisinger) - **Green**: p-comp. proj. (MCMC) --- ### Detailed Analysis #### Diagram (a) - **Textual Elements**: - "ΔWeight Calculation", "Weight < Capacity", "ΔValue Calculation", "Value Function", "Accept/Reject". - Arrows indicate sequential flow from RNG to Kernel to data collector. #### Chart (b) - **Data Series Trends**: 1. **CPU (DP) (Blue)**: Steep upward slope. At *n=10²*, ~10⁻³ s; at *n=10⁴*, ~10³ s. 2. **GPU (DP) (Red)**: Moderate upward slope. At *n=10²*, ~10⁻³ s; at *n=10⁴*, ~10¹ s. 3. **CPU (MCMC) (Purple, dashed)**: Slight upward curve. At *n=10²*, ~10⁻⁴ s; at *n=10⁴*, ~10⁰ s. 4. **p-comp. emul. (MCMC) (Orange)**: Flat then slight rise. At *n=10²*, ~10⁻⁵ s; at *n=10⁴*, ~10⁻² s. 5. **CPU (Pisinger) (Brown)**: Flat line. ~10⁻⁵ s across all *n*. 6. **p-comp. proj. (MCMC) (Green)**: Gradual upward curve. At *n=10²*, ~10⁻⁶ s; at *n=10⁴*, ~10⁻³ s. --- ### Key Observations 1. **CPU (DP)** exhibits exponential growth in time with problem size. 2. **p-comp. proj. (MCMC)** (green line) is the most efficient, scaling sub-linearly. 3. **CPU (Pisinger)** (brown line) remains constant, suggesting fixed computational cost. 4. **GPU (DP)** (red line) outperforms CPU (DP) but lags behind MCMC methods. --- ### Interpretation - **Diagram (a)**: The knapsack problem workflow involves random number generation (RNG), look-up tables (LUT), and lightweight feedback shift registers (LFSR) to drive kernel computations. The kernel evaluates weight/value constraints and decides whether to accept/reject solutions, feeding results to a data collector. - **Chart (b)**: Computational efficiency varies significantly by method. CPU (DP) becomes impractical for large *n* (e.g., 10⁴), while p-comp. proj. (MCMC) scales efficiently. The constant time for CPU (Pisinger) suggests a specialized optimization. - **Notable Anomaly**: The dashed purple line (CPU MCMC) initially outperforms GPU (DP) but diverges at larger *n*. This analysis highlights trade-offs between algorithmic design (diagram) and computational scalability (chart), emphasizing the importance of method selection for problem size. </details> ## C. Knapsack Problem Let us now look at a problem which requires random numbers to be generated with a probability determined by the outcome from the last sample generated by the same RNG. Every RNG then requires feedback from the very Kernel that processes its output. This belongs to the broad class of problems that are labeled as Markov Chain Monte Carlo (MCMC) . For an excellent summary and evaluation of MCMC sampling techniques we refer the reader to Ref. 41 . The knapsack is a textbook optimization problem described in terms of a set of items, m = 1 , · · N , the m th , each containing a value v m and weighing w m . The problem is to figure out which items to take ( s m = 1) and which to leave behind ( s m = 0) such that the total value V = ∑ m v m s m is a maximum, while keeping the total weight W = ∑ m w m s m below a capacity C . We could straightforwardly map it to the p -computer architecture (Fig. 1), using the RNG to propose solutions { s } at random, and the Kernel to evaluate V, W and decide to accept or reject. But this approach would take us toward the solution far too slowly. It is better to propose solutions intelligently looking at the previous accepted proposal, and making only a small change to it. For our examples we proposed a change of only two items each time. This intelligent proposal, however, requires feedback from the kernel which can take multiple clock cycles. One could wait between proposals, but the solution is faster if instead we continue to make proposals every clock cycle in the spirit of what is referred to as multiple-try Metropolis 42 . The results are shown in Fig. 3 4 and compared with CPU (Intel Xeon @ 2.3GHz) and GPU (Tesla T4 @ 1.59GHz) implementations, using the probabilistic algorithm. Also shown are two efficient deterministic algorithms, one based on dynamic programming (DP), and one due to Pisinger et al. 40,43 . Note that the probabilistic algorithm (MCMC) gives solutions that are within 1% of the correct solution, while the deterministic algorithms give the correct solution. For the Knapsack problem getting a solution that is 99% accurate should be sufficient for most real world applications. The p-computer provides orders of magnitude improvement over CPU implementation of the same MCMC algorithm. It is outperformed by the algorithm developed by Pisinger et al. 40,43 , which is specifically optimized for the Knapsack problem. However, we note that the pcomputer projection in Fig. 3 (b) is based on utilizing better hardware like s-MTJ's but there is also significant room for improvement of the p-computer by optimizing the Metropolis algorithm used here and/or by adding parallel tempering 5,6 . ## D. Ising model Another widely used model for optimization within MCMC is based on the concept of Boltzmann machines (BM) defined by an energy function E from which one can calculate the synaptic function I i $$I _ { i } = \beta ( E ( s _ { i } = 0 ) - E ( s _ { i } = 1 ) ) , \quad \ \ ( 4 )$$ that can be used to guide the sample generation from each RNG 'i' in sequence 44 according to Eq. (1). Alternatively the sample generation from each RNG FIG. 4. Example of Quantum Monte Carlo (QMC) - Transverse Field Ising model: (a) Mapping to the general p-computer framework. (b) Solving the transverse Ising model for quantum annealing. Subfigure (b) is adapted from B. Sutton, R. Faria, L. A. Ghantasala, R. Jaiswal, K. Y. Camsari and S. Datta, IEEE Access, vol. 8, pp. 157238-157252, 2020 licensed under a Creative Commons Attribution (CC BY) license 8 . <details> <summary>Image 4 Details</summary>  ### Visual Description ## Diagram: Transverse Field Ising Model and Correlation Analysis ### Overview The image contains two components: 1. **(a) Transverse Field Ising Model**: A schematic diagram illustrating the structure of a computational model for simulating the Ising model with transverse fields. 2. **(b) Correlations**: A line graph showing the decay of spin correlations as a function of lattice distance, comparing exact theoretical predictions to numerical sample averages. --- ### Components/Axes #### (a) Transverse Field Ising Model - **Diagram Elements**: - **Lattice**: A grid of 250 nodes (blue dots) representing spin sites. - **Replicas**: A smaller grid of 10 nodes (green dots) overlaid on the lattice. - **Components**: - **RNG (Random Number Generator)**: Feeds into a series of **LUT (Look-Up Tables)** and **LFSR (Linear Feedback Shift Register)** modules. - **Kernel**: Processes weighted LUT outputs via multiplication and accumulation. - **Data Collector**: Receives output from the Kernel. - **Flow**: `RNG → LUT/LFSR → Weighted LUT → Kernel → Data Collector`. #### (b) Correlations Graph - **Axes**: - **X-axis**: "Lattice distance (L)" with integer ticks from 2 to 14. - **Y-axis**: "⟨m_i^z m_{i+L}^z⟩" (spin correlation) ranging from -0.8 to 0.8. - **Legend**: - **Dashed line**: "Exact" (theoretical prediction). - **Solid dots**: "Sample average" (numerical results). --- ### Detailed Analysis #### (a) Transverse Field Ising Model - **Key Labels**: - "Replicas" (green dots) and "Lattice" (blue dots) are spatially distinct but interconnected. - **RNG** generates randomness, which is processed through **LUT** and **LFSR** modules. - **Weighted LUT** combines outputs before feeding into the **Kernel**, which multiplies and accumulates values for the **Data Collector**. #### (b) Correlations Graph - **Data Points**: - **Exact (dashed line)**: - Starts at ~0.6 for L=2, decaying exponentially to ~0.0 by L=14. - **Sample average (dots)**: - Mirrors the exact trend closely, with minor deviations (e.g., ~0.1 offset at L=2, converging to ~0.0 by L=10). - **Trend Verification**: - Both series show a **monotonic decay** of correlations with increasing lattice distance, consistent with the Ising model's expected behavior. --- ### Key Observations 1. **Correlation Decay**: - Spin correlations weaken rapidly with distance, dropping from ~0.6 (L=2) to near-zero (L≥10). 2. **Model Accuracy**: - The sample average closely tracks the exact solution, validating the model's numerical implementation. 3. **Diagram Structure**: - The RNG-LUT-LFSR-Kernel pipeline suggests a hardware/software co-design for efficient simulation. --- ### Interpretation - **Physical Insight**: The graph demonstrates that spin correlations in the Ising model decay exponentially with distance, a hallmark of critical phenomena near phase transitions. The transverse field (not explicitly shown in the graph) likely suppresses long-range order, accelerating correlation decay. - **Model Validation**: The close agreement between the sample average and exact solution confirms the model's fidelity in capturing the Ising Hamiltonian's dynamics. - **Computational Design**: The diagram in (a) implies a modular architecture where randomness (RNG) and deterministic logic (LUT/LFSR) are combined to efficiently compute spin correlations. The Kernel's role in weighting and accumulating suggests optimization for parallel processing. --- ### Spatial Grounding & Logic-Check - **Legend Placement**: Bottom-right corner, clearly associating dashed lines with "Exact" and dots with "Sample average." - **Color Consistency**: - Dashed line (gray) matches "Exact"; solid dots (blue) match "Sample average." - **Trend Logic-Check**: - The exponential decay of both series aligns with theoretical expectations for the Ising model. --- ### Content Details - **Graph Data Points**: - **Exact**: - L=2: ~0.6 | L=4: ~0.4 | L=6: ~0.2 | L=8: ~0.1 | L=10: ~0.05 | L=12: ~0.02 | L=14: ~0.01. - **Sample Average**: - L=2: ~0.5 | L=4: ~0.35 | L=6: ~0.15 | L=8: ~0.05 | L=10: ~0.0 | L=12: ~-0.02 | L=14: ~-0.01. --- ### Final Notes - **Language**: All text is in English. - **Missing Elements**: No explicit mention of transverse field strength or temperature parameters in the diagram/graph. - **Anomalies**: Minor discrepancies between sample average and exact values at small L (e.g., L=2), likely due to finite-size effects or sampling noise. </details> can be fixed and the synaptic function used to decide whether to accept or reject it within a MetropolisHastings framework 45 . Either way, samples will be generated with probabilities P α ∼ exp ( -βE α ). We can solve optimization problems by identifying E with the negative of the cost function that we are seeking to minimize. Using a large β we can ensure that the probability is nearly 1 for the configuration with the minimum value of E. In principle, the energy function is arbitrary, but much of the work is based on quadratic energy functions defined by a connection matrix W ij and a bias vector h i (see for example 8,10-17 ): $$E = - \sum _ { i j } W _ { i j } s _ { i } s _ { j } - \sum _ { i } h _ { i } s _ { i } , \quad ( 5 ) \quad o r a$$ For this quadratic energy function, Eq. (4) gives I i = β ( ∑ j W ij s j + h i ) , so that the Kernel has to perform a multiply and accumulate operation as shown in Fig. 4 (a). We refer the reader to Sutton et al. 8 for an example of the max-cut optimization problem on a two-dimensional 90 × 90 array implemented with a pcomputer . Eq. (4), however, is more generally applicable even if the energy expression is more complicated, or given by a table. The Kernel can be modified accordingly. For an example of a energy function with fourth order terms implemented on an eight bit p-computer , we refer the reader to Borders et al. 30 . A wide variety of problems can be mapped onto the BM with an appropriate choice of the energy function. For example, we could generate samples from a desired probability distribution P , by choosing βE = -nP . Another example is the implementation of logic gates by defining E to be zero for all { s } that belong to the truth table, and have some positive value for those that do not 24 . Unlike standard digital logic, such a BM-based implementation would provide invertible logic that not only provides the output for a given input, but also generates all possible inputs corresponding to a specified output 24,46,47 . ## E. Quantum Monte Carlo Finally let us briefly describe the feasibility of using p-computers to emulate quantum or q-computers. A qcomputer is based on qubits that are neither 0 or 1, but are described by a complex wavefunction whose squared magnitude gives the probability of measuring either a 0 or a 1. The state of an n -qubit computer is described by a wavefunction { ψ } with 2 n complex components, one for each possible configuration of the n qubits. In gate-based quantum computing (GQC) a set of qubits is placed in a known state at time t , operated on with d quantum gates to manipulate the wavefunction through unitary transformations [ U ( i ) ] $$\begin{array} { r l r l } & { a p ^ { - } } & \\ & { \left \{ \psi ( t + d ) \right \} = [ U ^ { ( d ) } ] \cdots \cdots [ U ^ { ( 1 ) } ] \{ \psi ( t ) \} } & & { ( G Q C ) \ ( 6 ) } \\ \end{array}$$ and measurements are made to obtain results with probabilities given by the squared magnitudes of the final wavefunctions. From the rules of matrix multiplication, the final wavefunction can be written as a sum over a very large number of terms: $$\psi _ { m } ( t + d ) = \sum _ { i , \cdot \cdot j , k } U _ { m , i } ^ { ( d ) } \cdots \cdots U _ { j , k } ^ { ( 1 ) } \, \psi _ { k } ( t ) \quad ( 7 )$$ Conceptually we could represent a system of n qubits and d gates with a system of ( n × d ) p-bits with 2 nd states which label the 2 nd terms in the summation in Eq.(7) 48 . Each of these terms is often referred to as a Feynman path and what we want is the sum of the amplitudes of all such paths: $$\psi _ { m } ( t + d ) = \sum _ { \alpha = 1 } ^ { 2 ^ { n d } } A _ { m } ^ { ( \alpha ) }$$ The essential idea of quantum Monte Carlo is to estimate this enormous sum from a few suitably chosen samples, not unlike the simple Monte Carlo stated earlier in Eq. (2). What makes it more difficult, however, is the socalled sign problem 19 which can be understood intuitively as follows. If all the quantities A ( α ) m are positive then it is relatively easy to estimate the sum from a few samples. But if some are positive while some are negative with lots of cancellations, then many more samples will be required. The same is true if the quantities are complex quantities that cancel each other. The matrices U that appear in GQC are unitary with complex elements which often leads to significant cancellation of Feynman paths, except in special cases when there may be complete constructive interference. In general this could make it necessary to use large numbers of samples for accurate estimation. A noiseless quantum computer would not have this problem, since qubits intuitively perform the entire sum exactly and yield samples according to the squared magnitude of the resulting wavefunction. However, real world quantum computers have noise and p-computers could be competitive for many problems. Adiabatic quantum computing (AQC) operates on very different physical principles but its mathematical description can also be viewed as summing the Feynman paths representing the multiplication of r matrices: $$[ e ^ { - \beta H / r } ] \cdots \, \cdot \cdot [ e ^ { - \beta H / r } ] \quad ( A Q C ) \quad ( 9 )$$ This is based on the Suzuki-Trotter method described in Camsari et al. 49 , where the number of replicas, r , is chosen large enough to ensure that if H = H 1 + H 2 , one can approximately write e -βH/r ≈ e -βH 1 /r e -βH 2 /r . The matrices e -βH/r in AQC are Hermitian, and their elements can be all positive 50 . A special class of Hamiltonians H having this property is called stoquastic and there is no sign problem since the amplitudes A ( α ) m in the Feynman sum in Eq. (8) all have the same sign. An example of such a stoquastic Hamiltonian is the transverse field Ising model (TFIM) commonly used for quantum annealing where a transverse field which is quantum in nature is introduced and slowly reduced to zero to recover the original classical problem. Fig. 4 adapted from Sutton et al. 8 , shows a n = 250 qubit problem mapped to a 2-D lattice of 250 × 10 = 2500 p-bits using r = 10 replicas to calculated average correlations between the z-directed spins on lattice sites separated by L . Very accurate results are obtained using N s = 10 5 samples. However,these samples were suitably spaced to ensure their independence, which is an important concern in problems involving feedback. Finally we note that quantum Monte Carlo methods, both GQC and AQC, involve selective summing of Feynman paths to evaluate matrix products. As such we might expect conceptual overlap with the very active field of randomized algorithms for linear algebra 51,52 , though the two fields seem very distinct at this time. ## IV. CONCLUDING REMARKS In summary, we have presented a generic architecture for a p-computer based on p-bits which take on values 0 and 1 with controlled probabilities, and can be implemented with specialized compact energy-efficient hardware. We emulate systems with thousands of p-bits to show that they can significantly accelerate the implementation of randomized algorithms that are widely used for many applications 53 . A few prototypical examples are presented such as Bayesian networks, optimization, Ising models and quantum Monte Carlo. ## ACKNOWLEDGMENTS The authors are grateful to Behtash Behin-Aein for helpful discussions and advice. We also thank Kerem Camsari and Shuvro Chowdhury for their feedback on the manuscript. The contents are based on the work done over the last 5-10 years in our group, some of which has been cited here, and it is a pleasure to acknowledge all who have contributed to our understanding. This work was supported in part by ASCENT, one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program sponsored by DARPA. ## DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. ## AUTHOR DECLARATIONS ## Conflict of interests One of the authors (SD) has a financial interest in Ludwig Computing. - 1 R. P. Feynman, 'Simulating physics with computers,' Int. J. Theor. Phys 21 (1982). - 2 B. A. Cipra, 'The Best of the 20th Century: Editors Name Top 10 Algorithms,' SIAM News 33 , 1-2 (2000). - 3 K. Y. Camsari, S. Salahuddin, and S. Datta, 'Implementing pbits with Embedded MTJ,' IEEE Electron Device Letters 38 , 1767-1770 (2017). - 4 J. Kaiser, R. Jaiswal, B. Behin-Aein, and S. 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