Noise-augmented chaotic Ising machines for combinatorial optimization and sampling

Abstract

Ising machines are hardware accelerators for combinatorial optimization and probabilistic sampling, using stochasticity to explore spin configurations and avoid local minima. We refine the previously proposed coupled chaotic bits (c-bits), which operate deterministically, by introducing noise. This improves performance in combinatorial optimization, achieving algorithmic scaling comparable to probabilistic bits (p-bits). We show that c-bits follow the quantum Boltzmann law in a 1D transverse field Ising model. Furthermore, c-bits exhibit critical dynamics similar to p-bits in 2D Ising and 3D spin glass models. Finally, we propose a noise-augmented c-bit approach via the adaptive parallel tempering algorithm (APT), which outperforms fully deterministic c-bits running simulated annealing. Analog Ising machines with coupled oscillators could draw inspiration from our approach, as running replicas at constant temperature eliminates the need for global modulation of coupling strengths. Ultimately, mixing stochasticity with deterministic c-bits yields a powerful hybrid computing scheme that can offer benefits in asynchronous, massively parallel hardware implementations.

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communications physics Article
https://doi.org/10.1038/s42005-025-01945-1
Noise-augmented chaotic Ising machines
for combinatorial optimization and
sampling
Check for updates
Kyle Lee , Shuvro Chowdhury & Kerem Y. Camsari
Ising machines are hardware accelerators for combinatorial optimization and probabilistic sampling,
using stochasticity to explore spin configurations and avoid local minima. We refine the previously
proposed coupled chaotic bits (c-bits), which operate deterministically, by introducing noise. This
improves performance in combinatorial optimization, achieving algorithmic scaling comparable to
probabilistic bits (p-bits). We show that c-bits follow the quantum Boltzmann law in a 1D transverse
field Ising model. Furthermore, c-bits exhibit critical dynamics similar to p-bits in 2D Ising and 3D
spin glass models. Finally, we propose a noise-augmented c-bit approach via the adaptive parallel
tempering algorithm (APT), which outperforms fully deterministic c-bits running simulated annealing.
Analog Ising machines with coupled oscillators could draw inspiration from our approach, as running
replicas at constant temperature eliminates the need for global modulation of coupling strengths.
Ultimately, mixing stochasticity with deterministic c-bits yields a powerful hybrid computing scheme
that can offer benefits in asynchronous, massively parallel hardware implementations.
As Moore’s law stagnates, there is growing interest in unconventional
computing schemes and hardware accelerators. Among these emerging
technologies are Ising machines1, employing devices such as probabilistic
bits (p-bits)2–5, coupled oscillators6–12, photonic devices13–16,andother
nonlinear elements17–28. Ising machines have traditionally focused on pro-
viding scaling and prefactor improvementsinsolvingcombinatorialopti-
mization problems29–31. Efforts to link Ising machines to generative machine
learning32–36 and quantum simulation problems are also underway37–39.
Here,weevaluateandimproveanearlierIsingmachineconceptbasedon
chaotic bits (c-bits), originally proposed in ref. 40. Chaotic bits resemble
coupled oscillators, following a set of deterministic neuron update rules
without explicit randomness. However, unlike oscillator-based Ising
Machines that encode the Ising spin in the continuous phase of oscillators,
c-bits make use of oscillating billiard balls that set a latch to a +1or −1 state.
As such, chaotic bits do not require cumbersome subharmonic injection
locking schemes to binarize naturally continuous phase variables.
The potential hardware implementation of chaotic Ising machines
without any explicit random number generators while retaining similarity
to the mathematics of p-bits is appealing. However, as we demonstrate in
this paper, c-bits without any randomness cannot straightforwardly employ
the most powerful Monte Carlo algorithms, such as Adaptive Parallel
Tempering, which offer better algorithmic scaling for the Circuit SAT and
3D spin glass problems. We propose and evaluate a hybrid computing
scheme wherein deterministic c-bits are augmented with stochasticity for
improved performance in combinatorial optimization and sampling. The
approach we propose may be applied to similar devices, such as certain
classes of coupled oscillators, where a continuous global modulation of
analog coupling elements may be inconvenient.
Theoretical work has shown convergence to the Boltzmann dis-
tribution for simple toy models of 2 c-bits41. However, it has not been
determined that larger c-bit networks necessarily sample from the
Boltzmann distribution as stochastic p-bits do. Experimental work sug-
gests that c-bit networks seem to closely approximate the Boltzmann
distribution for 2D Ising models of smallsizes42. Furthermore, it has been
shown that c-bits match stochastic p-bits for equilibrium statistics on the
2D ferromagnetic Ising model40 and the Potts model42. c-bits have been
used to determine the critical temperature (T
c
) and critical exponents (ν,
β, and γ) of the Ising universality class42. Simulated annealing has been
demonstrated with c-bits40,43. Additionally, CMOS and analog VLSI
implementations have been proposed44,45. Nonetheless, the theoretical
equivalence of p-bits to c-bits without explicit noise is far from clear.
Moreover, the initial phase randomization and rounding errors in the
solution of ODEs (or physical noise in real systems) might be responsible
for this striking correspondence.
Here, we subject c-bits to more stringent tests on three important
problem classes, comparing their performance to p-bits. These tests
Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA, USA. e-mail: kylelee@ucsb.edu;
camsari@ece.ucsb.edu
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involve a quantum problem, criticality in 3D spin glasses, and hard
combinatorial optimization in the form of integer factorization with
invertible logic gates. We conclude that while c-bits perform similarly to
p-bits in algorithms such as simulated annealing, c-bits also benefitfrom
added stochasticity, particularly through random swaps used in powerful
tempering algorithms.
Results and discussion
Probabilistic computers perform a discrete Markov Chain Monte Carlo
(MCMC) algorithm, simulated in software via Gibbs sampling (Fig. 1c).
p-bit networks are typically described by second-order interactions in
energy and stochastic neurons:
mi¼sgn ½tanhðβIiÞranduð1;1Þ ð1Þ
Ii¼X
j
Jijmjþhið2Þ
where m
i
∈{−1, +1} and rand
u
(−1, 1) is a random uniform distribution
between −1and1(Fig.1a, c). βis inverse temperature. J
ij
and h
i
represent
weights and biases. At equilibrium, p-bit networks sample from the
Boltzmann distribution, given by:
pðfmgÞ ¼ 1
Zexp βEðfmgÞ
 ð3Þ
EðfmgÞ ¼ 1
2X
i;j
JijmimjX
i
himið4Þ
where {m}isaspinconfiguration, Zis the partition function, and E({m}) is
the energy of a spin configuration. Chaotic bits follow deterministic update
rules, inspired by p-bit equations, which model an oscillating billiard ball
and a latch. We reformulate the original chaotic bit equations40,42 to
emphasize their similarity to p-bits:
dxi
dt ¼miþtanhðβIiÞ
xi¼þ1)set mito þ1
xi¼1)set mito 1ð5Þ
where x
i
represents the position of a billiard between the boundaries
x
i
=−1 and x
i
=+1. The billiard has a tunable slope, dx
i
/dt,whichis
dependent on the input I
i
. When the billiard x
i
reaches −1or +1, the
c-bit state m
i
is latched to that value, and the billiard changes direction
(Fig. 1b). Note that in numerical or hardware implementations of c-bits,
the billiard state x
i
in Eq. (5) can slightly exceed +1or −1 due to
discretization errors.
The c-bit definition can be motivated from the Boltzmann
distribution, analogous to p-bits and Gibbs sampling. Consider a
single spin m
i
with a fixed set of neighbors. For p-bits, combining
Eq. (2) and Eq. (3), the conditional probability of m
i
being ±1 is
expressed as:
pðmi¼þ1Þ¼ 1
Zexp½βðIiErestÞ
pðmi¼1Þ¼1
Zexp½βðIiErestÞ ð6Þ
where Zis the partition function and E
rest
is the part of the global energy (or
the environment) that is not dependent on p-bit i. When expressed as a ratio
of probabilities, Eq. (6) becomes:
pðmi¼þ1Þ
pðmi¼1Þ¼expð2βIiÞð7Þ
which describes the behavior of a single p-bit given fixed neighbors.
Fig. 1 | Overview of probabilistic bits and chaotic bits. a A probabilistic bit (p-bit)
contains a random number generator with uniform distribution between −1 and 1.
tanhðIiÞis a threshold for latching m
i
.bA chaotic bit (c-bit) features deterministic
billiard dynamics with a tunable slope. The billiard is periodic, given that I
i
is held
constant. When the billiard reaches −1or +1, m
i
is latched, and the billiard changes
direction. cp-bits and c-bits form similar, asynchronous architectures, employing
the same synaptic function. dA 20-bit factorizer conceptualized with p-bits,
represented as a graph with 310 nodes and 1200 edges. eThe 3D spin glass problem,
with coupling strengths J
ij
∈{−1, +1}. fThe adaptive parallel tempering (APT)
algorithm. Replicas of the same network run at different inverse temperature βin
parallel. Swap attempts are made between pairs of replicas with the closest β,
according to the Metropolis criterion (Eq. (17)).
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As a deterministic unit, c-bit updates do not involve probabilities, but
rather they try to achieve the same sampling ratio of +1’sto −1’sby
modifying the lifetime of up and down trajectories of the billiard ball,
conditioned on the state of its neighbors (Fig. 1b):
τðmi¼þ1Þ
τðmi¼1Þ¼
dxi
dt when mi¼1

dxi
dt when mi¼þ1

¼expð2βIiÞð8Þ
where τis the lifetime of a c-bit at its current state (m
i
= ±1). In other words,
c-bits are defined such that the time average of the c-bit state
hmii¼tanhðβIiÞ,analogoustoap-bit.Thekeypointisthatτare
deterministic, unlike a p-bit whose state is stochastically sampled at each
update.
The comparison of p-bits (Eq. (1)andEq.(2)) with c-bits (Eq. (5)and
Eq. (2)) is fundamental. In the case of p-bits, the fundamental theorem of
Markov Chains46,47 ensures that a network of p-bits updated sequentially via
Gibbs sampling will eventually reach the Boltzmann distribution, defined by
Eq. (3). Ultimately, this is what enables mapping problems of interest to
p-bit networks and then solving them with algorithms such as simulated
annealing and parallel tempering. Networks of c-bits cannot be proved to
sample exactly from the Boltzmann distribution due to the lack of prob-
abilistic and serial updates.
Note the following parallel to p-bits. It is well-known that in Gibbs
sampling, simultaneously updating connected nodes leads to pathological
oscillations48,49, preventing easy parallelization of the chain. For c-bits, it
seems that the differential equation formulationincontinuoustimepre-
vents simultaneous updates naturally, as long as the time difference between
subsequent billiard ball arrivals is greater than the time it takes to update a
latch. When c-bits are discretized in software, latches are set instantaneously
within a single time step, Δt, chosen to solve the differential equations.
To better understand how critical serial updating is for c-bits, we
construct a simple example, with 5 c-bits (or 5 p-bits), interconnected such
that their low energy states represent the truth table of a full adder, given by
the interactions50,51:
JFA ¼
0111 2
1011 2
110 1 2
11102
22220
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
with no bias terms (h= 0). The columns correspond to carry in, A, B, sum
and carry out bits, respectively. Figure 2a shows the analytically obtained
Boltzmann distribution from Eq. (3)atβ= 0.5, which can be numerically
approximated by sequential p-bit updates.
Figure 2b shows the probability distribution of 32 possible spin con-
figurationsina5p-bitfulladderwhenp-bitsareupdatedinparallelrather
than sequentially. The steady-state distribution corresponding to this
Markov chain can be obtained analytically5,47 and describes the numerical
sampling (shown in Fig. 2c) well.
Figure 2d shows that similar pathological oscillations between states 3
and 28 are observed in c-bit networks when the initial states are synchro-
nized by making all x
i
and m
i
equal in the beginning. These oscillations go
away if the initial phases are sufficiently separate and the integration step, Δt
is small. We conclude that c-bit networks implicitly rely on sequential
updating for correct operation, despite their continuous time and massively
parallel dynamics. While simultaneous c-bit updates in an integration step
Δtmay be considered an artifact of software discretization, hardware
implementations will have similar restrictions due to finite setup/hold times
and synapse delays associated with calculating I
i
.Inthe“Possible hardware
realizations”subsection of “Results and discussion,”we brieflydiscussthe
implications of this update mechanism for eventual hardware
implementations.
Note that p-bits and c-bits have different notions of time. There is no
clear correspondence between a p-bit Monte Carlo sweep (MCS), in which
Eqs. (1)and(2) are sequentially solved for each p-bit in the network, and the
time variable tof a c-bit network. In this study, we sample the state {m}ofa
continuously updating c-bit network at natural numbers t= {0, 1, 2, …,t
a
}.
Because of these different notions of time, in order to draw comparisons
between p-bit and c-bit algorithms, we shall employ power law and expo-
nential scaling arguments, which hold independent of prefactors. Further-
more, we apply similar algorithmic parameters to both p-bits and c-bits to
ensure fair comparisons.
Most existing work employs a definition of c-bit billiard velocity
(dx
i
/dt) that grows exponentially at cold temperatures (β≫1), with the
exception of one previous study42. This is problematic because optimi-
zation algorithms such as simulated annealing and adaptive
parallel tempering require cold temperatures which may result in
unbounded slopes. In this study, we employ a c-bit definition that has a
maximum billiard speed of 2 (see Eq. (5)), suitable for software and
hardware implementation. Unless otherwise specified, we employ
Euler’s method with a step of Δt= 0.1 to simulate the dynamics of
coupled c-bits.
Our bounded c-bit choice is also a potential improvement over pre-
vious hardware implementations, which may be suffering from accuracy
losses due to exponentially growing slopes. For example, the anomalous
increaseofenergyplotsinapriorFPGAimplementationislikelyduetothe
overflows and underflowsinslopes(ref.43).
Finally, in all of our experiments, the initial phases of the billiard balls
and the states of the latches are randomized. The asynchronous dynamics of
the billiard balls, coupled with this phase randomization ensures non-
simultaneous updates with quasi-random arrivals. In addition, rounding
errors and the choice of a naive integration scheme provide additional noise
in our simulations. We believe these seem to be the key reasons behind the
near-equivalence of c-bits to p-bits, where explicit and implicit inclusion of
noise seems to play a central role.
Sampling from the 1D transverse field Ising Hamiltonian
with c-bits
As a difficult sampling problem that requires high-quality pseudorandom
number generators52, we consider a 1D ferromagnetic chain of 8 qubits at
constant inverse temperature β= 10. This system is described by the
transverse Ising Hamiltonian in 1D,
HQ¼X
i<j
Jijσz
iσz
jX
i
hiσz
iΓxX
i
σx
ið9Þ
where σx
iand σz
iare Pauli spin matrices at site i.Notethateachoftheseis
2N×2
Nmatrix and is computed as σα
i¼Ii1
2σαINi
2¼I2
...σα... I2with α∈{x,z}where ⊗denotes the ‘Kronecker
product’,I
2
is the 2 × 2 identity matrix, σzand σxare Pauli spin matrices (only
at the i-th term of the product):
σz¼10
01

and σx¼01
10
 ð10Þ
The interaction weight J
ij
=+2fornearestneighbors(withaperiodic
boundary condition) and J
ij
=0otherwise.h
i
=+1, representing a symmetry
breaking magnetic field in the þ^
zdirection. The Suzuki-Trotter
transformation37,53 maps the 1D quantum Hamiltonian to the following
2D classical Hamiltonian:
HC¼X
R
k¼1X
i<j
Jij
Rmi;kmj;kþX
i
hi
Rmi;kþX
i
J?mi;kmi;kþ1
"#
ð11Þ
where J?¼1=2βln tanh βΓx=R

.Rrepresents the total number of
system replicas after applying the quantum-to-classical mapping. Note the
unusual dependence of this coupling to the inverse temperature, β.
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Foraninfinite number of replicas (R→∞), this quantum-to-classical
mapping is exact. In practice, theerror scalesas Oð1=R2Þfor a given β. Here,
we employ 250 replicas, thus our system contains N=8qubitsandR= 250
replicas, totaling 2000 classical spins. We simulate our classical system using
c-bits. As we vary the value of the transverse field Γ
x
, we measure the average
magnetization:
hmzi¼ 1
NR X
R
k¼1X
N
i¼1
mi;kð12Þ
In theory, the system should obey the quantum Boltzmann Law:
hmzi¼tr expðβHQÞPiσz
i

tr expðβHQÞ
 ð13Þ
Piσz
iin the numerator is the corresponding quantum operator for
measuring spins along the z-direction. Notice that the argument of the
‘exp (⋅)’in Eq. (13) is a matrix. Evaluating this equation becomes
intractable for large N, as it involves taking the exponential of a 2N×2
N
matrix. When Nis small like the system in our example, one can compute
exponential of a matrix directly by explicitly writing down H
Q
and
calculating its exponential. Also, to obtain numerically stable results at
low temperatures (high β), we first diagonalize the Hamiltonian and
subtract the ground-state energy from the diagonals which does not
affect any observable quantities.
Figure 3shows that numerical results from c-bits exhibit strong
agreement with the quantum Boltzmann law, obtained by solving Eq. (13)as
afunctionofΓ
x
.Asimilarresulthasbeenshowninapreviousworkusing
stochastic p-bits54. We observe that c-bits appear to show comparable
performance to p-bits, despite the fact that c-bits only useexplicitrando-
mization for the initialization of spins and phases at the start of each
simulation.
Critical scaling dynamics: c-bits vs p-bits
Next, we discuss the critical dynamics of c-bit networks for the 2D Ising
model and the 3D spin glass problem. The idea is to quench c-bit networks
at varying anneal rates and measure the residual energy as a proxy of
topological defects. The density of these defects is qualitatively predicted by
the Kibble-Zurek Mechanism (KZM)55, which relates equilibrium correla-
tion lengths and relaxation times to defect densities borne out of non-
equilibrium dynamics. In the present context, the study of KZM allows a
comparison of c-bit performance with that of p-bits. In addition, residual
energy may be a useful figure of merit for combinatorial optimization where
scaling comparisons between different algorithms can be made56.The
residual energy density at the critical point is defined as:
ρc
E¼hEEci
nð14Þ
where E
c
is equilibrium energy at the critical temperature and nis the
number of spins in the system. From the theory of phase transitions, phe-
nomenological scaling arguments can be made to describe the residual
energy in terms of an anneal with varying velocities (see Supplementary
Note 1 for details):
ρc
E/ακcð15Þ
where κ
c
is the exponent describing the scaling behavior of the residual
energy density at the critical point, and αis the velocity of the anneal
measured at the critical temperature for any schedule that starts from some
initial temperature and ends at the critical temperature:
α¼dλðtÞ
dt t¼tað16Þ
One way to make contact with known critical exponents and our numeri-
cally calculated exponents is to measure the number of defects via kinks57 in
Fig. 2 | Oscillatory behavior of parallel updates. a Distribution of spin config-
urations for a 5 p-bit full adder, obtained analytically from the Boltzmann dis-
tribution. bAnalytical distribution for p-bits employing parallel updates, as opposed
to sequential Gibbs sampling5.cNumerical results for parallel p-bit updates, col-
lected over 105Monte Carlo sweeps. dc-bits exhibit similar pathological behavior
when their phases x
i
and states m
i
are all initialized to the same value, +1. Samples
are collected over 105time steps.
Fig. 3 | Chaotic bits emulate a 1D transverse field Ising model. Chaotic bits
emulate a 1D ferromagnetic chain (J
ij
=+2) of 8 qubits described by the quantum
transverse Ising Hamiltonian (Eq. (9)). For emulation purposes, we use Suzuki-
Trotter decomposition with R= 250 replicas. We show average magnetization as a
function of the transverse field (Γ
x
) when the system is at constant inverse tem-
perature β= 10. A symmetry breaking magnetic field of h
i
= 1 is applied in the þ^
z
direction such that when Γ
x
= 0, all spins point in the þ^
zdirection. Red dots show the
results of 80 independent trials of c-bit simulations, and a single set of spin con-
figurations is recorded at the end of t
a
= 1000 time steps, for each trial. Blue triangles
show the average numerical result. The green dashed line is an analytical solution
from solving Eq. (13) as a function of Γ
x
.
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Ising models, but our purpose here is to provide a comparative analysis with
p-bits and c-bits, so direct verification of the exponents is not necessary.
First, we consider a 50 × 50 ferromagnetic Ising model lattice with
nonperiodic boundary conditions (J
ij
=+1). For a given annealing time t
a
,β
is geometrically increased from β= 0.001 to the critical temperature,
β¼Tc1. For this problem, we use the exact critical temperature calculated
at the thermodynamic limit58 as an approximate T
c
for the finite-size model:
Tc¼2=lnð1þffiffiffi
2
pÞ2:269. When annealing terminates at the critical
point, the energy of the spin configuration is recorded to calculate ρc
E.We
average over 3000 randomized trials.
Although there is no obvious correspondence between a Monte Carlo
sweep for behavioral models of p-bits and the time step of a c-bit, power-law
exponents can be compared objectively since they do not depend on pre-
factors. Figure 4a shows that c-bits and p-bits demonstrate similar critical
scaling dynamics on the 2D ferromagnetic Ising model. Linear regression is
performed on the 5 leftmost points in order to avoid finite-size effects. Both
p-bit and c-bit curves exhibit downward concavity, but different time scales
mean that this concavity does not manifest at the same time t
a
for p-bits and
c-bits. We provide the results of using Euler’smethodinstepsofΔt=0.1and
Δt= 0.01. The c-bit curve using Δt= 0.01 appears to follow a similar power-
law scaling but with a steeper slope ∣κ
c
∣. The reason for this discrepancy is not
clear. On the one hand, smaller Δtimplies higher theoretical solver preci-
sion, while on the other hand, smaller Δtleads to more round-off errors due
to double precision. It must also be noted that a smaller Δtrequires more
computational effort, and in hardware, Δtmay have a lower bound dictated
by the physical latch required to set the state of the oscillator.
Figure 4b shows a similar experiment conducted on select instances of
the 3D spin glass problem, studied by D-Wave56. We consider 300 different
spin glass instances on a cubic lattice with side length L=11and
J
ij
∈{−1, +1}. Analogous to Fig. 4a, for a given annealing time t
a
,we
geometrically increase βto the inverse critical temperature 1/T
c
.For3Dspin
glasses, T
c
is approximately calculated as ≈1.159. For each of 300 problem
instances, we take an ensemble average over 40 random seeds. We again
observe similar power-law scaling. Our p-bit results match the numerical
results in the literature, with κ
c
≈0.5156. However, the values of κ
c
do not
seem to be similar when comparing between p-bits and c-bits. The Δt
dependence of c-bit slopes also is an indication that the underlying
dynamics for c-bits might depend on separating near arrivals that latch a
Fig. 4 | Critical scaling dynamics on the 2D Ising and 3D spin glass models.
aResidual energy density at the critical point (ρc
E) as a function of simulated
annealing (SA) duration for a 50 × 50 ferromagnetic Ising model lattice with non-
periodic boundary conditions (J
ij
=+1). We perform linear regression on the 5
leftmost points of each data series to avoid finite-size effects. Error bars show 95%
bootstrap confidence intervals. Purple squares represent p-bits, yellow triangles
show c-bits (Δt= 0.1), and green circles show c-bits (Δt= 0.01). Time is measured in
Monte Carlo sweeps for p-bits and time is dimensionless for c-bits. The inset plot
shows short-term exponential divergence of c-bit systems with slightly perturbed
initial conditions. The slope represents a positive (maximum) Lyapunov exponent,
implying chaotic behavior. bρc
Evs. annealing duration for cubic spin-glass instances
(L= 11, J
ij
∈{−1, +1}). cρc
Eas a function of inverse annealing velocity for the 2D
ferromagnetic Ising model, using p-bits. Different annealing schedules are shown in
a data collapse by considering the quench rate at the critical point, α.dρc
Evs. inverse
annealing velocity for the 2D ferromagnetic Ising model, using c-bits.
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state (where Eq. (2)isalwaysassumedtobeinfinitely fast). We conclude that
while c-bits seem to obey a similar power law as p-bits, their dynamics seem
different in subtle ways.
One feature of chaotic dynamics is sensitivity to initial conditions. This
can be tested by calculating Lyapunov exponents, which involves measuring
the distance between a chosen initial condition and a slightly perturbed
version of it as a function of time. If the distance in phase space grows
exponentially in time, this is an indication of chaotic dynamics. In general,
there is a spectrum of Lyapunov exponents, but the dynamics are dominated
by the maximum Lyapunov exponent, which can be empirically measured.
In the inset plots of Fig. 4a and b, we show maximum Lyapunov exponents
for the 2D Ising model and the 3D spin glass problem, respectively. We
observe that in both cases the log distance acquires a linear slope and then
saturates. The linear and positive slope indicates chaotic dynamics before
saturating. The saturation of the distance is indicative of the finite size of the
phase space where a distance between two trajectories has an upper bound.
The specifics of the calculation are as follows: the phases xand the states
mare randomly initialized. We then introduce a perturbation in the initial
conditions by offsetting the phase x
i
of a randomly selected c-bit by a
magnitudeof0.1.WethenconsidertheEuclideandistancebetweenthe
phases of the perturbed and unperturbed systems: D¼jjxx0jj,wherex0
represents the perturbed system’s phases. The log Euclidean distance is
plotted over 200 time steps, with the system at constant inverse temperature
β= 1. The slope represents the Lyapunov exponent λ,asDexp λt.The
results are averaged over 100 trials, where each trial compares one unper-
turbed system xwith 10 perturbed versions, x0.
In Fig. 4c, d, we show how both c-bits and p-bits exhibit a clear power-
law relationship with critical exponents by comparing different annealing
schedules with different initial temperatures. As Eq. (15) predicts, the
simulations should exhibit universal behavior if plotted as a function of how
fast the anneal is performed near the critical point. When plotted as function
of annealing velocity, α(Eq. (16)), all c-bit and p-bit plots fall on top of each
other, demonstrating an excellent collapse. The different exponents of c-bits
and p-bits and the time-step dependence of the c-bit networks indicate that
c-bit networks may not be a drop-in replacement for p-bit networks whose
steady-state provably takes samples from the Boltzmann distribution47.
Adaptive parallel tempering
Algorithm 1.Adaptive Parallel Tempering with p-bits or c-bits
Critical scaling dynamics are intriguing from a physics perspective, but it is
notessentialthatc-bitsobeythesamephysicsasp-bitsinanoptimization
context. For optimization, we use an adaptive version of the parallel tem-
pering algorithm, abbreviated as APT60–62. A given p-bit or c-bit network is
duplicated into Nsystem replicas, with each replica at a different inverse
temperature β(Fig. 1f). In fixed time intervals (time steps per swap attempt),
a probabilistic swap attempt is conducted between adjacent replica pairs
according to the Metropolis criterion:
PðswapÞ¼min 1;expðΔβΔEÞ
ð17Þ
If this condition is met, then the adjacent replicas swap their spin states
{m}. Alternatively, replicas may swap their temperature values with a
rearrangement of replica indices, which may be more convenient in
dedicated hardware implementations. Intuitively, hot replicas (small β)
explore the state space of possible spin states {m}. The Metropolis cri-
terion makes low energy configurations more likely to swap to cold
replicas (high β). Over time, the coldest system replica tends toward the
ground energy, E
0
.
The APT algorithm can be applied using either p-bits or c-bits. While
theneuronupdateruleisstochasticforp-bits(Eq.(1)), it is deterministic for
c-bits (Eq. (5)). Replica swaps via the Metropolis criterion are always
probabilistic (Eq. (17)). Therefore, while p-bit APT is a purely probabilistic
algorithm, c-bit APT is a hybrid chaotic-probabilistic scheme, where neuron
updates are deterministic and replica-swap attempts are probabilistic. In
typical p-bit APT, the great majority of the random numbers are used for
neuron updates rather than replica-swap attempts. In contrast, c-bit APT
requires orders of magnitude fewer random numbers. However, this
potential benefit critically depends on the hardware cost of implementing c-
bits, which may be greater than that of p-bits. Given the relative ease of
creating pseudo random number generators through compact linear-
feedback-shift-register circuits63, we suspect that c-bits may surreptitiously
contain PRNG-like circuits in their implementation. While p-bits and c-bits
appear to be similar in their computational power, it is important to note
that only p-bits enjoy a mathematical certainty that they sample from the
Boltzmann distribution at steady state.Wefurtherdiscusstheseideasinthe
“Possiblehardwarerealizations”subsection of “Results and discussion.”
The algorithm is referred to as ‘adaptive’becauseweemployan
instance-specific preprocessing method that finds a suitable βschedule for a
given problem. This process ensures that replica-swap probabilities are
nearly uniform, avoiding bottlenecks during the exchange process (see
Supplementary Note 2). The complete APT algorithm is summarized in
pseudo code (Algorithm 1).
Optimization of the 3D spin glass problem
We consider 300 spin glass instances on a cubic lattice with side length L=7
and J
ij
∈{−1, +1}, borrowed from ref. 56. Purely probabilistic APT (p-bits)
and hybrid chaotic-probabilistic APT (c-bits) are performed using similar
parameters. We conduct APT preprocessing on 1 spin glass instance,
returning a βschedule for 14 system replicas. In principle, to achieve the best
performance, one could apply the APT preprocessing to each of 300 dif-
ferent problem instances and obtain 300 unique βschedules. Since this is a
comparative study between p-bits and c-bits, we forego this step. Tuning
parameters is not of particular concern.
Foragivensimulationtimet
a
, each of 300 spin glass instances is
optimized using APT. For each simulation, we record the lowest energy E
sampled across all replicas. We introduce residual energy as an optimization
performance metric, defined as:
ρf
E¼hEE0i
nð18Þ
where E
0
is the ground-state energy and nis the number of spins in the
system. Figure 5shows ρf
Evs. computational time, where the time axis
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Communications Physics | (2025) 8:35 6
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accounts for the total time steps summed across all N=14replicasthatrun
in parallel. We take an ensemble average over 40 random seeds that
randomize initial states (for p-bits and c-bits) and phases (for c-bits).
We devise a similar experiment for simulated annealing. In order to
draw a comparison to APT, simulated annealing is conducted 14 times in
parallel, with βincreasing linearly from 0 to 10 during simulation time t
a
.
We record the lowest energy sampled across all replicas to compute the
residual energy ρf
E.Wetakeanensembleaverageover40randomseeds.
To compare p-bit and c-bit performance, we employ similar power-law
scaling arguments as we did for critical scaling dynamics. ρf
E¼Atκf
a,where
Ais a constant accounting for the difference in p-bit and c-bit prefactors,
and κ
f
is the power-law scaling exponent. Figure 5shows that p-bit SA
exhibits a similar scaling exponent κ
f
to c-bit SA, while purely probabilistic
APT yields a similar κ
f
as hybrid probabilistic-chaotic APT. Furthermore,
the APT algorithms show κ
f
of larger magnitude than the SA algorithms,
indicating a faster convergence to the ground energy E
0
. Two conclusions
can be drawn from our results. First, in the optimization setting, there is a
strong similarity between c-bits and p-bits since they essentially show the
same scaling for both SA and APT. Second, the hybrid APT algorithm we
propose for c-bits shows superior performance over its fully deterministic
counterpart. Besides better algorithmic scaling, the APT algorithm we
propose enjoys another benefit: because coupling strengths are not adjusted
globally, overflow or underflow issues related to the lifetime of c-bits become
less of a concern, and they are a one-time problem to solve ( for instance, past
definitions of the c-bit have dx
i
/dt that grow exponentially as βincreases,
thus overflows may occur at cold temperatures40).
The hybrid chaotic-probabilistic APT algorithm we propose here may
be applicable to other approaches, such as the noise-injected oscillator-based
Ising machines proposed in ref. 33, that are designed to enable sampling
from the Boltzmann distribution. However, the simplicity of the c-bit and its
state-based (rather than phase) representation through explicit latches may
stillbemoreappealing.
Adaptive parallel tempering for semiprime factorization
As a final example, we consider a 10-bit × 10-bit invertible multiplier circuit
composed of p-AND gates and p-Full Adders2,5(Fig. 6a). This multiplier is
represented by an undirected network of n= 310 p-bits with 1200 weighted
connections, normalized such that J
ij
∈[−1, 1]. Of the 310 spins in the
system, 20 specific spins represent the product bits, while another set of
20 specific spins represents each of the two 10-bit factors. Using bipolar spin
states (m
i
∈{−1, +1}), traditional binary 0s are instead represented as
negative spins m
i
=−1. Clamping the multiplier’s 20-bit output to a
semiprime number configures the system such that its ground state, E
0
,
solves for the correct prime factors. To accommodate semiprime numbers
that are smaller than 20 bits, we pad the most significant bits with zeros. This
scheme is referred to as invertible Boolean logic, in which p-bit-based logic
gates are run in reverse, similar in spirit to those used in memcomputing2,64.
We employ identical parameters for both probabilistic AP T and hybrid
chaotic-probabilistic APT. Our APT preprocessing method finds a suitable
βprofile (Fig. 6c). We consider N= 17 system replicas, each running at a
different inverse temperature βfor a duration of t
a
=10
5time steps. Prob-
abilistic swap attempts are made between adjacent replica pairs once every
10 time steps, according to the Metropolis criterion (Equation (17)). These
parameters yield Nt
a
=17×10
5time steps of total computation summed
across replicas.
During an APT simulation of t
a
=10
5time steps, if the coldest replica
(highest β) detects the correct factors at any time step, the simulation is
considered a successful solve (Fig. 6b). Only the coldest replica is considered
to remove the effect of random search statistics which may play a role in
small problem sizes. For our comparative study, optimizing the algorithms’
performance is not necessary.
We adopt a commonly used time-to-solution (TTS) metric65:
hTTS i¼ðNtaÞlnð10:99Þ
ln½1pðNtaÞ
 ð19Þ
where Nt
a
is the total computational time summed across all replicas. The
time-to-solution represents the amount of computation necessary to factor
semiprimes with a 99% success probability (see Supplementary Note 3). For
a given product size (number of bits), we consider 180 unique semiprime
numbers. We attempt to factor each semiprime number 80 times. p(Nt
a
)is
measured as the number of successful solves divided by the total number of
attempted factorizations.
We devise a similar scheme for simulated annealing for the sake of
comparison. We anneal 17 system replicas in parallel. Each replica runs for
t
a
=10
5time steps, and βis linearly increased from 0 to 10. In order to avoid
random-find effects due to small problem si zes, we only check for the correct
factors at the last 1000 time steps. If any of the 17 replicas find the correct
factors, the trial is considered a successful solve. For a given problem size (#
bits), we again consider 180 unique semiprimes. We attempt to factor each
semiprime 120 times. p(Nt
a
) and time-to-solution are calculated in a fashion
analogous to APT.
Once again, in order to rule out implementation-dependent differences
in unit time between c-bits and p-bits, we perform a scaling analysis, where
TTS vs. problem size is assumed to have exponential dependence: TTS
¼AexpðkxÞ,wherexis the product size in bits, while Aand kare constants.
Using this method of comparison, Fig. 6d shows that purely prob-
abilistic APT exhibits similar algorithmic scaling to the proposed chaotic-
probabilistic hybrid scheme. Moreover, they both exhibit b etter scaling than
SA-based approaches. Pure chaotic SA, which has been commonly
implemented40, shows the worst scaling. Furthermore, both p-bits and c-bits
observe a boundary effect as the problem size approaches 20 bits, where the
time-to-solutionactuallydecreases(forallthealgorithms).Thisisafeature
particular to the fixed 20-bit factorizer circuit: since numbers larger than 20
bits cannot be represented by the circuit, the solution space is reduced as we
approach 20 bits. We exclude this data because it is uninformative for
algorithmic scaling.
The two computing schemes not only share similar algorithmic scaling
but also exhibit similarities in their underlying mechanisms. We consider a
specific 11-bit semiprime number and conduct APT for t
a
=10
5time steps.
We average over 100 random seeds. Figure 6e shows the probability of the
Fig. 5 | Optimization of the 3D spin glass problem. Residual energy density (ρf
E)as
a function of total computational time (Nt
a
) for cubic spin-glass instances with side
length L=7(J
ij
∈{−1, +1}). Curves scale as a power law: ρf
E/tκf
a. Time is
measured in Monte Carlo sweeps for p-bits and time is dimensionless for c-bits.
Error bars show 95% bootstrap confidence intervals. Results include p-bit simulated
annealing (SA, purple squares), probabilistic adaptive parallel tempering (APT, blue
circles), c-bit SA (yellow triangles), and c-bit APT with stochasticity (orange
diamonds).
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Metropolis criterion dictating a swap between a given pair of replicas. The
hybrid APT scheme follows a similar qualitative distribution as the prob-
abilistic APT scheme.
The choice of integer factorization as a benchmark problem is simply
due to the challenging combinatorial nature it poses to stochastic local
search algorithms66. Specialized factoring algorithms starting with trial
division and more complex field sieve approaches exhibit superior algo-
rithmic scaling. It is important to note, however, that the factoring repre-
sentation we use in this work is based on the more general NP-hard Circuit
SAT problem, for which no generic algorithmic speed up is known. As such,
searching for scaling and prefactor improvements by dedicated hardware is
highly desired.
Possible hardware realizations
Based on the original c-bit definition that is different from the one we used in
this paper, chaotic bit implementations have been made in digital and
analog CMOS44,45,67. We now make some remarks regarding hardware
implementations of chaotic networks; however, a concrete discussion of
hardware implementations is beyond our scope.
As a first evaluation of the algorithm we proposed, the results we
presented are hardware-agnostic based on scaling laws and algorithmic
exponents. However, there are restrictions that would be encountered in any
hardware implementation.
The first point is related to the synapse calculation time. In our software
implementation, the synapse (Eq. (2)) is updated every time step, so c-bits
states are always updated using up-to-date information from their neigh-
bors. Just as in asynchronous networks of p-bits49, the asynchronous
operation of c-bits require a fast local field calculation. This places stringent
requirements on the network topology since dense networks will require
more time to update the synapse.
The second point is related to the latch mechanism: as we showed in
Fig. 2if the phases of two different billiards are too close to each other in
time, the latches might erroneously update at the same time depending on
the way the latch is implemented, especially in dense network topologies.
Beyond digital implementations, c-bits can make use of “natural”lat-
ches. For example, the physics of spin-torque switching of a non-volatile
nanomagnet68 canfunctionasthelatch:themagnetwillswitchonlyifa
current threshold is reached, and the magnet will exhibit hysteresis until the
Fig. 6 | Semiprime factorization with adaptive parallel tempering. a p-bits are
utilized to construct AND gates and full adders, that are the fundamental gates for a
multiplier. The product bits are clamped to a semiprime number as in invertible
boolean logic2.bEnergy profile of the 5 coldest replicas during hybrid chaotic-
probabilistic adaptive parallel tempering (APT). E−E
0
represents distance from
ground-state energy. cβprofile obtained from APT preprocessing (see Supple-
mentary Note 2). dTime-to-solution vs. product size using a 10-bit × 10-bit
factorizer for semiprime numbers. p-bit simulated annealing (SA, purple squares)
yields a slope of 0.441, while p-bit APT (blue circles) yields a slope of 0.438. c-bit SA
(yellow triangles) exhibits a slope of 0.658, and hybrid APT (orange diamonds) has a
slope of 0.420. eAverage replica-swap probabilities from factoring a specific semi-
prime number over 100 trials (p-bit APT shown in blue, hybrid APT shown in
orange).
https://doi.org/10.1038/s42005-025-01945-1 Article
Communications Physics | (2025) 8:35 8
Content courtesy of Springer Nature, terms of use apply. Rights reserved

current threshold changes sign, much like the billiard-ball metaphor used
in c-bits.
A purported benefit of c-bits is the absence of explicit random numbers
in their update rules. However, it is not clear whether the cost of c-bit
hardware implementation is significantly cheaper than that of p-bits. First,
low-cost pseudo random number generators, such as LFSRs, do not sig-
nificantly increase the complexity of implementing a p-bit. For example, an
n-bit LFSR requires about 32 × ntransistors to implement63. In contrast,
constructing a c-bit digitally may be significantly more complex than this
cost due to the necessity of an oscillator with a tunable duty cycle. Addi-
tionally, there may be “hidden”sources of randomness, such as floating
point round-off errors in simulation or thermal noise in physical
implementation.
Augmenting c-bits with explicit randomness in powerful tempering
algorithms such as the one considered in this paper could offer further
benefits. We believe that while the lack of random numbers may be an
important c-bit advantage, comparisons must be made in concrete imple-
mentations in order to be meaningful (for example, against the most energy-
efficient and compact hardware solutions of p-bits, using magnetic
nanodevices3). We refrain from making preemptive assumptions about the
scalability and efficiency of c-bit hardware, as specific implementation
details are unknown. p-bit Monte Carlo sweeps are not 1-to-1 with c-bit’s
time variable t, so speculations are not particularly informative until
implementation. Nevertheless, we believe that the similar algorithmic
scaling performance of c-bits and p-bits across the wide variety of problems
considered in this paper is very promising, providing valuable insights into
the level of stochasticity required for effective probabilistic computing.
Conclusions
In this work, we evaluated and improved deterministic chaotic bit networks
that combine billiard dynamics with latches. We note that while algorithm
parameters were not excessively tuned, using similar parameters for p-bit
and c-bit versions of SA and APT enabled fair comparison of algorithmic
scaling performance. We demonstrated that augmenting deterministic
c-bits with stochasticity via probabilistic APT replica swaps exhibits algo-
rithmic scaling comparable to APT using fully stochastic networks of p-bits.
Additionally, noise-injected c-bit APT achieved better algorithmic scaling
than common simulated annealing-based algorithms. Beyond c-bits, the
proposed adaptive parallel tempering algorithm is applicable to certain
classes of oscillator Ising machines and similar nonlinear devices.
Moreover, we have shown that c-bits approximately sample from the
quantum Boltzmann law in a 1D TFIM model. On the 2D Ising and 3D spin
glass problems, c-bits qualitatively exhibit similar critical scaling dynamics
as p-bits, though a precise correspondence between their measured scaling
exponents κ
c
is unclear. While intriguing from a physics perspective, these
critical dynamics are not essential to optimization, for which c-bits show
strong promise. Combining stochastic p-bits and deterministic c-bits in
asynchronous and massively parallel hardware implementations could
create powerful domain-specific computers for combinatorial optimization
and probabilistic sampling.
Methods
Numerical simulation uses double precision (64-bit) in C++ programming
language. Random numbers are generated using the Mersenne Twister
engine. Spins (and initial billard states, x
i
, for c-bits) are randomly initialized
at the beginning of every SA and APT simulation. For c-bits, we employ
Euler’s method with a step of Δt= 0.1 unless otherwise stated. At a given
step, if the billiard x
i
is greater than or equal to +1orlessthanorequalto
−1, then m
i
is latched to that value.
For the 2D ferromagnetic Ising model, we obtain the critical energy
experimentally. Using p-bits, we conduct 106Monte Carlo sweeps at con-
stant critical temperature Tc¼2=lnð1þffiffiffi
2
pÞ. We record the average
energy over the last 105sweeps. We take an ensemble average over 40
random seeds.
For the 3D spin glass instances, using p-bits, we conduct 106Monte
Carlo sweeps at constant critical temperature T
c
=1.1,avaluefoundfroma
previous study59. For each of 300 problem instances, we record the average
energy over the last 105sweeps.
Data availability
The data used for generating the figures are available from the corre-
sponding author upon reasonable request.
Code availability
Relevant codes are available from the corresponding author upon reason-
able request.
Received: 9 August 2024; Accepted: 9 January 2025;
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Acknowledgements
We gratefully acknowledge discussionswith Sanaaya Lakdawala and Navid
Anjum Aadit, Nikhil Shukla and Corentin Delacour for insights regarding
Oscillator Ising Machines and Masoud Mohseni for insights on KZM. This
work is partially supported by an Office of Naval Research Young
Investigator Program grant, a National Science Foundation CCF 2106260
grant, and the Army Research Laboratory under grant number W911NF-24-
1-0228. Use was made of computational facilities purchased with funds
from the National Science Foundation (CNS-1725797) and administered by
the Center for Scientific Computing (CSC). The CSC is supported by the
California NanoSystems Institute and the Materials Research Science and
Engineering Center (MRSEC; NSF DMR 2308708) at UC Santa Barbara.
Author contributions
K.L. and K.Y.C. conceived the idea for the study. K.Y.C. supervised the
study. K.L.conducted numericalexperiments with assistance from S.C.K.L.
wrote the initial draft of the manuscript. Allauthors participatedin writing the
manuscript, analyzing the experiments and discussing the results.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information The online version contains
supplementary material available at
https://doi.org/10.1038/s42005-025-01945-1.
Correspondence and requests for materials should be addressed to
Kyle Lee or Kerem Y. Camsari.
Peer review information Communications Physics thanks Natalia Berloff,
Jérémie Laydevant and Roman Khymyn for their contribution to the peer
review of this work.
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