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Machine Learning techniques for state recognition and auto-tuning in quantum dots
Sandesh S. Kalantre, Justyna P. Zwolak, Stephen Ragole, Xingyao Wu, Neil M. Zimmerman, M. D. Stewart, Jacob M. Taylor
TL;DR
Quantum-dot devices require careful gate-voltage tuning, and the growing number of gates makes finding voltages for a desired configuration challenging. The paper trains neural networks to recognize device configurations and uses optimization to tune gate voltages. It reports over 90% accuracy on its stated problems and demonstrates state recognition and auto-tuning with experimental data.
Problem
Choosing gate voltages for a desired quantum-dot configuration becomes challenging as the number of gates grows, and heuristic methods are difficult beyond a modest gate count.
Source: §I · p. 1, §II · p. 2Method
The authors train neural networks on labeled data to recognize quantum-dot states, then optimize gate voltages toward a desired configuration.
Source: §I · pp. 1–2, B. Tuning the device · p. 8Results
Over 90% accuracy was reported across the stated problems, and experimental data showed successful single- and double-dot state recognition and auto-tuning.
Source: §I · p. 2, §IX · p. 11Takeaways & Limitations
State recognition paired with optimization provides an automated approach to navigating and tuning quantum-dot devices in gate-voltage space.
Source: B. Auto-tuning of the device to a… · p. 8Extending learning and auto-tuning to multiple dots faces higher-dimensional gate-voltage spaces, and pattern recognition beyond two dimensions has not been extensively studied.
Source: B. Further Problems · p. 10
Abstract
from arXiv · showhide
Recent progress in building large-scale quantum devices for exploring quantum computing and simulation paradigms has relied upon effective tools for achieving and maintaining good experimental parameters, i.e. tuning up devices. In many cases, including in quantum-dot based architectures, the parameter space grows substantially with the number of qubits, and may become a limit to scalability. Fortunately, machine learning techniques for pattern recognition and image classification using so-called deep neural networks have shown surprising successes for computer-aided understanding of complex systems. In this work, we use deep and convolutional neural networks to characterize states and charge configurations of semiconductor quantum dot arrays when one can only measure a current-voltage characteristic of transport (here conductance) through such a device. For simplicity, we model a semiconductor nanowire connected to leads and capacitively coupled to depletion gates using the Thomas-Fermi approximation and Coulomb blockade physics. We then generate labelled training data for the neural networks, and find at least $90\,\%$ accuracy for charge and state identification for single and double dots purely from the dependence of the nanowire's conductance upon gate voltages. Using these characterization networks, we can then optimize the parameter space to achieve a desired configuration of the array, a technique we call `auto-tuning'. Finally, we show how such techniques can be implemented in an experimental setting by applying our approach to an experimental data set, and outline further problems in this domain, from using charge sensing data to extensions to full one and two-dimensional arrays, that can be tackled with machine learning.
I. INTRODUCTION
As quantum-dot devices gain independently controlled gates, choosing voltages for a desired configuration becomes difficult to manage with experimental heuristics. The paper proposes neural-network state recognition followed by optimization of gate voltages, reporting over 90% accuracy on its stated problems.
I. INTRODUCTION
- Convolutional neural networks trained on simulated data recognize quantum-dot configurations, after which an optimizer searches gate voltages for a desired configuration.The paper calls finding gate values for a desired island configuration “auto-tuning.”Source: §I · p. 1
- Over 90% accuracy was reported for simple neural-network architectures across the stated charge- and state-identification and auto-tuning problems.Accuracy is the fraction of predictions agreeing with pre-assigned labels.Source: §I · p. 2
II. MOTIVATION
- Gate voltages form a multidimensional search space: barrier gates control tunneling, plunger gates control well depth, and the device state is the number of islands.The paper frames automated tuning as identifying the state and then optimizing toward the required configuration.Source: §II · pp. 2–3
- As gate count grows, selecting voltages for a required configuration becomes challenging, while heuristic tuning is difficult beyond a modest number of gates.Semiconductor devices may require careful setting of tens of gate voltages.Source: §I · p. 1, §II · p. 2
III. PHYSICAL MODEL OF A NANOWIRE
The model converts gate-set potential landscapes into electron transport and charge configurations, then checks whether it reproduces single- and double-dot signatures.
III. PHYSICAL MODEL OF A NANOWIRE
- The transport model calculates electron density and current from a gate-determined potential, using a capacitance model and fixed contact Fermi level.Source: §III · p. 3
- A Markov chain over island charge states uses energy-based thermal selection and WKB-times-attempt tunneling rates to calculate steady-state current.Source: §III · p. 3
- In the 3-gate single-dot simulation, Coulomb blockade appears and charge changes coincide with current peaks as VP is swept.The figure pairs simulated current and electron number as functions of VP.Source: Figure 2 · p. 3, §III · p. 3
- For the 5-gate double dot, current appears only at triple points and fixed-charge contours form a honeycomb.These signatures are shown in the simulated current map and charge-stability diagram.Source: Figure 3 · p. 4, §III · p. 4
- Rising peak heights reflect lowered tunnel barriers, while shrinking peak spacing reflects increasing dot capacitance with electron number.Source: §III · p. 4
- The authors describe this as a minimum qualitative model and say more sophisticated models are needed in future studies.Source: §III · pp. 3–4
IV. LEARNING COULOMB BLOCKADE
The section tests whether a neural network can infer single-dot charge from current versus plunger-gate voltage. A three-hidden-layer network reaches 91% accuracy, although unknown initial charge limits experimental applicability.
IV. LEARNING COULOMB BLOCKADE
- Training used 1000 dot realizations with varied gate positions, widths, and heights sampled around mean parameters.The variation is intended to model differences among experimental dots and provide generic training data.Source: §IV · p. 4
- 91% accuracy for charge-state values was achieved by a three-hidden-layer DNN mapping current versus VP to charge states.The figure illustrates the current input and predicted versus actual charge states.Source: §IV · p. 4, Figure 4 · p. 5
- The network uses 512-point input and output layers and hidden layers of 1024, 256, and 12 neurons, with outputs rounded to integers.Source: §IV · p. 5
- Successively narrowing hidden layers was critical to respectable accuracy, suggesting the network must ignore redundant information in current characteristics.Source: §IV · p. 5
- The network first learns average-dot charge boundaries and then identifies charge states for individual dot samples.Source: §IV · p. 5
- Unknown initial electron number creates a charge-offset problem that limits experimental applicability of this charge-identification setup.The authors still present it as an example that machine learning can be applied to charge identification.Source: §IV · p. 5
V. LEARNING STATE
For a 5-gate device, the section maps current across plunger voltages to dot configurations and reports 91% state agreement. It then frames auto-tuning as finding a voltage region with a desired configuration.
V. LEARNING STATE
- The four configurations are short circuit, barrier, single dot, and double dot; a probability vector represents their probabilities at each voltage point or averaged over a region.Source: §V · p. 5
- Training uses 1000 two-dimensional gate configurations, with state labels inferred from Thomas-Fermi electron-density islands separated by zero-density barriers.Different barrier-voltage combinations can produce the same configuration.Source: §V · pp. 5–6
- 91% state-value agreement was achieved when reproducing state maps across different devices from 100 × 100 current maps.Figure 5 pairs current and state maps across VP1 and VP2.Source: §V · p. 6, Figure 5 · p. 6
VI. AUTO-TUNING
- Auto-tuning means finding a range of gate voltages with a specific configuration through repeated state identification and voltage optimization.For m gates, the target is an m-dimensional cuboid in gate-voltage space.Source: §VI · p. 6
- The demonstrated 5-gate tuning case controls VP1 and VP2 with barrier gates fixed, starting in a double-dot region and targeting a single-dot region.Source: §VI · p. 6
A. State learning
The section frames state recognition as image classification over gate-voltage subregions, then uses the predicted state probabilities to tune gate voltages toward a desired configuration. The networks are also evaluated on experimental conductance data.
A. State learning
- The CNN classifies limited two-dimensional voltage subregions into short circuit, barrier, single-dot, or double-dot states.The highest-probability state is used as the prediction, and convolutional layers improved accuracy and training efficiency.Source: A. State learning · pp. 7–8
- 96% accuracy was achieved in predicting device state from 30 × 30 pixel voltage submaps.The training set used 50,000 submaps generated from 1,000 full two-dimensional current maps, with 90% used for training.Source: A. State learning · p. 7
B. Tuning the device
- Auto-tuning minimizes the distance between a subregion’s predicted probability vector and a desired vector over gate-voltage space.COBYLA required 15–30 CNN evaluations to reach the target subregion, depending on the starting region.Source: B. Tuning the device · p. 8
VII. WORKING WITH EXPERIMENTAL DATA
- Experimental evaluation used differential-conductance maps over two barrier voltages at varied, fixed plunger voltages from a three-gate device.The trained CNN weights came from simulated data, and barrier gates also served as generic plunger gates in the experiment.Source: §VII · p. 8
A. Identification of state in experimental Data
Experimental conductance maps show a gradual transition from single-dot to double-dot behavior as the plunger voltage changes, and the CNN distinguishes the two states. An optimizer also finds a double-dot setting from either a single-dot region or a no-current region.
A. Identification of state in experimental Data
- Experimental maps show a gradual transition from single-dot to double-dot current characteristics as the plunger voltage becomes more negative.The measurements varied the plunger voltage while mapping differential conductance over the two barrier gates.Source: A. Identification of state in… · p. 8, Figure 9 · p. 9
- The CNN distinguished single-dot from double-dot states in experimental current maps after the data were rescaled to match simulated data.The rescaling was needed because the model reproduced current values only qualitatively.Source: A. Identification of state in… · p. 8
B. Auto-tuning of the device to a double dot state
- The optimizer reached the required double-dot state from both a single-dot region and a no-current region by tuning two barrier gates and the plunger.The process took 40–50 iterations, depending on the initial region.Source: B. Auto-tuning of the device to a… · pp. 8–9
A. Neural Networks in an Experimental Setting
- The auto-tuner compares a predicted state-probability vector with a destination vector and iteratively adjusts the device parameters.The procedure explores subregions rather than the entire voltage space, saving experimental time without human intervention during tuning.Source: A. Neural Networks in an Experimental… · p. 9
B. Further Problems
The conclusion reports recognition of quantum-dot states from current characteristics and experimental auto-tuning in three-dimensional gate-voltage space. The further-problems section proposes approaches to multi-dot and charge tuning while noting limits of the current algorithmic scope.
B. Further Problems
- Multi-dot tuning raises challenges from higher-dimensional gate spaces, while pattern recognition beyond two dimensions has not been extensively studied.The authors propose inductive learning as a strategy that can be generalized to this setting.Source: B. Further Problems · p. 10
- Inductive learning would tune pairs of barrier gates and then use another neural network to tune the plunger, repeating across the array.The proposed strategy relies on gates far apart being loosely coupled.Source: B. Further Problems · p. 10
- Charge tuning would learn the capacitance matrix, estimate device charge from that matrix and gate voltages, and optimize toward required charge values.The matrix determines quantitative current features, including single-dot charging energy and double-dot honeycomb-hexagon size.Source: B. Further Problems · p. 10
- Further problems may require machine-learning algorithms beyond deep and convolutional neural networks.The conclusion also describes possible experimental integration and outlines additional tuning problems.Source: B. Further Problems · p. 10, §IX · p. 11
IX. CONCLUSION
- The trained networks distinguished single- and double-dot states on experimental data, and auto-tuning was demonstrated in three-dimensional gate-voltage space.The paper also describes a bare-bones transport model that qualitatively reproduces current-versus-gate-voltage characteristics.Source: §IX · pp. 10–11
Appendix A: Thomas-Fermi Capacitance Model
The model uses the Thomas-Fermi approximation to calculate electron density from a local electron-gas description. Gate potentials and Coulomb self-interaction modify the band minimum, so density is found self-consistently.
Appendix A: Thomas-Fermi Capacitance Model
- The Thomas-Fermi approximation applies homogeneous-electron-gas properties locally and is justified when density or potential varies slowly over an electron wavelength.Source: §1 · p. 12
- The model takes an external potential and Fermi level as given, and treats electron density on a finite one-dimensional grid as the quantity to determine.Source: §1 · p. 12
- The conduction-band density is calculated from the density of states and inverse temperature, with the band bottom set by the Fermi sea.Source: §1 · p. 12
- Gate potential and Coulomb self-interaction shift the band minimum, which is used with the density equation to calculate density self-consistently.The Coulomb kernel models effective one-dimensional interactions and avoids a numerical singularity at coincident positions.Source: §1 · pp. 12–13
- The density calculation assumes a constant two-dimensional electron-gas density of states and iterates from zero density until convergence.The Coulomb interaction strength is increased gradually during initial iterations to avoid numerical convergence pathologies.Source: §1 · p. 13
- Regions with chemical potential above the potential are treated as electron islands, while regions below it are treated as barriers.Source: §1 · p. 13
2. Calculation of a capacitance model
The capacitance model represents island charges and gate-induced charges with a capacitance matrix. It assigns electrostatic energies to integer-valued island charge configurations.
2. Calculation of a capacitance model
- For m conductors, a capacitance matrix relates island charges to conductor potentials, including capacitances between conductors and to ground.Source: §2 · p. 13
- The electrostatic energy uses the charge vector and induced charges from fixed gate voltages, with the inverse capacitance matrix defining the energy relation.Source: §2 · p. 13
- The model treats islands as conductors with discrete electron counts, and defines the capacitance model by the matrix C and induced-charge vector Z.Source: §2 · p. 13
- The induced charges Z are obtained by integrating electron density over each island, while inverse-capacitance elements are determined from the interaction potential and density.The energy expression includes a kinetic-energy contribution for each island and Coulomb interaction between islands.Source: §2 · pp. 13–14
- Determining the capacitance model requires finding both the inverse-capacitance elements and the induced charges.Source: §2 · p. 14
3. Calculation of equilibrium charge distribution
The model selects the lowest-energy integer charge configuration and uses a bounded Markov graph to represent tunneling between configurations. Edge rates combine energy-based selection and tunneling estimates to support transport simulation.
3. Calculation of equilibrium charge distribution
- The equilibrium charge configuration is the lowest-energy configuration among those closest to induced charges Z, with integer electron counts on each island.Source: §3 · p. 14
Appendix B: Markov Chain Model for mesoscopic transport in the semi-classical regime
- Transport is simulated with a Markov chain under infinitesimal contact bias, using elastic tunneling and WKB-estimated tunnel rates.Source: Appendix B · p. 14
1. Graph definitions and construction
- Graph nodes represent island charge configurations, and edges represent single-electron tunneling events; breadth-first construction starts from the configuration Z.The graph is restricted to order p = 1, while higher order can incorporate co-tunneling.Source: §1 · p. 14
2. Calculation of edge weights
- Each transition rate combines an energy-based selection rule with a WKB tunnel rate across the adjacent barrier.The WKB estimate treats the electron as having Fermi-level energy in an effective potential that includes inter-electron repulsion.Source: §2 · p. 14
- The edge rate also uses the classical travel timescale of an electron through a dot.Source: §2 · p. 14
3. Current calculation
Current is calculated from the stationary state of the Markov chain by summing transitions in a fixed direction. The gate model represents each top gate with a logarithmically varying potential profile.
3. Current calculation
- After assigning edge weights, the current is calculated from a stationary state of the Markov chain, summing transitions that transfer electrons in a fixed direction.The stationary state is found from the Markov matrix nullspace using singular value decomposition.Source: §3 · p. 15
Appendix C: Gate model
- Each top gate is modeled as a cylindrical conductor whose potential varies logarithmically with distance and is set by four parameters.The parameters are gate voltage V0, position x0, radius r0, and height h.Source: Appendix C · p. 15
- The gate model uses a screening term with σsc = 20 nm, equal to the separation between adjacent gates.Source: Appendix C · p. 15
Appendix D: Device Parameters
Device parameters were sampled around listed means to represent expected lab variation and make learning and tuning robust to device differences. The datasets varied one or two plunger-gate voltages, and neural-network computations ran on a CPU-only laptop.
Appendix D: Device Parameters
- Parameters were sampled from Gaussian distributions with standard deviations set to 0.05 times their means, aiming to represent lab devices and make learning and tuning robust to device differences and imperfections.Source: Appendix D · p. 15
- Common physical parameters included K0 = 10 meV, σ = 2 nm, g0 = 0.5 eV−1 nm−1, and ck = 1 meV nm, chosen for a few-electron regime of 1 to 10 electrons.Source: Appendix D · p. 15
1. 3-gate device dataset
- For the 3-gate dataset, current and charge were calculated versus plunger-gate voltage with both barrier gates fixed across a device extent of (−40, 40) nm.Source: §1 · p. 15
2. 5-gate device dataset
- For the 5-gate dataset, current and charge were calculated as functions of both plunger-gate voltages across a device extent of (−60, 60) nm.Source: §2 · p. 15
Appendix E: Computing environment and TensorFlow parameters
- Networks were built and trained with TensorFlow; computations ran without an explicit GPU on a 2015 MacBook Air with a 1.8GHz dual-core Intel Core i5 and 4 GB of RAM.Source: Appendix E · p. 15