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TensorFlow Quantum: A Software Framework for Quantum Machine Learning

Michael Broughton, Guillaume Verdon, Trevor McCourt, Antonio J. Martinez, Jae Hyeon Yoo, Sergei V. Isakov, Philip Massey, Ramin Halavati, Murphy Yuezhen Niu, Alexander Zlokapa, Evan Peters, Owen Lockwood, Andrea Skolik, Sofiene Jerbi, Vedran Dunjko, Martin Leib, Michael Streif, David Von Dollen, Hongxiang Chen, Shuxiang Cao, Roeland Wiersema, Hsin-Yuan Huang, Jarrod R. McClean, Ryan Babbush, Sergio Boixo, Dave Bacon, Alan K. Ho, Hartmut Neven, Masoud Mohseni

arXiv:2003.02989v2quant-phcond-mat.dis-nncs.LGcs.PL

TL;DR

Hybrid quantum-classical model research is difficult and error-prone because building, training, and validating models across classical and quantum processors requires substantial engineering. This paper introduces TensorFlow Quantum, an integrated TensorFlow–Cirq framework for designing, training, simulating, and testing such models, and demonstrates its use across diverse quantum learning tasks.

  • Problem

    Developing hybrid quantum-classical models requires substantial manual engineering across model construction, quantum datasets, training, and validation.

    Source: D. TensorFlow Quantum · p. 5

  • Method

    TensorFlow Quantum integrates Cirq with TensorFlow to support design, training, simulation, and testing of hybrid quantum-classical models across classical and quantum processors.

    Source: D. TensorFlow Quantum · p. 5

  • Results

    TFQ is demonstrated across basic and advanced applications including quantum classification, control, noisy-circuit simulation, Hamiltonian learning, thermal-state sampling, and reinforcement learning.

    Source: §2 · pp. 45, 47, §3 · p. 48

  • Takeaways & Limitations

    TFQ provides software tools for exploring hybrid quantum-classical models and quantum learning tasks.

    Source: D. TensorFlow Quantum · p. 5

    Training quantum models can be hindered by barren plateaus, whose exponentially flat landscapes impede gradient-based and gradient-free optimization.

    Source: B. Hybrid Quantum-Classical Models · p. 4

Abstract

from arXiv · show

We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulators. We provide an overview of the software architecture and building blocks through several examples and review the theory of hybrid quantum-classical neural networks. We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks including meta-learning, layerwise learning, Hamiltonian learning, sampling thermal states, variational quantum eigensolvers, classification of quantum phase transitions, generative adversarial networks, and reinforcement learning. We hope this framework provides the necessary tools for the quantum computing and machine learning research communities to explore models of both natural and artificial quantum systems, and ultimately discover new quantum algorithms which could potentially yield a quantum advantage.

I. INTRODUCTION … 1. Design Principles and Overview

This section introduces hybrid quantum-classical learning for quantum data and presents TensorFlow Quantum as an integration of Cirq and TensorFlow for differentiable, batched, hardware-agnostic quantum machine learning. It outlines TFQ’s tensor-based representation of quantum circuits and operators and its integrated software stack for execution across simulators and quantum hardware.

A. Quantum Machine Learning

  • First-generation quantum machine-learning algorithms can offer exponential advantages on certain quantum data, but classical-data encoding costs, structural assumptions, and unresolved speedup claims limit their applicability.These limitations motivate looking beyond classical data toward quantum-data applications.Source: A. Quantum Machine Learning · p. 3

B. Hybrid Quantum-Classical Models

  • Near-term quantum machine learning uses parameterized quantum circuits optimized with classical methods, but noisy hardware and barren plateaus motivate hybrid models that distribute computation across quantum and classical processors.Learning performance is assessed through representation capacity, training efficiency, inference tractability, and generalization power.Source: A. Quantum Machine Learning · p. 3, B. Hybrid Quantum-Classical Models · pp. 3–4

C. Quantum Data

  • Quantum data includes outputs of quantum experiments or devices, with applications spanning quantum simulations, communication, metrology, and control.These domains motivate software that can combine coherent quantum-data access with machine learning.Source: C. Quantum Data · p. 4

D. TensorFlow Quantum

  • TFQ addresses the engineering difficulty of constructing hybrid quantum-classical models, quantum datasets, and training pipelines by integrating Cirq with TensorFlow and enabling backpropagation across quantum and classical layers.The framework supports simulation, training, and testing, with eventual execution of quantum components on quantum processors.Source: D. TensorFlow Quantum · p. 5

II. SOFTWARE ARCHITECTURE & BUILDING BLOCKS

  • TFQ combines Cirq’s quantum-circuit capabilities with TensorFlow’s differentiable, distributed dataflow and Keras abstractions to support quantum machine-learning research across heterogeneous hardware.The software architecture addresses the need to bridge quantum computing and machine learning communities.Source: §II · p. 5, B. TensorFlow · p. 6

A. Cirq

  • Cirq provides open-source structures for qubits, gates, circuits, measurements, simulation, real-hardware execution, and NISQ-oriented compilation and scheduling.It also supports symbolic circuit parameters resolved at simulation or execution time.Source: A. Cirq · pp. 5–6

B. TensorFlow

  • TensorFlow represents computations as differentiable dataflow graphs whose tensor and operation abstractions enable backpropagation and parallel execution across CPUs, GPUs, and TPUs.TensorFlow 2 adds @tf.function and native Keras support for concise model construction.Source: B. TensorFlow · p. 6

C. Technical Hurdles in Combining Cirq with TensorFlow

  • TFQ rejects a states-as-edges architecture because static circuits hinder variational updates, quantum-memory latency impedes hardware execution, and duplicating Cirq would increase interface and maintenance burdens.Its design therefore prioritizes differentiability, circuit batching, and execution-backend agnosticism.Source: C. Technical Hurdles in Combining Cirq… · p. 7, §1 · p. 7

1. Design Principles and Overview

  • TFQ follows a minimalist design that reuses Cirq and TensorFlow functionality, while Keras layers provide most users a simplified interface over lower-level circuit-construction and execution ops.The tfq_append_circuit op is intended for cases where most of a circuit is static and only a small part changes during training or inference.Source: §1 · pp. 7–8
  • TFQ converts Cirq circuits and Pauli operators into TensorFlow string tensors, then uses ops for simulation, expectation values, or sampling to produce classical information.Parameterized circuits may use SymPy symbols, and expectation ops accept circuit tensors, symbol names, parameter values, and measurement operators.Source: §1 · pp. 7–8
  • Expectation ops are differentiable and support finite differencing, parameter-shift, stochastic, and user-defined gradient methods through the tfq.Differentiator interface.They can execute on Cirq or qsim simulators or on real devices, configured when the op is instantiated.Source: §1 · p. 8
  • TFQ enables a single TensorFlow Quantum program to run locally, at highly parallel distributed throughput, or on real QPU hardware through TensorFlow, Cirq, and hardware interfaces.The software stack connects TensorFlow and Keras abstractions with quantum layers, differentiators, custom execution ops, Cirq or qsim simulation, and QPU execution.Source: §1 · p. 8, Figure 4 · p. 8
  • The overview establishes TFQ’s hybrid modeling workflow, including a discriminator pipeline and a minimal single-qubit binary-classification example called Hello Many-Worlds.The paper defers more detailed descriptions of TFQ building blocks to section II E.Source: §1 · p. 8

2. The Abstract TFQ Pipeline for a specific hybrid discriminator model · 3. Hello Many-Worlds: Binary Classifier for Quantum Data

TFQ implements hybrid quantum-classical discrimination through a pipeline that prepares quantum data, evaluates quantum and classical models, computes costs, and updates parameters via backpropagation. A one-qubit binary-classification example instantiates this workflow using Bloch-sphere data, a parameterized rotation, Z measurement, classical post-processing, and Adam optimization.

2. The Abstract TFQ Pipeline for a specific hybrid discriminator model

  • TFQ’s abstract hybrid-discriminator pipeline prepares quantum datasets, evaluates quantum and classical models, computes a task-dependent cost, and updates parameters through gradient-based optimization.Quantum data are specified by unparameterized Cirq circuits, while quantum-model derivatives are exposed to TensorFlow backpropagation through the Differentiator interface.Source: §2 · pp. 8–9
  • Quantum models extract hidden information by disentangling input data so it becomes accessible through local measurements and classical post-processing.Measurements produce classical samples whose distributions depend on the quantum state and observable; TFQ also supports averaging across repeated runs for variational algorithms.Source: §2 · p. 9
  • The pipeline’s stages are organized as a TensorFlow/Keras model, allowing classical neural networks and losses to process measured quantum information and drive joint parameter optimization.TFQ-compatible quantum models can attach directly to tf.keras.layers.Layer objects, and the resulting model exposes tf.keras.losses functionality.Source: §2 · p. 9, Figure 5 · p. 9

3. Hello Many-Worlds: Binary Classifier for Quantum Data

  • The one-qubit example classifies two sampled quantum-data regions around random vectors in the X-Z plane of the Bloch sphere.The illustrated dataset uses θa = 1, θb = 4, and N = 200, with categories a and b represented by blue and orange states, respectively.Source: §3 · p. 9, Figure 6 · p. 10
  • The hybrid model applies a two-output SoftMax layer to measured quantum data, computes categorical cross entropy, and updates both quantum and classical parameters with Adam.Figure 7 summarizes the specialized workflow from quantum data through measurement, classical probabilities, loss computation, and optimization.Source: Figure 7 · p. 10, §3 · p. 10
  • The classifier uses a single parameterized Ry rotation and Z-axis measurement to reduce input superpositions and extract useful information for classification.The quantum model is the simplest universal quantum discriminator [43], and its output is attached to a small classical neural network within a Keras model.Source: §3 · p. 10
  • The trained hybrid model classifies new quantum datapoints by returning probabilities for categories a and b.The example is presented as runnable code in a Colab notebook or Python script after installing TFQ.Source: §3 · pp. 9–10

E. TFQ Building Blocks … 6. Basic Quantum Datasets

TFQ provides software components for representing, composing, executing, differentiating, and simplifying quantum models within TensorFlow, together with datasets for quantum-learning benchmarks. Its abstractions connect Cirq circuits to TensorFlow tensors, support classical loss computation and backpropagation, and expose labelled quantum datasets for classification and optimization.

E. TFQ Building Blocks

  • The building blocks are organized as Cirq-to-TensorFlow conversion, in-graph composition, circuit simulation and measurement, differentiation, and higher-level layers for quantum-machine-learning workflows.This component structure is presented as the framework’s route from quantum computations to trainable and more abstract models.Source: E. TFQ Building Blocks · p. 10

1. Quantum Computations as Tensors

  • TFQ converts Cirq circuits and PauliSum measurements into serialized tf.string tensors, enabling quantum data, models, and measurements to enter TensorFlow and Keras graphs.The tfq.convert_to_tensor function performs this conversion through custom serializers, motivating string-typed Keras inputs.Source: §1 · p. 11

2. Composing Quantum Models

  • TFQ composes quantum data and models with AddCircuit, then converts quantum outputs to classical information through sampling or measurement expectation values for loss functions.These outputs can feed functions producing a scalar loss, allowing quantum models to be optimized analogously to classical models.Source: §2 · p. 11, §3 · p. 12

3. Sampling and Expectation Values

  • TFQ exposes Sample and Expectation layers for circuit execution, producing ragged bitstring samples or measurement expectation values from parameterized circuits.Sampling supports variable circuit sizes and expectation execution can use either simulators or real devices; sampling is also central to quantum-supremacy applications.Source: §3 · pp. 11–12

4. Differentiating Quantum Circuits

  • TFQ integrates quantum-circuit differentiation with TensorFlow backpropagation, supporting built-in finite-difference and parameter-shift methods plus custom differentiators.Expectation and SampledExpectation layers use differentiator implementations to compute gradients; parameter-shift methods exploit ancillary circuits for suitable 2-local circuits [79].Source: §4 · p. 12

5. Simplified Layers

  • PQC and ControlledPQC simplify hybrid workflows by automatically managing parameterized-circuit variables or accepting parameter values supplied by external machinery.Both layers support hybrid backpropagation without requiring users to provide circuit-symbol lists and are used in later applications.Source: §5 · p. 12

6. Basic Quantum Datasets

  • TFQ supplies labelled quantum datasets, including excited cluster states for ground-versus-excited classification and TFI and XXZ chain data for phase classification and optimization benchmarking.The TFI dataset has 81 datapoints spanning g in [0.2, 1.8], while the XXZ dataset has 76 datapoints spanning Δ in [0.3, 1.8].Source: §6 · p. 13

F. High Performance Quantum Circuit Simulation with qsim … B. Sampling and Expectations

TFQ combines qsim’s gate-fusion and hardware-accelerated simulation with theory for quantum neural networks, sampling, and expectation estimation. Benchmarks show substantial speedups over Cirq, while the theory highlights measurement-sampling costs and stochastic-term selection for gradients.

F. High Performance Quantum Circuit Simulation with qsim

  • qsim accelerates circuit simulation by fusing neighboring gates and using optimized matrix-vector multiplication functions for multi-qubit gates within TFQ’s TensorFlow operations.The benchmark advantage is attributed to circuit preprocessing through gate fusion and performant C++ simulators.Source: F. High Performance Quantum Circuit… · p. 13, §4 · p. 14

2. Gate Fusion with qsim

  • The fusion algorithm greedily combines gates close in space and time into gates acting on at most F qubits, with F between 2 and 6 and typical optima depending on thread count.Typically F = 4 is optimal for many threads, while F = 2 or F = 3 is optimal for one or two threads.Source: §2 · pp. 13–14, Figure 8 · p. 14

3. Hardware-Acceleration

  • TFQ adapts simulation to available hardware by detecting SSE2, AVX2, AVX512, and compatible CUDA GPUs, with future TPU integration proposed.Fused circuits require simulators optimized for applying up to 2^F × 2^F matrices to state vectors.Source: §3 · p. 14

4. Benchmarks

  • At 30 qubits, qsim improves simulation time over Cirq by approximately 10-fold for dense circuits and roughly 100 times for sparse circuits, despite TensorFlow serialization overhead.Figure 9 reports approximately 10x CPU and approximately 50x GPU speedups for dense circuits, with well over 100x for sparse circuits on both CPU and GPU.Source: Figure 9 · p. 14, §4 · p. 14

5. Large-scale simulation for quantum machine learning

  • Using qsim on Google Cloud Platform, large-scale quantum-machine-learning simulations reached peak throughput of up to 1.1 petaflop/s across experiments of up to 30 qubits.Observed trends included approximately 300 teraflop/s for quantum simulation and 800 teraflop/s for classical analysis, totaling approximately two exaflop across experiments.Source: §5 · p. 15

6. Noise in qsim

  • qsim supports Cirq’s common noise channels and TFQ serialization, using trajectory simulation because it often requires less memory and is more efficient than density-matrix simulation for circuits that are not too noisy.Density-matrix simulation requires memory scaling as 2^2N, whereas trajectory simulation requires 2^N memory at any one time and averages over repeated runs.Source: §6 · p. 15

III. THEORY OF HYBRID QUANTUM-CLASSICAL MACHINE LEARNING

  • TFQ models quantum neural networks as products of parameterized unitaries and combines them with classical networks through expectation-based losses and hybrid quantum-classical backpropagation.Parameterized layers may use fixed and variational unitaries, with noncommuting generators handled through decompositions such as Trotter-Suzuki methods.Source: §III · p. 15, A. Quantum Neural Networks · pp. 15–16, Figure 10 · p. 16

B. Sampling and Expectations

  • Expectation values are estimated by sampling measurement outcomes, so finite samples introduce variance and quantum gradient estimation can require many measurements; stochastic term selection can reduce measurements per update.Hamiltonian expectations are decomposed into operator-term expectations, with noncommuting terms estimated over separate runs.Source: B. Sampling and Expectations · pp. 16–17

C. Gradients of Quantum Neural Networks … 4. Adjoint Gradient Backpropagation in Simulations

The section presents gradient-estimation methods needed to train hybrid quantum-classical models, spanning finite differences, analytic and stochastic parameter shifts, and adjoint differentiation for classical simulation. These methods trade query cost, precision, stochasticity, and hardware applicability in different ways.

C. Gradients of Quantum Neural Networks

  • Gradient computation is central to training quantum neural networks because hybrid quantum-classical optimization relies on gradient-based techniques such as stochastic gradient descent and backpropagation [92].Automatic differentiation underlies differentiability in frameworks including TensorFlow and JAX.Source: C. Gradients of Quantum Neural Networks · p. 17

1. Finite difference methods

  • Finite differences estimate gradients numerically, requiring 2M objective evaluations with O(ε2) error for M parameters under central differences.Forward differences require M + 1 queries with O(ε) error, while a five-point stencil requires 4M queries with O(ε4) error.Source: §1 · p. 17

2. Parameter shift methods

  • Parameter-shift rules can obtain analytic expectation-value gradients for suitable parametric unitaries, using shifts generally much larger than numerical-differentiation shifts and sometimes requiring fewer runs for precision.For a parametric unitary with Kℓ terms, the chain-rule procedure requires 2Kℓ expectation evaluations to obtain the gradient of θℓ.Source: §2 · p. 17

3. Stochastic Parameter Shift Gradient Estimation

  • Stochastic parameter-shift estimation reduces the prohibitive cost of evaluating all parameter- and Hamiltonian-term contributions by sampling terms according to coefficient-weighted distributions.TFQ independently supports parameter, Hamiltonian-term, and component sampling, corresponding to stochastic, doubly stochastic, and triply stochastic gradient descent.Source: §2 · pp. 17–18, §3 · p. 18
  • Stochastic parameter-shift gradients address the scaling of full analytic gradients, whose evaluation requires sums over parameters, generator terms, and cost-Hamiltonian terms.Sampling can extend from generator terms to Hamiltonian terms and parameter components, with the sampled gradient components averaged or masked during optimization.Source: §3 · p. 18

4. Adjoint Gradient Backpropagation in Simulations

  • Adjoint differentiation provides high-performance gradients for deep, many-parameter circuits in classical simulation, using reversible layers for a backward pass without storing state histories.It is unavailable on quantum hardware but supports rapid prototyping with simulator backends such as qsim in TFQ; compared with finite differences, it uses twice the memory rather than 2M forward passes for M parameters.Source: §4 · pp. 18–20
  • Adjoint backpropagation evaluates expectation-value derivatives recursively from the final layer, combining forward evolution with layer-wise uncomputation and contractions involving the observable.The method computes intermediate-layer gradients by reversing later unitary layers and recursively propagating the backward state and observable contraction.Source: §4 · p. 19

D. Hybrid Quantum-Classical Computational Graphs · 1. Hybrid Quantum-Classical Neural Networks · E. Autodifferentiation through hybrid quantum-classical backpropagation

The framework represents quantum neural networks as differentiable vector-valued blocks within directed hybrid quantum-classical computational graphs. It enables gradient backpropagation through these graphs by converting upstream gradients into an effective Hamiltonian and differentiating its expectation value.

D. Hybrid Quantum-Classical Computational Graphs

  • Hybrid quantum-classical computational graphs compose quantum and classical function blocks in a directed topology, with edges carrying classical information between abstracted components.Source: §1 · p. 20

1. Hybrid Quantum-Classical Neural Networks

  • HQCNN blocks map circuit parameters to vectors of expectation values, generalizing scalar cost functions into differentiable functions fθ: R^M→R^N suitable for composition.A Hamiltonian expectation is a linear combination of expectation values of multiple operators, so the QNN naturally produces a vector before scalar evaluation.Source: §1 · p. 20
  • Measurement histograms can be represented as expectation-value vectors using eigenstate projectors, while informationally complete observables can characterize the wavefunction at exponential qubit overhead.Source: §1 · pp. 20–21
  • Direct per-sample measurement feedback forms a stochastic map, but TFQ’s current implementation supports deterministic expectation-value nodes rather than native backpropagation through stochastic nodes.Single-sample feedback is described as relevant to applications such as quantum error correction, but is out of scope because Cirq lacks the required feature.Source: §1 · p. 21

E. Autodifferentiation through hybrid quantum-classical backpropagation

  • Quantum parameter-to-expectation maps compose with classical or quantum function blocks, forming meta-networks whose losses can be differentiated through sequential function composition.Source: E. Autodifferentiation through hybrid… · p. 21
  • Backpropagation contracts the downstream gradient with the QNN Jacobian by defining an effective Hamiltonian weighted by upstream components, then differentiating its expectation value with respect to circuit parameters.This produces the QNN parameter gradient, after which standard classical backpropagation continues through preceding function blocks.Source: E. Autodifferentiation through hybrid… · p. 21
  • TFQ enables quantum-circuit backpropagation through a Differentiator interface offering finite-difference, regular parameter-shift, stochastic parameter-shift, and user-defined gradient methods.Source: E. Autodifferentiation through hybrid… · p. 22

IV. BASIC QUANTUM APPLICATIONS … 2. Implementations

TFQ demonstrates hybrid quantum-classical learning through reproducible applications, focusing here on cluster-state error classification with quantum convolution, pooling, and classical postprocessing. The multi-filter hybrid classifier trains fastest, while the purely quantum classifier trains slowest under identical optimization settings.

IV. BASIC QUANTUM APPLICATIONS

  • TFQ examples are paired with browser-runnable Colab notebooks that reproduce the demonstrated results and illustrate framework features through code snippets.The notebooks provide full implementation details beyond the snippets shown in the paper.Source: §IV · p. 22

1. Background

  • QCNNs exploit translational symmetry by tying spatial parameters, while hierarchical convolution and pooling progressively coarse-grain entanglement and compress quantum information.The HQCNN extends this design with classical postprocessing after a shallow QCNN layer and multiple translationally invariant feature maps.Source: §1 · pp. 22–23
  • The work emphasizes quantum-data classification because near-term quantum advantage for QNNs classifying classical data lacks strong evidence, using cluster-state errors as a concrete task.The approach targets information hidden in quantum correlations while using shallow circuits hybridized with classical neural networks.Source: §1 · pp. 22–23

2. Implementations

  • The classifier detects preparation errors in eight-qubit cluster states, using Rx(θ) excitations labeled 1 above a threshold and -1 otherwise.Correctly prepared cluster states form the initial input, and excitation circuits generate varied error states.Source: §2 · p. 23
  • QCNN-inspired layers apply tied two-qubit convolution unitaries to nearest-neighbor pairs and pooling unitaries that disentangle and reduce two qubits to one.The pooling operation supports variational selection of control and target bases through a CNOT with general one-qubit unitaries.Source: §2 · pp. 23–24
  • The implementation compares a purely quantum CNN, a truncated-QCNN hybrid with dense processing, and a three-filter hybrid combining multiple quantum outputs.The purely quantum model pools from eight qubits to one and reads out Pauli-Z, while truncated hybrids measure remaining-qubit operators for classical processing.Source: §2 · pp. 24–25
  • The three-quantum-filter hybrid classifier trains fastest, whereas the purely quantum classifier trains slowest under the same optimization settings, demonstrating benefits of hybrid architectures for quantum-data classification.The comparison uses mean squared error loss across training epochs.Source: §2 · p. 25, Figure 18 · p. 25

B. Hybrid Machine Learning for Quantum Control · 1. Time-Constant Hamiltonian Control

TFQ supports quantum-control optimization by combining hybrid quantum-classical models, simulators, expectation-based backpropagation, and classical optimizers. For time-invariant Hamiltonians, a supervised feed-forward model learns robust single-qubit gate controls and converges with excellent validation generalization.

B. Hybrid Machine Learning for Quantum Control

  • TFQ addresses quantum-control problems including learning dynamics, optimizing control signals, and mitigating errors on realistic quantum devices.Its required tools include hybrid quantum-classical models, batch circuit simulation, expectation-based backpropagation, and gradient-based or non-gradient-based optimizers.Source: B. Hybrid Machine Learning for Quantum… · p. 26
  • TFQ tailors neural architectures to quantum dynamics, using a DNN for noise-free single-qubit control and an RNN with long-term memory for stochastic non-Markovian noise.The examples cover constant-Hamiltonian and time-dependent-Hamiltonian evolution.Source: B. Hybrid Machine Learning for Quantum… · p. 26

1. Time-Constant Hamiltonian Control

  • For time-invariant Hamiltonians, open-loop optimization enables offline control because the optimal solution is independent of instance-by-instance control actualization.A controller F maps inputs x to control vectors g, whose system outputs H(g) are compared with desired outputs through an aggregate error objective.Source: §1 · p. 26
  • A parameterized feed-forward network approximates the control map when analytical inversion is unavailable or costly, learning weights and biases so H(g_i) approximates each target y_i.The architecture is shown schematically in Figure 19.Source: §1 · p. 26, Figure 19 · p. 26
  • Supervised learning improves practical robustness by training on experimentally measured control pairs that reflect systematic offsets and classical-electronics noise, with TFQ connecting neural outputs to laboratory circuits through ControlledPQC.This is motivated both by the computational cost of inversion and by deviations of realistic systems from ideal behavior.Source: §1 · p. 26
  • The demonstrated task decomposes unknown single-qubit unitaries into hardware-friendly fixed-axis rotations using a hybrid feed-forward network.Inputs are rotation parameters [φ, θ1, θ2], while targets are experimentally observable Pauli expectation values [⟨X̂⟩, ⟨Ŷ⟩, ⟨Ẑ⟩].Source: §1 · p. 26
  • The hybrid implementation combines a three-gate single-qubit quantum circuit with a classical network mapping target parameters to the control vector g = {β, γ, δ}.The classical network uses Dense layers of widths 128, 128, 64, and 3, while ControlledPQC produces expectation outputs.Source: §1 · p. 27
  • After around 100 epochs, the hybrid model converges with excellent validation generalization on training and validation datasets of 5000 examples each.The result is shown as mean square error versus training epoch.Source: Figure 20 · p. 27, §1 · p. 27

2. Time-dependent Hamiltonian Control … 3. Training Models with Noise

The section demonstrates TFQ workflows for learning control under time-dependent Hamiltonians, simulating noisy circuits, and training quantum models despite mild noise. These examples combine recurrent prediction, trajectory-based noise simulation, and hybrid quantum-classical classification.

2. Time-dependent Hamiltonian Control

  • Time-dependent control is harder because the optimal control includes a potentially complex hidden fluctuation δ(t) that must be learned from changing system dynamics.The framework targets compensation of realistic low-frequency noise, including 1/f α-like Hamiltonian-amplitude errors.Source: §2 · p. 27
  • An RNN can learn low-frequency noise parameters by predicting future X-observable time series from prior expectation-value measurements.The workflow supports offline training from prerecorded series or online controller tuning.Source: §2 · pp. 27–28
  • Within 30 epochs, the LSTM converged to the accuracy supported by expectation-value measurements on validation data.The model predicts future observables from recorded expectation-value time series.Source: §2 · pp. 27–28, Figure 21 · p. 28

C. Simulating Noisy Circuits

  • The noisy-circuit example is available through a browser-based Colab workflow associated with the QCNN tutorial.This subsection provides an executable entry point rather than additional experimental findings.Source: C. Simulating Noisy Circuits · p. 28

1. Background

  • Noise testing is motivated by environmental interference, fabrication imperfections, TLS, and cosmic rays before large-scale error correction is available.The stated TFQ requirements include serializing Cirq circuits with channels and performing quantum-trajectory simulation with qsim.Source: §1 · p. 28

2. Noise in Cirq and TFQ

  • Because noise can insert, delete, or replace gates, noisy execution is represented as an ensemble of possible circuit realizations described by a density operator.The section illustrates this effect with a circuit whose Hadamard and CNot gates are corrupted or deleted.Source: §2 · pp. 28–29, Figure 22 · p. 28
  • TFQ simulates noisy circuits with trajectory methods and exposes noisy backends for sampling, sampled expectations, and expectation calculations.Cirq channels provide practical approximations when all error modes and probabilities are unknown.Source: §2 · p. 29

3. Training Models with Noise

  • Similar accuracy was achieved when the quantum classifier was trained on noisy versus noiseless data, showing learning can succeed under mild noise.The classifier distinguishes critical metallic and insulating phases of the XXZ chain.Source: §3 · p. 30, Figure 24 · p. 30

D. Quantum Approximate Optimization … A. Meta-learning for Variational Quantum Optimization

The section presents TFQ implementations of QAOA for MaxCut and advanced meta-learning workflows, showing how hybrid quantum-classical tools support optimization research. It also describes an RNN optimizer that learns QAOA parameter updates and generalizes from smaller to larger MaxCut instances.

D. Quantum Approximate Optimization

  • QAOA minimizes a pseudo-Boolean cost function by preparing an initial product state, applying tunable mixer and cost-Hamiltonian evolutions, and selecting low-energy measurement outcomes.The framework targets approximate minima for binary-variable objectives, including quadratic problems such as MaxCut.Source: §1 · p. 30, §2 · p. 31

1. Background

  • The tutorial focuses on 2-local MaxCut and demonstrates QAOA with p = 1 for a discrete optimization problem.QAOA was originally proposed for MaxCut [12] and has been extended to higher-order and continuous-variable Hamiltonians [49].Source: §1 · p. 30

2. Implementation

  • The TFQ implementation constructs Hadamard, mixing, and cost Hamiltonians for a random 3-regular ten-node graph, then represents the circuit in a Keras PQC model.The example uses TensorFlow gradient evaluation and a classical optimizer to train the variational parameters.Source: §1 · p. 30, §2 · p. 31

V. ADVANCED QUANTUM APPLICATIONS

  • TFQ advanced applications combine hybrid quantum-classical models and advanced optimizers to reduce time-to-working-prototype from weeks to days or hours compared with alternative tools.The authors present these examples as demonstrations of research workflows enabled by TFQ building blocks.Source: §V · pp. 31–32

A. Meta-learning for Variational Quantum Optimization

  • Meta-learning replaces fixed QAOA parameter updates with an LSTM RNN that receives the current parameters, estimated cost expectation, and hidden state before proposing the next parameters.The RNN repeatedly sends proposed parameters to the QPU for evaluation, forming a quantum-classical optimization loop.Source: A. Meta-learning for Variational… · p. 32
  • The RNN is trained on random QAOA MaxCut instances using cumulative observed improvement, with a five-step horizon requiring rapid approximate optimization.The objective compares each proposed value with the best value found earlier in the optimization history.Source: A. Meta-learning for Variational… · p. 32
  • After training on random 10-qubit MaxCut problems, the RNN learned a heuristic that generalized to a 12-qubit MaxCut instance.The section identifies this workflow as learning to learn with quantum neural networks, building a neural-network-based QAOA optimizer, and lowering required iterations.Source: A. Meta-learning for Variational… · pp. 32–33, Figure 26 · p. 33

B. Vanishing Gradients and Adaptive Layerwise Training Strategies · 1. Random Quantum Circuits and Barren Plateaus · 2. Layerwise quantum circuit learning

Randomly initialized deep quantum circuits can suffer barren plateaus, where gradients vanish exponentially and training becomes impractical. Layerwise training addresses both plateau initialization and training-induced drift by progressively adding and selectively optimizing circuit parameters.

1. Random Quantum Circuits and Barren Plateaus

  • Barren plateaus produce exponentially small gradients in sufficiently deep randomly initialized circuits, making straightforward training impossible and worsening with qubit count and cost-function scope [128].A unitary 2-design may arise at depth O(n^(1/d)), potentially as low as O(log(n)) with all-to-all connectivity.Source: §1 · p. 33
  • A vanishing gradient such as 10−7 may require approximately 10^14 quantum-classical samples, whereas classical readout scales as O(log 1/ϵ) rather than O(1/ϵ^α).This readout-complexity gap makes quantum-classical optimization diffuse essentially randomly when gradients become very small.Source: §1 · p. 33
  • Identity or blockwise initialization can avoid an initial plateau but may bias general-dataset results, while high learning rates can later diffuse the circuit onto a plateau.These effects severely limit learning rates and training efficiency in quantum neural networks.Source: §1 · p. 33, §2 · p. 34
  • Layerwise learning trains parameter subsets to permit larger learning rates, improve training efficiency, and reduce randomization into barren plateaus, especially with appropriately designed local cost functions.The framework advocates combining layerwise strategies with local objectives because fidelity-based objectives exhibit dramatically worse gradient problems [128].Source: §1 · p. 33

2. Layerwise quantum circuit learning

  • Phase one progressively constructs a circuit by adding layers after fixed training intervals, while retaining always-active initial layers and freezing older layers according to hyperparameters s, el, p, and q [133].For example, p = 2 and q = 4 adds two layers every el iterations while training only the latest four layers.Source: §2 · p. 34
  • Phase two alternates optimization across larger parameter partitions, containing training randomization while balancing overly large partitions against partitions too small to reduce the loss.Training continues until convergence or a predefined iteration limit.Source: §2 · p. 34
  • TFQ supports adaptive layerwise experimentation through dynamic circuit construction, parameter manipulation, reduced active parameters, and parameter-shift gradients without manually implementing training procedures or optimizers.The example uses randomly generated rotation-and-CZ layers and demonstrates growth from a shallow circuit when the ideal structure is unknown.Source: §2 · pp. 34–35

C. Hamiltonian Learning with Quantum Graph Recurrent Neural Networks · 1. Motivation: Learning Quantum Dynamics with a Quantum Representation · 2. Implementation

The section presents Hamiltonian learning as an inverse quantum-dynamics problem and implements it with a structured Quantum Graph Recurrent Neural Network in TFQ. The method uses Trotterized, graph-parameterized Hamiltonian evolution, low-energy training states, randomized time steps, and fidelity-based optimization to recover an unknown Ising Hamiltonian.

1. Motivation: Learning Quantum Dynamics with a Quantum Representation

  • Quantum simulation may offer an advantage for unitary time evolution because classical simulation overhead scales exponentially with evolution depth.This motivates investigating whether quantum machine learning can extend such advantages to learning quantum dynamics.Source: §1 · p. 35
  • Hamiltonian learning seeks a Hamiltonian whose time evolution reproduces unknown black-box dynamics from quantum states observed at different time steps.The section approaches this inverse problem with a quantum neural network.Source: §1 · pp. 35–36
  • The QGRNN imposes physics-informed structure by tying parameters across Trotterized time-evolution layers and encoding local couplings through a graph-based Hamiltonian parameterization [136].This recurrent structure targets constant dynamics over time and can learn effective Hamiltonian topology and coupling strengths from time-indexed quantum states.Source: §1 · p. 36

2. Implementation

  • The implementation represents a general QGNN as repeated exponentials of generally 2-local graph Hamiltonians, while the target is an Ising Hamiltonian with edge couplings and site biases.The recurrent QGNN uses Trotterized generators whose parameters describe couplings and local bias terms, approximating the target dynamics as Trotter steps increase and step sizes decrease.Source: §2 · p. 36
  • Training data begin with approximate low-energy states obtained by VQE, then apply target and QGRNN Trotterized evolutions at randomized time steps using minibatched gradient training.The example uses 15 randomized timesteps to form paired true and predicted quantum-state data.Source: §2 · p. 37
  • The training loss maximizes average state fidelity, estimated with a swap test whose observer-qubit expectation is measured through TFQ and optimized with a custom Keras loss.The implementation uses Adam with learning rate 0.05 and 500 epochs.Source: §2 · p. 38
  • The time-randomized gradient-based QGRNN optimization learns the target Hamiltonian and its couplings to a high degree of accuracy.Figure 27 contrasts the true Ising Hamiltonian, a random initial graph, and the learned Hamiltonian.Source: §2 · p. 38, Figure 27 · p. 38

D. Generative Modelling of Quantum Mixed States with Hybrid Quantum-Probabilistic Models · 1. Background · 2. Variational Quantum Thermalizer

The section motivates hybrid quantum-probabilistic models for representing and generating quantum mixed-state statistics, combining classical distributions with quantum neural networks. It then presents variational quantum thermalization, training the hybrid model by minimizing free energy and demonstrating convergence to a known thermal-state density matrix.

1. Background

  • Mixed states combine classical probability distributions with quantum wavefunctions and can exhibit both classical and quantum correlations, including entanglement.The section targets models that learn to represent and mimic mixed-state statistics arising in finite-temperature, open, and subsystem settings.Source: §1 · pp. 38–39
  • The proposed hybrid representation uses a classical probabilistic latent model and a parameterized quantum neural network to reduce quantum-circuit depth while adding quantum correlations.The visible state is generated by applying the quantum unitary to the classical latent distribution, whose parameters remain classical.Source: §1 · p. 39
  • The framework combines TensorFlow Probability [140] with TFQ to train hybrid models for thermal-state generation and learning mixed-state statistics from data.Required functionality includes sample-based circuit simulation and parameter-shift gradient computation.Source: §1 · p. 39

2. Variational Quantum Thermalizer

  • Variational quantum thermalization approximates the Gibbs state σ_β = Z_β^-1e^-βH by optimizing hybrid parameters θ and φ.The target is defined for a Hamiltonian H and inverse temperature β = 1/T.Source: §2 · p. 39
  • Minimizing quantum relative entropy is equivalent to minimizing free energy, whose terms are the model’s energy expectation and entropy.The visible-state entropy equals the classical entropy of the latent distribution, making the entropy term useful during optimization.Source: §2 · p. 39
  • Gradients are estimated using TFQ parameter-shift differentiation for quantum parameters and covariance-based sampling gradients for classical energy-based-model parameters.The required expectations can be evaluated by sampling the classical model and quantum-device outputs.Source: §2 · p. 40
  • Because the partition function need not be estimated, the method can use expressive latent energy-based models beyond factorized distributions, including Boltzmann machines.The factorized Bernoulli model is a special bias-only Boltzmann-machine case; a fully connected classical Ising model can be sampled with Metropolis-Hastings [141].Source: §2 · pp. 40–41
  • The factorized-Bernoulli VQT density matrix converges to the known exact thermal-state result for the 2D Heisenberg model, as shown in Fig. 28.The model is trained on a 2×2 grid for 100 epochs; Fig. 28 shows its final density matrix.Source: §2 · pp. 40–41, Figure 28 · p. 41

3. Quantum Generative Learning from Quantum Data · E. Subspace-Search Variational Quantum Eigensolver for Excited States: Integration with OpenFermion · 1. Background

The section presents quantum generative learning from mixed-state data using a quantum-probabilistic model with an energy-based classical latent distribution, and introduces TFQ’s implementation of SSVQE for ground- and excited-state calculations. It emphasizes gradient-based training, TFQ differentiators, custom objectives, and OpenFermion integration as enabling tools.

3. Quantum Generative Learning from Quantum Data

  • TFQ supports learning a model state ρ_θφ that approximates a supplied mixed quantum state σ_D using an energy-based quantum-probabilistic model.The model is trained from copies of the data state rather than from its Hamiltonian.Source: §3 · p. 41
  • Quantum cross entropy is used as the loss because minimizing it suffices to minimize the reverse quantum relative entropy D(σ_D∥ρ_θφ).Source: §3 · p. 41
  • The latent distribution is obtained by applying the inverse QNN U†(φ) to σ_D and measuring in the standard basis, yielding σ_φ(x).Source: §3 · p. 41
  • QNN parameters use standard TFQ differentiators such as the parameter-shift rule, while energy-based parameters follow positive data-gradient and negative model-gradient terms.The classical gradient is ∂_θL_xe(θ,φ)=E_x∼σ_φ(x)[∇_θE_θ(x)]−E_y∼p_θ(y)[∇_θE_θ(y)].Source: §3 · p. 41
  • The data-driven demonstration fixes VQT parameters, samples bitstrings from a learned Bernoulli energy-based model, and feeds them through the learned VQT unitary to reproduce the data-state statistics.The setup treats the fixed VQT parameters as a quantum datasource for QMHL.Source: §3 · p. 41

1. Background

  • VQE variationally estimates the ground-state energy, whereas SSVQE extends the approach to higher-energy states by evaluating orthogonal initial states.The background motivates excited-state calculations because higher energy states can be chemically and physically important, and identifies SSVQE as the state-of-the-art example studied in TFQ.Source: §1 · pp. 41–42
  • TFQ enables SSVQE experimentation through rapid adjoint differentiation, custom layers and objectives, and native OpenFermion–Cirq integration for quantum-chemistry workflows.The stated targets are implementing SSVQE and finding H2 ground- and first-excited-state energies at increasing bond lengths; required functionality includes OpenFermion integration and custom TFQ layers.Source: §1 · p. 42

2. Implementation · F. Classification of Quantum Many-Body Phase Transitions · 1. Background

The implementation builds VQE and SSVQE components in TensorFlow Quantum, then applies an end-to-end quantum classifier to label quantum many-body phases using prepared ground-state data. The examples include energy prediction for ground and excited states and phase-transition classification in the transverse-field Ising model.

2. Implementation

  • The SSVQE implementation uses custom VQE layers with shared parameters, distinct readout operators, orthogonal inputs, and a Pauli X gate before U(θ).The Hamiltonian is represented as a sparse sum of Pauli strings, typically with L = O(N^4) terms.Source: §2 · pp. 42–43
  • The SSVQE training loop directly minimizes the model output with TensorFlow gradient computation and optimizer updates until the loss change falls below a tolerance.The routine then returns the predicted ground- and excited-state energies.Source: §2 · p. 43
  • The molecular inputs are generated with OpenFermion and PySCF, transformed from fermionic operators to qubits by Jordan-Wigner mapping, and evaluated across bond lengths.The example uses k = 2 states and iterates from bond length 0.2 to 2.0 in increments of 0.1.Source: §2 · p. 43
  • Figure 29 compares SSVQE-predicted ground- and first-excited-state energies with their true values, while the implementation requires several minutes to run.The workload includes 48 parameters per layer and optimization over 20 different Hamiltonians, even with the Adjoint differentiator.Source: §2 · p. 43, Figure 29 · p. 44

F. Classification of Quantum Many-Body Phase Transitions

  • Quantum many-body phase classification uses supervised learning to predict phase labels from quantum states, avoiding direct reliance on observables estimated by quantum Monte Carlo or tensor-network methods.The section demonstrates end-to-end training of such a quantum classifier in TensorFlow Quantum using a data set of quantum many-body states [155, 156, 157].Source: §1 · p. 44

1. Background

  • The classifier comprises fixed state preparation, a trainable hardware-efficient variational circuit, and measurements combined into a scalar predictor for phase labels.The model trains with a hinge loss and requires batching quantum data plus hybrid quantum-classical optimization.Source: §1 · p. 44
  • The example studies the two-dimensional transverse-field Ising model on a torus, whose transition occurs at g ≈ 3.04 between ordered and disordered phases.TFQ provides a data-set module containing circuits for different quantum many-body ground states, supporting this workflow.Source: §1 · p. 44

2. Implementation … 2. Noise Suppression with Adversarial Learning

The paper implements TFQ examples spanning quantum phase classification and EQ-GAN noise suppression, showing that adversarially learned discriminator corrections improve noisy swap-test performance. EQ-GAN also addresses quantum-state generation and approximate quantum memory through entangling adversarial learning.

2. Implementation

  • For 4 × 4 two-dimensional TFIM data over g ∈[2.5, 3.5], the classifier uses 51 quantum circuits preparing ground states with > 0.999 fidelity and adds a hardware-efficient ansatz.The ansatz includes parameterized X, nearest-neighbor XX, and Z layers.Source: §2 · p. 45
  • The TFQ classifier uses hinge loss, Adam with learning rate 0.01, hinge accuracy, and early stopping, with its output inflection near g ≈3.04 matching the TFIM phase transition.Early stopping terminates training when loss no longer decreases consistently, and the phase-transition result is shown in Fig. 31.Source: §2 · p. 45, Figure 31 · p. 46

1. Background

  • The EQ-GAN examples target two applications: suppressing noise during quantum-state generation and preparing approximate quantum memory to accelerate quantum-neural-network training.The framework considers both purely quantum datasets and quantum states constructed from classical datasets.Source: §1 · p. 46
  • The entangling quantum GAN lets its discriminator entangle true and generated data, theoretically converges to the global optimal Nash equilibrium, and succeeds on instances where QuGAN failed [76].This design targets QuGAN’s oscillations from mode collapse and non-unique Nash equilibria [76].Source: §1 · p. 46

2. Noise Suppression with Adversarial Learning

  • EQ-GAN adversarially learns a discriminator closest to the true swap test despite drifting CZ and single-qubit Z-phase errors that challenge noisy gradient-based training.The relevant gate parameters can drift and oscillate over O(10) minutes, while additional single-qubit Z-phase compensations mitigate large rotation-angle deviations.Source: §2 · p. 47
  • EQ-GAN learns single-qubit Z-rotation corrections after a noisy CZ gate, obtaining significantly better overlap with the target superposition than a perfect swap test.The experiment learned 1/√2(|0⟩+|1⟩) using generator X and Z rotations and an adversarially parameterized discriminator.Source: §2 · p. 47, Figure 32 · p. 47
  • Around 4× lower error was achieved on a calibrated quantum device than with a direct standard swap test [76].The adversarial swap test was evaluated on quantum hardware, not only under simulation.Source: §2 · p. 47

3. Approximate Quantum Random Access Memory … 3. Implementation

This section presents variational QRAM as an approximate approach to encoding classical data into quantum superpositions and develops an implementation of a policy-gradient reinforcement-learning agent using data-reuploading PQCs. It also describes parallelized model evaluations and fast automatic differentiation to support practical training on a regular laptop.

3. Approximate Quantum Random Access Memory

  • Preparing arbitrary classical data superpositions generally requires O(n) operations, motivating EQ-GANs that approximately learn the corresponding quantum state.Exact encoding and Gaussian-state preparation can require very deep circuits, so shallow ansatzes are used to approximate the empirical distribution.Source: §3 · pp. 47–48
  • At equal gradient-evaluation budgets, encoding classes as superpositions in variational QRAM reaches better final accuracy than individually encoded data, despite approximation.The variational QRAM is trained with an EQ-GAN and closely reproduces a two-peak dataset using a shallow ansatz.Source: §3 · p. 48, Figure 33 · p. 48

1. Background

  • Quantum reinforcement learning replaces conventional policy or value models with PQCs, with data re-uploading highlighted as an expressive encoding strategy for large or continuous state spaces.The background frames RL as an agent maximizing expected episodic rewards through repeated state, action, and reward interactions.Source: §1 · pp. 48–49

2. Policy-Gradient Reinforcement Learning with PQCs

  • The policy-gradient method parameterizes action probabilities from PQC observable expectations and updates circuit parameters by gradient ascent, using a loss formulation that avoids numerical differentiation.The policy is evaluated on batches of sampled episodes, with action-specific trainable weights augmenting observable expectations.Source: §2 · p. 49

3. Implementation

  • The implementation combines alternating variational, entangling, and data-encoding layers with custom Keras components for re-uploading, observable post-processing, episodic sampling, discounted returns, and gradient updates.The CartPole-v1 example uses a 4-qubit PQC for a four-dimensional continuous state and two-action discrete space, while tf.GradientTape computes policy-gradient updates.Source: §3 · pp. 49–51
  • Training for 500 episodes takes about 15 minutes on a regular laptop for a PQC acting on 4 qubits with 5 re-uploading layers.The execution benefits from parallelized model evaluations and tfq.differentiators.Adjoint() automatic differentiation.Source: §3 · p. 51

4. Value-Based Reinforcement Learning with PQCs · 5. Quantum Environments · VI. CLOSING REMARKS

The paper extends PQC reinforcement learning to value-based methods and discusses when quantum environments may enable learning advantages. It closes by positioning TensorFlow Quantum as an evolving, openly licensed framework for quantum applications.

4. Value-Based Reinforcement Learning with PQCs

  • The accompanying tutorial implements both the policy-based approach and the value-based algorithm introduced in [179].The tutorial is identified as docs/tutorials/quantum_reinforcement_learning.ipynb.Source: §4 · p. 51
  • Value-based PQC reinforcement learning approximates Q(s, a) with expectation values ⟨Oa⟩s,θ,λ, using deep-RL mechanisms such as replay memory and a target model.The loss is derived from Q-learning, with further details in [179].Source: §4 · p. 51

5. Quantum Environments

  • Although initial results prove that some environments can be efficiently tackled only by quantum agents [176], those environments are described as artificial and contrived.The open question is whether natural environments yield large quantum learning advantages over classical agents.Source: §5 · p. 51
  • Variational quantum methods have shown advantages when an agent’s data comes from measurements of a quantum system [176].This setting is presented as an intuitive candidate for natural quantum reinforcement-learning advantages.Source: §5 · p. 51
  • Quantum control environments can provide genuinely quantum states that are processed directly by a PQC, as explored in recent work [185].This extends beyond agents receiving classical measurement data from quantum systems.Source: §5 · p. 51

VI. CLOSING REMARKS

  • TensorFlow Quantum is intended to accelerate quantum machine-learning applications as hardware develops, and the framework is expected to evolve with research needs and hardware availability.The closing remarks place TFQ alongside OpenFermion, released in 2017 to accelerate quantum simulation algorithms for chemistry and materials science.Source: §VI · p. 52
  • TFQ is open-sourced under the commercially friendly Apache 2 license, permitting royalty-free embedding in future commercial products.The project community is directed to github.com/tensorflow/quantum/.Source: §VI · p. 52

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