Quantum Machine Learning with SQUID

1 Introduction

• Quantum-enhanced machine learning seeks potential quantum speed-ups [5] [7] [8] and recognition of statistical patterns difficult for purely classical schemes [10].Recent work also addresses trainability [11] and quantum-model generalization error [16].Source: §1 · p. 1
• Near-term quantum devices cannot efficiently load datasets such as MNIST into quantum memory, while encoding and decoding add classical–quantum overhead.These resource constraints motivate alternatives to direct quantum data loading.Source: §1 · p. 1
• Small synthetic datasets make benchmarks artificial, whereas classical preprocessing complicates performance attribution and cross-study comparison.Different preprocessing choices also create overhead for comparing quantum approaches or conducting meta-analysis.Source: §1 · p. 1
• SQUID standardizes hybrid quantum–classical model design on top of PyTorch, requiring user-defined forward- and back-propagation functions for quantum models.The framework is intended to improve reproducibility and support extensible quantum model implementations.Source: §1 · p. 1
• Standard preprocessing across applications and approaches makes experiments more directly comparable and reduces the coding burden for new researchers.The framework also connects classical and quantum components in a way resembling quantum-inspired algorithms [1].Source: §1 · p. 1

2 The squid framework

• SQUID jointly trains classical networks with a quantum network to learn encoders and decoders for hybrid classification models.This architecture can learn feature maps for representing large datasets on near-term devices and incorporate classical information known a priori.Source: §2 · p. 2
• The framework assumes training examples contain classical vectors vclass and quantum states |vquant⟩, using classical bit-encoding and quantum amplitude encoding.The amplitude-encoding choice also subsumes quantum training vectors represented as distinct computational-basis bit strings.Source: §2 · p. 2

2.1 Framework Design

• MainModel composes a classical encoder, a middle quantum or classical model, and a classical decoder, reducing to PyTorch nn.Sequential when all three are classical.Figure 1 represents model nodes as classical neurons or quantum state vectors and edges as transformations.Source: §2.1 · p. 2, Figure 1 · p. 2

2.2 Propagating through the Main Model

• When the middle model is quantum, SQUID stops PyTorch autograd at the tensor-to-array boundary and uses the quantum model’s backward function to return gradients.A fake loss formed from the encoder output and supplied gradient lets PyTorch propagate gradients through the remaining parameters.Source: §2.2 · p. 3

2.3 Complexity of Gradient Evaluation

• The quantum-gradient procedure estimates Q1 output probabilities and M-dimensional parameter gradients using parameter-shift circuits, with sampling error controlled probabilistically.On future fault-tolerant devices, the analysis states that quadratic speedups in both ϵ and M may be possible at the cost of longer circuit depth.Source: §2.3 · pp. 3–4

2.4 Available classical models

• SQUID provides extensible classical models and constructs quantum models from nearest-neighbor one- and two-qubit unitary layers.Quantum outputs are computational-basis probabilities, but only a small subset is typically selected because the number of outcomes grows exponentially with register size.Source: §2.4 · p. 4, §2.5 · pp. 4–5

2.5 Available quantum models

• SQUID replaces a fixed encoding–variational split with a global unitary whose parameters come from the classical encoder, while retaining the ability to reproduce a factorized form when useful.The encoder output dimension must remain below the generic unitary parameter bound M < (4^N − 1), although efficiently decomposable circuits can require only polynomially many parameters.Source: §2.5 · p. 5

3 Example applications

• The MNIST experiment classifies digits 3 and 7 using an encoder, a classifier or quantum subroutine, and a decoder, with quantum outputs ranging from one to 2^N probabilities.The study uses 48 independent optimization runs and compares the maximum output dimension M1 = 2^N with a minimum setting M1 = 2.Source: §3 · pp. 5–6, Figure 2 · p. 6
• Classical networks exceed 99% MNIST accuracy, with larger hidden layers improving training fit but not producing a statistically significant improvement in final accuracy.The validation-accuracy distribution remains largely stable as parameter count increases, while training accuracy can reach 100% after more than 80 epochs.Source: §3 · p. 6, Figure 3 · p. 6

3.1 Separable Quantum Models

• Full-output separable quantum models show a mild accuracy increase with model size but remain below smaller classical networks, whereas single-output models deteriorate as size increases.For six-qubit models, more than half of full-output runs exceed 99% accuracy, while restricting the output prevents most runs from reaching that threshold.Source: §3.1 · p. 7, Figure 5 · p. 7

3.2 Quantum models with entanglement

• Quantum models with entanglement never exceed classical accuracy, and restricted-output models lose training accuracy as they grow larger.The optimizer struggles to find good parameters for large restricted-output models, indicating optimization difficulty rather than simple overfitting.Source: §3.2 · p. 7, Figure 6 · p. 8

4.1 SQUID Helpers Package

• SQUID Helpers uses YAML configuration files to simplify bootstrapping multiple runs, test cases, and result aggregation.Source: §4.1 · p. 8
• Deterministic seed changes during bootstrap runs make each result exactly reproducible, while configurations and aggregated outputs are saved together.Source: §4.1 · p. 8
• The Helpers workflow performs an early dimension-checking forward pass, then runs standard training with the Main Model backward call.Source: §4.1 · p. 8

4.2 Future Work

• SQUID can abstract communication with specific quantum backends, but each backend requires a custom QuantumModel subclass implementing its API.Source: §4.2 · p. 8
• Current SQUID support is limited to classification and regression, while more advanced scenarios would require changes to SQUID and especially SQUID Helpers.Source: §4.2 · p. 8
• Future extensions include additional quantum-computing frameworks and backend-agnostic operation across multiple quantum devices.Source: §4.2 · p. 8

• SQUID embeds quantum algorithms within multi-layer classical–quantum architectures and uses PyTorch automatic differentiation for efficient optimization.Source: §5 · p. 9
• The framework’s decoder tunes measurement settings to balance limited probability measurements against the exponential repetition cost of full computational-basis sampling.Source: §5 · p. 9
• SQUID supports flexible interchange of classical and quantum components and could be adapted to classically efficient entangled representations such as tensor-network states [37].Source: §5 · p. 9
• Assessing practical viability requires sample-complexity analysis, including finite-statistics and decoherence effects for target devices.Source: §5 · p. 9

• A variational quantum model applies a parameterized unitary W(w) to an initial state, with gradients obtained from derivatives of its constituent one- and two-qubit unitaries.Source: A Gradient evaluation for variational… · p. 12
• The measured classical output is represented by probabilities for computational-basis states, using explicitly introduced projectors.Source: A Gradient evaluation for variational… · p. 12
• Parameter derivatives can be evaluated through parameter shifts by π/2 of the corresponding unitary angles.Source: A Gradient evaluation for variational… · p. 12
• For K ≪ 2^n requested outputs, explicit probability estimation uses 2K circuits, while shifted-unitary gradients require 2KM executions and 2K(M + 1) observables.Source: A Gradient evaluation for variational… · pp. 12–13
• When all K = 2^n outputs are required, decomposing computational-basis projectors leads to KM circuit executions, whereas unitary-expectation alternatives can reduce independent circuits.Source: A Gradient evaluation for variational… · pp. 12–13

• Table 2 reports classical feed-forward MNIST models, including training and validation 90% accuracy-percentile boundaries with estimated errors given as 68% confidence intervals.Source: B Additional information on the MNIST… · p. 13, Table 2 · p. 13
• Tables 3 and 4 report separable and entangled quantum models, respectively, with model labels indicating the number of qubits used.Source: B Additional information on the MNIST… · p. 13, Table 3 · p. 13, Table 4 · p. 14
• Figure 7 compares training and validation accuracy over epochs for maximal output M1 = 16 and restricted output M1 = 1 models.Source: B Additional information on the MNIST… · p. 13, Figure 7 · p. 14
• Figure 8 compares classical, full-output quantum, and single-output quantum accuracy as functions of parameter count and summarizes accuracy distributions for the largest N = 4-qubit model.Source: B Additional information on the MNIST… · p. 13, Figure 8 · p. 14